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Statistical Treatment of Data – Explained & Example

• By DiscoverPhDs
• September 8, 2020

‘Statistical treatment’ is when you apply a statistical method to a data set to draw meaning from it. Statistical treatment can be either descriptive statistics, which describes the relationship between variables in a population, or inferential statistics, which tests a hypothesis by making inferences from the collected data.

Introduction to Statistical Treatment in Research

Every research student, regardless of whether they are a biologist, computer scientist or psychologist, must have a basic understanding of statistical treatment if their study is to be reliable.

This is because designing experiments and collecting data are only a small part of conducting research. The other components, which are often not so well understood by new researchers, are the analysis, interpretation and presentation of the data. This is just as important, if not more important, as this is where meaning is extracted from the study .

What is Statistical Treatment of Data?

Statistical treatment of data is when you apply some form of statistical method to a data set to transform it from a group of meaningless numbers into meaningful output.

Statistical treatment of data involves the use of statistical methods such as:

• regression,
• conditional probability,
• standard deviation and
• distribution range.

These statistical methods allow us to investigate the statistical relationships between the data and identify possible errors in the study.

In addition to being able to identify trends, statistical treatment also allows us to organise and process our data in the first place. This is because when carrying out statistical analysis of our data, it is generally more useful to draw several conclusions for each subgroup within our population than to draw a single, more general conclusion for the whole population. However, to do this, we need to be able to classify the population into different subgroups so that we can later break down our data in the same way before analysing it.

Statistical Treatment Example – Quantitative Research

For a statistical treatment of data example, consider a medical study that is investigating the effect of a drug on the human population. As the drug can affect different people in different ways based on parameters such as gender, age and race, the researchers would want to group the data into different subgroups based on these parameters to determine how each one affects the effectiveness of the drug. Categorising the data in this way is an example of performing basic statistical treatment.

Type of Errors

A fundamental part of statistical treatment is using statistical methods to identify possible outliers and errors. No matter how careful we are, all experiments are subject to inaccuracies resulting from two types of errors: systematic errors and random errors.

Systematic errors are errors associated with either the equipment being used to collect the data or with the method in which they are used. Random errors are errors that occur unknowingly or unpredictably in the experimental configuration, such as internal deformations within specimens or small voltage fluctuations in measurement testing instruments.

These experimental errors, in turn, can lead to two types of conclusion errors: type I errors and type II errors . A type I error is a false positive which occurs when a researcher rejects a true null hypothesis. On the other hand, a type II error is a false negative which occurs when a researcher fails to reject a false null hypothesis.

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Research Paper Statistical Treatment of Data: A Primer

We can all agree that analyzing and presenting data effectively in a research paper is critical, yet often challenging.

This primer on statistical treatment of data will equip you with the key concepts and procedures to accurately analyze and clearly convey research findings.

You'll discover the fundamentals of statistical analysis and data management, the common quantitative and qualitative techniques, how to visually represent data, and best practices for writing the results - all framed specifically for research papers.

If you are curious on how AI can help you with statistica analysis for research, check Hepta AI .

Introduction to Statistical Treatment in Research

Statistical analysis is a crucial component of both quantitative and qualitative research. Properly treating data enables researchers to draw valid conclusions from their studies. This primer provides an introductory guide to fundamental statistical concepts and methods for manuscripts.

Understanding the Importance of Statistical Treatment

Careful statistical treatment demonstrates the reliability of results and ensures findings are grounded in robust quantitative evidence. From determining appropriate sample sizes to selecting accurate analytical tests, statistical rigor adds credibility. Both quantitative and qualitative papers benefit from precise data handling.

Objectives of the Primer

This primer aims to equip researchers with best practices for:

Statistical tools to apply during different research phases

Techniques to manage, analyze, and present data

Methods to demonstrate the validity and reliability of measurements

By covering fundamental concepts ranging from descriptive statistics to measurement validity, it enables both novice and experienced researchers to incorporate proper statistical treatment.

Navigating the Primer: Key Topics and Audience

The primer spans introductory topics including:

Research planning and design

Data collection, management, analysis

Result presentation and interpretation

While useful for researchers at any career stage, earlier-career scientists with limited statistical exposure will find it particularly valuable as they prepare manuscripts.

How do you write a statistical method in a research paper?

Statistical methods are a critical component of research papers, allowing you to analyze, interpret, and draw conclusions from your study data. When writing the statistical methods section, you need to provide enough detail so readers can evaluate the appropriateness of the methods you used.

