null and alternative hypothesis for multiple linear regression

Statistics Made Easy

Understanding the Null Hypothesis for Linear Regression

Linear regression is a technique we can use to understand the relationship between one or more predictor variables and a response variable .

If we only have one predictor variable and one response variable, we can use simple linear regression , which uses the following formula to estimate the relationship between the variables:

ŷ = β 0 + β 1 x

• ŷ: The estimated response value.
• β 0 : The average value of y when x is zero.
• β 1 : The average change in y associated with a one unit increase in x.
• x: The value of the predictor variable.

Simple linear regression uses the following null and alternative hypotheses:

• H 0 : β 1 = 0
• H A : β 1 ≠ 0

The null hypothesis states that the coefficient β 1 is equal to zero. In other words, there is no statistically significant relationship between the predictor variable, x, and the response variable, y.

The alternative hypothesis states that β 1 is not equal to zero. In other words, there is a statistically significant relationship between x and y.

If we have multiple predictor variables and one response variable, we can use multiple linear regression , which uses the following formula to estimate the relationship between the variables:

ŷ = β 0 + β 1 x 1 + β 2 x 2 + … + β k x k

• β 0 : The average value of y when all predictor variables are equal to zero.
• β i : The average change in y associated with a one unit increase in x i .
• x i : The value of the predictor variable x i .

Multiple linear regression uses the following null and alternative hypotheses:

• H 0 : β 1 = β 2 = … = β k = 0
• H A : β 1 = β 2 = … = β k ≠ 0

The null hypothesis states that all coefficients in the model are equal to zero. In other words, none of the predictor variables have a statistically significant relationship with the response variable, y.

The alternative hypothesis states that not every coefficient is simultaneously equal to zero.

The following examples show how to decide to reject or fail to reject the null hypothesis in both simple linear regression and multiple linear regression models.

Example 1: Simple Linear Regression

Suppose a professor would like to use the number of hours studied to predict the exam score that students will receive in his class. He collects data for 20 students and fits a simple linear regression model.

The following screenshot shows the output of the regression model:

The fitted simple linear regression model is:

Exam Score = 67.1617 + 5.2503*(hours studied)

To determine if there is a statistically significant relationship between hours studied and exam score, we need to analyze the overall F value of the model and the corresponding p-value:

• Overall F-Value:  47.9952
• P-value:  0.000

Since this p-value is less than .05, we can reject the null hypothesis. In other words, there is a statistically significant relationship between hours studied and exam score received.

Example 2: Multiple Linear Regression

Suppose a professor would like to use the number of hours studied and the number of prep exams taken to predict the exam score that students will receive in his class. He collects data for 20 students and fits a multiple linear regression model.

The fitted multiple linear regression model is:

Exam Score = 67.67 + 5.56*(hours studied) – 0.60*(prep exams taken)

To determine if there is a jointly statistically significant relationship between the two predictor variables and the response variable, we need to analyze the overall F value of the model and the corresponding p-value:

• Overall F-Value:  23.46
• P-value:  0.00

Since this p-value is less than .05, we can reject the null hypothesis. In other words, hours studied and prep exams taken have a jointly statistically significant relationship with exam score.

Note: Although the p-value for prep exams taken (p = 0.52) is not significant, prep exams combined with hours studied has a significant relationship with exam score.

Additional Resources

Understanding the F-Test of Overall Significance in Regression How to Read and Interpret a Regression Table How to Report Regression Results How to Perform Simple Linear Regression in Excel How to Perform Multiple Linear Regression in Excel

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Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

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Null and Alternative hypothesis for multiple linear regression

I have 1 dependent variable and 3 independent variables.

I run multiple regression, and find that the p value for one of the independent variables is higher than 0.05 (95% is my confidence level).

I take that variable out and run it again. Both remaining independent variables have $p$-value less than 0.05 so I conclude I have my model.

Am I correct in thinking that initially, my null hypothesis is

$$H_0= β_1=β_2 = \dots =β_{k-1} = 0$$

and that the alternative hypothesis is

$$H_1=\textrm{At least one } β \neq 0 \textrm{ whilst } p<0.05$$

And that after the first regression, I do not reject, as one variable does not meet my confidence level needs...

So I run it again, and then reject the null as all $p$-values are significant?

Is what I have written accurate?

Edit: Thanks to Bob Jansen for improving this aesthetics of this post.

2 Answers 2

The hypothesis $H_0: β_1=β_2=\dots =β_{k−1}=0$ is normally tested by the $F$-test for the regression.

You are carrying out 3 independent tests of your coefficients (Do you also have a constant in the regression or is the constant one of your three variables?) If you do three independent tests at a 5% level you have a probability of over 14% of finding one of the coefficients significant at the 5% level even if all coefficients are truly zero (the null hypothesis). This is often ignored but be careful. Even so, If the coefficient is close to significant I would think about the underlying theory before coming to a decision.

If you add dummies you will have a beta for each dummy

• $\begingroup$ Thanks for your response. I don't have a constant, all of my p-values are very significant (the least is a dummy variable at 0.039). What would my null hypothesis be? My knowledge is that I'm seeking p-values because that'd give me my model. I don't understand the technicalities of it and want to learn it :) $\endgroup$ –  Harry Jan 7, 2015 at 22:36
• $\begingroup$ I think you meant to say 14% of committing a type one error (probability of 0.14 of finding at least one of the coefficient significant when there true value is actually the null hypothesis value) $\endgroup$ –  Kamster Jan 8, 2015 at 0:36
• $\begingroup$ @Kamster Thanks. You are correct and I have amended my answer. $\endgroup$ –  user1483 Jan 21, 2015 at 21:26

These are independent variables so the hypothesis applies to each parameter independently.

• $\begingroup$ +1: Yes, you are right - but the rest of it should be fine $\endgroup$ –  vonjd Jan 2, 2015 at 21:18
• $\begingroup$ sorry, could you clarify? How do I change the equation so it applies to each parameter independently? And also, what is the effect of adding 3 dummy variables. Is it simply 2 more betas? Or do they require their own symbol $\endgroup$ –  Harry Jan 4, 2015 at 0:32
• $\begingroup$ It just means that you have an H_0 and an H_1 for every parameter. $\endgroup$ –  vonjd Jan 4, 2015 at 11:33
• $\begingroup$ Ok I see. Do you know the procedure for dummy variables? Are they just additional beta? Or is it more accurate to refer to them as delta? $\endgroup$ –  Harry Jan 4, 2015 at 11:43
• $\begingroup$ Maybe I have this wrong but isn't it true if you remain your individual significance levels at 0.05 that the probability of type one error (ie the probability that reject null hypothesis when it is actually true; significance level) will be greater than or equal 0.14 $\endgroup$ –  Kamster Jan 8, 2015 at 0:43

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• Null and Alternative Hypotheses | Definitions & Examples

Null & Alternative Hypotheses | Definitions, Templates & Examples

Published on May 6, 2022 by Shaun Turney . Revised on June 22, 2023.

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :

• Null hypothesis ( H 0 ): There’s no effect in the population .
• Alternative hypothesis ( H a or H 1 ) : There’s an effect in the population.

Table of contents

Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, similarities and differences between null and alternative hypotheses, how to write null and alternative hypotheses, other interesting articles, frequently asked questions.

The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”:

• The null hypothesis ( H 0 ) answers “No, there’s no effect in the population.”
• The alternative hypothesis ( H a ) answers “Yes, there is an effect in the population.”

The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample. It’s critical for your research to write strong hypotheses .

You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.

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The null hypothesis is the claim that there’s no effect in the population.

If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.

Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept . Be careful not to say you “prove” or “accept” the null hypothesis.

Null hypotheses often include phrases such as “no effect,” “no difference,” or “no relationship.” When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).

You can never know with complete certainty whether there is an effect in the population. Some percentage of the time, your inference about the population will be incorrect. When you incorrectly reject the null hypothesis, it’s called a type I error . When you incorrectly fail to reject it, it’s a type II error.

Examples of null hypotheses

The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.

*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .

The alternative hypothesis ( H a ) is the other answer to your research question . It claims that there’s an effect in the population.

Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.

The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

Alternative hypotheses often include phrases such as “an effect,” “a difference,” or “a relationship.” When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes < or >). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.

Examples of alternative hypotheses

The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.

Null and alternative hypotheses are similar in some ways:

• They’re both answers to the research question.
• They both make claims about the population.
• They’re both evaluated by statistical tests.

However, there are important differences between the two types of hypotheses, summarized in the following table.

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To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.

General template sentences

The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:

Does independent variable affect dependent variable ?

• Null hypothesis ( H 0 ): Independent variable does not affect dependent variable.
• Alternative hypothesis ( H a ): Independent variable affects dependent variable.

Test-specific template sentences

Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.

Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.

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

• Normal distribution
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

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.

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.

The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (“ x affects y because …”).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses . In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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5.3 - the multiple linear regression model, notation for the population model.

