define hypothesis matrix

Linear Mixed Models in Linguistics and Psychology: A Comprehensive Introduction

6.2 the hypothesis matrix illustrated with a three-level factor.

Consider an example with the three word classes nouns, verbs, and adjectives. We load simulated data from a lexical decision task with response times as dependent variable. The research question is: do response times differ as a function of the between-subject factor word class with three levels: nouns, verbs, and adjectives? We here make the ad-hoc assumption that nouns may have longer response times and that adjectives may have shorter response times. Here, we specify word class as a between-subject factor. In cognitive science experiments, word class will usually vary within subjects and between items. However, the within- or between-subjects status of an effect is independent of its contrast coding; we assume the manipulation to be between subjects for ease of exposition. The concepts presented here extend to repeated measures designs that are often analyzed using hierarchical Bayesian (linear mixed) models.

As shown in Table  6.2 , the estimated means reflect our assumptions about the true means in the data simulation: Response times are longest for nouns and shortest for adjectives. In the following sections, we use this data-set to illustrate sum . Furthermore, we will use an additional data set to illustrate repeated , polynomial , and custom contrasts. In practice, usually only one set of contrasts is selected when the expected pattern of means is formulated during the design of the experiment.

6.2.1 Sum contrasts

For didactic purposes, the next sections describe sum contrasts. Suppose that the expectation is that nouns are responded to slower and verbs words are responded to faster than the GM response time. Then, the research question could be: Do nouns differ from the GM and do adjectives differ from the GM? And if so, are they above or below the GM? We want to estimate the following two quantities:

\[\begin{equation} \beta_1 = \mu_1 - \frac{\mu_1+\mu_2+\mu_3}{3} = \mu_1 - GM \end{equation}\]

\[\begin{equation} \beta_2 = \mu_2 - \frac{\mu_1+\mu_2+\mu_3}{3} = \mu_2 - GM \end{equation}\]

This translates into the following two hypotheses:

\[\begin{equation} H_{0_1}: \mu_1 = \frac{\mu_1+\mu_2+\mu_3}{3} = GM \end{equation}\]

\[\begin{equation} H_{0_2}: \mu_2 = \frac{\mu_1+\mu_2+\mu_3}{3} = GM \end{equation}\]

\(H_{0_1}\) can also be written as:

\[\begin{align} \label{h01} & \mu_1 =\frac{\mu_1+\mu_2+\mu_3}{3}\\ \Leftrightarrow & \mu_1 - \frac{\mu_1+\mu_2+\mu_3}{3} = 0\\ \Leftrightarrow & \frac{2}{3} \mu_1 - \frac{1}{3}\mu_2 - \frac{1}{3}\mu_3 = 0 \end{align}\]

This corresponds to estimating the quantity \(\beta_1 = \frac{2}{3} \mu_1 - \frac{1}{3}\mu_2 - \frac{1}{3}\mu_3\) . Here, the weights \(2/3, -1/3, -1/3\) are informative about how to combine the condition means to estimate the linear model coefficient and to define the null hypothesis.

\(H_{0_2}\) is also rewritten as:

\[\begin{align}\label{h02} & \mu_2 = \frac{\mu_1+\mu_2+\mu_3}{3}\\ \Leftrightarrow & \mu_2 - \frac{\mu_1+\mu_2+\mu_3}{3} = 0 \\ \Leftrightarrow & -\frac{1}{3}\mu_1 + \frac{2}{3} \mu_2 - \frac{1}{3} \mu_3 = 0 \end{align}\]

This corresponds to estimating the quantity \(\beta_2 = -\frac{1}{3}\mu_1 + \frac{2}{3} \mu_2 - \frac{1}{3} \mu_3\) . Here, the weights are \(-1/3, 2/3, -1/3\) , and they again indicate how to combine the condition means for estimating the regression coefficient and for defining the null hypothesis.

6.2.2 The hypothesis matrix

The weights of the condition means are not only useful to define parameter estimates and hypotheses. They also provide the starting step in a very powerful method which allows the researcher to generate the contrasts that are needed to test these hypotheses in a linear model. That is, what we did so far is to explain some kinds of different contrast codings that exist and what the hypotheses are that they test. That is, if a certain data-set is given and the goal is to estimate certain comparisons (and test certain hypotheses), then the procedure would be to check whether any of the contrasts that we encountered above happen to estimate these comparisons and test exactly the hypotheses of interest. Sometimes it suffices to use one of these existing contrasts. However, at other times, our research questions may not correspond exactly to any of the contrasts in the default set of standard contrasts provided in R. For these cases, or simply for more complex designs, it is very useful to know how contrast matrices are created. Indeed, a relatively simple procedure exists in which we write our comparisons or hypotheses formally, extract the weights of the condition means from the comparisons/hypotheses, and then automatically generate the correct contrast matrix that we need in order to estimate these comparisons or test these hypotheses in a linear model. Using this powerful method, it is not necessary to find a match to a contrast matrix provided by the family of functions in R starting with the prefix contr . Instead, it is possible to simply define the comparisons that one wants to estimate or the hypotheses that one wants to test, and to obtain the correct contrast matrix for these in an automatic procedure. Here, for pedagogical reasons, we show some examples of how to apply this procedure in cases where the comparisons/hypotheses do correspond to some of the existing contrasts.

Defining a custom contrast matrix involves four steps:

• Write down the estimated comparisons or hypotheses
• Extract the weights and write them into what we will call a hypothesis matrix
• Apply the generalized matrix inverse to the hypothesis matrix to create the contrast matrix
• Assign the contrast matrix to the factor and run the linear (mixed) model

Let us apply this four-step procedure to our example of the sum contrast. The first step, writing down the estimated parameters and hypotheses, is shown above. The second step involves writing down the weights that each comparison / hypothesis gives to condition means. The weights for the first comparison or null hypothesis are wH01=c(+2/3, -1/3, -1/3) , and the weights for the second comparison or null hypothesis are wH02=c(-1/3, +2/3, -1/3) .

Before writing these into a hypothesis matrix, we also define the estimated quantity and the null hypothesis for the intercept term. The intercept parameter estimates the mean across all conditions:

\[\begin{align} \alpha = \frac{\mu_1 + \mu_2 + \mu_3}{3} \\ \alpha = \frac{1}{3} \mu_1 + \frac{1}{3}\mu_2 + \frac{1}{3}\mu_3 \end{align}\]

This corresponds to the null hypothesis that the mean across all conditions is zero:

\[\begin{align} H_{0_0}: &\frac{\mu_1 + \mu_2 + \mu_3}{3} = 0 \\ H_{0_0}: &\frac{1}{3} \mu_1 + \frac{1}{3}\mu_2 + \frac{1}{3}\mu_3 = 0 \end{align}\]

This estimate and null hypothesis has weights of \(1/3\) for all condition means. The weights from all three model parameters / hypotheses that were defined are now combined and written into a matrix that we refer to as the hypothesis matrix ( Hc ):

Each set of weights is first entered as a row into the matrix (command rbind() ). This has mathematical reasons (see Schad et al. 2020b ) . However, we then switch rows and columns of the matrix for easier readability using the command t() (this transposes the matrix, i.e., switches rows and columns). The command fractions() turns the decimals into fractions to improve readability.

Now that the condition weights have been written into the hypothesis matrix, the third step of the procedure is implemented: a matrix operation called the ‘generalized matrix inverse’ 2 is used to obtain the contrast matrix that is needed to test these hypotheses in a linear model. In R this next step is done using the function ginv() from the MASS package. Here, we define a function ginv2() for nicer formatting of the output. 3

Applying the generalized inverse to the hypothesis matrix results in the new matrix XcSum . This is the contrast matrix \(X_c\) that estimates exactly those comparisons and tests exactly those hypotheses that were specified earlier:

This contrast matrix corresponds exactly to the sum contrasts described above. In the case of the sum contrast, the contrast matrix looks very different from the hypothesis matrix. The contrast matrix in sum contrasts codes with \(+1\) the condition that is to be compared to the GM. The condition that is never compared to the GM is coded as \(-1\) . This is a very different interpretation than the two-condition sum contrast we saw in earlier chapters!

The important point here is that unless we know the relationship between the hypothesis matrix and the contrast matrix, the meaning of the coefficients is completely opaque.

To verify this custom-made contrast matrix, it is compared to the sum contrast matrix as generated by the R function contr.sum() in the stats package. The resulting contrast matrix is identical to the result when adding the intercept term, a column of ones, to the contrast matrix:

In order to estimate model parameters, step four in our procedure involves assigning sum contrasts to the factor F in our example data, and running a linear (mixed) model. This allows us to estimate the regression coefficients associated with each contrast. We compare these to the data shown above (Table  6.2 ) to test whether the regression coefficients actually correspond to the differences of condition means, as intended. To define the contrast, it is necessary to remove the intercept term, as this is automatically added by the linear model function lm() .

The linear model regression coefficients show the GM response time of \(450\) ms in the intercept. Remember that the first regression coefficient FcH01 was designed to estimate the extent to which nouns are responded to slower than the GM. The regression coefficient FcH01 (‘Estimate’) of \(50\) reflects the difference between adjectives ( \(400\) ms) and the GM of \(450\) ms. The estimate of interest was in how response times for adjectives differ from the GM. The fact that the second regression coefficient FcH02 is close to \(-50\) indicates that response times for adjectives ( \(400\) ms) are 50 ms faster than the GM of \(450\) ms. Verbs are estimated to have \(50\) ms longer reading times than the GM (the second slope labeled FcH02).

We have now not only derived contrasts, parameter estimates, and hypotheses for the sum contrast, we have also used a powerful and highly general procedure that is used to generate contrasts for many kinds of different hypotheses and experimental designs.

6.2.3 Generating contrasts: The hypr package

To work with the 4-step procedure, i.e., to flexibly design contrasts to estimate specific comparisons, we have developed the R package hypr ( Rabe et al. 2020 ) . This package allows the user to specify comparisons as null hypotheses, and based on these null hypotheses, it automatically generates contrast matrices that allow the user to estimate these comparisons and test these hypotheses in linear models. It thus considerably simplifies the implementation of the 4-step procedure outlined above.

To illustrate the functionality of the hypr package, we will use the two comparisons and associated null hypotheses that we had defined and analyzed in the previous section:

These null hypotheses are effectively comparisons between condition means or between bundles of condition means. That is, \(\mu_1\) is compared to the GM and \(\mu_2\) is compared to the grand mean. These two comparisons/hypotheses can be directly entered into R using the hypr() function from the hypr package. To do so, we use some labels to indicate factor levels. E.g., nouns , verbs , and adjectives can represent factor levels \(\mu_1\) , \(\mu_2\) , and \(\mu_3\) . The first comparison/hypothesis specifies that \(\mu_1 = \frac{\mu_1+\mu_2+\mu_3}{3}\) . This can be written as a formula in R: nouns ~ (nouns + verbs + adjectives)/3 . The second comparison/hypothesis is that \(\mu_2 = \frac{\mu_1+\mu_2+\mu_3}{3}\) , which can be written in R as verbs ~ (nouns + verbs + adjectives)/3 .

The results show that the null hypotheses or comparisons between condition means have been re-written into a form where \(0\) is coded on the left side of the equation, and the condition means together with associated weights are written on the right side of the equation. This presentation makes it easy to see the weights of the condition means to code a certain null hypothesis or comparison. The next part of the results shows the hypothesis matrix, which contains the weights from the condition means. Thus, hypr takes as input comparisons between condition means, which also define null hypotheses, and automatically extracts the corresponding weights and encodes them into the hypothesis matrix. hypr moreover applies the generalized matrix inverse to obtain the contrast matrix from the hypothesis matrix. Note that the different steps correspond exactly to the steps we had carried out manually in the preceding section. hypr automatically performs these steps for us. We can now extract the contrast matrix by a simple function call:

We can assign this contrast to our factor as we did before, and again fit a linear model:

In the hypr , the focus lies on estimation of contrasts that code comparisons between condition means or groups of condition means. Thus, the null hypotheses that one specifies implicitly imply the estimation of a difference between condition means or bundles of condition means. The output of the hypr() function (see the first section of the results) makes this clear - these formulate the null hypotheses in a way that also illustrates the estimation of model parameters. I.e., the null hypothesis H0.b1: 0 = 2/3*m1 - 1/3*m2 - 1/3*m3 corresponds to a parameter estimate of b1 = 2/3*m1 - 1/3*m2 - 1/3*m3 . The resulting contrasts will then allow us to estimate the specified differences between condition means or bundles of condition means.

At this point, there is no need to understand in detail what this means. We refer the interested reader to Schad et al. ( 2020b ) . For a quick overview, we recommend a vignette explaining the generalized inverse in the matlib package ( Friendly, Fox, and Chalmers 2020 ) . ↩︎

The function from the package is used to make the output more easily readable, and the function is used to keep row and column names. ↩︎

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Hypothesis tests

Associated data.

• • Hypothesis tests are used to assess whether a difference between two samples represents a real difference between the populations from which the samples were taken.
• • A null hypothesis of ‘no difference’ is taken as a starting point, and we calculate the probability that both sets of data came from the same population. This probability is expressed as a p -value.
• • When the null hypothesis is false, p- values tend to be small. When the null hypothesis is true, any p- value is equally likely.

Learning objectives

By reading this article, you should be able to:

• • Explain why hypothesis testing is used.
• • Use a table to determine which hypothesis test should be used for a particular situation.
• • Interpret a p- value.

A hypothesis test is a procedure used in statistics to assess whether a particular viewpoint is likely to be true. They follow a strict protocol, and they generate a ‘ p- value’, on the basis of which a decision is made about the truth of the hypothesis under investigation. All of the routine statistical ‘tests’ used in research— t- tests, χ 2 tests, Mann–Whitney tests, etc.—are all hypothesis tests, and in spite of their differences they are all used in essentially the same way. But why do we use them at all?

Comparing the heights of two individuals is easy: we can measure their height in a standardised way and compare them. When we want to compare the heights of two small well-defined groups (for example two groups of children), we need to use a summary statistic that we can calculate for each group. Such summaries (means, medians, etc.) form the basis of descriptive statistics, and are well described elsewhere. 1 However, a problem arises when we try to compare very large groups or populations: it may be impractical or even impossible to take a measurement from everyone in the population, and by the time you do so, the population itself will have changed. A similar problem arises when we try to describe the effects of drugs—for example by how much on average does a particular vasopressor increase MAP?

To solve this problem, we use random samples to estimate values for populations. By convention, the values we calculate from samples are referred to as statistics and denoted by Latin letters ( x ¯ for sample mean; SD for sample standard deviation) while the unknown population values are called parameters , and denoted by Greek letters (μ for population mean, σ for population standard deviation).

Inferential statistics describes the methods we use to estimate population parameters from random samples; how we can quantify the level of inaccuracy in a sample statistic; and how we can go on to use these estimates to compare populations.

Sampling error

There are many reasons why a sample may give an inaccurate picture of the population it represents: it may be biased, it may not be big enough, and it may not be truly random. However, even if we have been careful to avoid these pitfalls, there is an inherent difference between the sample and the population at large. To illustrate this, let us imagine that the actual average height of males in London is 174 cm. If I were to sample 100 male Londoners and take a mean of their heights, I would be very unlikely to get exactly 174 cm. Furthermore, if somebody else were to perform the same exercise, it would be unlikely that they would get the same answer as I did. The sample mean is different each time it is taken, and the way it differs from the actual mean of the population is described by the standard error of the mean (standard error, or SEM ). The standard error is larger if there is a lot of variation in the population, and becomes smaller as the sample size increases. It is calculated thus:

where SD is the sample standard deviation, and n is the sample size.

As errors are normally distributed, we can use this to estimate a 95% confidence interval on our sample mean as follows:

We can interpret this as meaning ‘We are 95% confident that the actual mean is within this range.’

Some confusion arises at this point between the SD and the standard error. The SD is a measure of variation in the sample. The range x ¯ ± ( 1.96 × SD ) will normally contain 95% of all your data. It can be used to illustrate the spread of the data and shows what values are likely. In contrast, standard error tells you about the precision of the mean and is used to calculate confidence intervals.

One straightforward way to compare two samples is to use confidence intervals. If we calculate the mean height of two groups and find that the 95% confidence intervals do not overlap, this can be taken as evidence of a difference between the two means. This method of statistical inference is reasonably intuitive and can be used in many situations. 2 Many journals, however, prefer to report inferential statistics using p -values.