Here are some key things to include when describing statistical methods in a research paper:

Type of Statistical Tests Used

Specify the types of statistical tests performed on the data, including:

Parametric vs nonparametric tests

Descriptive statistics (means, standard deviations)

Inferential statistics (t-tests, ANOVA, regression, etc.)

Statistical significance level (often p < 0.05)

For example: We used t-tests and one-way ANOVA to compare means across groups, with statistical significance set at p < 0.05.

Analysis of Subgroups

If you examined subgroups or additional variables, describe the methods used for these analyses.

For example: We stratified data by gender and used chi-square tests to analyze differences between subgroups.

Software and Versions

List any statistical software packages used for analysis, including version numbers. Common programs include SPSS, SAS, R, and Stata.

For example: Data were analyzed using SPSS version 25 (IBM Corp, Armonk, NY).

The key is to give readers enough detail to assess the rigor and appropriateness of your statistical methods. The methods should align with your research aims and design. Keep explanations clear and concise using consistent terminology throughout the paper.

What are the 5 statistical treatment in research?

The five most common statistical treatments used in academic research papers include:

The mean, or average, is used to describe the central tendency of a dataset. It provides a singular value that represents the middle of a distribution of numbers. Calculating means allows researchers to characterize typical observations within a sample.

Standard Deviation

Standard deviation measures the amount of variability in a dataset. A low standard deviation indicates observations are clustered closely around the mean, while a high standard deviation signifies the data is more spread out. Reporting standard deviations helps readers contextualize means.

Regression Analysis

Regression analysis models the relationship between independent and dependent variables. It generates an equation that predicts changes in the dependent variable based on changes in the independents. Regressions are useful for hypothesizing causal connections between variables.

Hypothesis Testing

Hypothesis testing evaluates assumptions about population parameters based on statistics calculated from a sample. Common hypothesis tests include t-tests, ANOVA, and chi-squared. These quantify the likelihood of observed differences being due to chance.

Sample Size Determination

Sample size calculations identify the minimum number of observations needed to detect effects of a given size at a desired statistical power. Appropriate sampling ensures studies can uncover true relationships within the constraints of resource limitations.

These five statistical analysis methods form the backbone of most quantitative research processes. Correct application allows researchers to characterize data trends, model predictive relationships, and make probabilistic inferences regarding broader populations. Expertise in these techniques is fundamental for producing valid, reliable, and publishable academic studies.

How do you know what statistical treatment to use in research?

The selection of appropriate statistical methods for the treatment of data in a research paper depends on three key factors:

The Aim and Objective of the Study

The aim and objectives that the study seeks to achieve will determine the type of statistical analysis required.

Descriptive research presenting characteristics of the data may only require descriptive statistics like measures of central tendency (mean, median, mode) and dispersion (range, standard deviation).

Studies aiming to establish relationships or differences between variables need inferential statistics like correlation, t-tests, ANOVA, regression etc.

Predictive modeling research requires methods like regression, discriminant analysis, logistic regression etc.

Thus, clearly identifying the research purpose and objectives is the first step in planning appropriate statistical treatment.

Type and Distribution of Data

The type of data (categorical, numerical) and its distribution (normal, skewed) also guide the choice of statistical techniques.

Parametric tests have assumptions related to normality and homogeneity of variance.

Non-parametric methods are distribution-free and better suited for non-normal or categorical data.

Testing data distribution and characteristics is therefore vital.

Nature of Observations

Statistical methods also differ based on whether the observations are paired or unpaired.

Analyzing changes within one group requires paired tests like paired t-test, Wilcoxon signed-rank test etc.

Comparing between two or more independent groups needs unpaired tests like independent t-test, ANOVA, Kruskal-Wallis test etc.

Thus the nature of observations is pivotal in selecting suitable statistical analyses.

In summary, clearly defining the research objectives, testing the collected data, and understanding the observational units guides proper statistical treatment and interpretation.

What is statistical techniques in research paper?

Statistical methods are essential tools in scientific research papers. They allow researchers to summarize, analyze, interpret and present data in meaningful ways.

Some key statistical techniques used in research papers include:

Descriptive statistics: These provide simple summaries of the sample and the measures. Common examples include measures of central tendency (mean, median, mode), measures of variability (range, standard deviation) and graphs (histograms, pie charts).