• A population model for a multiple linear regression model that relates a y -variable to p -1 x -variables is written as

\(\begin{equation} y_{i}=\beta_{0}+\beta_{1}x_{i,1}+\beta_{2}x_{i,2}+\ldots+\beta_{p-1}x_{i,p-1}+\epsilon_{i}. \end{equation} \)

• We assume that the \(\epsilon_{i}\) have a normal distribution with mean 0 and constant variance \(\sigma^{2}\). These are the same assumptions that we used in simple regression with one x -variable.
• The subscript i refers to the \(i^{\textrm{th}}\) individual or unit in the population. In the notation for the x- variables, the subscript following i simply denotes which x -variable it is.
• The word "linear" in "multiple linear regression" refers to the fact that the model is linear in the parameters , \(\beta_0, \beta_1, \ldots, \beta_{p-1}\). This simply means that each parameter multiplies an x -variable, while the regression function is a sum of these "parameter times x -variable" terms. Each x -variable can be a predictor variable or a transformation of predictor variables (such as the square of a predictor variable or two predictor variables multiplied together). Allowing non-linear transformation of predictor variables like this enables the multiple linear regression model to represent non-linear relationships between the response variable and the predictor variables. We'll explore predictor transformations further in Lesson 9 . Note that even \(\beta_0\) represents a "parameter times x -variable" term if you think of the x-variable that is multiplied by \(\beta_0\) as being the constant function "1."
• The model includes p-1 x-variables, but p regression parameters (beta) because of the intercept term \(\beta_0\).

Estimates of the Model Parameters

• The estimates of the \(\beta\) parameters are the values that minimize the sum of squared errors for the sample. The exact formula for this is given in the next section on matrix notation.
• The letter b is used to represent a sample estimate of a \(\beta\) parameter. Thus \(b_{0}\) is the sample estimate of \(\beta_{0}\), \(b_{1}\) is the sample estimate of \(\beta_{1}\), and so on.
• \(\textrm{MSE}=\frac{\textrm{SSE}}{n-p}\) estimates \(\sigma^{2}\), the variance of the errors. In the formula, n = sample size, p = number of \(\beta\) parameters in the model (including the intercept) and \(\textrm{SSE}\) = sum of squared errors. Notice that for simple linear regression p = 2. Thus, we get the formula for MSE that we introduced in the context of one predictor.
• \(S=\sqrt{MSE}\) estimates \(\sigma\) and is known as the regression standard error or the residual standard error .
• In the case of two predictors, the estimated regression equation yields a plane (as opposed to a line in the simple linear regression setting). For more than two predictors, the estimated regression equation yields a hyperplane.

Interpretation of the Model Parameters

• Each \(\beta\) parameter represents the change in the mean response, E( y ), per unit increase in the associated predictor variable when all the other predictors are held constant.
• For example, \(\beta_1\) represents the estimated change in the mean response, E( y ), per unit increase in \(x_1\) when \(x_2\), \(x_3\), ..., \(x_{p-1}\) are held constant.
• The intercept term, \(\beta_0\), represents the estimated mean response, E( y ), when all the predictors \(x_1\), \(x_2\), ..., \(x_{p-1}\), are all zero (which may or may not have any practical meaning).

Predicted Values and Residuals

• A predicted value is calculated as \(\hat{y}_{i}=b_{0}+b_{1}x_{i,1}+b_{2}x_{i,2}+\ldots+b_{p-1}x_{i,p-1}\), where the b values come from statistical software and the x -values are specified by us.
• A residual ( error ) term is calculated as \(e_{i}=y_{i}-\hat{y}_{i}\), the difference between an actual and a predicted value of y .
• A plot of residuals (vertical) versus predicted values (horizontal) ideally should resemble a horizontal random band. Departures from this form indicate difficulties with the model and/or data.
• Other residual analyses can be done exactly as we did in simple regression. For instance, we might wish to examine a normal probability plot (NPP) of the residuals. Additional plots to consider are plots of residuals versus each x -variable separately. This might help us identify sources of curvature or nonconstant variance. We'll explore this further in Lesson 7 .

ANOVA Table

Coefficient of Determination, R-squared, and Adjusted R-squared

• As in simple linear regression, \(R^2=\frac{SSR}{SSTO}=1-\frac{SSE}{SSTO}\), and represents the proportion of variation in \(y\) (about its mean) "explained" by the multiple linear regression model with predictors, \(x_1, x_2, ...\).
• If we start with a simple linear regression model with one predictor variable, \(x_1\), then add a second predictor variable, \(x_2\), \(SSE\) will decrease (or stay the same) while \(SSTO\) remains constant, and so \(R^2\) will increase (or stay the same). In other words, \(R^2\) always increases (or stays the same) as more predictors are added to a multiple linear regression model, even if the predictors added are unrelated to the response variable . Thus, by itself, \(R^2\) cannot be used to help us identify which predictors should be included in a model and which should be excluded.
• An alternative measure, adjusted \(R^2\), does not necessarily increase as more predictors are added, and can be used to help us identify which predictors should be included in a model and which should be excluded. Adjusted \(R^2=1-\left(\frac{n-1}{n-p}\right)(1-R^2)\), and, while it has no practical interpretation, is useful for such model building purposes. Simply stated, when comparing two models used to predict the same response variable, we generally prefer the model with the higher value of adjusted \(R^2\) – see Lesson 10 for more details.

Significance Testing of Each Variable

Within a multiple regression model, we may want to know whether a particular x -variable is making a useful contribution to the model. That is, given the presence of the other x -variables in the model, does a particular x -variable help us predict or explain the y -variable? For instance, suppose that we have three x -variables in the model. The general structure of the model could be

\(\begin{equation} y=\beta _{0}+\beta _{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\epsilon. \end{equation}\)

As an example, to determine whether variable \(x_{1}\) is a useful predictor variable in this model, we could test

\(\begin{align*} \nonumber H_{0}&\colon\beta_{1}=0 \\ \nonumber H_{A}&\colon\beta_{1}\neq 0 \end{align*}\)

If the null hypothesis above were the case, then a change in the value of \(x_{1}\) would not change y , so y and \(x_{1}\) are not linearly related (taking into account \(x_2\) and \(x_3\)). Also, we would still be left with variables \(x_{2}\) and \(x_{3}\) being present in the model. When we cannot reject the null hypothesis above, we should say that we do not need variable \(x_{1}\) in the model given that variables \(x_{2}\) and \(x_{3}\) will remain in the model. In general, the interpretation of a slope in multiple regression can be tricky. Correlations among the predictors can change the slope values dramatically from what they would be in separate simple regressions. To carry out the test, statistical software will report p -values for all coefficients in the model. Each p -value will be based on a t -statistic calculated as

\(t^{*}=\dfrac{ (\text{sample coefficient} - \text{hypothesized value})}{\text{standard error of coefficient}}\)

For our example above, the t -statistic is:

\(\begin{equation*} t^{*}=\dfrac{b_{1}-0}{\textrm{se}(b_{1})}=\dfrac{b_{1}}{\textrm{se}(b_{1})}. \end{equation*}\)

Note that the hypothesized value is usually just 0, so this portion of the formula is often omitted.

Multiple linear regression, in contrast to simple linear regression, involves multiple predictors and so testing each variable can quickly become complicated. For example, suppose we apply two separate tests for two predictors, say \(x_1\) and \(x_2\), and both tests have high p-values. One test suggests \(x_1\) is not needed in a model with all the other predictors included, while the other test suggests \(x_2\) is not needed in a model with all the other predictors included. But, this doesn't necessarily mean that both \(x_1\) and \(x_2\) are not needed in a model with all the other predictors included. It may well turn out that we would do better to omit either \(x_1\) or \(x_2\) from the model, but not both. How then do we determine what to do? We'll explore this issue further in Lesson 6 .

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Writing hypothesis for linear multiple regression models

I struggle writing hypothesis because I get very much confused by reference groups in the context of regression models.

For my example I'm using the mtcars dataset. The predictors are wt (weight), cyl (number of cylinders), and gear (number of gears), and the outcome variable is mpg (miles per gallon).

Say all your friends think you should buy a 6 cylinder car, but before you make up your mind you want to know how 6 cylinder cars perform miles-per-gallon-wise compared to 4 cylinder cars because you think there might be a difference.

Would this be a fair null hypothesis (since 4 cylinder cars is the reference group)?: There is no difference between 6 cylinder car miles-per-gallon performance and 4 cylinder car miles-per-gallon performance.

Would this be a fair model interpretation ?: 6 cylinder vehicles travel fewer miles per gallon (p=0.010, β -4.00, CI -6.95 - -1.04) as compared to 4 cylinder vehicles when adjusting for all other predictors, thus rejecting the null hypothesis.

Sorry for troubling, and thanks in advance for any feedback!

• multiple-regression
• linear-model
• interpretation

Yes, you already got the right answer to both of your questions.

• Your null hypothesis in completely fair. You did it the right way. When you have a factor variable as predictor, you omit one of the levels as a reference category (the default is usually the first one, but you also can change that). Then all your other levels’ coefficients are tested for a significant difference compared to the omitted category. Just like you did.