Inference testing using a null hypothesis

In 1925, the British statistician R.A. Fisher described a technique for comparing groups using a null hypothesis , a method which has dominated statistical comparison ever since. The technique itself is rather straightforward, but often gets lost in the mechanics of how it is done. To illustrate, imagine we want to compare the HR of two different groups of people. We take a random sample from each group, which we call our data. Then:

• (i) Assume that both samples came from the same group. This is our ‘null hypothesis’.
• (ii) Calculate the probability that an experiment would give us these data, assuming that the null hypothesis is true. We express this probability as a p- value, a number between 0 and 1, where 0 is ‘impossible’ and 1 is ‘certain’.
• (iii) If the probability of the data is low, we reject the null hypothesis and conclude that there must be a difference between the two groups.

Formally, we can define a p- value as ‘the probability of finding the observed result or a more extreme result, if the null hypothesis were true.’ Standard practice is to set a cut-off at p <0.05 (this cut-off is termed the alpha value). If the null hypothesis were true, a result such as this would only occur 5% of the time or less; this in turn would indicate that the null hypothesis itself is unlikely. Fisher described the process as follows: ‘Set a low standard of significance at the 5 per cent point, and ignore entirely all results which fail to reach this level. A scientific fact should be regarded as experimentally established only if a properly designed experiment rarely fails to give this level of significance.’ 3 This probably remains the most succinct description of the procedure.

A question which often arises at this point is ‘Why do we use a null hypothesis?’ The simple answer is that it is easy: we can readily describe what we would expect of our data under a null hypothesis, we know how data would behave, and we can readily work out the probability of getting the result that we did. It therefore makes a very simple starting point for our probability assessment. All probabilities require a set of starting conditions, in much the same way that measuring the distance to London needs a starting point. The null hypothesis can be thought of as an easy place to put the start of your ruler.

If a null hypothesis is rejected, an alternate hypothesis must be adopted in its place. The null and alternate hypotheses must be mutually exclusive, but must also between them describe all situations. If a null hypothesis is ‘no difference exists’ then the alternate should be simply ‘a difference exists’.

Hypothesis testing in practice

The components of a hypothesis test can be readily described using the acronym GOST: identify the Groups you wish to compare; define the Outcome to be measured; collect and Summarise the data; then evaluate the likelihood of the null hypothesis, using a Test statistic .

When considering groups, think first about how many. Is there just one group being compared against an audit standard, or are you comparing one group with another? Some studies may wish to compare more than two groups. Another situation may involve a single group measured at different points in time, for example before or after a particular treatment. In this situation each participant is compared with themselves, and this is often referred to as a ‘paired’ or a ‘repeated measures’ design. It is possible to combine these types of groups—for example a researcher may measure arterial BP on a number of different occasions in five different groups of patients. Such studies can be difficult, both to analyse and interpret.

In other studies we may want to see how a continuous variable (such as age or height) affects the outcomes. These techniques involve regression analysis, and are beyond the scope of this article.

The outcome measures are the data being collected. This may be a continuous measure, such as temperature or BMI, or it may be a categorical measure, such as ASA status or surgical specialty. Often, inexperienced researchers will strive to collect lots of outcome measures in an attempt to find something that differs between the groups of interest; if this is done, a ‘primary outcome measure’ should be identified before the research begins. In addition, the results of any hypothesis tests will need to be corrected for multiple measures.

The summary and the test statistic will be defined by the type of data that have been collected. The test statistic is calculated then transformed into a p- value using tables or software. It is worth looking at two common tests in a little more detail: the χ 2 test, and the t -test.

Categorical data: the χ 2 test

The χ 2 test of independence is a test for comparing categorical outcomes in two or more groups. For example, a number of trials have compared surgical site infections in patients who have been given different concentrations of oxygen perioperatively. In the PROXI trial, 4 685 patients received oxygen 80%, and 701 patients received oxygen 30%. In the 80% group there were 131 infections, while in the 30% group there were 141 infections. In this study, the groups were oxygen 80% and oxygen 30%, and the outcome measure was the presence of a surgical site infection.

The summary is a table ( Table 1 ), and the hypothesis test compares this table (the ‘observed’ table) with the table that would be expected if the proportion of infections in each group was the same (the ‘expected’ table). The test statistic is χ 2 , from which a p- value is calculated. In this instance the p -value is 0.64, which means that results like this would occur 64% of the time if the null hypothesis were true. We thus have no evidence to reject the null hypothesis; the observed difference probably results from sampling variation rather than from an inherent difference between the two groups.

Table 1

Summary of the results of the PROXI trial. Figures are numbers of patients.

Continuous data: the t- test

The t- test is a statistical method for comparing means, and is one of the most widely used hypothesis tests. Imagine a study where we try to see if there is a difference in the onset time of a new neuromuscular blocking agent compared with suxamethonium. We could enlist 100 volunteers, give them a general anaesthetic, and randomise 50 of them to receive the new drug and 50 of them to receive suxamethonium. We then time how long it takes (in seconds) to have ideal intubation conditions, as measured by a quantitative nerve stimulator. Our data are therefore a list of times. In this case, the groups are ‘new drug’ and suxamethonium, and the outcome is time, measured in seconds. This can be summarised by using means; the hypothesis test will compare the means of the two groups, using a p- value calculated from a ‘ t statistic’. Hopefully it is becoming obvious at this point that the test statistic is usually identified by a letter, and this letter is often cited in the name of the test.

The t -test comes in a number of guises, depending on the comparison being made. A single sample can be compared with a standard (Is the BMI of school leavers in this town different from the national average?); two samples can be compared with each other, as in the example above; or the same study subjects can be measured at two different times. The latter case is referred to as a paired t- test, because each participant provides a pair of measurements—such as in a pre- or postintervention study.

A large number of methods for testing hypotheses exist; the commonest ones and their uses are described in Table 2 . In each case, the test can be described by detailing the groups being compared ( Table 2 , columns) the outcome measures (rows), the summary, and the test statistic. The decision to use a particular test or method should be made during the planning stages of a trial or experiment. At this stage, an estimate needs to be made of how many test subjects will be needed. Such calculations are described in detail elsewhere. 5

Table 2

The principle types of hypothesis test. Tests comparing more than two samples can indicate that one group differs from the others, but will not identify which. Subsequent ‘post hoc’ testing is required if a difference is found.

Controversies surrounding hypothesis testing

Although hypothesis tests have been the basis of modern science since the middle of the 20th century, they have been plagued by misconceptions from the outset; this has led to what has been described as a crisis in science in the last few years: some journals have gone so far as to ban p -value s outright. 6 This is not because of any flaw in the concept of a p -value, but because of a lack of understanding of what they mean.

Possibly the most pervasive misunderstanding is the belief that the p- value is the chance that the null hypothesis is true, or that the p- value represents the frequency with which you will be wrong if you reject the null hypothesis (i.e. claim to have found a difference). This interpretation has frequently made it into the literature, and is a very easy trap to fall into when discussing hypothesis tests. To avoid this, it is important to remember that the p- value is telling us something about our sample , not about the null hypothesis. Put in simple terms, we would like to know the probability that the null hypothesis is true, given our data. The p- value tells us the probability of getting these data if the null hypothesis were true, which is not the same thing. This fallacy is referred to as ‘flipping the conditional’; the probability of an outcome under certain conditions is not the same as the probability of those conditions given that the outcome has happened.

A useful example is to imagine a magic trick in which you select a card from a normal deck of 52 cards, and the performer reveals your chosen card in a surprising manner. If the performer were relying purely on chance, this would only happen on average once in every 52 attempts. On the basis of this, we conclude that it is unlikely that the magician is simply relying on chance. Although simple, we have just performed an entire hypothesis test. We have declared a null hypothesis (the performer was relying on chance); we have even calculated a p -value (1 in 52, ≈0.02); and on the basis of this low p- value we have rejected our null hypothesis. We would, however, be wrong to suggest that there is a probability of 0.02 that the performer is relying on chance—that is not what our figure of 0.02 is telling us.

To explore this further we can create two populations, and watch what happens when we use simulation to take repeated samples to compare these populations. Computers allow us to do this repeatedly, and to see what p- value s are generated (see Supplementary online material). 7 Fig 1 illustrates the results of 100,000 simulated t -tests, generated in two set of circumstances. In Fig 1 a , we have a situation in which there is a difference between the two populations. The p- value s cluster below the 0.05 cut-off, although there is a small proportion with p >0.05. Interestingly, the proportion of comparisons where p <0.05 is 0.8 or 80%, which is the power of the study (the sample size was specifically calculated to give a power of 80%).

The p- value s generated when 100,000 t -tests are used to compare two samples taken from defined populations. ( a ) The populations have a difference and the p- value s are mostly significant. ( b ) The samples were taken from the same population (i.e. the null hypothesis is true) and the p- value s are distributed uniformly.

Figure 1 b depicts the situation where repeated samples are taken from the same parent population (i.e. the null hypothesis is true). Somewhat surprisingly, all p- value s occur with equal frequency, with p <0.05 occurring exactly 5% of the time. Thus, when the null hypothesis is true, a type I error will occur with a frequency equal to the alpha significance cut-off.

Figure 1 highlights the underlying problem: when presented with a p -value <0.05, is it possible with no further information, to determine whether you are looking at something from Fig 1 a or Fig 1 b ?

Finally, it cannot be stressed enough that although hypothesis testing identifies whether or not a difference is likely, it is up to us as clinicians to decide whether or not a statistically significant difference is also significant clinically.

Hypothesis testing: what next?

As mentioned above, some have suggested moving away from p -values, but it is not entirely clear what we should use instead. Some sources have advocated focussing more on effect size; however, without a measure of significance we have merely returned to our original problem: how do we know that our difference is not just a result of sampling variation?

One solution is to use Bayesian statistics. Up until very recently, these techniques have been considered both too difficult and not sufficiently rigorous. However, recent advances in computing have led to the development of Bayesian equivalents of a number of standard hypothesis tests. 8 These generate a ‘Bayes Factor’ (BF), which tells us how more (or less) likely the alternative hypothesis is after our experiment. A BF of 1.0 indicates that the likelihood of the alternate hypothesis has not changed. A BF of 10 indicates that the alternate hypothesis is 10 times more likely than we originally thought. A number of classifications for BF exist; greater than 10 can be considered ‘strong evidence’, while BF greater than 100 can be classed as ‘decisive’.

Figures such as the BF can be quoted in conjunction with the traditional p- value, but it remains to be seen whether they will become mainstream.

Declaration of interest

The author declares that they have no conflict of interest.

The associated MCQs (to support CME/CPD activity) will be accessible at www.bjaed.org/cme/home by subscribers to BJA Education .

Jason Walker FRCA FRSS BSc (Hons) Math Stat is a consultant anaesthetist at Ysbyty Gwynedd Hospital, Bangor, Wales, and an honorary senior lecturer at Bangor University. He is vice chair of his local research ethics committee, and an examiner for the Primary FRCA.

Matrix codes: 1A03, 2A04, 3J03

Supplementary data to this article can be found online at https://doi.org/10.1016/j.bjae.2019.03.006 .

Supplementary material

The following is the Supplementary data to this article:

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• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

• An independent variable is something the researcher changes or controls.
• A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias  will affect your results.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1. Ask a question

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2. Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in  if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

• H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
• H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

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A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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What Is A Research (Scientific) Hypothesis? A plain-language explainer + examples

By:  Derek Jansen (MBA)  | Reviewed By: Dr Eunice Rautenbach | June 2020

If you’re new to the world of research, or it’s your first time writing a dissertation or thesis, you’re probably noticing that the words “research hypothesis” and “scientific hypothesis” are used quite a bit, and you’re wondering what they mean in a research context .

“Hypothesis” is one of those words that people use loosely, thinking they understand what it means. However, it has a very specific meaning within academic research. So, it’s important to understand the exact meaning before you start hypothesizing. 

Research Hypothesis 101

• What is a hypothesis ?
• What is a research hypothesis (scientific hypothesis)?
• Requirements for a research hypothesis
• Definition of a research hypothesis
• The null hypothesis

What is a hypothesis?

Let’s start with the general definition of a hypothesis (not a research hypothesis or scientific hypothesis), according to the Cambridge Dictionary:

Hypothesis: an idea or explanation for something that is based on known facts but has not yet been proved.

In other words, it’s a statement that provides an explanation for why or how something works, based on facts (or some reasonable assumptions), but that has not yet been specifically tested . For example, a hypothesis might look something like this:

Hypothesis: sleep impacts academic performance.

This statement predicts that academic performance will be influenced by the amount and/or quality of sleep a student engages in – sounds reasonable, right? It’s based on reasonable assumptions , underpinned by what we currently know about sleep and health (from the existing literature). So, loosely speaking, we could call it a hypothesis, at least by the dictionary definition.

But that’s not good enough…

Unfortunately, that’s not quite sophisticated enough to describe a research hypothesis (also sometimes called a scientific hypothesis), and it wouldn’t be acceptable in a dissertation, thesis or research paper . In the world of academic research, a statement needs a few more criteria to constitute a true research hypothesis .

What is a research hypothesis?

A research hypothesis (also called a scientific hypothesis) is a statement about the expected outcome of a study (for example, a dissertation or thesis). To constitute a quality hypothesis, the statement needs to have three attributes – specificity , clarity and testability .

Let’s take a look at these more closely.

Need a helping hand?

Hypothesis Essential #1: Specificity & Clarity

A good research hypothesis needs to be extremely clear and articulate about both what’ s being assessed (who or what variables are involved ) and the expected outcome (for example, a difference between groups, a relationship between variables, etc.).

Let’s stick with our sleepy students example and look at how this statement could be more specific and clear.

Hypothesis: Students who sleep at least 8 hours per night will, on average, achieve higher grades in standardised tests than students who sleep less than 8 hours a night.

As you can see, the statement is very specific as it identifies the variables involved (sleep hours and test grades), the parties involved (two groups of students), as well as the predicted relationship type (a positive relationship). There’s no ambiguity or uncertainty about who or what is involved in the statement, and the expected outcome is clear.

Contrast that to the original hypothesis we looked at – “Sleep impacts academic performance” – and you can see the difference. “Sleep” and “academic performance” are both comparatively vague , and there’s no indication of what the expected relationship direction is (more sleep or less sleep). As you can see, specificity and clarity are key.

Hypothesis Essential #2: Testability (Provability)

A statement must be testable to qualify as a research hypothesis. In other words, there needs to be a way to prove (or disprove) the statement. If it’s not testable, it’s not a hypothesis – simple as that.

For example, consider the hypothesis we mentioned earlier:

Hypothesis: Students who sleep at least 8 hours per night will, on average, achieve higher grades in standardised tests than students who sleep less than 8 hours a night.  

We could test this statement by undertaking a quantitative study involving two groups of students, one that gets 8 or more hours of sleep per night for a fixed period, and one that gets less. We could then compare the standardised test results for both groups to see if there’s a statistically significant difference. 

Again, if you compare this to the original hypothesis we looked at – “Sleep impacts academic performance” – you can see that it would be quite difficult to test that statement, primarily because it isn’t specific enough. How much sleep? By who? What type of academic performance?

So, remember the mantra – if you can’t test it, it’s not a hypothesis 🙂

Defining A Research Hypothesis

You’re still with us? Great! Let’s recap and pin down a clear definition of a hypothesis.

A research hypothesis (or scientific hypothesis) is a statement about an expected relationship between variables, or explanation of an occurrence, that is clear, specific and testable.

So, when you write up hypotheses for your dissertation or thesis, make sure that they meet all these criteria. If you do, you’ll not only have rock-solid hypotheses but you’ll also ensure a clear focus for your entire research project.

What about the null hypothesis?

You may have also heard the terms null hypothesis , alternative hypothesis, or H-zero thrown around. At a simple level, the null hypothesis is the counter-proposal to the original hypothesis.

For example, if the hypothesis predicts that there is a relationship between two variables (for example, sleep and academic performance), the null hypothesis would predict that there is no relationship between those variables.

At a more technical level, the null hypothesis proposes that no statistical significance exists in a set of given observations and that any differences are due to chance alone.

And there you have it – hypotheses in a nutshell. 

If you have any questions, be sure to leave a comment below and we’ll do our best to help you. If you need hands-on help developing and testing your hypotheses, consider our private coaching service , where we hold your hand through the research journey.

17 Comments

Very useful information. I benefit more from getting more information in this regard.

Very great insight,educative and informative. Please give meet deep critics on many research data of public international Law like human rights, environment, natural resources, law of the sea etc

In a book I read a distinction is made between null, research, and alternative hypothesis. As far as I understand, alternative and research hypotheses are the same. Can you please elaborate? Best Afshin

This is a self explanatory, easy going site. I will recommend this to my friends and colleagues.