Inferential statistics: These help make inferences and predictions about a population from a sample. Common techniques include estimation of parameters, hypothesis testing, correlation and regression analysis.

Analysis of variance (ANOVA): This technique allows researchers to compare means across multiple groups and determine statistical significance.

Factor analysis: This technique identifies underlying relationships between variables and latent constructs. It allows reducing a large set of variables into fewer factors.

Structural equation modeling: This technique estimates causal relationships using both latent and observed factors. It is widely used for testing theoretical models in social sciences.

Proper statistical treatment and presentation of data are crucial for the integrity of any quantitative research paper. Statistical techniques help establish validity, account for errors, test hypotheses, build models and derive meaningful insights from the research.

Fundamental Concepts and Data Management

Exploring basic statistical terms.

Understanding key statistical concepts is essential for effective research design and data analysis. This includes defining key terms like:

Statistics : The science of collecting, organizing, analyzing, and interpreting numerical data to draw conclusions or make predictions.

Variables : Characteristics or attributes of the study participants that can take on different values.

Measurement : The process of assigning numbers to variables based on a set of rules.

Sampling : Selecting a subset of a larger population to estimate characteristics of the whole population.

Data types : Quantitative (numerical) or qualitative (categorical) data.

Descriptive vs. inferential statistics : Descriptive statistics summarize data while inferential statistics allow making conclusions from the sample to the larger population.

Ensuring Validity and Reliability in Measurement

When selecting measurement instruments, it is critical they demonstrate:

Validity : The extent to which the instrument measures what it intends to measure.

Reliability : The consistency of measurement over time and across raters.

Researchers should choose instruments aligned to their research questions and study methodology .

Data Management Essentials

Proper data management requires:

Ethical collection procedures respecting autonomy, justice, beneficence and non-maleficence.

Handling missing data through deletion, imputation or modeling procedures.

Data cleaning by identifying and fixing errors, inconsistencies and duplicates.

Data screening via visual inspection and statistical methods to detect anomalies.

Data Management Techniques and Ethical Considerations

Ethical data management includes:

Obtaining informed consent from all participants.

Anonymization and encryption to protect privacy.

Secure data storage and transfer procedures.

Responsible use of statistical tools free from manipulation or misrepresentation.

Adhering to ethical guidelines preserves public trust in the integrity of research.

Statistical Methods and Procedures

This section provides an introduction to key quantitative analysis techniques and guidance on when to apply them to different types of research questions and data.

Descriptive Statistics and Data Summarization

Descriptive statistics summarize and organize data characteristics such as central tendency, variability, and distributions. Common descriptive statistical methods include:

Measures of central tendency (mean, median, mode)

Measures of variability (range, interquartile range, standard deviation)

Graphical representations (histograms, box plots, scatter plots)

Frequency distributions and percentages

These methods help describe and summarize the sample data so researchers can spot patterns and trends.

Inferential Statistics for Generalizing Findings

While descriptive statistics summarize sample data, inferential statistics help generalize findings to the larger population. Common techniques include:

Hypothesis testing with t-tests, ANOVA

Correlation and regression analysis

Nonparametric tests

These methods allow researchers to draw conclusions and make predictions about the broader population based on the sample data.

Selecting the Right Statistical Tools

Choosing the appropriate analyses involves assessing:

The research design and questions asked

Type of data (categorical, continuous)

Data distributions

Statistical assumptions required

Matching the correct statistical tests to these elements helps ensure accurate results.

Statistical Treatment of Data for Quantitative Research

For quantitative research, common statistical data treatments include:

Testing data reliability and validity

Checking assumptions of statistical tests

Transforming non-normal data

Identifying and handling outliers

Applying appropriate analyses for the research questions and data type

Examples and case studies help demonstrate correct application of statistical tests.

Approaches to Qualitative Data Analysis

Qualitative data is analyzed through methods like:

Thematic analysis

Content analysis

Discourse analysis

Grounded theory

These help researchers discover concepts and patterns within non-numerical data to derive rich insights.

Data Presentation and Research Method

Crafting effective visuals for data presentation.

When presenting analyzed results and statistics in a research paper, well-designed tables, graphs, and charts are key for clearly showcasing patterns in the data to readers. Adhering to formatting standards like APA helps ensure professional data presentation. Consider these best practices:

Choose the appropriate visual type based on the type of data and relationship being depicted. For example, bar charts for comparing categorical data, line graphs to show trends over time.

Label the x-axis, y-axis, legends clearly. Include informative captions.