If you would like to compare 6-cylinder cars with 8-cylinder car, then you would have to change the reference category. In your hypothesis you just could had added at the end (or as a footnote): "when adjusting for weight and gear", but it is fine the way you did it.

• Your model interpretation is correct : It is perfect the way you did it. You could even had said: "the best estimate is that 6 cylinder vehicles travel 4 miles per gallon less than 4 cylinder vehicles (p-value: 0.010; CI: -6.95, -1.04), when adjusting for weight and gear, thus rejecting the null hypothesis".

Let's assume that your hypothesis was related to gears, and you were comparing 4-gear vehicles with 3-gear vehicles. Then your result would be β: 0.65; p-value: 0.67; CI: -2.5, 3.8. You would say that: "There is no statistically significant difference between three and four gear cars in fuel consumption, when adjusting for weight and engine power, thus failing to reject the null hypothesis".

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• Null and Alternative Hypotheses | Definitions & Examples

Null and Alternative Hypotheses | Definitions & Examples

Published on 5 October 2022 by Shaun Turney . Revised on 6 December 2022.

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :

• Null hypothesis (H 0 ): There’s no effect in the population .
• Alternative hypothesis (H A ): There’s an effect in the population.

The effect is usually the effect of the independent variable on the dependent variable .

Table of contents

Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, differences between null and alternative hypotheses, how to write null and alternative hypotheses, frequently asked questions about null and alternative hypotheses.

The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”, the null hypothesis (H 0 ) answers “No, there’s no effect in the population.” On the other hand, the alternative hypothesis (H A ) answers “Yes, there is an effect in the population.”

The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample.

You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.

The null hypothesis is the claim that there’s no effect in the population.

If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.

Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept. Be careful not to say you “prove” or “accept” the null hypothesis.

Null hypotheses often include phrases such as “no effect”, “no difference”, or “no relationship”. When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).

Examples of null hypotheses

The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.

*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .

The alternative hypothesis (H A ) is the other answer to your research question . It claims that there’s an effect in the population.

Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.

The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

Alternative hypotheses often include phrases such as “an effect”, “a difference”, or “a relationship”. When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes > or <). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.

Examples of alternative hypotheses

The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.

Null and alternative hypotheses are similar in some ways:

• They’re both answers to the research question
• They both make claims about the population
• They’re both evaluated by statistical tests.

However, there are important differences between the two types of hypotheses, summarized in the following table.

To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.

The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:

Does independent variable affect dependent variable ?

• Null hypothesis (H 0 ): Independent variable does not affect dependent variable .
• Alternative hypothesis (H A ): Independent variable affects dependent variable .

Test-specific

Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.

Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.

The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (‘ x affects y because …’).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses. In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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Multiple Regression

Now we're going to look at the rest of the data that we collected about the weight lifters. We will still have one response (y) variable, clean, but we will have several predictor (x) variables, age, body, and snatch. We're not going to use total because it's just the sum of snatch and clean.

The heaviest weights (in kg) that men who weigh more than 105 kg were able to lift are given in the table.

Data Dictionary

Regression Model

If there are k predictor variables, then the regression equation model is y = β 0 + β 1 x 1 + β 2 x 2 + ... + β k x k + ε.

The x 1 , x 2 , ..., x k represent the k predictor variables. Those parameters are the same as before, β 0 is the y-intercept or constant, β 1 is the coefficient on the first predictor variable, β 2 is the coefficient on the second predictor variable, and so on. ε is the error term or the residual that can't be explained by the model. Those parameters are estimated by b 0 , b 1 , b 2 , ..., b k .

This gives us a regression equation used for prediction of y = b 0 + b 1 x 1 + b 2 x 2 + ...+ b k x k .

Basically, everything we did with simple linear regression will just be extended to involve k predictor variables instead of just one.

Regression Analysis Explained

Round 1: all predictor variables included.

Minitab was used to perform the regression analysis. This is not really something you want to try by hand.

Response Variable: clean Predictor Variables: age, body, snatch

Regression Equation

There's the regression equation. You can use it for estimation purposes, but you really should look further down the page to see if the equation is a good predictor or not.

Table of Coefficients

Notice how the coefficients column (labeled "Coef") are again the coefficients that you find in the regression equation. The constant 32.88 is b 0 , the coefficient on age is b 1 = 1.0257, and so on.

Also notice that we have four test statistics and four p-values. That means that there were four hypothesis tests going on and four null hypotheses. The null hypothesis in each case is that the population parameter for that particular coefficient (or constant) is zero. If the coefficient is zero, then that variable drops out of the model and it doesn't contribute significantly to the model.

Here's a summary of the table of coefficients. We're making our decision at an α = 0.05 level of significance, so if the p-value < 0.05, we'll reject the null hypothesis and retain it otherwise.

A note about the T test statistics. They are once again the coefficient divided by the standard error of the coefficient, but this time that don't have n-2 degrees of freedom. If you remember what we wrote during simple linear regression, the df for each of these tests was actually the sample size minus the number of parameters being estimated. Well, in this case, we have four (4) parameters we're estimating, the constant and the three coefficients. Since our sample size was n = 14, our df = 14 - 4 = 10 for these tests.

A further note - don't just blindly get rid of every variable that doesn't appear to contribute to the model. This will be explained later, but there are correlations between variables that don't show themselves here.

Analysis of Variance

This is why we're really here, but if we take what we learned in simple linear regression and apply it, it's not that difficult to understand.

Notice how the total line is exactly the same as it was for the simple linear regression? That's because the response variable, clean, is still the same. All that has happened is that the amount of variation due to each source has changed.

Here's the table we saw with simple linear regression with the comments specific to simple linear regression removed. The same instructions work here with multiple regression.

The df(Regression) is one less than the number of parameters being estimated. There are k predictor variables and so there are k parameters for the coefficients on those variables. There is always one additional parameter for the constant so there are k+1 parameters. But the df is one less than the number of parameters, so there are k+1 - 1 = k degrees of freedom. That is, the df(Regression) = # of predictor variables.

The df(Residual) is the sample size minus the number of parameters being estimated, so it becomes df(Residual) = n - (k+1) or df(Residual) = n - k - 1. It's often easier just to use subtraction once you know the total and the regression degrees of freedom.

The df(Total) is still one less than the sample size as it was before. df(Total) = n - 1.

The table still works like all ANOVA tables. A variance is a variation divided by degrees of freedom, that is MS = SS / df. The F test statistic is the ratio of two sample variances with the denominator always being the error variance. So F = MS(Regression) / MS(Residual).

Even the hypothesis test here is an extension of simple linear regression. There, the null hypothesis was H 0 : β 1 = 0 versus the alternative hypothesis H 1 : β 1 ≠ 0.

In multiple regression, the hypotheses read like this:

H 0 : β 1 = β 2 = ... = β k = 0 H 1 : At least one β is not zero

The null hypothesis claims that there is no significant correlation at all. That is, all of the coefficients are zero and none of the variables belong in the model.

The alternative hypothesis is not that every variable belongs in the model but that at least one of the variables belongs in the model. If you remember back to probability, the complement of "none" is "at least one" and that's what we're seeing here.

In this case, because our p-value is 0.000, we would reject that there is no correlation at all and say that we do have a good model for prediction.

Summary Line

Recall that all the values on the summary line (plus some other useful ones) can be computed from the ANOVA table.

First, the MS(Total) is not given in the table, but we need it for other things. MS(Total) = SS(Total) / df(Total), it is not simply the sum of the other two MS values. MS(Total) = 4145.1 / 13 = 318.85. This is the value of the sample variance for the response variable clean. That is, s 2 = 318.85 and the sample standard deviation would be the square root of 318.85 or s = 17.86.

The value labeled S = 7.66222 is actually s e , the standard error of the estimate, and is the square root of the error variance, MS(Residual). The square root of 58.7 is 7.66159, but the difference is due to rounding errors.

The R-Sq is the multiple R 2 and is R 2 = ( SS(Total) - SS(Residual) ) / SS(Total).

R 2 = ( 4145.1 - 587.1 ) / 4145.1 = 0.858 = 85.8%

The R-Sq(adj) is the adjuster R 2 and is Adj-R 2 = ( MS(Total) - MS(Residual) ) / MS(Total).

Adj-R 2 = ( 318.85 - 58.7 ) / 318.85 = 0.816 = 81.6%

R-Squared vs Adjusted R-Squared

There is a problem with the R 2 for multiple regression. Yes, it is still the percent of the total variation that can be explained by the regression equation, but the largest value of R 2 will always occur when all of the predictor variables are included, even if those predictor variables don't significantly contribute to the model. R 2 will only go down (or stay the same) as variables are removed, but never increase.

The Adjusted-R 2 uses the variances instead of the variations. That means that it takes into consideration the sample size and the number of predictor variables. The value of the adjusted-R 2 can actually increase with fewer variables or smaller sample sizes. You should always look at the adjusted-R 2 when comparing models with different sample sizes or number of predictor variables, not the R 2 . If you have a tie for two models that have the same adjusted-R 2 , then take the one with the fewer variables as it's a simpler model.