Very good definition. How can I cite your definition in my thesis? Thank you. Is nul hypothesis compulsory in a research?

It’s a counter-proposal to be proven as a rejection

Please what is the difference between alternate hypothesis and research hypothesis?

It is a very good explanation. However, it limits hypotheses to statistically tasteable ideas. What about for qualitative researches or other researches that involve quantitative data that don’t need statistical tests?

In qualitative research, one typically uses propositions, not hypotheses.

could you please elaborate it more

I’ve benefited greatly from these notes, thank you.

This is very helpful

well articulated ideas are presented here, thank you for being reliable sources of information

Excellent. Thanks for being clear and sound about the research methodology and hypothesis (quantitative research)

I have only a simple question regarding the null hypothesis. – Is the null hypothesis (Ho) known as the reversible hypothesis of the alternative hypothesis (H1? – How to test it in academic research?

this is very important note help me much more

Hi” best wishes to you and your very nice blog” 

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Home » What is a Hypothesis – Types, Examples and Writing Guide

What is a Hypothesis – Types, Examples and Writing Guide

Table of Contents

Definition:

Hypothesis is an educated guess or proposed explanation for a phenomenon, based on some initial observations or data. It is a tentative statement that can be tested and potentially proven or disproven through further investigation and experimentation.

Hypothesis is often used in scientific research to guide the design of experiments and the collection and analysis of data. It is an essential element of the scientific method, as it allows researchers to make predictions about the outcome of their experiments and to test those predictions to determine their accuracy.

Types of Hypothesis

Types of Hypothesis are as follows:

Research Hypothesis

A research hypothesis is a statement that predicts a relationship between variables. It is usually formulated as a specific statement that can be tested through research, and it is often used in scientific research to guide the design of experiments.

Null Hypothesis

The null hypothesis is a statement that assumes there is no significant difference or relationship between variables. It is often used as a starting point for testing the research hypothesis, and if the results of the study reject the null hypothesis, it suggests that there is a significant difference or relationship between variables.

Alternative Hypothesis

An alternative hypothesis is a statement that assumes there is a significant difference or relationship between variables. It is often used as an alternative to the null hypothesis and is tested against the null hypothesis to determine which statement is more accurate.

Directional Hypothesis

A directional hypothesis is a statement that predicts the direction of the relationship between variables. For example, a researcher might predict that increasing the amount of exercise will result in a decrease in body weight.

Non-directional Hypothesis

A non-directional hypothesis is a statement that predicts the relationship between variables but does not specify the direction. For example, a researcher might predict that there is a relationship between the amount of exercise and body weight, but they do not specify whether increasing or decreasing exercise will affect body weight.

Statistical Hypothesis

A statistical hypothesis is a statement that assumes a particular statistical model or distribution for the data. It is often used in statistical analysis to test the significance of a particular result.

Composite Hypothesis

A composite hypothesis is a statement that assumes more than one condition or outcome. It can be divided into several sub-hypotheses, each of which represents a different possible outcome.

Empirical Hypothesis

An empirical hypothesis is a statement that is based on observed phenomena or data. It is often used in scientific research to develop theories or models that explain the observed phenomena.

Simple Hypothesis

A simple hypothesis is a statement that assumes only one outcome or condition. It is often used in scientific research to test a single variable or factor.

Complex Hypothesis

A complex hypothesis is a statement that assumes multiple outcomes or conditions. It is often used in scientific research to test the effects of multiple variables or factors on a particular outcome.

Applications of Hypothesis

Hypotheses are used in various fields to guide research and make predictions about the outcomes of experiments or observations. Here are some examples of how hypotheses are applied in different fields:

• Science : In scientific research, hypotheses are used to test the validity of theories and models that explain natural phenomena. For example, a hypothesis might be formulated to test the effects of a particular variable on a natural system, such as the effects of climate change on an ecosystem.
• Medicine : In medical research, hypotheses are used to test the effectiveness of treatments and therapies for specific conditions. For example, a hypothesis might be formulated to test the effects of a new drug on a particular disease.
• Psychology : In psychology, hypotheses are used to test theories and models of human behavior and cognition. For example, a hypothesis might be formulated to test the effects of a particular stimulus on the brain or behavior.
• Sociology : In sociology, hypotheses are used to test theories and models of social phenomena, such as the effects of social structures or institutions on human behavior. For example, a hypothesis might be formulated to test the effects of income inequality on crime rates.
• Business : In business research, hypotheses are used to test the validity of theories and models that explain business phenomena, such as consumer behavior or market trends. For example, a hypothesis might be formulated to test the effects of a new marketing campaign on consumer buying behavior.
• Engineering : In engineering, hypotheses are used to test the effectiveness of new technologies or designs. For example, a hypothesis might be formulated to test the efficiency of a new solar panel design.

How to write a Hypothesis

Here are the steps to follow when writing a hypothesis:

Identify the Research Question

The first step is to identify the research question that you want to answer through your study. This question should be clear, specific, and focused. It should be something that can be investigated empirically and that has some relevance or significance in the field.

Conduct a Literature Review

Before writing your hypothesis, it’s essential to conduct a thorough literature review to understand what is already known about the topic. This will help you to identify the research gap and formulate a hypothesis that builds on existing knowledge.

Determine the Variables

The next step is to identify the variables involved in the research question. A variable is any characteristic or factor that can vary or change. There are two types of variables: independent and dependent. The independent variable is the one that is manipulated or changed by the researcher, while the dependent variable is the one that is measured or observed as a result of the independent variable.

Formulate the Hypothesis

Based on the research question and the variables involved, you can now formulate your hypothesis. A hypothesis should be a clear and concise statement that predicts the relationship between the variables. It should be testable through empirical research and based on existing theory or evidence.

Write the Null Hypothesis

The null hypothesis is the opposite of the alternative hypothesis, which is the hypothesis that you are testing. The null hypothesis states that there is no significant difference or relationship between the variables. It is important to write the null hypothesis because it allows you to compare your results with what would be expected by chance.

Refine the Hypothesis

After formulating the hypothesis, it’s important to refine it and make it more precise. This may involve clarifying the variables, specifying the direction of the relationship, or making the hypothesis more testable.

Examples of Hypothesis

Here are a few examples of hypotheses in different fields:

• Psychology : “Increased exposure to violent video games leads to increased aggressive behavior in adolescents.”
• Biology : “Higher levels of carbon dioxide in the atmosphere will lead to increased plant growth.”
• Sociology : “Individuals who grow up in households with higher socioeconomic status will have higher levels of education and income as adults.”
• Education : “Implementing a new teaching method will result in higher student achievement scores.”
• Marketing : “Customers who receive a personalized email will be more likely to make a purchase than those who receive a generic email.”
• Physics : “An increase in temperature will cause an increase in the volume of a gas, assuming all other variables remain constant.”
• Medicine : “Consuming a diet high in saturated fats will increase the risk of developing heart disease.”

Purpose of Hypothesis

The purpose of a hypothesis is to provide a testable explanation for an observed phenomenon or a prediction of a future outcome based on existing knowledge or theories. A hypothesis is an essential part of the scientific method and helps to guide the research process by providing a clear focus for investigation. It enables scientists to design experiments or studies to gather evidence and data that can support or refute the proposed explanation or prediction.

The formulation of a hypothesis is based on existing knowledge, observations, and theories, and it should be specific, testable, and falsifiable. A specific hypothesis helps to define the research question, which is important in the research process as it guides the selection of an appropriate research design and methodology. Testability of the hypothesis means that it can be proven or disproven through empirical data collection and analysis. Falsifiability means that the hypothesis should be formulated in such a way that it can be proven wrong if it is incorrect.

In addition to guiding the research process, the testing of hypotheses can lead to new discoveries and advancements in scientific knowledge. When a hypothesis is supported by the data, it can be used to develop new theories or models to explain the observed phenomenon. When a hypothesis is not supported by the data, it can help to refine existing theories or prompt the development of new hypotheses to explain the phenomenon.

When to use Hypothesis

Here are some common situations in which hypotheses are used:

• In scientific research , hypotheses are used to guide the design of experiments and to help researchers make predictions about the outcomes of those experiments.
• In social science research , hypotheses are used to test theories about human behavior, social relationships, and other phenomena.
• I n business , hypotheses can be used to guide decisions about marketing, product development, and other areas. For example, a hypothesis might be that a new product will sell well in a particular market, and this hypothesis can be tested through market research.

Characteristics of Hypothesis

Here are some common characteristics of a hypothesis:

• Testable : A hypothesis must be able to be tested through observation or experimentation. This means that it must be possible to collect data that will either support or refute the hypothesis.
• Falsifiable : A hypothesis must be able to be proven false if it is not supported by the data. If a hypothesis cannot be falsified, then it is not a scientific hypothesis.
• Clear and concise : A hypothesis should be stated in a clear and concise manner so that it can be easily understood and tested.
• Based on existing knowledge : A hypothesis should be based on existing knowledge and research in the field. It should not be based on personal beliefs or opinions.
• Specific : A hypothesis should be specific in terms of the variables being tested and the predicted outcome. This will help to ensure that the research is focused and well-designed.
• Tentative: A hypothesis is a tentative statement or assumption that requires further testing and evidence to be confirmed or refuted. It is not a final conclusion or assertion.
• Relevant : A hypothesis should be relevant to the research question or problem being studied. It should address a gap in knowledge or provide a new perspective on the issue.

Advantages of Hypothesis

Hypotheses have several advantages in scientific research and experimentation:

• Guides research: A hypothesis provides a clear and specific direction for research. It helps to focus the research question, select appropriate methods and variables, and interpret the results.
• Predictive powe r: A hypothesis makes predictions about the outcome of research, which can be tested through experimentation. This allows researchers to evaluate the validity of the hypothesis and make new discoveries.
• Facilitates communication: A hypothesis provides a common language and framework for scientists to communicate with one another about their research. This helps to facilitate the exchange of ideas and promotes collaboration.
• Efficient use of resources: A hypothesis helps researchers to use their time, resources, and funding efficiently by directing them towards specific research questions and methods that are most likely to yield results.
• Provides a basis for further research: A hypothesis that is supported by data provides a basis for further research and exploration. It can lead to new hypotheses, theories, and discoveries.
• Increases objectivity: A hypothesis can help to increase objectivity in research by providing a clear and specific framework for testing and interpreting results. This can reduce bias and increase the reliability of research findings.

Limitations of Hypothesis

Some Limitations of the Hypothesis are as follows:

• Limited to observable phenomena: Hypotheses are limited to observable phenomena and cannot account for unobservable or intangible factors. This means that some research questions may not be amenable to hypothesis testing.
• May be inaccurate or incomplete: Hypotheses are based on existing knowledge and research, which may be incomplete or inaccurate. This can lead to flawed hypotheses and erroneous conclusions.
• May be biased: Hypotheses may be biased by the researcher’s own beliefs, values, or assumptions. This can lead to selective interpretation of data and a lack of objectivity in research.
• Cannot prove causation: A hypothesis can only show a correlation between variables, but it cannot prove causation. This requires further experimentation and analysis.
• Limited to specific contexts: Hypotheses are limited to specific contexts and may not be generalizable to other situations or populations. This means that results may not be applicable in other contexts or may require further testing.
• May be affected by chance : Hypotheses may be affected by chance or random variation, which can obscure or distort the true relationship between variables.

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How to Write a Great Hypothesis

Hypothesis Definition, Format, Examples, and Tips

Verywell / Alex Dos Diaz

• The Scientific Method

Hypothesis Format

Falsifiability of a hypothesis.

• Operationalization

Hypothesis Types

Hypotheses examples.

• Collecting Data

A hypothesis is a tentative statement about the relationship between two or more variables. It is a specific, testable prediction about what you expect to happen in a study. It is a preliminary answer to your question that helps guide the research process.

Consider a study designed to examine the relationship between sleep deprivation and test performance. The hypothesis might be: "This study is designed to assess the hypothesis that sleep-deprived people will perform worse on a test than individuals who are not sleep-deprived."

At a Glance

A hypothesis is crucial to scientific research because it offers a clear direction for what the researchers are looking to find. This allows them to design experiments to test their predictions and add to our scientific knowledge about the world. This article explores how a hypothesis is used in psychology research, how to write a good hypothesis, and the different types of hypotheses you might use.

The Hypothesis in the Scientific Method

In the scientific method , whether it involves research in psychology, biology, or some other area, a hypothesis represents what the researchers think will happen in an experiment. The scientific method involves the following steps:

• Forming a question
• Performing background research
• Creating a hypothesis
• Designing an experiment
• Collecting data
• Analyzing the results
• Drawing conclusions
• Communicating the results

The hypothesis is a prediction, but it involves more than a guess. Most of the time, the hypothesis begins with a question which is then explored through background research. At this point, researchers then begin to develop a testable hypothesis.

Unless you are creating an exploratory study, your hypothesis should always explain what you  expect  to happen.

In a study exploring the effects of a particular drug, the hypothesis might be that researchers expect the drug to have some type of effect on the symptoms of a specific illness. In psychology, the hypothesis might focus on how a certain aspect of the environment might influence a particular behavior.

Remember, a hypothesis does not have to be correct. While the hypothesis predicts what the researchers expect to see, the goal of the research is to determine whether this guess is right or wrong. When conducting an experiment, researchers might explore numerous factors to determine which ones might contribute to the ultimate outcome.

In many cases, researchers may find that the results of an experiment  do not  support the original hypothesis. When writing up these results, the researchers might suggest other options that should be explored in future studies.

In many cases, researchers might draw a hypothesis from a specific theory or build on previous research. For example, prior research has shown that stress can impact the immune system. So a researcher might hypothesize: "People with high-stress levels will be more likely to contract a common cold after being exposed to the virus than people who have low-stress levels."

In other instances, researchers might look at commonly held beliefs or folk wisdom. "Birds of a feather flock together" is one example of folk adage that a psychologist might try to investigate. The researcher might pose a specific hypothesis that "People tend to select romantic partners who are similar to them in interests and educational level."

Elements of a Good Hypothesis

So how do you write a good hypothesis? When trying to come up with a hypothesis for your research or experiments, ask yourself the following questions:

• Is your hypothesis based on your research on a topic?
• Can your hypothesis be tested?
• Does your hypothesis include independent and dependent variables?

Before you come up with a specific hypothesis, spend some time doing background research. Once you have completed a literature review, start thinking about potential questions you still have. Pay attention to the discussion section in the  journal articles you read . Many authors will suggest questions that still need to be explored.

How to Formulate a Good Hypothesis

To form a hypothesis, you should take these steps:

• Collect as many observations about a topic or problem as you can.
• Evaluate these observations and look for possible causes of the problem.
• Create a list of possible explanations that you might want to explore.
• After you have developed some possible hypotheses, think of ways that you could confirm or disprove each hypothesis through experimentation. This is known as falsifiability.

In the scientific method ,  falsifiability is an important part of any valid hypothesis. In order to test a claim scientifically, it must be possible that the claim could be proven false.

Students sometimes confuse the idea of falsifiability with the idea that it means that something is false, which is not the case. What falsifiability means is that  if  something was false, then it is possible to demonstrate that it is false.

One of the hallmarks of pseudoscience is that it makes claims that cannot be refuted or proven false.

The Importance of Operational Definitions

A variable is a factor or element that can be changed and manipulated in ways that are observable and measurable. However, the researcher must also define how the variable will be manipulated and measured in the study.

Operational definitions are specific definitions for all relevant factors in a study. This process helps make vague or ambiguous concepts detailed and measurable.

For example, a researcher might operationally define the variable " test anxiety " as the results of a self-report measure of anxiety experienced during an exam. A "study habits" variable might be defined by the amount of studying that actually occurs as measured by time.

These precise descriptions are important because many things can be measured in various ways. Clearly defining these variables and how they are measured helps ensure that other researchers can replicate your results.

Replicability

One of the basic principles of any type of scientific research is that the results must be replicable.

Replication means repeating an experiment in the same way to produce the same results. By clearly detailing the specifics of how the variables were measured and manipulated, other researchers can better understand the results and repeat the study if needed.

Some variables are more difficult than others to define. For example, how would you operationally define a variable such as aggression ? For obvious ethical reasons, researchers cannot create a situation in which a person behaves aggressively toward others.

To measure this variable, the researcher must devise a measurement that assesses aggressive behavior without harming others. The researcher might utilize a simulated task to measure aggressiveness in this situation.