Use consistent, readable fonts and sizing. Avoid clutter with unnecessary elements. White space can aid readability.

Order data logically. Such as largest to smallest values, or chronologically.

Include clear statistical notations, like error bars, where applicable.

Following academic standards for visuals lends credibility while making interpretation intuitive for readers.

Writing the Results Section with Clarity

When writing the quantitative Results section, aim for clarity by balancing statistical reporting with interpretation of findings. Consider this structure:

Open with an overview of the analysis approach and measurements used.

Break down results by logical subsections for each hypothesis, construct measured etc.

Report exact statistics first, followed by interpretation of their meaning. For example, “Participants exposed to the intervention had significantly higher average scores (M=78, SD=3.2) compared to controls (M=71, SD=4.1), t(115)=3.42, p = 0.001. This suggests the intervention was highly effective for increasing scores.”

Use present verb tense. And scientific, formal language.

Include tables/figures where they aid understanding or visualization.

Writing results clearly gives readers deeper context around statistical findings.

Highlighting Research Method and Design

With a results section full of statistics, it's vital to communicate key aspects of the research method and design. Consider including:

Brief overview of study variables, materials, apparatus used. Helps reproducibility.

Descriptions of study sampling techniques, data collection procedures. Supports transparency.

Explanations around approaches to measurement, data analysis performed. Bolsters methodological rigor.

Noting control variables, attempts to limit biases etc. Demonstrates awareness of limitations.

Covering these methodological details shows readers the care taken in designing the study and analyzing the results obtained.

Acknowledging Limitations and Addressing Biases

Honestly recognizing methodological weaknesses and limitations goes a long way in establishing credibility within the published discussion section. Consider transparently noting:

Measurement errors and biases that may have impacted findings.

Limitations around sampling methods that constrain generalizability.

Caveats related to statistical assumptions, analysis techniques applied.

Attempts made to control/account for biases and directions for future research.

Rather than detracting value, acknowledging limitations demonstrates academic integrity regarding the research performed. It also gives readers deeper insight into interpreting the reported results and findings.

Conclusion: Synthesizing Statistical Treatment Insights

Recap of statistical treatment fundamentals.

Statistical treatment of data is a crucial component of high-quality quantitative research. Proper application of statistical methods and analysis principles enables valid interpretations and inferences from study data. Key fundamentals covered include:

Descriptive statistics to summarize and describe the basic features of study data

Inferential statistics to make judgments of the probability and significance based on the data

Using appropriate statistical tools aligned to the research design and objectives

Following established practices for measurement techniques, data collection, and reporting

Adhering to these core tenets ensures research integrity and allows findings to withstand scientific scrutiny.

Key Takeaways for Research Paper Success

When incorporating statistical treatment into a research paper, keep these best practices in mind:

Clearly state the research hypothesis and variables under examination

Select reliable and valid quantitative measures for assessment

Determine appropriate sample size to achieve statistical power

Apply correct analytical methods suited to the data type and distribution

Comprehensively report methodology procedures and statistical outputs

Interpret results in context of the study limitations and scope

Following these guidelines will bolster confidence in the statistical treatment and strengthen the research quality overall.

Encouraging Continued Learning and Application

As statistical techniques continue advancing, it is imperative for researchers to actively further their statistical literacy. Regularly reviewing new methodological developments and learning advanced tools will augment analytical capabilities. Persistently putting enhanced statistical knowledge into practice through research projects and manuscript preparations will cement competencies. Statistical treatment mastery is a journey requiring persistent effort, but one that pays dividends in research proficiency.

Antonio Carlos Filho @acfilho_dev

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• Indian J Anaesth
• v.60(9); 2016 Sep

Basic statistical tools in research and data analysis

Zulfiqar ali.

Department of Anaesthesiology, Division of Neuroanaesthesiology, Sheri Kashmir Institute of Medical Sciences, Soura, Srinagar, Jammu and Kashmir, India

S Bala Bhaskar

1 Department of Anaesthesiology and Critical Care, Vijayanagar Institute of Medical Sciences, Bellary, Karnataka, India

Statistical methods involved in carrying out a study include planning, designing, collecting data, analysing, drawing meaningful interpretation and reporting of the research findings. The statistical analysis gives meaning to the meaningless numbers, thereby breathing life into a lifeless data. The results and inferences are precise only if proper statistical tests are used. This article will try to acquaint the reader with the basic research tools that are utilised while conducting various studies. The article covers a brief outline of the variables, an understanding of quantitative and qualitative variables and the measures of central tendency. An idea of the sample size estimation, power analysis and the statistical errors is given. Finally, there is a summary of parametric and non-parametric tests used for data analysis.