Regression Analysis Repeated

Round 2: remove a predictor variable.

Do you remember earlier in this document when it appeared that neither age (p-value = 0.059) or body weight (p-value = 0.530) belonged in the model? Well now it's time to remove some variables.

We don't want to remove all the variables at once, though, because there might be some correlation between the predictor variables, so we'll pick the one that contributes the least to the model. This is the one with the largest p-value, so we'll get rid of body weight first.

Here are the results from Minitab.

Response Variable: clean Predictor Variables: age, snatch

Notice there are now 2 regression df in the ANOVA because we have two predictor variables. Also notice that the p-value on age is only marginally above the significance level so we may want to use it.

But the thing I want to look at here is the values of R-Sq and R-Sq(adj).

Notice that the R 2 has gone down but the Adjusted-R 2 has actually gone up from when we included all three variables. That is, we have a better model with only two variables than we did with three. That means that the model is easier to work with since there's not as much information to keep track of or substitute into the equation to make a prediction.

Round 3: Eliminating Another Variable

We said that the p-value for the age was slightly above 0.05, so we could say that age doesn't contribute greatly to the model. Let's throw it out and see how things are affected. At this point, we'll be back to the simple linear regression that we did earlier since we only have one predictor variable.

Here is the summary table again

Wow! Notice the big drops in both the R 2 and Adjusted-R 2 . For that reason, we're going to stick with the two variable model and use a competitor's age and the weight they can snatch to predict how much they can lift in the clean and jerk.

Last modified August 29, 2004 11:21 AM

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13.5 Testing the Significance of the Overall Model

Learning objectives.

• Conduct and interpret an overall model test on a multiple regression model.

Previously, we learned that the population model for the multiple regression equation is

[latex]\begin{eqnarray*} y & = & \beta_0+\beta_1x_1+\beta_2x_2+\cdots+\beta_kx_k +\epsilon \end{eqnarray*}[/latex]

where [latex]x_1,x_2,\ldots,x_k[/latex] are the independent variables, [latex]\beta_0,\beta_1,\ldots,\beta_k[/latex] are the population parameters of the regression coefficients, and [latex]\epsilon[/latex] is the error variable.  The error variable [latex]\epsilon[/latex] accounts for the variability in the dependent variable that is not captured by the linear relationship between the dependent and independent variables.  The value of [latex]\epsilon[/latex] cannot be determined, but we must make certain assumptions about [latex]\epsilon[/latex] and the errors/residuals in the model in order to conduct a hypothesis test on how well the model fits the data.  These assumptions include:

• The model is linear.
• The errors/residuals have a normal distribution.
• The mean of the errors/residuals is 0.
• The variance of the errors/residuals is constant.
• The errors/residuals are independent.

Because we do not have the population data, we cannot verify that these conditions are met.  We need to assume that the regression model has these properties in order to conduct hypothesis tests on the model.

Testing the Overall Model

We want to test if there is a relationship between the dependent variable and the set of independent variables.  In other words, we want to determine if the regression model is valid or invalid.

• Invalid Model .  There is no relationship between the dependent variable and the set of independent variables.  In this case, all of the regression coefficients [latex]\beta_i[/latex] in the population model are zero.  This is the claim for the null hypothesis in the overall model test:  [latex]H_0: \beta_1=\beta_2=\cdots=\beta_k=0[/latex].
• Valid Model.   There is a relationship between the dependent variable and the set of independent variables.  In this case, at least one of the regression coefficients [latex]\beta_i[/latex] in the population model is not zero.  This is the claim for the alternative hypothesis in the overall model test:  [latex]H_a: \mbox{at least one } \beta_i \neq 0[/latex].

The overall model test procedure compares the means of explained and unexplained variation in the model in order to determine if the explained variation (caused by the relationship between the dependent variable and the set of independent variables) in the model is larger than the unexplained variation (represented by the error variable [latex]\epsilon[/latex]).  If the explained variation is larger than the unexplained variation, then there is a relationship between the dependent variable and the set of independent variables, and the model is valid.  Otherwise, there is no relationship between the dependent variable and the set of independent variables, and the model is invalid.

The logic behind the overall model test is based on two independent estimates of the variance of the errors:

• One estimate of the variance of the errors, [latex]MSR[/latex], is based on the mean amount of explained variation in the dependent variable [latex]y[/latex].
• One estimate of the variance of the errors, [latex]MSE[/latex], is based on the mean amount of unexplained variation in the dependent variable [latex]y[/latex].

The overall model test compares these two estimates of the variance of the errors to determine if there is a relationship between the dependent variable and the set of independent variables.  Because the overall model test involves the comparison of two estimates of variance, an [latex]F[/latex]-distribution is used to conduct the overall model test, where the test statistic is the ratio of the two estimates of the variance of the errors.

The mean square due to regression , [latex]MSR[/latex], is one of the estimates of the variance of the errors.  The [latex]MSR[/latex] is the estimate of the variance of the errors determined by the variance of the predicted [latex]\hat{y}[/latex]-values from the regression model and the mean of the [latex]y[/latex]-values in the sample, [latex]\overline{y}[/latex].  If there is no relationship between the dependent variable and the set of independent variables, then the [latex]MSR[/latex] provides an unbiased estimate of the variance of the errors.  If there is a relationship between the dependent variable and the set of independent variables, then the [latex]MSR[/latex] provides an overestimate of the variance of the errors.

[latex]\begin{eqnarray*} SSR & = & \sum \left(\hat{y}-\overline{y}\right)^2 \\  \\ MSR & =& \frac{SSR}{k} \end{eqnarray*}[/latex]

The mean square due to error , [latex]MSE[/latex], is the other estimate of the variance of the errors.  The [latex]MSE[/latex] is the estimate of the variance of the errors determined by the error [latex](y-\hat{y})[/latex] in using the regression model to predict the values of the dependent variable in the sample.  The [latex]MSE[/latex] always provides an unbiased estimate of the variance of errors, regardless of whether or not there is a relationship between the dependent variable and the set of independent variables.

[latex]\begin{eqnarray*} SSE & = & \sum \left(y-\hat{y}\right)^2\\  \\ MSE & =& \frac{SSE}{n -k-1} \end{eqnarray*}[/latex]

The overall model test depends on the fact that the [latex]MSR[/latex] is influenced by the explained variation in the dependent variable, which results in the [latex]MSR[/latex] being either an unbiased or overestimate of the variance of the errors.  Because the [latex]MSE[/latex] is based on the unexplained variation in the dependent variable, the [latex]MSE[/latex] is not affected by the relationship between the dependent variable and the set of independent variables, and is always an unbiased estimate of the variance of the errors.

The null hypothesis in the overall model test is that there is no relationship between the dependent variable and the set of independent variables.  The alternative hypothesis is that there is a relationship between the dependent variable and the set of independent variables.  The [latex]F[/latex]-score for the overall model test is the ratio of the two estimates of the variance of the errors, [latex]\displaystyle{F=\frac{MSR}{MSE}}[/latex] with [latex]df_1=k[/latex] and [latex]df_2=n-k-1[/latex].  The p -value for the test is the area in the right tail of the [latex]F[/latex]-distribution to the right of the [latex]F[/latex]-score.

• If there is no relationship between the dependent variable and the set of independent variables, both the [latex]MSR[/latex] and the [latex]MSE[/latex] are unbiased estimates of the variance of the errors.  In this case, the [latex]MSR[/latex] and the [latex]MSE[/latex] are close in value, which results in an [latex]F[/latex]-score close to 1 and a large p -value.  The conclusion of the test would be that the null hypothesis is true.
• If there is a relationship between the dependent variable and the set of independent variables, the [latex]MSR[/latex] is an overestimate of the variance of the errors.  In this case, the [latex]MSR[/latex] is significantly larger than the [latex]MSE[/latex], which results in a large [latex]F[/latex]-score and a small p -value.  The conclusion of the test would be that the alternative hypothesis is true.

Steps to Conduct a Hypothesis Test on the Overall Regression Model

[latex]\begin{eqnarray*} H_0: &  &  \beta_1=\beta_2=\cdots=\beta_k=0 \\ \\ \end{eqnarray*}[/latex]

[latex]\begin{eqnarray*}  H_a: &  &  \mbox{at least one } \beta_i \mbox{ is not 0} \\ \\ \end{eqnarray*}[/latex]

• Collect the sample information for the test and identify the significance level [latex]alpha[/latex].

[latex]\begin{eqnarray*}F & = & \frac{MSR}{MSE} \\ \\ df_1 & = & k \\ \\  df_2 &  = & n-k-1 \\ \\ \end{eqnarray*}[/latex]

• The results of the sample data are significant.  There is sufficient evidence to conclude that the null hypothesis [latex]H_0[/latex] is an incorrect belief and that the alternative hypothesis [latex]H_a[/latex] is most likely correct.
• The results of the sample data are not significant.  There is not sufficient evidence to conclude that the alternative hypothesis [latex]H_a[/latex] may be correct.
• Write down a concluding sentence specific to the context of the question.