Hypothesis Checklist

• Does your hypothesis focus on something that you can actually test?
• Does your hypothesis include both an independent and dependent variable?
• Can you manipulate the variables?
• Can your hypothesis be tested without violating ethical standards?

The hypothesis you use will depend on what you are investigating and hoping to find. Some of the main types of hypotheses that you might use include:

• Simple hypothesis : This type of hypothesis suggests there is a relationship between one independent variable and one dependent variable.
• Complex hypothesis : This type suggests a relationship between three or more variables, such as two independent and dependent variables.
• Null hypothesis : This hypothesis suggests no relationship exists between two or more variables.
• Alternative hypothesis : This hypothesis states the opposite of the null hypothesis.
• Statistical hypothesis : This hypothesis uses statistical analysis to evaluate a representative population sample and then generalizes the findings to the larger group.
• Logical hypothesis : This hypothesis assumes a relationship between variables without collecting data or evidence.

A hypothesis often follows a basic format of "If {this happens} then {this will happen}." One way to structure your hypothesis is to describe what will happen to the  dependent variable  if you change the  independent variable .

The basic format might be: "If {these changes are made to a certain independent variable}, then we will observe {a change in a specific dependent variable}."

A few examples of simple hypotheses:

• "Students who eat breakfast will perform better on a math exam than students who do not eat breakfast."
• "Students who experience test anxiety before an English exam will get lower scores than students who do not experience test anxiety."​
• "Motorists who talk on the phone while driving will be more likely to make errors on a driving course than those who do not talk on the phone."
• "Children who receive a new reading intervention will have higher reading scores than students who do not receive the intervention."

Examples of a complex hypothesis include:

• "People with high-sugar diets and sedentary activity levels are more likely to develop depression."
• "Younger people who are regularly exposed to green, outdoor areas have better subjective well-being than older adults who have limited exposure to green spaces."

Examples of a null hypothesis include:

• "There is no difference in anxiety levels between people who take St. John's wort supplements and those who do not."
• "There is no difference in scores on a memory recall task between children and adults."
• "There is no difference in aggression levels between children who play first-person shooter games and those who do not."

Examples of an alternative hypothesis:

• "People who take St. John's wort supplements will have less anxiety than those who do not."
• "Adults will perform better on a memory task than children."
• "Children who play first-person shooter games will show higher levels of aggression than children who do not." 

Collecting Data on Your Hypothesis

Once a researcher has formed a testable hypothesis, the next step is to select a research design and start collecting data. The research method depends largely on exactly what they are studying. There are two basic types of research methods: descriptive research and experimental research.

Descriptive Research Methods

Descriptive research such as  case studies ,  naturalistic observations , and surveys are often used when  conducting an experiment is difficult or impossible. These methods are best used to describe different aspects of a behavior or psychological phenomenon.

Once a researcher has collected data using descriptive methods, a  correlational study  can examine how the variables are related. This research method might be used to investigate a hypothesis that is difficult to test experimentally.

Experimental Research Methods

Experimental methods  are used to demonstrate causal relationships between variables. In an experiment, the researcher systematically manipulates a variable of interest (known as the independent variable) and measures the effect on another variable (known as the dependent variable).

Unlike correlational studies, which can only be used to determine if there is a relationship between two variables, experimental methods can be used to determine the actual nature of the relationship—whether changes in one variable actually  cause  another to change.

The hypothesis is a critical part of any scientific exploration. It represents what researchers expect to find in a study or experiment. In situations where the hypothesis is unsupported by the research, the research still has value. Such research helps us better understand how different aspects of the natural world relate to one another. It also helps us develop new hypotheses that can then be tested in the future.

Thompson WH, Skau S. On the scope of scientific hypotheses .  R Soc Open Sci . 2023;10(8):230607. doi:10.1098/rsos.230607

Taran S, Adhikari NKJ, Fan E. Falsifiability in medicine: what clinicians can learn from Karl Popper [published correction appears in Intensive Care Med. 2021 Jun 17;:].  Intensive Care Med . 2021;47(9):1054-1056. doi:10.1007/s00134-021-06432-z

Eyler AA. Research Methods for Public Health . 1st ed. Springer Publishing Company; 2020. doi:10.1891/9780826182067.0004

Nosek BA, Errington TM. What is replication ?  PLoS Biol . 2020;18(3):e3000691. doi:10.1371/journal.pbio.3000691

Aggarwal R, Ranganathan P. Study designs: Part 2 - Descriptive studies .  Perspect Clin Res . 2019;10(1):34-36. doi:10.4103/picr.PICR_154_18

Nevid J. Psychology: Concepts and Applications. Wadworth, 2013.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

• 5.4 - A Matrix Formulation of the Multiple Regression Model

Note: This portion of the lesson is most important for those students who will continue studying statistics after taking Stat 462. We will only rarely use the material within the remainder of this course.

A matrix formulation of the multiple regression model

In the multiple regression setting, because of the potentially large number of predictors, it is more efficient to use matrices to define the regression model and the subsequent analyses. Here, we review basic matrix algebra, as well as learn some of the more important multiple regression formulas in matrix form.

As always, let's start with the simple case first. Consider the following simple linear regression function:

\[y_i=\beta_0+\beta_1x_i+\epsilon_i  \;\;\;\;\;\;\; \text {for } i=1, ... , n\]

If we actually let i = 1, ..., n , we see that we obtain n equations:

\[\begin{align} y_1 & =\beta_0+\beta_1x_1+\epsilon_1 \\ y_2 & =\beta_0+\beta_1x_2+\epsilon_2 \\ \vdots \\ y_n & = \beta_0+\beta_1x_n+\epsilon_n \end{align}\]

Well, that's a pretty inefficient way of writing it all out! As you can see, there is a pattern that emerges. By taking advantage of this pattern, we can instead formulate the above simple linear regression function in matrix notation:

That is, instead of writing out the n equations, using matrix notation, our simple linear regression function reduces to a short and simple statement:

\[Y=X\beta+\epsilon\]

Now, what does this statement mean? Well, here's the answer:

• X is an n × 2 matrix .
• Y is an n × 1 column vector , β is a 2 × 1 column vector, and ε is an n × 1 column vector.
• The matrix X and vector β are multiplied together using the techniques of matrix multiplication .
• And, the vector X β is added to the vector ε using the techniques of matrix addition .

Now, that might not mean anything to you, if you've never studied matrix algebra — or if you have and you forgot it all! So, let's start with a quick and basic review.

Definition of a matrix

An r × c matrix is a rectangular array of symbols or numbers arranged in r rows and c columns. A matrix is almost always denoted by a single capital letter in boldface type.

Here are three examples of simple matrices. The matrix A is a 2 × 2 square matrix containing numbers:

\[A=\begin{bmatrix}  1&2 \\ 6 & 3 \end{bmatrix}\]

The matrix B is a 5 × 3 matrix containing numbers:

\[B=\begin{bmatrix} 1 & 80 &3.4\\ 1 & 92 & 3.1\\ 1 & 65 &2.5\\ 1 &71  & 2.8\\ 1 & 40 & 1.9 \end{bmatrix}\]

And, the matrix X is a 6 × 3 matrix containing a column of 1's and two columns of various x variables:

\[X=\begin{bmatrix} 1 & x_{11}&x_{12}\\ 1 & x_{21}& x_{22}\\ 1 & x_{31}&x_{32}\\ 1 &x_{41}& x_{42}\\ 1 & x_{51}& x_{52}\\ 1 & x_{61}& x_{62}\\ \end{bmatrix}\]

Definition of a vector and a scalar

A column vector is an r × 1 matrix, that is, a matrix with only one column. A vector is almost often denoted by a single lowercase letter in boldface type. The following vector q is a 3 × 1 column vector containing numbers:

\[q=\begin{bmatrix} 2\\ 5\\ 8\end{bmatrix}\]

A row vector is an 1 × c matrix, that is, a matrix with only one row. The vector h is a 1 × 4 row vector containing numbers:

\[h=\begin{bmatrix} 21 &46  & 32 & 90 \end{bmatrix}\]

A 1 × 1 "matrix" is called a scalar , but it's just an ordinary number, such as 29 or σ 2 .

Matrix multiplication

Recall that X β that appears in the regression function:

is an example of matrix multiplication. Now, there are some restrictions — you can't just multiply any two old matrices together. Two matrices can be multiplied together only if the number of columns of the first matrix equals the number of rows of the second matrix. Then, when you multiply the two matrices:

• the number of rows of the resulting matrix equals the number of rows of the first matrix, and
• the number of columns of the resulting matrix equals the number of columns of the second matrix.

For example, if A is a 2 × 3 matrix and B is a 3 × 5 matrix, then the matrix multiplication AB is possible. The resulting matrix C = AB has 2 rows and 5 columns. That is, C is a 2 × 5 matrix. Note that the matrix multiplication BA is not possible.

For another example, if X is an n × ( k +1) matrix and β is a  ( k +1)  × 1 column vector, then the matrix multiplication X β is possible. The resulting matrix X β has n rows and 1 column. That is, X β is an n × 1 column vector.

Okay, now that we know when we can multiply two matrices together, how do we do it? Here's the basic rule for multiplying A by B to get C = A B :

The entry in the i th row and j th column of C is the inner product — that is, element-by-element products added together — of the i th row of A with the j th column of B.

For example:

\[C=AB=\begin{bmatrix} 1&9&7 \\ 8&1&2 \end{bmatrix}\begin{bmatrix} 3&2&1&5 \\ 5&4&7&3 \\ 6&9&6&8 \end{bmatrix}=\begin{bmatrix} 90&101&106&88 \\ 41&38&27&59 \end{bmatrix}\]

That is, the entry in the first row and first column of C , denoted c 11 , is obtained by:

\[c_{11}=1(3)+9(5)+7(6)=90\]

And, the entry in the first row and second column of C , denoted c 12 , is obtained by:

\[c_{12}=1(2)+9(4)+7(9)=101\]

And, the entry in the second row and third column of C , denoted c 23 , is obtained by:

\[c_{23}=8(1)+1(7)+2(6)=27\]

You might convince yourself that the remaining five elements of C have been obtained correctly.

Matrix addition

Recall that X β + ε that appears in the regression function:

is an example of matrix addition. Again, there are some restrictions — you can't just add any two old matrices together. Two matrices can be added together only if they have the same number of rows and columns. Then, to add two matrices, simply add the corresponding elements of the two matrices. That is:

• Add the entry in the first row, first column of the first matrix with the entry in the first row, first column of the second matrix.
• Add the entry in the first row, second column of the first matrix with the entry in the first row, second column of the second matrix.
• And, so on.

\[C=A+B=\begin{bmatrix}  2&4&-1\\  1&8&7\\ 3&5&6 \end{bmatrix}+\begin{bmatrix} 7 & 5 & 2\\ 9 & -3 & 1\\ 2 & 1 & 8 \end{bmatrix}=\begin{bmatrix} 9 & 9 & 1\\ 10 & 5 & 8\\ 5 & 6 & 14 \end{bmatrix}\]

\[c_{11}=2+7=9\]

\[c_{12}=4+5=9\]

You might convince yourself that the remaining seven elements of C have been obtained correctly.

Least squares estimates in matrix notation

Here's the punchline: the  ( k +1)  × 1 vector containing the estimates of the  ( k +1)  parameters of the regression function can be shown to equal:

\[ b=\begin{bmatrix} b_0 \\ b_1 \\ \vdots \\ b_{k} \end{bmatrix}= (X^{'}X)^{-1}X^{'}Y \]

• ( X ' X ) –1 is the inverse of the X ' X matrix, and
• X ' is the transpose of the X matrix.

As before, that might not mean anything to you, if you've never studied matrix algebra — or if you have and you forgot it all! So, let's go off and review inverses and transposes of matrices.

Definition of the transpose of a matrix

The transpose of a matrix A is a matrix, denoted A' or A T , whose rows are the columns of A and whose columns are the rows of A — all in the same order. For example, the transpose of the 3 × 2 matrix A :

\[A=\begin{bmatrix}  1&5 \\  4&8 \\  7&9 \end{bmatrix}\]

is the 2 × 3 matrix A ':

\[A^{'}=A^T=\begin{bmatrix}  1& 4 & 7\\ 5 & 8 & 9 \end{bmatrix}\]

And, since the X matrix in the simple linear regression setting is:

\[X=\begin{bmatrix} 1 & x_1\\ 1 & x_2\\ \vdots & \vdots\\ 1 & x_n \end{bmatrix}\]

the X ' X matrix in the simple linear regression setting must be:

\[X^{'}X=\begin{bmatrix} 1 & 1 & \cdots & 1\\ x_1 & x_2 & \cdots & x_n \end{bmatrix}\begin{bmatrix} 1 & x_1\\ 1 & x_2\\ \vdots &  x_n\\  1&   \end{bmatrix}=\begin{bmatrix} n & \sum_{i=1}^{n}x_i \\ \sum_{i=1}^{n}x_i  & \sum_{i=1}^{n}x_{i}^{2} \end{bmatrix}\]

Definition of the identity matrix

The square n × n identity matrix, denoted I n , is a matrix with 1's on the diagonal and 0's elsewhere. For example, the 2 × 2 identity matrix is:

\[I_2=\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\]

The identity matrix plays the same role as the number 1 in ordinary arithmetic:

\[\begin{bmatrix} 9 & 7\\ 4& 6 \end{bmatrix}\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}=\begin{bmatrix} 9& 7\\ 4& 6 \end{bmatrix}\]

That is, when you multiply a matrix by the identity, you get the same matrix back.

Definition of the inverse of a matrix

The inverse A -1 of a square (!!) matrix A is the unique matrix such that:

\[A^{-1}A = I = AA^{-1}\]

That is, the inverse of A is the matrix A -1 that you have to multiply A by in order to obtain the identity matrix I . Note that I am not just trying to be cute by including (!!) in that first sentence. The inverse only exists for square matrices!

Now, finding inverses is a really messy venture. The good news is that we'll always let computers find the inverses for us. In fact, we won't even know that statistical software is finding inverses behind the scenes!

Ugh! All of these definitions! Let's take a look at an example just to convince ourselves that, yes, indeed the least squares estimates are obtained by the following matrix formula:

\[b=\begin{bmatrix} b_0\\ b_1\\ \vdots\\ b_{p-1} \end{bmatrix}=(X^{'}X)^{-1}X^{'}Y\]

Let's see if we can obtain the same answer using the above matrix formula. We previously showed that:

\[X^{'}X=\begin{bmatrix} n & \sum_{i=1}^{n}x_i \\ \sum_{i=1}^{n}x_i  & \sum_{i=1}^{n}x_{i}^{2} \end{bmatrix}\]

We can easily calculate some parts of this formula:

That is, the 2 × 2 matrix X ' X is:

\[X^{'}X=\begin{bmatrix} 7 & 38.5\\ 38.5& 218.75 \end{bmatrix}\]

And, the 2 × 1 column vector X ' Y is:

\[X^{'}Y=\begin{bmatrix} \sum_{i=1}^{n}y_i\\ \sum_{i=1}^{n}x_iy_i \end{bmatrix}=\begin{bmatrix} 347\\ 1975 \end{bmatrix}\]

So, we've determined X ' X and X ' Y. Now, all we need to do is to find the inverse ( X ' X ) -1 . As mentioned before, it is very messy to determine inverses by hand. Letting computer software do the dirty work for us, it can be shown that the inverse of X ' X is:

\[(X^{'}X)^{-1}=\begin{bmatrix} 4.4643 & -0.78571\\ -0.78571& 0.14286 \end{bmatrix}\]

And so, putting all of our work together, we obtain the least squares estimates:

\[b=(X^{'}X)^{-1}X^{'}Y=\begin{bmatrix} 4.4643 & -0.78571\\ -0.78571& 0.14286 \end{bmatrix}\begin{bmatrix} 347\\ 1975 \end{bmatrix}=\begin{bmatrix} -2.67\\ 9.51 \end{bmatrix}\]

That is, the estimated intercept is b 0 = -2.67 and the estimated slope is b 1 = 9.51. Aha! Our estimates are the same as those reported above ( within rounding error)!

Further Matrix Results for Multiple Linear Regression

Matrix notation applies to other regression topics, including fitted values, residuals, sums of squares, and inferences about regression parameters. One important matrix that appears in many formulas is the so-called "hat matrix," \(H = X(X^{'}X)^{-1}X^{'}\), since it puts the hat on \(Y\)!