INTRODUCTION

Statistics is a branch of science that deals with the collection, organisation, analysis of data and drawing of inferences from the samples to the whole population.[ 1 ] This requires a proper design of the study, an appropriate selection of the study sample and choice of a suitable statistical test. An adequate knowledge of statistics is necessary for proper designing of an epidemiological study or a clinical trial. Improper statistical methods may result in erroneous conclusions which may lead to unethical practice.[ 2 ]

Variable is a characteristic that varies from one individual member of population to another individual.[ 3 ] Variables such as height and weight are measured by some type of scale, convey quantitative information and are called as quantitative variables. Sex and eye colour give qualitative information and are called as qualitative variables[ 3 ] [ Figure 1 ].

Classification of variables

Quantitative variables

Quantitative or numerical data are subdivided into discrete and continuous measurements. Discrete numerical data are recorded as a whole number such as 0, 1, 2, 3,… (integer), whereas continuous data can assume any value. Observations that can be counted constitute the discrete data and observations that can be measured constitute the continuous data. Examples of discrete data are number of episodes of respiratory arrests or the number of re-intubations in an intensive care unit. Similarly, examples of continuous data are the serial serum glucose levels, partial pressure of oxygen in arterial blood and the oesophageal temperature.

A hierarchical scale of increasing precision can be used for observing and recording the data which is based on categorical, ordinal, interval and ratio scales [ Figure 1 ].

Categorical or nominal variables are unordered. The data are merely classified into categories and cannot be arranged in any particular order. If only two categories exist (as in gender male and female), it is called as a dichotomous (or binary) data. The various causes of re-intubation in an intensive care unit due to upper airway obstruction, impaired clearance of secretions, hypoxemia, hypercapnia, pulmonary oedema and neurological impairment are examples of categorical variables.

Ordinal variables have a clear ordering between the variables. However, the ordered data may not have equal intervals. Examples are the American Society of Anesthesiologists status or Richmond agitation-sedation scale.

Interval variables are similar to an ordinal variable, except that the intervals between the values of the interval variable are equally spaced. A good example of an interval scale is the Fahrenheit degree scale used to measure temperature. With the Fahrenheit scale, the difference between 70° and 75° is equal to the difference between 80° and 85°: The units of measurement are equal throughout the full range of the scale.

Ratio scales are similar to interval scales, in that equal differences between scale values have equal quantitative meaning. However, ratio scales also have a true zero point, which gives them an additional property. For example, the system of centimetres is an example of a ratio scale. There is a true zero point and the value of 0 cm means a complete absence of length. The thyromental distance of 6 cm in an adult may be twice that of a child in whom it may be 3 cm.

STATISTICS: DESCRIPTIVE AND INFERENTIAL STATISTICS

Descriptive statistics[ 4 ] try to describe the relationship between variables in a sample or population. Descriptive statistics provide a summary of data in the form of mean, median and mode. Inferential statistics[ 4 ] use a random sample of data taken from a population to describe and make inferences about the whole population. It is valuable when it is not possible to examine each member of an entire population. The examples if descriptive and inferential statistics are illustrated in Table 1 .

Example of descriptive and inferential statistics

Descriptive statistics

The extent to which the observations cluster around a central location is described by the central tendency and the spread towards the extremes is described by the degree of dispersion.

Measures of central tendency

The measures of central tendency are mean, median and mode.[ 6 ] Mean (or the arithmetic average) is the sum of all the scores divided by the number of scores. Mean may be influenced profoundly by the extreme variables. For example, the average stay of organophosphorus poisoning patients in ICU may be influenced by a single patient who stays in ICU for around 5 months because of septicaemia. The extreme values are called outliers. The formula for the mean is

where x = each observation and n = number of observations. Median[ 6 ] is defined as the middle of a distribution in a ranked data (with half of the variables in the sample above and half below the median value) while mode is the most frequently occurring variable in a distribution. Range defines the spread, or variability, of a sample.[ 7 ] It is described by the minimum and maximum values of the variables. If we rank the data and after ranking, group the observations into percentiles, we can get better information of the pattern of spread of the variables. In percentiles, we rank the observations into 100 equal parts. We can then describe 25%, 50%, 75% or any other percentile amount. The median is the 50 th percentile. The interquartile range will be the observations in the middle 50% of the observations about the median (25 th -75 th percentile). Variance[ 7 ] is a measure of how spread out is the distribution. It gives an indication of how close an individual observation clusters about the mean value. The variance of a population is defined by the following formula:

where σ 2 is the population variance, X is the population mean, X i is the i th element from the population and N is the number of elements in the population. The variance of a sample is defined by slightly different formula:

where s 2 is the sample variance, x is the sample mean, x i is the i th element from the sample and n is the number of elements in the sample. The formula for the variance of a population has the value ‘ n ’ as the denominator. The expression ‘ n −1’ is known as the degrees of freedom and is one less than the number of parameters. Each observation is free to vary, except the last one which must be a defined value. The variance is measured in squared units. To make the interpretation of the data simple and to retain the basic unit of observation, the square root of variance is used. The square root of the variance is the standard deviation (SD).[ 8 ] The SD of a population is defined by the following formula:

where σ is the population SD, X is the population mean, X i is the i th element from the population and N is the number of elements in the population. The SD of a sample is defined by slightly different formula:

where s is the sample SD, x is the sample mean, x i is the i th element from the sample and n is the number of elements in the sample. An example for calculation of variation and SD is illustrated in Table 2 .

Example of mean, variance, standard deviation

Normal distribution or Gaussian distribution

Most of the biological variables usually cluster around a central value, with symmetrical positive and negative deviations about this point.[ 1 ] The standard normal distribution curve is a symmetrical bell-shaped. In a normal distribution curve, about 68% of the scores are within 1 SD of the mean. Around 95% of the scores are within 2 SDs of the mean and 99% within 3 SDs of the mean [ Figure 2 ].

Normal distribution curve

Skewed distribution

It is a distribution with an asymmetry of the variables about its mean. In a negatively skewed distribution [ Figure 3 ], the mass of the distribution is concentrated on the right of Figure 1 . In a positively skewed distribution [ Figure 3 ], the mass of the distribution is concentrated on the left of the figure leading to a longer right tail.

Curves showing negatively skewed and positively skewed distribution

Inferential statistics

In inferential statistics, data are analysed from a sample to make inferences in the larger collection of the population. The purpose is to answer or test the hypotheses. A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. Hypothesis tests are thus procedures for making rational decisions about the reality of observed effects.

Probability is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty).

In inferential statistics, the term ‘null hypothesis’ ( H 0 ‘ H-naught ,’ ‘ H-null ’) denotes that there is no relationship (difference) between the population variables in question.[ 9 ]

Alternative hypothesis ( H 1 and H a ) denotes that a statement between the variables is expected to be true.[ 9 ]

The P value (or the calculated probability) is the probability of the event occurring by chance if the null hypothesis is true. The P value is a numerical between 0 and 1 and is interpreted by researchers in deciding whether to reject or retain the null hypothesis [ Table 3 ].

P values with interpretation

If P value is less than the arbitrarily chosen value (known as α or the significance level), the null hypothesis (H0) is rejected [ Table 4 ]. However, if null hypotheses (H0) is incorrectly rejected, this is known as a Type I error.[ 11 ] Further details regarding alpha error, beta error and sample size calculation and factors influencing them are dealt with in another section of this issue by Das S et al .[ 12 ]

Illustration for null hypothesis

PARAMETRIC AND NON-PARAMETRIC TESTS

Numerical data (quantitative variables) that are normally distributed are analysed with parametric tests.[ 13 ]

Two most basic prerequisites for parametric statistical analysis are:

• The assumption of normality which specifies that the means of the sample group are normally distributed
• The assumption of equal variance which specifies that the variances of the samples and of their corresponding population are equal.

However, if the distribution of the sample is skewed towards one side or the distribution is unknown due to the small sample size, non-parametric[ 14 ] statistical techniques are used. Non-parametric tests are used to analyse ordinal and categorical data.

Parametric tests

The parametric tests assume that the data are on a quantitative (numerical) scale, with a normal distribution of the underlying population. The samples have the same variance (homogeneity of variances). The samples are randomly drawn from the population, and the observations within a group are independent of each other. The commonly used parametric tests are the Student's t -test, analysis of variance (ANOVA) and repeated measures ANOVA.