The calculation of the [latex]MSR[/latex], the [latex]MSE[/latex], and the [latex]F[/latex]-score for the overall model test can be time consuming, even with the help of software like Excel.  However, the required [latex]F[/latex]-score and p -value for the test can be found on the regression summary table, which we learned how to generate in Excel in a previous section.

The human resources department at a large company wants to develop a model to predict an employee’s job satisfaction from the number of hours of unpaid work per week the employee does, the employee’s age, and the employee’s income.  A sample of 25 employees at the company is taken and the data is recorded in the table below.  The employee’s income is recorded in $1000s and the job satisfaction score is out of 10, with higher values indicating greater job satisfaction.

Previously, we found the multiple regression equation to predict the job satisfaction score from the other variables:

[latex]\begin{eqnarray*} \hat{y} & = & 4.7993-0.3818x_1+0.0046x_2+0.0233x_3 \\ \\ \hat{y} & = & \mbox{predicted job satisfaction score} \\ x_1 & = & \mbox{hours of unpaid work per week} \\ x_2 & = & \mbox{age} \\ x_3 & = & \mbox{income (\$1000s)}\end{eqnarray*}[/latex]

At the 5% significance level, test the validity of the overall model to predict the job satisfaction score.

Hypotheses:

[latex]\begin{eqnarray*} H_0: & & \beta_1=\beta_2=\beta_3=0 \\   H_a: & & \mbox{at least one } \beta_i \mbox{ is not 0} \end{eqnarray*}[/latex]

The regression summary table generated by Excel is shown below:

The  p -value for the overall model test is in the middle part of the table under the ANOVA heading in the Significance F column of the Regression row .  So the  p -value=[latex]0.0017[/latex].

Conclusion:  

Because p -value[latex]=0.0017 \lt 0.05=\alpha[/latex], we reject the null hypothesis in favour of the alternative hypothesis.  At the 5% significance level there is enough evidence to suggest that there is a relationship between the dependent variable “job satisfaction” and the set of independent variables “hours of unpaid work per week,” “age”, and “income.”

• The null hypothesis [latex]\beta_1=\beta_2=\beta_3=0[/latex] is the claim that all of the regression coefficients are zero.  That is, the null hypothesis is the claim that there is no relationship between the dependent variable and the set of independent variables, which means that the model is not valid.
• The alternative hypothesis is the claim that at least one of the regression coefficients is not zero.  The alternative hypothesis is the claim that at least one of the independent variables is linearly related to the dependent variable, which means that the model is valid.  The alternative hypothesis does not say that all of the regression coefficients are not zero, only that at least one of them is not zero.  The alternative hypothesis does not tell us which independent variables are related to the dependent variable.
• The p -value for the overall model test is located in the middle part of the table under the Significance F column heading in the Regression row (right underneath the ANOVA heading ).  You will notice a p -value column heading at the bottom of the table in the rows corresponding to the independent variables.  These p -values in the bottom part of the table are not related to the overall model test we are conducting here.  These p -values in the independent variable rows are the p -values we will need when we conduct tests on the individual regression coefficients in the next section.
• The p -value of 0.0017 is a small probability compared to the significance level, and so is unlikely to happen assuming the null hypothesis is true.  This suggests that the assumption that the null hypothesis is true is most likely incorrect, and so the conclusion of the test is to reject the null hypothesis in favour of the alternative hypothesis.  In other words, at least one of the regression coefficients is not zero and at least one independent variable is linearly related to the dependent variable.

Watch this video: Basic Excel Business Analytics #51: Testing Significance of Regression Relationship with p-value by ExcelIsFun [20:44]

Concept Review

The overall model test determines if there is a relationship between the dependent variable and the set of independent variable.  The test compares two estimates of the variance of the errors ([latex]MSR[/latex] and [latex]MSE[/latex]).  The ratio of these two estimates of the variance of the errors is the [latex]F[/latex]-score from an [latex]F[/latex]-distribution with [latex]df_1=k[/latex] and [latex]df_2=n-k-1[/latex].  The p -value for the test is the area in the right tail of the [latex]F[/latex]-distribution.  The p -value can be found on the regression summary table generated by Excel.

The overall model hypothesis test is a well established process:

• Write down the null and alternative hypotheses in terms of the regression coefficients.  The null hypothesis is the claim that there is no relationship between the dependent variable and the set of independent variables.  The alternative hypothesis is the claim that there is a relationship between the dependent variable and the set of independent variables.
• Collect the sample information for the test and identify the significance level.
• The p -value is the area in the right tail of the [latex]F[/latex]-distribution.  Use the regression summary table generated by Excel to find the p -value.
• Compare the  p -value to the significance level and state the outcome of the test.

Introduction to Statistics Copyright © 2022 by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

Understanding the Null Hypothesis for Linear Regression

Linear regression is a technique we can use to understand the relationship between one or more predictor variables and a response variable .

If we only have one predictor variable and one response variable, we can use simple linear regression , which uses the following formula to estimate the relationship between the variables:

ŷ = β 0 + β 1 x

• ŷ: The estimated response value.
• β 0 : The average value of y when x is zero.
• β 1 : The average change in y associated with a one unit increase in x.
• x: The value of the predictor variable.

Simple linear regression uses the following null and alternative hypotheses:

• H 0 : β 1 = 0
• H A : β 1 ≠ 0

The null hypothesis states that the coefficient β 1 is equal to zero. In other words, there is no statistically significant relationship between the predictor variable, x, and the response variable, y.

The alternative hypothesis states that β 1 is not equal to zero. In other words, there is a statistically significant relationship between x and y.

If we have multiple predictor variables and one response variable, we can use multiple linear regression , which uses the following formula to estimate the relationship between the variables:

ŷ = β 0 + β 1 x 1 + β 2 x 2 + … + β k x k

• β 0 : The average value of y when all predictor variables are equal to zero.
• β i : The average change in y associated with a one unit increase in x i .
• x i : The value of the predictor variable x i .

Multiple linear regression uses the following null and alternative hypotheses:

• H 0 : β 1 = β 2 = … = β k = 0
• H A : β 1 = β 2 = … = β k ≠ 0

The null hypothesis states that all coefficients in the model are equal to zero. In other words, none of the predictor variables have a statistically significant relationship with the response variable, y.

The alternative hypothesis states that not every coefficient is simultaneously equal to zero.

The following examples show how to decide to reject or fail to reject the null hypothesis in both simple linear regression and multiple linear regression models.

Example 1: Simple Linear Regression

Suppose a professor would like to use the number of hours studied to predict the exam score that students will receive in his class. He collects data for 20 students and fits a simple linear regression model.

The following screenshot shows the output of the regression model:

The fitted simple linear regression model is:

Exam Score = 67.1617 + 5.2503*(hours studied)

To determine if there is a statistically significant relationship between hours studied and exam score, we need to analyze the overall F value of the model and the corresponding p-value:

• Overall F-Value:  47.9952
• P-value:  0.000

Since this p-value is less than .05, we can reject the null hypothesis. In other words, there is a statistically significant relationship between hours studied and exam score received.

Example 2: Multiple Linear Regression

Suppose a professor would like to use the number of hours studied and the number of prep exams taken to predict the exam score that students will receive in his class. He collects data for 20 students and fits a multiple linear regression model.

The fitted multiple linear regression model is:

Exam Score = 67.67 + 5.56*(hours studied) – 0.60*(prep exams taken)

To determine if there is a jointly statistically significant relationship between the two predictor variables and the response variable, we need to analyze the overall F value of the model and the corresponding p-value:

• Overall F-Value:  23.46
• P-value:  0.00

Since this p-value is less than .05, we can reject the null hypothesis. In other words, hours studied and prep exams taken have a jointly statistically significant relationship with exam score.

Note: Although the p-value for prep exams taken (p = 0.52) is not significant, prep exams combined with hours studied has a significant relationship with exam score.

Additional Resources

Understanding the F-Test of Overall Significance in Regression How to Read and Interpret a Regression Table How to Report Regression Results How to Perform Simple Linear Regression in Excel How to Perform Multiple Linear Regression in Excel

The Complete Guide: How to Report Regression Results

R vs. r-squared: what’s the difference, related posts, how to normalize data between -1 and 1, vba: how to check if string contains another..., how to interpret f-values in a two-way anova, how to create a vector of ones in..., how to determine if a probability distribution is..., what is a symmetric histogram (definition & examples), how to find the mode of a histogram..., how to find quartiles in even and odd..., how to calculate sxy in statistics (with example), how to calculate expected value of x^3.

Correlation and Regression with R

•   1  
• |   2  
• |   3  
• |   4  
• |   5  
• |   6  
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• Contributing Authors:
• Learning Objectives
• The Dataset
• Correlation
• Pearson Correlation
• Spearman's rank correlation
• Some Notes on Correlation
• Simple Linear Regression
• Introduction
• Simple Linear Regression Model Fitting
• Other Functions for Fitted Linear Model Objects

Multiple Linear Regression

Model specification and output, model with categorical variables or factors.