Linear Dependence

There is just one more really critical topic that we should address here, and that is linear dependence. We say that the columns of the matrix A :

\[A=\begin{bmatrix}  1& 2 & 4 &1 \\ 2 & 1 & 8 & 6\\ 3 & 6 & 12 & 3 \end{bmatrix}\]

are linearly dependent , since (at least) one of the columns can be written as a linear combination of another, namely the third column is 4 × the first column. If none of the columns can be written as a linear combination of the other columns, then we say the columns are linearly independent .

Unfortunately, linear dependence is not always obvious. For example, the columns in the following matrix A :

\[A=\begin{bmatrix}  1& 4 & 1 \\ 2 & 3 & 1\\ 3 & 2 & 1 \end{bmatrix}\]

are linearly dependent, because the first column plus the second column equals 5 × the third column.

Now, why should we care about linear dependence? Because the inverse of a square matrix exists only if the columns are linearly independent. Since the vector of regression estimates b depends on ( X ' X ) -1 , the parameter estimates b 0 , b 1 , and so on cannot be uniquely determined if some of the columns of X are linearly dependent! That is, if the columns of your X matrix — that is, two or more of your predictor variables — are linearly dependent (or nearly so), you will run into trouble when trying to estimate the regression equation.

For example, suppose for some strange reason we multiplied the predictor variable soap by 2 in the dataset soapsuds.txt. That is, we'd have two predictor variables, say soap1 (which is the original soap ) and soap2 (which is 2 × the original soap):

If we tried to regress y = suds on x 1 = soap1 and x 2 = soap2 , we see that statistical software spits out trouble:

In short, the first moral of the story is "don't collect your data in such a way that the predictor variables are perfectly correlated." And, the second moral of the story is "if your software package reports an error message concerning high correlation among your predictor variables, then think about linear dependence and how to get rid of it."

Start Here!

• Welcome to STAT 462!
• Search Course Materials
• Lesson 1: Statistical Inference Foundations
• Lesson 2: Simple Linear Regression (SLR) Model
• Lesson 3: SLR Evaluation
• Lesson 4: SLR Assumptions, Estimation & Prediction
• 5.1 - Example on IQ and Physical Characteristics
• 5.2 - Example on Underground Air Quality
• 5.3 - The Multiple Linear Regression Model
• 5.5 - Three Types of MLR Parameter Tests
• 5.6 - The General Linear F-Test
• 5.7 - MLR Parameter Tests
• 5.8 - Partial R-squared
• 5.9- Further MLR Examples
• Lesson 6: MLR Assumptions, Estimation & Prediction
• Lesson 7: Transformations & Interactions
• Lesson 8: Categorical Predictors
• Lesson 9: Influential Points
• Lesson 10: Regression Pitfalls
• Lesson 11: Model Building
• Lesson 12: Logistic, Poisson & Nonlinear Regression
• Website for Applied Regression Modeling, 2nd edition
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What Is a Hypothesis? (Science)

If...,Then...

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A hypothesis (plural hypotheses) is a proposed explanation for an observation. The definition depends on the subject.

In science, a hypothesis is part of the scientific method. It is a prediction or explanation that is tested by an experiment. Observations and experiments may disprove a scientific hypothesis, but can never entirely prove one.

In the study of logic, a hypothesis is an if-then proposition, typically written in the form, "If X , then Y ."

In common usage, a hypothesis is simply a proposed explanation or prediction, which may or may not be tested.

Writing a Hypothesis

Most scientific hypotheses are proposed in the if-then format because it's easy to design an experiment to see whether or not a cause and effect relationship exists between the independent variable and the dependent variable . The hypothesis is written as a prediction of the outcome of the experiment.

Null Hypothesis and Alternative Hypothesis

Statistically, it's easier to show there is no relationship between two variables than to support their connection. So, scientists often propose the null hypothesis . The null hypothesis assumes changing the independent variable will have no effect on the dependent variable.

In contrast, the alternative hypothesis suggests changing the independent variable will have an effect on the dependent variable. Designing an experiment to test this hypothesis can be trickier because there are many ways to state an alternative hypothesis.

For example, consider a possible relationship between getting a good night's sleep and getting good grades. The null hypothesis might be stated: "The number of hours of sleep students get is unrelated to their grades" or "There is no correlation between hours of sleep and grades."

An experiment to test this hypothesis might involve collecting data, recording average hours of sleep for each student and grades. If a student who gets eight hours of sleep generally does better than students who get four hours of sleep or 10 hours of sleep, the hypothesis might be rejected.

But the alternative hypothesis is harder to propose and test. The most general statement would be: "The amount of sleep students get affects their grades." The hypothesis might also be stated as "If you get more sleep, your grades will improve" or "Students who get nine hours of sleep have better grades than those who get more or less sleep."

In an experiment, you can collect the same data, but the statistical analysis is less likely to give you a high confidence limit.

Usually, a scientist starts out with the null hypothesis. From there, it may be possible to propose and test an alternative hypothesis, to narrow down the relationship between the variables.

Example of a Hypothesis

Examples of a hypothesis include:

• If you drop a rock and a feather, (then) they will fall at the same rate.
• Plants need sunlight in order to live. (if sunlight, then life)
• Eating sugar gives you energy. (if sugar, then energy)
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hypothesis , something supposed or taken for granted, with the object of following out its consequences (Greek hypothesis , “a putting under,” the Latin equivalent being suppositio ).

In planning a course of action, one may consider various alternatives , working out each in detail. Although the word hypothesis is not typically used in this case, the procedure is virtually the same as that of an investigator of crime considering various suspects. Different methods may be used for deciding what the various alternatives may be, but what is fundamental is the consideration of a supposal as if it were true, without actually accepting it as true. One of the earliest uses of the word in this sense was in geometry . It is described by Plato in the Meno .

The most important modern use of a hypothesis is in relation to scientific investigation . A scientist is not merely concerned to accumulate such facts as can be discovered by observation: linkages must be discovered to connect those facts. An initial puzzle or problem provides the impetus , but clues must be used to ascertain which facts will help yield a solution. The best guide is a tentative hypothesis, which fits within the existing body of doctrine. It is so framed that, with its help, deductions can be made that under certain factual conditions (“initial conditions”) certain other facts would be found if the hypothesis were correct.

The concepts involved in the hypothesis need not themselves refer to observable objects. However, the initial conditions should be able to be observed or to be produced experimentally, and the deduced facts should be able to be observed. William Harvey ’s research on circulation in animals demonstrates how greatly experimental observation can be helped by a fruitful hypothesis. While a hypothesis can be partially confirmed by showing that what is deduced from it with certain initial conditions is actually found under those conditions, it cannot be completely proved in this way. What would have to be shown is that no other hypothesis would serve. Hence, in assessing the soundness of a hypothesis, stress is laid on the range and variety of facts that can be brought under its scope. Again, it is important that it should be capable of being linked systematically with hypotheses which have been found fertile in other fields.

If the predictions derived from the hypothesis are not found to be true, the hypothesis may have to be given up or modified. The fault may lie, however, in some other principle forming part of the body of accepted doctrine which has been utilized in deducing consequences from the hypothesis. It may also lie in the fact that other conditions, hitherto unobserved, are present beside the initial conditions, affecting the result. Thus the hypothesis may be kept, pending further examination of facts or some remodeling of principles. A good illustration of this is to be found in the history of the corpuscular and the undulatory hypotheses about light .

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What is a contrast matrix?

What exactly is contrast matrix (a term, pertaining to an analysis with categorical predictors) and how exactly is contrast matrix specified? I.e. what are columns, what are rows, what are the constraints on that matrix and what does number in column j and row i mean? I tried to look into the docs and web but it seems that everyone uses it yet there's no definition anywhere. I could backward-engineer the available pre-defined contrasts, but I think the definition should be available without that.

• categorical-data
• categorical-encoding

• 2 $\begingroup$ "Contrast matrix" is used to represent categorical IVs (factors) in modeling. In particularly, it is used to recode a factor into a set of "contrast variables" (dummy variables being just an example). Each type of contrast variables has its own corresponding contrast matrix. See for example my own related question , not answered yet. $\endgroup$ –  ttnphns Commented Dec 3, 2013 at 7:21
• 9 $\begingroup$ @ttnphns Sorry but you keep doing what all the docs and webs do: you explain what are contrast matrices used for, without addressing the question what the contrast matrix is. This is the purpose of a definition . $\endgroup$ –  Tomas Commented Dec 3, 2013 at 10:39
• 6 $\begingroup$ Of course it is related, but deriving "what it is" from "what it is needed for" is a detective's job, which shouldn't be needed. That's reverse engineering. Things should be documented. $\endgroup$ –  Tomas Commented Dec 3, 2013 at 11:00
• 3 $\begingroup$ ats.ucla.edu/stat/r/library/contrast_coding.htm is a good R -oriented resource on coding methods. $\endgroup$ –  whuber ♦ Commented Jul 4, 2016 at 17:44
• 1 $\begingroup$ @Curious, just to let you know: I awarded 100 bounty to ttnphns, but I will start another bounty (or ask somebody else to do it) in order to award Gus_est as well. I have also written my own answer, just in case you prefer to have a shorter one :-) $\endgroup$ –  amoeba Commented Jul 8, 2016 at 23:46

5 Answers 5

In their nice answer, @Gus_est, undertook a mathematical explanation of the essence of the contrast coefficient matrix L (notated there a C ). $\bf Lb=k$ is the fundamental formula for testing hypotheses in univariate general linear modeling (where $\bf b$ are parameters and $\bf k$ are estimable function representing a null hypothesis), and that answer shows some necessary formulas used in modern ANOVA programs.

My answer is styled very differently. It is for a data analyst who sees himself rather an "engineer" than a "mathematician", so the answer will be a (superficial) "practical" or "didactic" account and will focus to answer just topics (1) what do the contrast coefficients mean and (2) how can they help to perform ANOVA via linear regression program.

ANOVA as regression with dummy variables: introducing contrasts .

Let us imagine ANOVA with dependent variable Y and categorical factor A having 3 levels (groups). Let us glance at the ANOVA from the linear regression point of view, that is - via turning the factor into the set of dummy (aka indicator aka treatment aka one-hot ) binary variables. This is our independent set X . (Probably everybody has heard that it is possible to do ANOVA this way - as linear regression with dummy predictors.)

Since one of the three groups is redundant, only two dummy variables will enter the linear model. Let's appoint Group3 to be redundant, or reference. The dummy predictors constituting X are an example of contrast variables , i.e. elementary variables representing categories of a factor. X itself is often called design matrix. We can now input the dataset in a multiple linear regression program which will center the data and find the regression coefficients (parameters) $\bf b= (X'X)^{-1}X'y=X^+y$, where "+" designates pseudoinverse.

Equivalent pass will be not to do the centering but rather add constant term of the model as the first column of 1 s in X , then estimate the coefficients same way as above $\bf b= (X'X)^{-1}X'y=X^+y$. So far so good.

Let us define matrix C to be the aggregation (summarization) of the independent variables design matrix X . It simply shows us the coding scheme observed there, - the contrast coding matrix (= basis matrix): $\bf C= {\it{aggr}} X$.

The colums are the variables (columns) of X - the elementary contrast variables A1 A2, dummy in this instance, and the rows are all the groups/levels of the factor. So was our coding matrix C for indicator or dummy contrast coding scheme.

Now, $\bf C^+=L$ is called the contrast coefficient matrix , or L-matrix. Since C is square, $\bf L=C^+=C^{-1}$. The contrast matrix, corresponding to our C - that is for indicator contrasts of our example - is therefore:

L-matrix is the matrix showing contrast coefficients . Note that sum of contrast coefficients in every row (except row Constant) is $0$. Every such row is called a contrast . Rows correspond to the contrast variables and columns correspond to the groups, factor levels.

The significance of contrast coefficients is that they help understand what each effect (each parameter b estimated in the regression with our X , coded as it is) represent in the sense of the difference (the group comparison). We immediately see, following the coefficients, that the estimated Constant will equal the Y mean in the reference group; that parameter b1 (i.e. of dummy variable A1) will equal the difference: Y mean in group1 minus Y mean in group3; and parameter b2 is the difference: mean in group2 minus mean in group3.

Note : Saying "mean" right above (and further below) we mean estimated (predicted by the model) mean for a group, not the observed mean in a group.

An instructive remark : When we do a regression by binary predictor variables, the parameter of such a variable says about the difference in Y between variable=1 and variable=0 groups. However, in the situation when the binary variables are the set of k-1 dummy variables representing a k -level factor, the meaning of the parameter gets narrower : it shows the difference in Y between variable=1 and (not just variable=0 but even) reference_variable=1 groups.

Like $\bf X^+$ (after multiplied by $\bf y$) brings us values of b , similarly $\bf(\it{aggr} \bf X)^+$ brings in meanings of b .

OK, we've given the definition of contrast coefficient matrix L . Since $\bf L=C^+=C^{-1}$, symmetrically $\bf C=L^+=L^{-1}$, which means that if you were given or have constructed a contrast matrix L based on categorical factor(s) - to test that L in your analysis, then you have clue for how to code correctly your contrast predictor variables X in order to test the L via an ordinary regression software (i.e. the one processing just "continuous" variables the standard OLS way, and not recognizing categorical factors at all). In our present example the coding was - indicator (dummy) type variables.

ANOVA as regression: other contrast types .

Let us briefly observe other contrast types (= coding schemes, = parameterization styles) for a categorical factor A .

Deviation or effect contrasts . C and L matrices and parameter meaning:

By deviation coding, each group of the factor is being compared with the unweighted grand mean, while Constant is that grand mean. This is what you get in regression with contrast predictors X coded in deviation or effect "manner".

Simple contrasts . This contrasts/coding scheme is a hybrid of indicator and deviation types, it gives the meaning of Constant as in deviation type and the meaning of the other parameters as in indicator type:

Helmert contrasts . Compares each group (except reference) with the unweighted mean of the subsequent groups, and Constant is the unweighted grand mean. C and L matrces:

Difference or reverse Helmert contrasts . Compares each group (except reference) with the unweighted mean of the previous groups, and Constant is the unweighted grand mean.

Repeated contrasts . Compares each group (except reference) with the next group, and Constant is the unweighted grand mean.

The Question asks: how exactly is contrast matrix specified? Looking at the types of contrasts outlined so far it is possible to grasp how. Each type has its logic how to "fill in" the values in L . The logic reflects what each parameter means - what are the two combinations of groups it is planned to compare.

Polynomial contrasts . These are a bit special, nonlinear. The first effect is a linear one, the second is quadratic, next is cubic. I'm leaving here unaccounted the question how their C and L matrices are to be constructed and if they are the inverse of each other. Please consult with profound @Antoni Parellada's explanations of this type of contrast: 1 , 2 .

In balanced designs, Helmert, reverse Helmert, and polynomial contrasts are always orthogonal contrasts . Other types considered above are not orthogonal contrasts. Orthogonal (under balancedness) is the contrast where in contrast matrix L sum in each row (except Const) is zero and sum of products of the corresponding elements of each pair of rows is zero.

Here is the angle similarity measures (cosine and Pearson correlation) under different contrast types, except polynomial which I didn't test. Let us have single factor A with k levels, and it was then recoded into the set of k-1 contrast variables of a specific type. What are the values in the correlation or cosine matrix between these contrast variables?

I'm giving the table for information and leaving it uncommented. It is of some importance for a deeper glance into general linear modeling.

User-defined contrasts . This is what we compose to test a custom comparison hypothesis. Normally sum in every but the first row of L should be 0 which means that two groups or two compositions of groups are being compared in that row (i.e. by that parameter).

Where are the model parameters after all ?

Are they the rows or the columns of L ? Throughout the text above I was saying that parameters correspond to the rows of L , as the rows represent contrast-variables, the predictors. While the columns are levels of a factor, the groups. That may appear to fall in contradiction with such, for example, theoretical block from @Gus_est answer, where clearly the columns correspond to the parameters:

$H_0: \begin{bmatrix} 0 & 1 & -1 & \phantom{-}0 & \phantom{-}0 \\ 0 & 0 & \phantom{-}1 & -1 & \phantom{-}0 \\ 0 & 0 & \phantom{-}0 & \phantom{-}1 & -1 \end{bmatrix} \begin{bmatrix} \beta_0 \\ \beta_1 \\ \beta_2 \\ \beta_3 \\ \beta_4 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$

Actually, there is no contradiction and the answer to the "problem" is: both rows and columns of the contrast coefficient matrix correspond to the parameters! Just recall that contrasts (contrast variables), the rows, were initially created to represent nothing else than the factor levels: they are the levels except the omitted reference one. Compare please these two equivalent spelling of the L-matrix for the simple contrast:

The first one is what I've shown before, the second is more "theoretical" (for general linear model algebra) layout. Simply, a column corresponding to Constant term was added. Parameter coefficients b label the rows and columns. Parameter b3, as redundant, will be set to zero. You may pseudoinverse the second layout to get the coding matrix C , where inside in the bottom-right part you will find still the correct codes for contrast variables A1 and A2. That will be so for any contrast type described (except for indicator type - where the pseudoinverse of such rectangular layout won't give correct result; this is probably why simple contrast type was invented for convenience: contrast coefficients identical to indicator type, but for row Constant).