• Regression Diagnostics
• Model Assumptions
• Diagnostic Plots
• More Diagnostics

Intro to R Contents

Common R Commands

In reality, most regression analyses use more than a single predictor. Specification of a multiple regression analysis is done by setting up a model formula with plus (+) between the predictors:

> lm2<-lm(pctfat.brozek~age+fatfreeweight+neck,data=fatdata)

which corresponds to the following multiple linear regression model:

pctfat.brozek = β 0 + β 1 *age + β 2 *fatfreeweight + β 3 *neck + ε

This tests the following hypotheses:

• H 0 : There is no linear association between pctfat.brozek and age, fatfreeweight and neck.
• H a : Here is a linear association between pctfat.brozek and age, fatfreeweight and neck.

> summary(lm2)

lm(formula = pctfat.brozek ~ age + fatfreeweight + neck, data = fatdata)

         Min        1Q       Median        3Q        Max

-16.67871  -3.62536   0.07768   3.65100  16.99197

Coefficients:

               Estimate    Std. Error t value Pr(>|t|)   

(Intercept)    -53.01330   5.99614   -8.841   < 2e-16 ***

age            0.03832    0.03298    1.162    0.246   

fatfreeweight  -0.23200    0.03086  -7.518    1.02e-12 ***

neck            2.72617    0.22627  12.049   < 2e-16 ***

Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 5.901 on 248 degrees of freedom

Multiple R-squared: 0.4273,     Adjusted R-squared: 0.4203

F-statistic: 61.67 on 3 and 248 DF,  p-value: < 2.2e-16

Global Null Hypothesis

• When testing the null hypothesis that there is no linear association between Brozek percent fat, age, fatfreeweight, and neck, we reject the null hypothesis (F 3,248 = 61.67, p-value < 2.2e-16). Age, fatfreeweight and neck explain 42.73% of the variability in Brozek percent fat.

Main Effects Hypothesis

• When testing the null hypothesis that there is no linear association between Brozek percent fat and age after adjusting for fatfreeweight and neck, we fail to reject the null hypothesis (t = 1.162, df = 248, p-value = 0.246).  For a one-unit change in age, on average, the Brozek percent fat increases by 0.03, after adjusting for fatfreeweight and neck.
• When testing the null hypothesis that there is no linear association between Brozek percent fat and fatfreeweight after adjusting for age and neck, we reject the null hypothesis (t = -7.518, df = 248, p-value =1.02e-12).  For a one-unit increase in fatfreeweight, Brozek percent fat decreases by 0.23 units after adjusting for age and neck.
• When testing the null hypothesis that there is no linear association between Brozek percent fat and neck after adjusting for fatfreeweight and age, we reject the null hypothesis (t = 12.049, df = 248, p-value < 2e-16).  For a one-unit increase in neck there is a 2.73 increase in Brozek percent fat, after adjusting for age and fatfreeweight.

Sometimes, we may be also interested in using categorical variables as predictors. According to the information posted in the website of National Heart Lung and Blood Institute ( http://www.nhlbi.nih.gov/health/public/heart/obesity/lose_wt/risk.htm ), individuals with body mass index (BMI) greater than or equal to 25 are classified as overweight or obesity. In our dataset, the variable adiposity is equivalent to BMI.

 With the variable bmi you generated from the previous exercise, we go ahead to model our data.

> lm3 <- lm(pctfat.brozek ~ age + fatfreeweight + neck + factor(bmi), data = fatdata)

> summary(lm3)

lm(formula = pctfat.brozek ~ age + fatfreeweight + neck + factor(bmi),

    data = fatdata)

     Min       1Q   Median       3Q      Max

-13.4222  -3.0969  -0.2637   2.7280  13.3875

                                  Estimate Std. Error t value Pr(>|t|)   

(Intercept)                      -21.31224    6.32852  -3.368 0.000879 ***

age                                0.01698    0.02887   0.588 0.556890   

fatfreeweight                     -0.23488    0.02691  -8.727 3.97e-16 ***

neck                               1.83080    0.22152   8.265 8.63e-15 ***

factor(bmi)overweight or obesity   7.31761    0.82282   8.893  < 2e-16 ***

Residual standard error: 5.146 on 247 degrees of freedom

Multiple R-squared:  0.5662,     Adjusted R-squared:  0.5591

F-statistic: 80.59 on 4 and 247 DF,  p-value: < 2.2e-16

Note that although factor bmi has two levels, the result only shows one level: "overweight or obesity", which is called the "treatment effect" . In other words, the level "normal or underweight" is considered as baseline or reference group and the estimate of factor(bmi) overweight or obesity 7.3176 is the effect difference of these two levels on percent body fat.

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8.7: Overall F-test in multiple linear regression

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• Mark Greenwood
• Montana State University

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In the MLR summary, there is an \(F\) -test and p-value reported at the bottom of the output. For the model with Elevation and Maximum Temperature , the last row of the model summary is:

This test is called the overall F-test in MLR and is very similar to the \(F\) -test in a reference-coded One-Way ANOVA model. It tests the null hypothesis that involves setting every coefficient except the \(y\) -intercept to 0 (so all the slope coefficients equal 0). We saw this reduced model in the One-Way material when we considered setting all the deviations from the baseline group to 0 under the null hypothesis. We can frame this as a comparison between a full and reduced model as follows:

• Full Model: \(y_i = \beta_0 + \beta_1x_{1i} + \beta_2x_{2i}+\cdots + \beta_Kx_{Ki}+\varepsilon_i\)
• Reduced Model: \(y_i = \beta_0 + 0x_{1i} + 0x_{2i}+\cdots + 0x_{Ki}+\varepsilon_i\)

The reduced model estimates the same values for all \(y\text{'s}\) , \(\widehat{y}_i = \bar{y} = b_0\) and corresponds to the null hypothesis of:

\(\boldsymbol{H_0:}\) No explanatory variables should be included in the model: \(\beta_1 = \beta_2 = \cdots = \beta_K = 0\) .

The full model corresponds to the alternative:

\(\boldsymbol{H_A:}\) At least one explanatory variable should be included in the model: Not all \(\beta_k\text{'s} = 0\) for \((k = 1,\ldots,K)\) .

Note that \(\beta_0\) is not set to 0 in the reduced model (under the null hypothesis) – it becomes the true mean of \(y\) for all values of the \(x\text{'s}\) since all the predictors are multiplied by coefficients of 0.

The test statistic to assess these hypotheses is \(F = \text{MS}_{\text{model}}/\text{MS}_E\) , which is assumed to follow an \(F\) -distribution with \(K\) numerator df and \(n-K-1\) denominator df under the null hypothesis. The output provides us with \(F(2, 20) = 56.43\) and a p-value of \(5.979*10^{-9}\) (p-value \(<0.00001\) ) and strong evidence against the null hypothesis. Thus, there is strong evidence against the null hypothesis that the true slopes for the two predictors are 0 and so we would conclude that at least one of the two slope coefficients ( Max.Temp ’s or Elevation ’s) is different from 0 in the population of SNOTEL sites in Montana on this date. While this test is a little bit interesting and a good indicator of something interesting existing in the model, the moment you see this result, you want to know more about each predictor variable. If neither predictor variable is important, we will discover that in the \(t\) -tests for each coefficient and so our general recommendation is to start there.

The overall F-test, then, is really about testing whether there is something good in the model somewhere. And that certainly is important but it is also not too informative. There is one situation where this test is really interesting, when there is only one predictor variable in the model (SLR). In that situation, this test provides exactly the same p-value as the \(t\) -test. \(F\) -tests will be important when we are mixing categorical and quantitative predictor variables in our MLR models (Section 8.12), but the overall \(F\) -test is of very limited utility.

What is The Null Hypothesis & When Do You Reject The Null Hypothesis

Julia Simkus

Editor at Simply Psychology

BA (Hons) Psychology, Princeton University

Julia Simkus is a graduate of Princeton University with a Bachelor of Arts in Psychology. She is currently studying for a Master's Degree in Counseling for Mental Health and Wellness in September 2023. Julia's research has been published in peer reviewed journals.

Learn about our Editorial Process

Saul Mcleod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

On This Page:

A null hypothesis is a statistical concept suggesting no significant difference or relationship between measured variables. It’s the default assumption unless empirical evidence proves otherwise.

The null hypothesis states no relationship exists between the two variables being studied (i.e., one variable does not affect the other).

The null hypothesis is the statement that a researcher or an investigator wants to disprove.

Testing the null hypothesis can tell you whether your results are due to the effects of manipulating ​ the dependent variable or due to random chance. 

How to Write a Null Hypothesis

Null hypotheses (H0) start as research questions that the investigator rephrases as statements indicating no effect or relationship between the independent and dependent variables.

It is a default position that your research aims to challenge or confirm.

For example, if studying the impact of exercise on weight loss, your null hypothesis might be:

There is no significant difference in weight loss between individuals who exercise daily and those who do not.

Examples of Null Hypotheses

When Do We Reject The Null Hypothesis? 