Contrast type and ANOVA table results .

ANOVA table shows effects as combined (aggregated) - for example main effect of factor A , whereas contrasts correspond to elementary effects, of contrast variables - A1, A2, and (omitted, reference) A3. The parameter estimates for the elementary terms depend on the type of the contrast selected, but the combined result - its mean square and significance level - is the same, whatever the type is. Omnibus ANOVA (say, one-way) null hypothesis that all the three means of A are equal may be put out in a number of equivalent statements, and each will correspond to a specific contrast type: $(\mu_1=\mu_2, \mu_2=\mu_3)$ = repeated type; $(\mu_1=\mu_{23}, \mu_2=\mu_3)$ = Helmert type; $(\mu_1=\mu_{123}, \mu_2=\mu_{123})$ = Deviation type; $(\mu_1=\mu_3, \mu_2=\mu_3)$ = indicator or simple types.

ANOVA programs implemented via general linear model paradigm can display both ANOVA table (combined effects: main, interactions) and parameter estimates table (elementary effects b ). Some programs may output the latter table correspondent to the contrast type as bid by the user, but most will output always the parameters correspondent to one type - often, indicator type, because ANOVA programs based on general linear model parameterize specifically dummy variables (most convenient to do) and then switch over for contrasts by special "linking" formulae interpreting the fixed dummy input to a (arbitrary) contrast.

Whereas in my answer - showing ANOVA as regression - the "link" is realized as early as at the level of the input X , which called to introduce the notion of the appropriarte coding schema for the data.

A few examples showing testing of ANOVA contrasts via usual regression .

Showing in SPSS the request a contrast type in ANOVA and getting the same result via linear regression. We have some dataset with Y and factors A (3 levels, reference=last) and B (4 levels, reference=last); find the data below later on.

Deviation contrasts example under full factorial model (A, B, A*B). Deviation type requested for both A and B (we might choose to demand different type for each factor, for your information).

Contrast coefficient matrix L for A and for B:

Request ANOVA program ( GLM in SPSS) to do analysis of variance and to output explicit results for deviation contrasts:

Deviation contrast type compared A=1 vs Grand unweighted Mean and A=2 with that same Mean. Red ellipses ink the difference estimates and their p-values. The combined effect over the factor A is inked by red rectangle. For factor B, everyting is analogously inked in blue. Displaying also the ANOVA table. Note there that the combined contrast effects equal the main effects in it.

Let us now create physically contrast variables dev_a1, dev_a2, dev_b1, dev_b2, dev_b3 and run regression. Invert the L -matrices to obtain the coding C matrices:

The column of ones (Constant) is omitted: because we'll use regular regression program (which internally centers variables, and is also intolerant to singularity) variable Constant won't be needed. Now create data X : actually no manual recoding of the factors into these values is needed, the one-stroke solution is $\bf X=DC$, where $\bf D$ is the indicator (dummy) variables, all k columns ( k is the number of levels in a factor).

Having created the contrast variables, multiply among those from different factors to get variables to represent interactions (our ANOVA model was full factorial): dev_a1b1, dev_a1b2, dev_a1b3, dev_a2b1, dev_a2b2, dev_a2b3. Then run multiple linear regression with all the predictors.

As expected, dev_a1 is the same as effect as was the contrast "Level 1 vs Mean"; dev_a2 is the same as was "Level 2 vs Mean", etc etc, - compare the inked parts with the ANOVA contrast analysis above.

Note that if we were not using interaction variables dev_a1b1, dev_a1b2... in regression the results will coincide with results of main-effects-only ANOVA contrast analysis.

Simple contrasts example under the same full factorial model (A, B, A*B).

ANOVA results for simple contrasts:

The overall results (ANOVA table) is the same as with deviation contrasts (not displaying now).

Create physically contrast variables sim_a1, sim_a2, sim_b1, sim_b2, sim_b3. The coding matrices by inverting of the L-matrices are (w/o Const column):

Create the data $\bf X=DC$ and add there the interaction contrast variables sim_a1b1, sim_a1b2, ... etc, as the products of the main effects contrast variables. Perform the regression.

As before, we see that the results of regression and ANOVA match. A regression parameter of a simple contrast variable is the difference (and significance test of it) between that level of the factor and the reference (the last, in our example) level of it.

The two-factor data used in the examples:

User defined contrast example. Let us have single factor F with 5 levels. I will create and test a set of custom orthogonal contrasts, in ANOVA and in regression.

The picture shows the process (one of possible) of combining/splitting among the 5 groups to obtain 4 orthogonal contrasts, and the L matrix of contrast coefficints resultant from that process is on the right. All the contrasts are orthogonal to each other: $\bf LL'$ is diagonal. (This example schema was years ago copied from D. Howell's book on Statistics for psychologist.)

Let us submit the matrix to SPSS' ANOVA procedure to test the contrasts. Well, we might submit even any one row (contrast) from the matrix, but we'll submit the whole matrix because - as in previous examples - we'll want to receive the same results via regression, and regression program will need the complete set of contrast variables (to be aware that they belong together to one factor!). We'll add the constant row to L, just as we did before, although if we don't need to test for the intercept we may safely omit it.

The overall contrasts effect (in the bottom of the pic) is not the same as the expected overall ANOVA effect:

but it is simply the artefact of our inserting Constant term into the L matrix. For, SPSS already implies Constant when user-defined contrasts are specified. Remove the constant row from L and we'll get the same contrasts results (matrix K on the pic above) except that L0 contrast won't be displayed. And the overall contrast effect will match the overall ANOVA:

OK, now create the contrast variables physically and submit them to regression. $\bf C=L^+$, $\bf X=DC$.

Observe the identity of results. The data used in this example:

Contrasts in other than (M)ANOVA analyses .

Wherever nominal predictors appear, the question of contrast (which contrast type to select for which predictor) arise. Some programs solve it behind the scene internally when the overall, omnibus results won't depend on the type selected. If you want a specific type to see more "elementary" results, you have to select. You select (or, rather, compose) a contrast also when you are testing a custom comparison hypothesis.

(M)ANOVA and Loglinear analysis, Mixed and sometimes Generalized linear modeling include options to treat predictors via different types of contrasts. But as I've tried to show, it is possible to create contrasts as contrast variables explicitly and by hand. Then, if you don't have ANOVA package at hand, you might do it - in many respects with as good luck - with multiple regression.

• 1 $\begingroup$ please do not restrict this answer just to anova if possible. The [anova] tag was added by @amoeba by the time when you answered my question, but I don't want the answer to be restricted just to anova. $\endgroup$ –  Tomas Commented Jul 6, 2016 at 10:07
• $\begingroup$ Thanks for writing and updating this answer! I have several questions, here is the first one. In your answer you introduced "contrast coding matrix" ($C$) and "contrast coefficient matrix" ($L$). (By the way, are these standard terms? When I google "contrast coding matrix", I get only 5 hits, two of which lead to this very page). The OP, however, asks about "contrast matrix" and also gives several examples of those as used in R ( see also this manual ). Am I right in understanding that this "contrast matrix" is your $C$ (not $L$)? $\endgroup$ –  amoeba Commented Jul 6, 2016 at 22:37
• $\begingroup$ @amoeba, I'm not familiar with "contrast matrix" and almost sure it stands for "contrast coefficient matrix" or L-matrix, which is an official or at least wide spread term in (M)ANOVA/GLM. "Contrast coding matrix" term is much less mentioned as it is simply the aggrigated view of the design matrix X; I've seen "basis matrix" word used in papers of one SPSS's senior statistician Dave Nichols. Absolutely, L (official label) and C (arbitrary label?) matrices are so closely related that one can hardly discuss one w/o the other. I suppose that "contrast matrix" should be considered as this pair. $\endgroup$ –  ttnphns Commented Jul 6, 2016 at 23:57
• 2 $\begingroup$ Yes, I agree. By now I am convinced that "contrast matrix" is a term that is only used in the R community and refers to the coding scheme. I checked the textbook that Gus_est refers to and they never use the term "contrast matrix", they only talk about "contrasts" (see my last comment under his answer). The OP clearly was asking about the "contrast matrix" in the R sense. $\endgroup$ –  amoeba Commented Jul 7, 2016 at 11:32
• 1 $\begingroup$ That L will determine what are you going to test, you aren't free anymore to choose what to test : No, I disagree with that. As far as I understand, one can "manually" perform a test that is not tied to the coding scheme. The formulas for that are provided in the Gus'es answer. I am not saying it's convenient in practice, I am just saying that it's possible. I think what you are saying is that C-matrix determines the meaning of each beta coefficient and the corresponding p-values will be for $\beta_i=0$. This is clear. But one can still "manually" test e.g. if $\beta_1-\beta_2/2-\beta_3/2=0$. $\endgroup$ –  amoeba Commented Jul 7, 2016 at 14:21

I'll use lower-case letters for vectors and upper-case letters for matrices.

In case of a linear model of the form: $$ \mathbf{y}=\mathbf{X} \boldsymbol{\beta} + \boldsymbol{\varepsilon} $$ where $\bf{X}$ is a $n \times (k+1)$ matrix of rank $k+1 \leq n$ , and we assume $\boldsymbol{\varepsilon} \sim \mathcal N(0,\sigma^2)$ .

We can estimate $\hat{\boldsymbol{\beta}}$ by $(\mathbf{X}^\top\mathbf{X})^{-1}\mathbf{X}^\top \mathbf{y}$ , since the inverse of $\mathbf{X}^\top \mathbf{X}$ exists.

Now, take an ANOVA case in which $\mathbf{X}$ is not full-rank anymore. The implication of this is that we don't have $(\mathbf{X}^\top\mathbf{X})^{-1}$ and we have to settle for the generalized inverse $(\mathbf{X}^\top\mathbf{X})^{-}$ .

One of the problems of using this generalized inverse is that it's not unique. Another problem is that we cannot find an unbiased estimator for $\boldsymbol{\beta}$ , since $$\hat{\boldsymbol{\beta}}=(\mathbf{X}^\top\mathbf{X})^{-}\mathbf{X}^\top\mathbf{y} \implies E(\hat{\boldsymbol{\beta}})=(\mathbf{X}^\top\mathbf{X})^{-}\mathbf{X}^\top\mathbf{X}\boldsymbol{\beta}.$$

So, we cannot estimate an unique and unbiased $\boldsymbol{\beta}$ . There are various approaches to work around the lack of uniqueness of the parameters in an ANOVA case with non-full-rank $\mathbf{X}$ . One of them is to work with the overparameterized model and define linear combinations of the $\boldsymbol{\beta}$ 's that are unique and can be estimated.

We have that a linear combination of the $\boldsymbol{\beta}$ 's, say $\mathbf{g}^\top \boldsymbol{\beta}$ , is estimable if there exists a vector $\mathbf{a}$ such that $E(\mathbf{a}^\top \mathbf{y})=\mathbf{g}^\top \boldsymbol{\beta}$ .

The contrasts are a special case of estimable functions in which the sum of the coefficients of $\mathbf{g}$ is equal to zero.

And, contrasts come up in the context of categorical predictors in a linear model. (if you check the manual linked by @amoeba, you see that all their contrast coding are related to categorical variables). Then, answering @Curious and @amoeba, we see that they arise in ANOVA, but not in a "pure" regression model with only continuous predictors (we can also talk about contrasts in ANCOVA, since we have some categorical variables in it).

Now, in the model $$\mathbf{y}=\mathbf{X} \boldsymbol{\beta} + \boldsymbol{\varepsilon}$$ where $\mathbf{X}$ is not full-rank, and $E(\mathbf{y})=\mathbf{X}^\top \boldsymbol{\beta}$ , the linear function $\mathbf{g}^\top \boldsymbol{\beta}$ is estimable iff there exists a vector $\mathbf{a}$ such that $\mathbf{a}^\top \mathbf{X}=\mathbf{g}^\top$ . That is, $\mathbf{g}^\top$ is a linear combination of the rows of $\mathbf{X}$ . Also, there are many choices of the vector $\mathbf{a}$ , such that $\mathbf{a}^\top \mathbf{X}=\mathbf{g}^\top$ , as we can see in the example below.

Consider the one-way model: $$y_{ij}=\mu + \alpha_i + \varepsilon_{ij}, \quad i=1,2 \, , j=1,2,3.$$

\begin{align} \mathbf{X} = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & 0 & 1 \\ 1 & 0 & 1 \end{bmatrix} \, , \quad \boldsymbol{\beta}=\begin{bmatrix} \mu \\ \tau_1 \\ \tau_2 \end{bmatrix} \end{align}

And suppose $\mathbf{g}^\top = [0, 1, -1]$ , so we want to estimate $[0, 1, -1] \boldsymbol{\beta}=\tau_1-\tau_2$ .

We can see that there are different choices of the vector $\mathbf{a}$ that yield $\mathbf{a}^\top \mathbf{X}=\mathbf{g}^\top$ : take $\mathbf{a}^\top=[0 , 0,1,-1,0,0]$ ; or $\mathbf{a}^\top = [1,0,0,0,0,-1]$ ; or $\mathbf{a}^\top = [2,-1,0,0,1,-2]$ .

Take the two-way model: $$ y_{ij}=\mu+\alpha_i+\beta_j+\varepsilon_{ij}, \, i=1,2, \, j=1,2$$ .

\begin{align} \mathbf{X} = \begin{bmatrix} 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 0 & 1\\ 1 & 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 \end{bmatrix} \, , \quad \boldsymbol{\beta}=\begin{bmatrix} \mu \\ \alpha_1 \\ \alpha_2 \\ \beta_1 \\ \beta_2 \end{bmatrix} \end{align}

We can define the estimable functions by taking linear combinations of the rows of $\mathbf{X}$ .

Subtracting Row 1 from Rows 2, 3, and 4 (of $\mathbf{X}$ ): $$ \begin{bmatrix} 1 & \phantom{-}1 & 0 & \phantom{-}1 & 0 \\ 0 & 0 & 0 & -1 & 1\\ 0 & -1 & 1 & \phantom{-}0 & 0 \\ 0 & -1 & 1 & -1 & 1 \end{bmatrix} $$

And taking Rows 2 and 3 from the fourth row: $$ \begin{bmatrix} 1 & \phantom{-}1 & 0 & \phantom{-}1 & 0 \\ 0 & 0 & 0 & -1 & 1\\ 0 & -1 & 1 & \phantom{-}0 & 0 \\ 0 & \phantom{-}0 & 0 & \phantom{-}0 & 0 \end{bmatrix} $$

Multiplying this by $\boldsymbol{\beta}$ yields: \begin{align} \mathbf{g}_1^\top \boldsymbol{\beta} &= \mu + \alpha_1 + \beta_1 \\ \mathbf{g}_2^\top \boldsymbol{\beta} &= \beta_2 - \beta_1 \\ \mathbf{g}_3^\top \boldsymbol{\beta} &= \alpha_2 - \alpha_1 \end{align}

So, we have three linearly independent estimable functions. Now, only $\mathbf{g}_2^\top \boldsymbol{\beta}$ and $\mathbf{g}_3^\top \boldsymbol{\beta}$ can be considered contrasts, since the sum of its coefficients (or, the row sum of the respective vector $\mathbf{g}$ ) is equal to zero.

Going back to a one-way balanced model $$y_{ij}=\mu + \alpha_i + \varepsilon_{ij}, \quad i=1,2, \ldots, k \, , j=1,2,\ldots,n.$$

And suppose we want to test the hypothesis $H_0: \alpha_1 = \ldots = \alpha_k$ .

In this setting the matrix $\mathbf{X}$ is not full-rank, so $\boldsymbol{\beta}=(\mu,\alpha_1,\ldots,\alpha_k)^\top$ is not unique and not estimable. To make it estimable we can multiply $\boldsymbol{\beta}$ by $\mathbf{g}^\top$ , as long as $\sum_{i} g_i = 0$ . In other words, $\sum_{i} g_i \alpha_i$ is estimable iff $\sum_{i} g_i = 0$ .