We reject the null hypothesis when the data provide strong enough evidence to conclude that it is likely incorrect. This often occurs when the p-value (probability of observing the data given the null hypothesis is true) is below a predetermined significance level.

If the collected data does not meet the expectation of the null hypothesis, a researcher can conclude that the data lacks sufficient evidence to back up the null hypothesis, and thus the null hypothesis is rejected. 

Rejecting the null hypothesis means that a relationship does exist between a set of variables and the effect is statistically significant ( p > 0.05).

If the data collected from the random sample is not statistically significance , then the null hypothesis will be accepted, and the researchers can conclude that there is no relationship between the variables. 

You need to perform a statistical test on your data in order to evaluate how consistent it is with the null hypothesis. A p-value is one statistical measurement used to validate a hypothesis against observed data.

Calculating the p-value is a critical part of null-hypothesis significance testing because it quantifies how strongly the sample data contradicts the null hypothesis.

The level of statistical significance is often expressed as a  p  -value between 0 and 1. The smaller the p-value, the stronger the evidence that you should reject the null hypothesis.

Usually, a researcher uses a confidence level of 95% or 99% (p-value of 0.05 or 0.01) as general guidelines to decide if you should reject or keep the null.

When your p-value is less than or equal to your significance level, you reject the null hypothesis.

In other words, smaller p-values are taken as stronger evidence against the null hypothesis. Conversely, when the p-value is greater than your significance level, you fail to reject the null hypothesis.

In this case, the sample data provides insufficient data to conclude that the effect exists in the population.

Because you can never know with complete certainty whether there is an effect in the population, your inferences about a population will sometimes be incorrect.

When you incorrectly reject the null hypothesis, it’s called a type I error. When you incorrectly fail to reject it, it’s called a type II error.

Why Do We Never Accept The Null Hypothesis?

The reason we do not say “accept the null” is because we are always assuming the null hypothesis is true and then conducting a study to see if there is evidence against it. And, even if we don’t find evidence against it, a null hypothesis is not accepted.

A lack of evidence only means that you haven’t proven that something exists. It does not prove that something doesn’t exist. 

It is risky to conclude that the null hypothesis is true merely because we did not find evidence to reject it. It is always possible that researchers elsewhere have disproved the null hypothesis, so we cannot accept it as true, but instead, we state that we failed to reject the null. 

One can either reject the null hypothesis, or fail to reject it, but can never accept it.

Why Do We Use The Null Hypothesis?

We can never prove with 100% certainty that a hypothesis is true; We can only collect evidence that supports a theory. However, testing a hypothesis can set the stage for rejecting or accepting this hypothesis within a certain confidence level.

The null hypothesis is useful because it can tell us whether the results of our study are due to random chance or the manipulation of a variable (with a certain level of confidence).

A null hypothesis is rejected if the measured data is significantly unlikely to have occurred and a null hypothesis is accepted if the observed outcome is consistent with the position held by the null hypothesis.

Rejecting the null hypothesis sets the stage for further experimentation to see if a relationship between two variables exists. 

Hypothesis testing is a critical part of the scientific method as it helps decide whether the results of a research study support a particular theory about a given population. Hypothesis testing is a systematic way of backing up researchers’ predictions with statistical analysis.

It helps provide sufficient statistical evidence that either favors or rejects a certain hypothesis about the population parameter. 

Purpose of a Null Hypothesis 

• The primary purpose of the null hypothesis is to disprove an assumption. 
• Whether rejected or accepted, the null hypothesis can help further progress a theory in many scientific cases.
• A null hypothesis can be used to ascertain how consistent the outcomes of multiple studies are.

Do you always need both a Null Hypothesis and an Alternative Hypothesis?

The null (H0) and alternative (Ha or H1) hypotheses are two competing claims that describe the effect of the independent variable on the dependent variable. They are mutually exclusive, which means that only one of the two hypotheses can be true. 

While the null hypothesis states that there is no effect in the population, an alternative hypothesis states that there is statistical significance between two variables. 

The goal of hypothesis testing is to make inferences about a population based on a sample. In order to undertake hypothesis testing, you must express your research hypothesis as a null and alternative hypothesis. Both hypotheses are required to cover every possible outcome of the study. 

What is the difference between a null hypothesis and an alternative hypothesis?

The alternative hypothesis is the complement to the null hypothesis. The null hypothesis states that there is no effect or no relationship between variables, while the alternative hypothesis claims that there is an effect or relationship in the population.

It is the claim that you expect or hope will be true. The null hypothesis and the alternative hypothesis are always mutually exclusive, meaning that only one can be true at a time.

What are some problems with the null hypothesis?

One major problem with the null hypothesis is that researchers typically will assume that accepting the null is a failure of the experiment. However, accepting or rejecting any hypothesis is a positive result. Even if the null is not refuted, the researchers will still learn something new.

Why can a null hypothesis not be accepted?

We can either reject or fail to reject a null hypothesis, but never accept it. If your test fails to detect an effect, this is not proof that the effect doesn’t exist. It just means that your sample did not have enough evidence to conclude that it exists.

We can’t accept a null hypothesis because a lack of evidence does not prove something that does not exist. Instead, we fail to reject it.

Failing to reject the null indicates that the sample did not provide sufficient enough evidence to conclude that an effect exists.

If the p-value is greater than the significance level, then you fail to reject the null hypothesis.

Is a null hypothesis directional or non-directional?

A hypothesis test can either contain an alternative directional hypothesis or a non-directional alternative hypothesis. A directional hypothesis is one that contains the less than (“<“) or greater than (“>”) sign.

A nondirectional hypothesis contains the not equal sign (“≠”).  However, a null hypothesis is neither directional nor non-directional.

A null hypothesis is a prediction that there will be no change, relationship, or difference between two variables.

The directional hypothesis or nondirectional hypothesis would then be considered alternative hypotheses to the null hypothesis.

Gill, J. (1999). The insignificance of null hypothesis significance testing.  Political research quarterly ,  52 (3), 647-674.

Krueger, J. (2001). Null hypothesis significance testing: On the survival of a flawed method.  American Psychologist ,  56 (1), 16.

Masson, M. E. (2011). A tutorial on a practical Bayesian alternative to null-hypothesis significance testing.  Behavior research methods ,  43 , 679-690.

Nickerson, R. S. (2000). Null hypothesis significance testing: a review of an old and continuing controversy.  Psychological methods ,  5 (2), 241.

Rozeboom, W. W. (1960). The fallacy of the null-hypothesis significance test.  Psychological bulletin ,  57 (5), 416.

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A comparative analysis of polynomial regression and artificial neural networks for prediction of lighting consumption.

1. Introduction

• Identifying Influencing Factors
• Historical data: The historical dataset provides a valuable record of past consumption patterns. By analyzing historical data, it is possible to identify trends, seasonal variations and potential anomalies.
• The climate: the influence of temperature, humidity and sunlight exposure on energy consumption for heating, cooling and lighting is a significant factor.
• Building characteristics: the size, insulation levels and energy-efficient features of a building play a pivotal role in determining its overall energy needs.
• The patterns of occupancy: the number of occupants and their activities have a significant impact on energy consumption, particularly in relation to lighting, heating and cooling, and appliance usage [ 1 , 3 , 5 , 6 ].
• Regression analysis is a statistical technique that can be employed to identify linear relationships between the influencing factors and the target variable (electricity consumption);
• Machine learning techniques, such as neural networks and support vector machines, are capable of capturing more complex, non-linear relationships.
• Research Problem:
• Research Motivation:

2. Prediction of Electricity Consumption

• Multiple methods: different studies have explored different methods for prediction, including data-driven algorithms, support vector machines, neural networks and regression analysis.
• Comparison of methods: studies have compared the accuracy and efficiency of different methods, often finding that SVM or decision tree models perform well.
• Focus on building vs. lighting: most research has focused on predicting overall building consumption, neglecting lighting specifically, which is influenced by daylight and not just outside temperature.
• Predicting lighting consumption: a few studies have directly addressed lighting predictions using a SVM, historical data analysis or an ANN, based on weather data.
• Importance of lighting: lighting accounts for a significant portion (15–20%) of total building energy use and offers potential savings through control algorithms and efficient light sources.
• Synergy with daylight: the effective use of daylight can significantly reduce lighting energy requirements.
• Studies have proposed simplified models and machine learning algorithms to predict daylight performance in buildings, particularly during the early design stage.
• They have explored ways to overcome limitations in current methods and design buildings that prioritize daylight access, such as schools with skylights.
• Research has investigated the relationship between daylight and occupant comfort, finding that lighting uniformity and natural ventilation play a significant role.
• Benefits of research:
• The present study focuses on the prediction of energy consumption in a building based on illumination levels. The objective of the research is to develop models that could forecast energy consumption based on a specific variable (illumination levels).
• The variable of interest is illumination levels, as the primary factor influencing energy consumption.
• A comparison of the two methods was conducted. The research compared two main methodologies: regression analysis and neural networks.
• The efficacy of the models was gauged in terms of their accuracy, resilience (performance with varying inputs) and interpretability (regression) as evaluation criteria in the context of energy consumption prediction.

2.1. Basic Approaches to Prediction: Regression and Artificial Neural Networks

• Regression analysis is a statistical method used to model the dependence between variables. In the context of forecasting, it estimates the target variable based on known input parameters. Its advantages include ease of implementation and interpretation of results. However, regression analysis has limited flexibility and cannot capture complex non-linear dependencies.
• ANNs provide a sophisticated approach for prediction. They are inspired by the workings of the human brain and can learn from complex data. Their flexible structure enables them to model even non-linear dependencies, capturing complex relationships between variables. However, a disadvantage of ANNs is that they can be challenging to implement and in the interpretation of results.

2.1.1. Regression

• Linear regression
• Simple linear regression
• Multiple linear regression
• Polynomial regression

2.1.2. The Backpropagation Neural Network

• Learning arbitrary non-linear mappings: unlike simpler models, BPNNs can capture intricate, non-linear patterns within data, making them suitable for a wide range of real-world applications [ 41 ].
• Self-organizing learning: BPNNs exhibit a degree of self-organization, meaning they can autonomously learn complex relationships within data without explicit programming of these relationships [ 37 ].
• Efficient training with backpropagation: The backpropagation algorithm enables efficient training of BPNNs, even for large datasets. It works by propagating the error signal backward through the network, allowing the network to adjust its internal weights and biases in a way that minimizes errors [ 42 , 43 ].
• Parallel processing capabilities: the architecture of BPNNs facilitates parallel processing, making them well-suited for handling large datasets on computing systems with multiple cores or processors [ 42 ].
• Resilience and fault tolerance: studies have shown that BPNNs exhibit a certain level of resilience and fault tolerance, meaning they can maintain functionality even in the presence of minor errors or disruptions [ 44 , 45 , 46 ].
• Layered architecture: BPNNs are characterized by a layered architecture, typically consisting of the following layers: ➢ Input layer: this layer receives the initial input data, representing the features or independent variables used for training. ➢ Hidden layers: these intermediate layers, which are present in varying numbers depending on the network complexity, perform the core information processing and feature extraction within the network. ➢ Output layer: the final layer produces the network’s predicted output, corresponding to the target variable or desired outcome.
• Interconnected neuron nodes: Each layer, except the input layer, comprises interconnected neuron nodes. These nodes act as the fundamental processing units of the network, performing calculations to transform received information.
• Weighted inputs: Each neuron receives input values from other nodes in the preceding layer. These inputs are multiplied by corresponding weights, which act as learnable parameters that determine the influence of each input on the neuron’s output.
• Activation function: The weighted sum of inputs is then processed through a non-linear activation function. This function introduces non-linearity into the network, enabling it to model complex relationships that cannot be captured by linear models.
• Threshold and output: The output of the activation function is further compared to a threshold value. If the result exceeds the threshold, the neuron is considered “activated” and its output is typically set to one. Otherwise, the output remains zero.
• Initialization: The initial step involves assigning random weights to the connections between neurons in different layers. These weights typically fall within a predefined range, such as [−1, 1].
• Input processing: The input layer receives the data from a training sample. This data represents the features or independent variables used for training.
• Layer-wise computation: Information propagates forward through the network layer by layer. At each layer, each neuron receives weighted inputs from the previous layer, applies a non-linear activation function and calculates its output value. This process transforms and transmits the information through the network.
• Output calculation: The final layer processes the information received from the previous layer and generates the network’s final output, representing the predicted value for the given training sample. This completes a single forward propagation cycle [ 35 , 40 ].
• Error calculation: Once the output is obtained, the network compares it to the desired target value (known from the training data). If there is a difference (error), the backpropagation phase begins.
• Error propagation: The error signal is propagated backward through the network, layer by layer. This involves calculating the contribution of each neuron’s error to the overall network error [ 43 ].
• Weight adjustment: Using an optimization algorithm (e.g., gradient descent), the weights connecting the neurons are adjusted in a way that minimizes the overall error. This adjustment process aims to improve the network’s ability to map the input data to the desired output in future training iterations. The weight update rule is given by Equation (10) [ 33 ], as follows:
• Iteration: the training process continues until the network’s output error reaches a predefined threshold or a specified number of iterations is completed.

2.2. Quality of Model

2.2.1. correlation coefficient.

• r = 0 indicates that there is no linear relationship between the variables.
• r = 1 indicates that there is a strong positive linear relationship between the variables.
• r   = −1 indicates that there is a strong negative linear relationship between the variables.

2.2.2. Coefficient of Determination

2.2.3. the root mean square error, 3. methodology, 4. prediction model, 4.1. the building description and dataset acquisition, 4.2. prediction model utilizing regression analysis, 4.2.1. application of polynomial regression to consumption prediction, 4.2.2. accuracy verification and comparison of individual models.

• Verification for 140 lx
• Verification for 155 lx
• Verification for 175 lx

4.2.3. Comparing the Accuracy of Individual Models for Light Intensity

4.3. prediction model utilizing artificial neural networks.

• Training: The ANN is trained using the specified training data. During this process, the network learns the underlying patterns and relationships within the data. The training process involves optimizing the network parameters to minimize the difference between predicted and actual energy consumption.
• Validation: The performance of the trained ANN is evaluated using the validation and test datasets. Evaluation metrics such as RMSE, R 2 and r are used to assess the model’s ability to accurately predict changing energy consumption for previously unseen data.

4.3.1. Application of Neural Network Model to Consumption Prediction

4.3.2. accuracy verification and comparison of individual models, 4.3.3. comparing the accuracy of individual models for light intensity, 5. determination and verification of the prediction curve for internal lighting in the interval of 100–200 lx, 5.1. regression model, verification of the accuracy of the prediction function, 5.2. neural network model, verifying the accuracy of the prediction function.

• Accurate approximation by the neural network: In this particular case, the neural network (NN) was able to accurately approximate the real value of energy consumption. This means that the predicted value without correction was already so accurate that the effect of the correction was minimal.
• Confirmation of the correctness of the approach: This phenomenon perfectly demonstrates and confirms the correctness of the chosen approach and solution. The prediction function predicts the value with the required accuracy without the need for correction.
• Proof of the correction function: At the same time, the concept of correcting the initial values is confirmed. Although, in this case, the correction was minimal, its implementation ensures that the results are provably correct, even in cases where the predicted value is less accurate.

6. Discussion

• Limitations
• Causes of incorrect operation
• Practical application
• Accuracy: it can accurately predict energy consumption even at high outdoor light levels.
• Speed: the calculation is done by multiplying constants, which is extremely fast with today’s computing power.
• Stability: it is robust to errors and unstable behavior in extreme conditions.
• Adaptability: the ANN model is adaptive and can adapt to different types of lighting systems.
• Robustness: the model is robust and resistant to errors in the measured data.
• The disadvantages of the ANN model are as follows:
• Training: training the model can take quite a long time, depending on the size and complexity of the training data set.
• Difficulty: implementing and understanding the ANN model can be more challenging for some users.
• Fixed lighting levels: setting lighting levels to the standardized values required for laboratory operations.
• Occupancy detection: utilizing occupancy sensors to detect the presence or absence of laboratory personnel.
• User overrides: providing manual controls for laboratory personnel to override the automated system if necessary for specific tasks or experiments.

7. Conclusions

• Improved operational planning: by accurately predicting lighting consumption, building managers can more effectively prepare operational strategies.
• Improved integration of renewable resources: the ability to predict lighting demand enables the better integration of renewable energy sources.
• Improved building energy efficiency: the accurate prediction of lighting consumption enables building managers to implement strategies that optimize energy use, resulting in a more energy-efficient building.
• Reduced greenhouse gas emissions: as a direct result of increased energy efficiency, greenhouse gas emissions associated with building operations are also reduced.
• Extending the data collection period to capture seasonal variations in lighting demand.
• Including additional relevant data points, such as weather conditions or occupancy patterns.
• Exploring data augmentation techniques to artificially increase the size and diversity of the training data.
• Combination of lighting system with other systems, e.g., control of the shutters and detection of persons.

Author Contributions

Data availability statement, conflicts of interest.

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Share and Cite

Belany, P.; Hrabovsky, P.; Sedivy, S.; Cajova Kantova, N.; Florkova, Z. A Comparative Analysis of Polynomial Regression and Artificial Neural Networks for Prediction of Lighting Consumption. Buildings 2024 , 14 , 1712. https://doi.org/10.3390/buildings14061712

Belany P, Hrabovsky P, Sedivy S, Cajova Kantova N, Florkova Z. A Comparative Analysis of Polynomial Regression and Artificial Neural Networks for Prediction of Lighting Consumption. Buildings . 2024; 14(6):1712. https://doi.org/10.3390/buildings14061712

Belany, Pavol, Peter Hrabovsky, Stefan Sedivy, Nikola Cajova Kantova, and Zuzana Florkova. 2024. "A Comparative Analysis of Polynomial Regression and Artificial Neural Networks for Prediction of Lighting Consumption" Buildings 14, no. 6: 1712. https://doi.org/10.3390/buildings14061712

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