Why this is true?

We know that $\mathbf{g}^\top \boldsymbol{\beta}=(0,g_1,\ldots,g_k) \boldsymbol{\beta} = \sum_{i} g_i \alpha_i$ is estimable iff there exists a vector $\mathbf{a}$ such that $\mathbf{g}^\top = \mathbf{a}^\top \mathbf{X}$ . Taking the distinct rows of $\mathbf{X}$ and $\mathbf{a}^\top=[a_1,\ldots,a_k]$ , then: $$[0,g_1,\ldots,g_k]=\mathbf{g}^\top=\mathbf{a}^\top \mathbf{X} = \left(\sum_i a_i,a_1,\ldots,a_k \right)$$

And the result follows.

If we would like to test a specific contrast, our hypothesis is $H_0: \sum g_i \alpha_i = 0$ . For instance: $H_0: 2 \alpha_1 = \alpha_2 + \alpha_3$ , which can be written as $H_0: \alpha_1 = \frac{\alpha_2+\alpha_3}{2}$ , so we are comparing $\alpha_1$ to the average of $\alpha_2$ and $\alpha_3$ .

This hypothesis can be expressed as $H_0: \mathbf{g}^\top \boldsymbol{\beta}=0$ , where ${\mathbf{g}}^\top = (0,g_1,g_2,\ldots,g_k)$ . In this case, $q=1$ and we test this hypothesis with the following statistic: $$F=\cfrac{\left[\mathbf{g}^\top \hat{\boldsymbol{\beta}}\right]^\top \left[\mathbf{g}^\top(\mathbf{X}^\top\mathbf{X})^{-}\mathbf{g} \right]^{-1}\mathbf{g}^\top \hat{\boldsymbol{\beta}}}{SSE/k(n-1)}.$$

If $H_0: \alpha_1 = \alpha_2 = \ldots = \alpha_k$ is expressed as $\mathbf{G}\boldsymbol{\beta}=\boldsymbol{0}$ where the rows of the matrix $$\mathbf{G} = \begin{bmatrix} \mathbf{g}_1^\top \\ \mathbf{g}_2^\top \\ \vdots \\ \mathbf{g}_k^\top \end{bmatrix}$$ are mutually orthogonal contrasts ( ${\mathbf{g}_i^\top\mathbf{g}}_j = 0$ ), then we can test $H_0: \mathbf{G}\boldsymbol{\beta}=\boldsymbol{0}$ using the statistic $F=\cfrac{\frac{\mbox{SSH}}{\mbox{rank}(\mathbf{G})}}{\frac{\mbox{SSE}}{k(n-1)}}$ , where $\mbox{SSH}=\left[\mathbf{G}\hat{\boldsymbol{\beta}}\right]^\top \left[\mathbf{G}(\mathbf{X}^\top\mathbf{X})^{-1} \mathbf{G}^\top \right]^{-1}\mathbf{G}\hat{\boldsymbol{\beta}}$ .

To understand this better, let's use $k=4$ , and suppose we want to test $H_0: \alpha_1 = \alpha_2 = \alpha_3 = \alpha_4,$ which can be expressed as $$H_0: \begin{bmatrix} \alpha_1 - \alpha_2 \\ \alpha_1 - \alpha_3 \\ \alpha_1 - \alpha_4 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$

Or, as $H_0: \mathbf{G}\boldsymbol{\beta}=\boldsymbol{0}$ : $$H_0: \underbrace{\begin{bmatrix} 0 & 1 & -1 & \phantom{-}0 & \phantom{-}0 \\ 0 & 1 & \phantom{-}0 & -1 & \phantom{-}0 \\ 0 & 1 & \phantom{-}0 & \phantom{-}1 & -1 \end{bmatrix}}_{{\mathbf{G}}, \mbox{our contrast matrix}} \begin{bmatrix} \mu \\ \alpha_1 \\ \alpha_2 \\ \alpha_3 \\ \alpha_4 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$

So, we see that the three rows of our contrast matrix are defined by the coefficients of the contrasts of interest. And each column gives the factor level that we are using in our comparison.

Pretty much all I've written was taken\copied (shamelessly) from Rencher & Schaalje, "Linear Models in Statistics", chapters 8 and 13 (examples, wording of theorems, some interpretations), but other things like the term "contrast matrix" (which, indeed, doesn't appear in this book) and its definition given here were my own.

Relating OP's contrast matrix to my answer

One of OP's matrix (which can also be found in this manual ) is the following:

In this case, our factor has 4 levels, and we can write the model as follows: This can be written in matrix form as: \begin{align} \begin{bmatrix} y_{11} \\ y_{21} \\ y_{31} \\ y_{41} \end{bmatrix} = \begin{bmatrix} \mu \\ \mu \\ \mu \\ \mu \end{bmatrix} + \begin{bmatrix} a_1 \\ a_2 \\ a_3 \\ a_4 \end{bmatrix} + \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{21} \\ \varepsilon_{31} \\ \varepsilon_{41} \end{bmatrix} \end{align}

Or \begin{align} \begin{bmatrix} y_{11} \\ y_{21} \\ y_{31} \\ y_{41} \end{bmatrix} = \underbrace{\begin{bmatrix} 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 1 & 0\\ 1 & 0 & 0 & 0 & 1\\ \end{bmatrix}}_{\mathbf{X}} \underbrace{\begin{bmatrix} \mu \\ a_1 \\ a_2 \\ a_3 \\ a_4 \end{bmatrix}}_{\boldsymbol{\beta}} + \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{21} \\ \varepsilon_{31} \\ \varepsilon_{41} \end{bmatrix} \end{align}

Now, for the dummy coding example on the same manual, they use $a_1$ as the reference group. Thus, we subtract Row 1 from every other row in matrix $\mathbf{X}$ , which yields the $\widetilde{\mathbf{X}}$ :

\begin{align} \begin{bmatrix} 1 & \phantom{-}1 & 0 & 0 & 0 \\ 0 & -1 & 1 & 0 & 0\\ 0 & -1 & 0 & 1 & 0\\ 0 & -1 & 0 & 0 & 1 \end{bmatrix} \end{align}

If you observe the numeration of the rows and columns in the contr.treatment(4) matrix, you'll see that they consider all rows and only the columns related to the factors 2, 3, and 4. If we do the same in the above matrix yields: \begin{align} \begin{bmatrix} 0 & 0 & 0 \\ 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \end{align}

This way, the contr.treatment(4) matrix is telling us that they are comparing factors 2, 3 and 4 to factor 1, and comparing factor 1 to the constant (this is my understanding of the above).

And, defining $\mathbf{G}$ (i.e. taking only the rows that sum to 0 in the above matrix): \begin{align} \begin{bmatrix} 0 & -1 & 1 & 0 & 0\\ 0 & -1 & 0 & 1 & 0\\ 0 & -1 & 0 & 0 & 1 \end{bmatrix} \end{align}

We can test $H_0: \mathbf{G}\boldsymbol{\beta}=0$ and find the estimates of the contrasts.

And the estimates are the same.

Relating @ttnphns' answer to mine.

On their first example, the setup has a categorical factor A having three levels. We can write this as the model (suppose, for simplicity, that $j=1$ ): $$y_{ij}=\mu+a_i+\varepsilon_{ij}\, , \quad \mbox{for } i=1,2,3$$

And suppose we want to test $H_0: a_1 = a_2 = a_3$ , or $H_0: a_1 - a_3 = a_2 - a_3=0$ , with $a_3$ as our reference group/factor.

This can be written in matrix form as: \begin{align} \begin{bmatrix} y_{11} \\ y_{21} \\ y_{31} \end{bmatrix} = \begin{bmatrix} \mu \\ \mu \\ \mu \end{bmatrix} + \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} + \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{21} \\ \varepsilon_{31} \end{bmatrix} \end{align}

Or \begin{align} \begin{bmatrix} y_{11} \\ y_{21} \\ y_{31} \end{bmatrix} = \underbrace{\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ \end{bmatrix}}_{\mathbf{X}} \underbrace{\begin{bmatrix} \mu \\ a_1 \\ a_2 \\ a_3 \end{bmatrix}}_{\boldsymbol{\beta}} + \begin{bmatrix} \varepsilon_{11} \\ \varepsilon_{21} \\ \varepsilon_{31} \end{bmatrix} \end{align}

Now, if we subtract Row 3 from Row 1 and Row 2, we have that $\mathbf{X}$ becomes (I will call it $\widetilde{\mathbf{X}}$ :

\begin{align} \widetilde{\mathbf{X}} =\begin{bmatrix} 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 1 & 0 & 0 & \phantom{-}1 \\ \end{bmatrix} \end{align}

Compare the last 3 columns of the above matrix with @ttnphns' matrix $\mathbf{L}$ . Despite of the order, they are quite similar. Indeed, if multiply $\widetilde{\mathbf{X}} \boldsymbol{\beta}$ , we get:

\begin{align} \begin{bmatrix} 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 1 & 0 & 0 & \phantom{-}1 \\ \end{bmatrix} \begin{bmatrix} \mu \\ a_1 \\ a_2 \\ a_3 \end{bmatrix} = \begin{bmatrix} a_1 - a_3 \\ a_2 - a_3 \\ \mu + a_3 \end{bmatrix} \end{align}

So, we have the estimable functions: $\mathbf{c}_1^\top \boldsymbol{\beta} = a_1-a_3$ ; $\mathbf{c}_2^\top \boldsymbol{\beta} = a_2-a_3$ ; $\mathbf{c}_3^\top \boldsymbol{\beta} = \mu + a_3$ .

Since $H_0: \mathbf{c}_i^\top \boldsymbol{\beta} = 0$ , we see from the above that we are comparing our constant to the coefficient for the reference group (a_3); the coefficient of group1 to the coefficient of group3; and the coefficient of group2 to the group3. Or, as @ttnphns said: "We immediately see, following the coefficients, that the estimated Constant will equal the Y mean in the reference group; that parameter b1 (i.e. of dummy variable A1) will equal the difference: Y mean in group1 minus Y mean in group3; and parameter b2 is the difference: mean in group2 minus mean in group3."

Moreover, observe that (following the definition of contrast: estimable function+row sum =0), that the vectors $\mathbf{c}_1$ and $\mathbf{c}_2$ are contrasts. And, if we create a matrix $\mathbf{G}$ of constrasts, we have:

\begin{align} \mathbf{G} = \begin{bmatrix} 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \end{bmatrix} \end{align}

Our contrast matrix to test $H_0: \mathbf{G}\boldsymbol{\beta}=0$

We will use the same data as @ttnphns' "User defined contrast example" (I'd like to mention that the theory that I've written here requires a few modifications to consider models with interactions, that's why I chose this example. However, the definitions of contrasts and - what I call - contrast matrix remain the same).

So, we have the same results.

It seems to me that there isn't one defining concept of what a contrast matrix is.

If you take the definition of contrast, given by Scheffe ("The Analysis of Variance", page 66), you'll see that it's an estimable function whose coefficients sum to zero. So, if we wish to test different linear combinations of the coefficients of our categorical variables, we use the matrix $\mathbf{G}$ . This is a matrix in which the rows sum to zero, that we use to multiply our matrix of coefficients by in order to make those coefficients estimable. Its rows indicate the different linear combinations of contrasts that we are testing and its columns indicate which factors (coefficients) are being compared.

As the matrix $\mathbf{G}$ above is constructed in a way that each of its rows is composed by a contrast vector (which sum to 0), for me it makes sense to call $\mathbf{G}$ a "contrast matrix" (Monahan - "A primer on linear models" - also uses this terminology).

However, as beautifully explained by @ttnphns, softwares are calling something else as "contrast matrix", and I couldn't find a direct relationship between the matrix $\mathbf{G}$ and the built-in commands/matrices from SPSS (@ttnphns) or R (OP's question), only similarities. But I believe that the nice discussion/collaboration presented here will help clarify such concepts and definitions.

• 2 $\begingroup$ please do not restrict this answer just to anova if possible. The [anova] tag was added by @amoeba by the time when you answered my question, but I don't want the answer to be restricted just to anova. $\endgroup$ –  Tomas Commented Jul 6, 2016 at 10:07
• 2 $\begingroup$ Thanks a lot for such a big update. I removed some of my comments above that were obsolete by now (you can remove some of yours, e.g. the first one). However, by now it is clear to me that "contrast matrix" in your (and Monahan's) sense is something entirely different from "contrast matrix" in the sense it's used in this R manual and also in the original question here (what ttnphns calls C-matrix). I think it would make sense if you make a note somewhere in your answer about this difference. $\endgroup$ –  amoeba Commented Jul 8, 2016 at 0:17
• $\begingroup$ I'm having troubles with understanding starting right from Example1. What is $i$ an $j$ in your notation $y_{ij}$? What is $a_i$ and what do the columns od $X$ represent? Is that Constant term (column of ones) and the two dummy variables? $\endgroup$ –  ttnphns Commented Jul 8, 2016 at 9:58
• $\begingroup$ @ttnphns: $i$ is indexing group (there are two groups in Example 1), $j$ is indexing data point inside each group. $\mu$ is a constant and $\alpha_i$ are constants for each group such that $\mu+\alpha_i$ are group means (so $\mu$ can be total mean and $\alpha_i$ can be deviation of the group means from the total mean). Columns of $X$ are constant term and two dummies, yes. $\endgroup$ –  amoeba Commented Jul 8, 2016 at 14:23
• 1 $\begingroup$ @fabiob technically, ANOVA is not GLM. But, I think I understand your question. When you have an overparameterized model you can: i) reparameterize the model to get a full-rank matrix; ii) define side conditions (which gives you a full-rank matrix); or iii) work with the model as is and define lin. comb. of the parameters that are unique and can be estimated (these are called contrasts). Since the question is about a contrast matrix, I used iii). However, we also have ANOVA cases where X is full-rank. I updated my answer to clarify that I'm talking about the special non-full-rank case. $\endgroup$ –  ogustavo Commented Feb 3, 2020 at 21:05

"Contrast matrix" is not a standard term in the statistical literature. It can have [at least] two related by distinct meanings:

A matrix specifying a particular null hypothesis in an ANOVA regression (unrelated to the coding scheme), where each row is a contrast . This is not a standard usage of the term. I used full text search in Christensen Plane Answers to Complex Questions , Rutherford Introducing ANOVA and ANCOVA; GLM Approach , and Rencher & Schaalje Linear Models in Statistics . They all talk a lot about "contrasts" but never ever mention the term "contrast matrix". However, as @Gus_est found, this term is used in Monahan's A Primer on Linear Models .

A matrix specifying the coding scheme for the design matrix in an ANOVA regression. This is how the term "contrast matrix" is used in the R community (see e.g. this manual or this help page ).

The answer by @Gus_est explores the first meaning. The answer by @ttnphns explores the second meaning (he calls it "contrast coding matrix" and also discusses "contrast coefficient matrix" which is a standard term in SPSS literature).

My understanding is that you were asking about meaning #2, so here goes the definition:

"Contrast matrix" in the R sense is $k\times k$ matrix $\mathbf C$ where $k$ is the number of groups, specifying how group membership is encoded in the design matrix $\mathbf X$ . Specifically, if a $m$ -th observation belongs to the group $i$ then $X_{mj}=C_{ij}$ .

Note: usually the first column of $\mathbf C$ is the column of all ones (corresponding to the intercept column in the design matrix). When you call R commands like contr.treatment(4) , you get matrix $\mathbf C$ without this first column.

I am planning to extend this answer to make an extended comment on how the answers by @ttnphns and @Gus_est fit together.

• $\begingroup$ The answer by @Gus_est explores the first meaning. The answer by @ttnphns explores the second meaning. I protest. (And am surprised to hear - after we both had a long conversation on the definitions in the comments to mty answer.) I invited two terms: contrast coefficient matrix (where rows are the contrasts, linear combibnation of means) aka L-matrix, and contrast coding schema matrix, aka C matrix. Both are related, I discussed both. $\endgroup$ –  ttnphns Commented Jul 8, 2016 at 16:58
• $\begingroup$ (cont.) Contrast coefficent L matrix is a standard term in ANOVA / General linear model, used in texts and in SPSS docs, for example . The coding schemes see here . $\endgroup$ –  ttnphns Commented Jul 8, 2016 at 16:59
• $\begingroup$ You were asking about meaning #2 We actually are not sure what meaning of the term the OP implied. The OP displayed some examples of contrast coding schemes, - it doesn't necessarily mean s/he wasn't interested in L matrices. $\endgroup$ –  ttnphns Commented Jul 8, 2016 at 17:11
• 1 $\begingroup$ I'm happy that we kinda speak the same language now. It seems so, at least. It would be great for everybody, especially a visitor reader, if you accomplish your answer, showing how Gus' and ttnphns' reports convert to the same result. If you want to accomplish. $\endgroup$ –  ttnphns Commented Jul 9, 2016 at 12:04
• 1 $\begingroup$ (cont.) Of course the L matrix in both "approaches " is the same (and no mysterious G matrix is needed). Show that two equivalent paths (L is arbitrary, X is dummies): L -> XC -> regression -> result and X -> [regression -> adjusting to test for L] -> result leave the same result. The 2nd path is how an ANOVA program will do (the bracketed part []); the 1st path is a didactic demonstration how contrasts are solvable via only regression program. $\endgroup$ –  ttnphns Commented Jul 9, 2016 at 12:12

A contrast compares two groups by comparing their difference with zero. In a contrast matrix the rows are the contrasts and must add to zero, the columns are the groups. For example:

Let's say you have 4 groups A,B,C,D that you want to compare, then the contrast matrix would be:

Group: A B C D A vs B: 1 -1 0 0 C vs D: 0 0 -1 1 A,B vs D,C: 1 1 -1 -1

Paraphrasing from Understanding Industrial Experimentation :

If there's a group of k objects to be compared, with k subgroups averages, a contrast is defined on this set of k objects by any set of k coefficients, [c1, c2, c3, ... cj, ..., ck] that sum to zero.

Let C be a contrast then,

$$ C = c_{1}\mu_{1} + c_{2}\mu_{2} + ... c_{j}\mu_{j} + ... c_{k}\mu_{k} $$

$$ C = \sum_{j=1}^{k} c_{j}\mu{j} $$

with the constraint $$ \sum_{j=1}^{k} c_{j} = 0 $$

Those subgroups that are assigned a coefficient of zero will be excluded from the comparison.(*)

It is the signs of the coefficients that actually define the comparison, not the values chosen. The absolute values of the coefficients can be anything as long as the sum of the coefficients is zero.

(*)Each statistical software has a different way of indicating which subgroups will be excluded/included.

I wanted to add some more basic information to the previous (great) responses, and clarify a little (also for myself) how contrast coding works in R, and why we need to calculate the inverse of the contrast coding matrix to understand which comparisons are performed.

I'll start with the description of the linear model and contrasts in terms of matrix algebra, and then go through an example in R.

The cell means model for ANOVA is:

\begin{equation} y = X\mu + \epsilon = X\begin{pmatrix} \mu1 \\\mu2 \\\mu3 \\\mu4 \end{pmatrix} + \epsilon \end{equation}

With X as the design matrix and u as the vector of means. An example is this, where we have 4 groups coded in each column:

\begin{equation} X=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix} \end{equation}

In this case, we can estimate the means by the least square method, using the equations:

\begin{equation} \hat{\mu} =(X^{\prime }X)^{-1}\ X^{\prime }y\\ \end{equation}

This is all good, but let's imagine we want specific comparisons rather than the means, like differences in means compared to a reference group. In the case of 4 groups, we could express this as a matrix C of comparisons, multiplied by the vector of means:

\begin{equation} C\mu = \begin{pmatrix} \phantom{..} 1 & 0 & 0 &0 \\ -1 & 1 & 0 & 0\\ -1 & 0 & 1 & 0\\ -1 & 0 & 0 & 1\\ \end{pmatrix}\ \begin{pmatrix}\mu1 \\\mu2 \\\mu3 \\\mu4 \end{pmatrix} = \begin{pmatrix} \mu1 \\\mu2-\mu1 \\\mu3-\mu1 \\\mu4-\mu1 \end{pmatrix} \end{equation}

The first group serves as reference, and we calculate the deviations from it. The matrix C serves to describe the comparisons, it is the contrast matrix. Technically here these are not contrasts, because the sum in each row should be zero by definition, but that will serve our purpose, and this is the matrix referred to in the contr.treatment() function in R (its inverse, see below).

The matrix C defines the contrasts. We want to evaluate contrasts from the data, in the context of the same model. We note that:

\begin{equation} y \ =\ X\mu \ +\ \epsilon \ =\ XI\mu \ +\ \epsilon \ =\ X \ (C^{-1}C)\ \ \mu \ +\ \epsilon = \ (X C^{-1}) \ (C \mu) \ + \epsilon \end{equation}

Therefore we can use the first term in parentheses to evaluate the second term (our comparisons), using the least squares method, just as we did for the original equation above. This is why we use the inverse of the contrast matrix C, and it needs to be square and full rank in this case.

We use the least square method to evaluate the contrasts, with the same equation as above, using the modified design matrix:

\begin{equation} (X C^{-1}) \end{equation}

And we evaluate: \begin{equation} C\mu \end{equation} using the method of least squares. The coefficients for this model can be evaluated as before using least squares, replacing the original design matrix by the new one. Or naming $X_{1} = (X C^{-1})$ the modified design matrix: \begin{equation} \hat{C\mu} = (X_{1}^{'}X_{1})^{-1}X_{1}^{'}y=\\C\hat{\mu}= \begin{pmatrix} \hat{\mu1} \\\hat{\mu2}-\hat{\mu1} \\\hat{\mu3}-\hat{\mu1} \\\hat{\mu4}-\hat{\mu1} \end{pmatrix} \end{equation}

Using the modified design matrix (with the inverse of the contrast matrix) and the least squares method, we evaluate the desired constrasts. Of course, to get the original contrast matrix, we need to invert the contrast coding matrix used in R.

Let's try and make it work on an example in R:

We have four teams A, B, C, D and the scores of each individual. Let's make the design matrix X for the cell means model:

We can find the means of each group by the least squares equation \begin{equation} \hat{\mu} =(X^{\prime }X)^{-1}\ X^{\prime }y\\ \end{equation} in R:

But we want comparisons of means to the first group (treatment contrasts). We use the matrix C of contrasts defined earlier. Based on what was said before, what we really want is the inverse of C, to evaluate the contrasts. R has a built-in function for this, called contr.treament(), where we specificy the number of factors. We build the inverse of C, the contrast coding matrix, this way:

if we invert this matrix, we get C, the comparisons we want:

Now we construct the modified design matrix for the model:

And we solve for the contrasts, either by plugging the modified design matrix into the least squares equation, or using the lm() function:

We get the mean of the first group and the deviations for the others, as defined in the contrast matrix C.

We can define any type of contrast in this way, either using the built-in functions contr.treatment(), contr.sum() etc or by specifying which comparisons we want. For its contrasts arguments, lm() expects the inverse of C without the intercept column, solve(C)[,-1], and it adds the intercept column to generate $C^{-1}$ , and uses it for the modified design matrix. There are many refinements on this scheme (orthogonal contrasts, more complex contrasts, not full rank design matrix etc), but this is the gist of it (cf also here for reference: https://cran.r-project.org/web/packages/codingMatrices/vignettes/codingMatrices.pdf ).

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Hypothesis in Machine Learning

The concept of a hypothesis is fundamental in Machine Learning and data science endeavours. In the realm of machine learning, a hypothesis serves as an initial assumption made by data scientists and ML professionals when attempting to address a problem. Machine learning involves conducting experiments based on past experiences, and these hypotheses are crucial in formulating potential solutions.

It’s important to note that in machine learning discussions, the terms “hypothesis” and “model” are sometimes used interchangeably. However, a hypothesis represents an assumption, while a model is a mathematical representation employed to test that hypothesis. This section on “Hypothesis in Machine Learning” explores key aspects related to hypotheses in machine learning and their significance.

Table of Content

How does a Hypothesis work?

Hypothesis space and representation in machine learning, hypothesis in statistics, faqs on hypothesis in machine learning.

A hypothesis in machine learning is the model’s presumption regarding the connection between the input features and the result. It is an illustration of the mapping function that the algorithm is attempting to discover using the training set. To minimize the discrepancy between the expected and actual outputs, the learning process involves modifying the weights that parameterize the hypothesis. The objective is to optimize the model’s parameters to achieve the best predictive performance on new, unseen data, and a cost function is used to assess the hypothesis’ accuracy.

In most supervised machine learning algorithms, our main goal is to find a possible hypothesis from the hypothesis space that could map out the inputs to the proper outputs. The following figure shows the common method to find out the possible hypothesis from the Hypothesis space:

Hypothesis Space (H)

Hypothesis space is the set of all the possible legal hypothesis. This is the set from which the machine learning algorithm would determine the best possible (only one) which would best describe the target function or the outputs.

Hypothesis (h)

A hypothesis is a function that best describes the target in supervised machine learning. The hypothesis that an algorithm would come up depends upon the data and also depends upon the restrictions and bias that we have imposed on the data.

The Hypothesis can be calculated as:

[Tex]y = mx + b [/Tex]

• m = slope of the lines
• b = intercept

To better understand the Hypothesis Space and Hypothesis consider the following coordinate that shows the distribution of some data:

Say suppose we have test data for which we have to determine the outputs or results. The test data is as shown below:

We can predict the outcomes by dividing the coordinate as shown below:

So the test data would yield the following result:

But note here that we could have divided the coordinate plane as:

The way in which the coordinate would be divided depends on the data, algorithm and constraints.

• All these legal possible ways in which we can divide the coordinate plane to predict the outcome of the test data composes of the Hypothesis Space.
• Each individual possible way is known as the hypothesis.

Hence, in this example the hypothesis space would be like:

The hypothesis space comprises all possible legal hypotheses that a machine learning algorithm can consider. Hypotheses are formulated based on various algorithms and techniques, including linear regression, decision trees, and neural networks. These hypotheses capture the mapping function transforming input data into predictions.

Hypothesis Formulation and Representation in Machine Learning

Hypotheses in machine learning are formulated based on various algorithms and techniques, each with its representation. For example:

• Linear Regression : [Tex] h(X) = \theta_0 + \theta_1 X_1 + \theta_2 X_2 + … + \theta_n X_n[/Tex]
• Decision Trees : [Tex]h(X) = \text{Tree}(X)[/Tex]
• Neural Networks : [Tex]h(X) = \text{NN}(X)[/Tex]

In the case of complex models like neural networks, the hypothesis may involve multiple layers of interconnected nodes, each performing a specific computation.

Hypothesis Evaluation:

The process of machine learning involves not only formulating hypotheses but also evaluating their performance. This evaluation is typically done using a loss function or an evaluation metric that quantifies the disparity between predicted outputs and ground truth labels. Common evaluation metrics include mean squared error (MSE), accuracy, precision, recall, F1-score, and others. By comparing the predictions of the hypothesis with the actual outcomes on a validation or test dataset, one can assess the effectiveness of the model.

Hypothesis Testing and Generalization:

Once a hypothesis is formulated and evaluated, the next step is to test its generalization capabilities. Generalization refers to the ability of a model to make accurate predictions on unseen data. A hypothesis that performs well on the training dataset but fails to generalize to new instances is said to suffer from overfitting. Conversely, a hypothesis that generalizes well to unseen data is deemed robust and reliable.

The process of hypothesis formulation, evaluation, testing, and generalization is often iterative in nature. It involves refining the hypothesis based on insights gained from model performance, feature importance, and domain knowledge. Techniques such as hyperparameter tuning, feature engineering, and model selection play a crucial role in this iterative refinement process.

In statistics , a hypothesis refers to a statement or assumption about a population parameter. It is a proposition or educated guess that helps guide statistical analyses. There are two types of hypotheses: the null hypothesis (H0) and the alternative hypothesis (H1 or Ha).

• Null Hypothesis(H 0 ): This hypothesis suggests that there is no significant difference or effect, and any observed results are due to chance. It often represents the status quo or a baseline assumption.
• Aternative Hypothesis(H 1 or H a ): This hypothesis contradicts the null hypothesis, proposing that there is a significant difference or effect in the population. It is what researchers aim to support with evidence.

Q. How does the training process use the hypothesis?

The learning algorithm uses the hypothesis as a guide to minimise the discrepancy between expected and actual outputs by adjusting its parameters during training.

Q. How is the hypothesis’s accuracy assessed?

Usually, a cost function that calculates the difference between expected and actual values is used to assess accuracy. Optimising the model to reduce this expense is the aim.

Q. What is Hypothesis testing?

Hypothesis testing is a statistical method for determining whether or not a hypothesis is correct. The hypothesis can be about two variables in a dataset, about an association between two groups, or about a situation.

Q. What distinguishes the null hypothesis from the alternative hypothesis in machine learning experiments?

The null hypothesis (H0) assumes no significant effect, while the alternative hypothesis (H1 or Ha) contradicts H0, suggesting a meaningful impact. Statistical testing is employed to decide between these hypotheses.

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What kind of Experience do you want to share?

A statement that could be true, which might then be tested.

Example: Sam has a hypothesis that "large dogs are better at catching tennis balls than small dogs". We can test that hypothesis by having hundreds of different sized dogs try to catch tennis balls.

Sometimes the hypothesis won't be tested, it is simply a good explanation (which could be wrong). Conjecture is a better word for this.

Example: you notice the temperature drops just as the sun rises. Your hypothesis is that the sun warms the air high above you, which rises up and then cooler air comes from the sides.

Note: when someone says "I have a theory" they should say "I have a hypothesis", because in mathematics a theory is actually well proven.

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Definition of hypothesis

Did you know.

The Difference Between Hypothesis and Theory

A hypothesis is an assumption, an idea that is proposed for the sake of argument so that it can be tested to see if it might be true.

In the scientific method, the hypothesis is constructed before any applicable research has been done, apart from a basic background review. You ask a question, read up on what has been studied before, and then form a hypothesis.

A hypothesis is usually tentative; it's an assumption or suggestion made strictly for the objective of being tested.

A theory , in contrast, is a principle that has been formed as an attempt to explain things that have already been substantiated by data. It is used in the names of a number of principles accepted in the scientific community, such as the Big Bang Theory . Because of the rigors of experimentation and control, it is understood to be more likely to be true than a hypothesis is.

In non-scientific use, however, hypothesis and theory are often used interchangeably to mean simply an idea, speculation, or hunch, with theory being the more common choice.

Since this casual use does away with the distinctions upheld by the scientific community, hypothesis and theory are prone to being wrongly interpreted even when they are encountered in scientific contexts—or at least, contexts that allude to scientific study without making the critical distinction that scientists employ when weighing hypotheses and theories.

The most common occurrence is when theory is interpreted—and sometimes even gleefully seized upon—to mean something having less truth value than other scientific principles. (The word law applies to principles so firmly established that they are almost never questioned, such as the law of gravity.)

This mistake is one of projection: since we use theory in general to mean something lightly speculated, then it's implied that scientists must be talking about the same level of uncertainty when they use theory to refer to their well-tested and reasoned principles.

The distinction has come to the forefront particularly on occasions when the content of science curricula in schools has been challenged—notably, when a school board in Georgia put stickers on textbooks stating that evolution was "a theory, not a fact, regarding the origin of living things." As Kenneth R. Miller, a cell biologist at Brown University, has said , a theory "doesn’t mean a hunch or a guess. A theory is a system of explanations that ties together a whole bunch of facts. It not only explains those facts, but predicts what you ought to find from other observations and experiments.”

While theories are never completely infallible, they form the basis of scientific reasoning because, as Miller said "to the best of our ability, we’ve tested them, and they’ve held up."

• proposition
• supposition

hypothesis , theory , law mean a formula derived by inference from scientific data that explains a principle operating in nature.

hypothesis implies insufficient evidence to provide more than a tentative explanation.

theory implies a greater range of evidence and greater likelihood of truth.

law implies a statement of order and relation in nature that has been found to be invariable under the same conditions.

Examples of hypothesis in a Sentence

These examples are programmatically compiled from various online sources to illustrate current usage of the word 'hypothesis.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Word History

Greek, from hypotithenai to put under, suppose, from hypo- + tithenai to put — more at do

1641, in the meaning defined at sense 1a

Phrases Containing hypothesis

• counter - hypothesis
• nebular hypothesis
• null hypothesis
• planetesimal hypothesis
• Whorfian hypothesis

Articles Related to hypothesis

This is the Difference Between a...

This is the Difference Between a Hypothesis and a Theory

In scientific reasoning, they're two completely different things

Dictionary Entries Near hypothesis

hypothermia

hypothesize

Cite this Entry

“Hypothesis.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/hypothesis. Accessed 10 Sep. 2024.

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