Student's t -test

Student's t -test is used to test the null hypothesis that there is no difference between the means of the two groups. It is used in three circumstances:

where X = sample mean, u = population mean and SE = standard error of mean

where X 1 − X 2 is the difference between the means of the two groups and SE denotes the standard error of the difference.

• To test if the population means estimated by two dependent samples differ significantly (the paired t -test). A usual setting for paired t -test is when measurements are made on the same subjects before and after a treatment.

The formula for paired t -test is:

where d is the mean difference and SE denotes the standard error of this difference.

The group variances can be compared using the F -test. The F -test is the ratio of variances (var l/var 2). If F differs significantly from 1.0, then it is concluded that the group variances differ significantly.

Analysis of variance

The Student's t -test cannot be used for comparison of three or more groups. The purpose of ANOVA is to test if there is any significant difference between the means of two or more groups.

In ANOVA, we study two variances – (a) between-group variability and (b) within-group variability. The within-group variability (error variance) is the variation that cannot be accounted for in the study design. It is based on random differences present in our samples.

However, the between-group (or effect variance) is the result of our treatment. These two estimates of variances are compared using the F-test.

A simplified formula for the F statistic is:

where MS b is the mean squares between the groups and MS w is the mean squares within groups.

Repeated measures analysis of variance

As with ANOVA, repeated measures ANOVA analyses the equality of means of three or more groups. However, a repeated measure ANOVA is used when all variables of a sample are measured under different conditions or at different points in time.

As the variables are measured from a sample at different points of time, the measurement of the dependent variable is repeated. Using a standard ANOVA in this case is not appropriate because it fails to model the correlation between the repeated measures: The data violate the ANOVA assumption of independence. Hence, in the measurement of repeated dependent variables, repeated measures ANOVA should be used.

Non-parametric tests

When the assumptions of normality are not met, and the sample means are not normally, distributed parametric tests can lead to erroneous results. Non-parametric tests (distribution-free test) are used in such situation as they do not require the normality assumption.[ 15 ] Non-parametric tests may fail to detect a significant difference when compared with a parametric test. That is, they usually have less power.

As is done for the parametric tests, the test statistic is compared with known values for the sampling distribution of that statistic and the null hypothesis is accepted or rejected. The types of non-parametric analysis techniques and the corresponding parametric analysis techniques are delineated in Table 5 .

Analogue of parametric and non-parametric tests

Median test for one sample: The sign test and Wilcoxon's signed rank test

The sign test and Wilcoxon's signed rank test are used for median tests of one sample. These tests examine whether one instance of sample data is greater or smaller than the median reference value.

This test examines the hypothesis about the median θ0 of a population. It tests the null hypothesis H0 = θ0. When the observed value (Xi) is greater than the reference value (θ0), it is marked as+. If the observed value is smaller than the reference value, it is marked as − sign. If the observed value is equal to the reference value (θ0), it is eliminated from the sample.

If the null hypothesis is true, there will be an equal number of + signs and − signs.

The sign test ignores the actual values of the data and only uses + or − signs. Therefore, it is useful when it is difficult to measure the values.

Wilcoxon's signed rank test

There is a major limitation of sign test as we lose the quantitative information of the given data and merely use the + or – signs. Wilcoxon's signed rank test not only examines the observed values in comparison with θ0 but also takes into consideration the relative sizes, adding more statistical power to the test. As in the sign test, if there is an observed value that is equal to the reference value θ0, this observed value is eliminated from the sample.

Wilcoxon's rank sum test ranks all data points in order, calculates the rank sum of each sample and compares the difference in the rank sums.

Mann-Whitney test

It is used to test the null hypothesis that two samples have the same median or, alternatively, whether observations in one sample tend to be larger than observations in the other.

Mann–Whitney test compares all data (xi) belonging to the X group and all data (yi) belonging to the Y group and calculates the probability of xi being greater than yi: P (xi > yi). The null hypothesis states that P (xi > yi) = P (xi < yi) =1/2 while the alternative hypothesis states that P (xi > yi) ≠1/2.

Kolmogorov-Smirnov test

The two-sample Kolmogorov-Smirnov (KS) test was designed as a generic method to test whether two random samples are drawn from the same distribution. The null hypothesis of the KS test is that both distributions are identical. The statistic of the KS test is a distance between the two empirical distributions, computed as the maximum absolute difference between their cumulative curves.

Kruskal-Wallis test

The Kruskal–Wallis test is a non-parametric test to analyse the variance.[ 14 ] It analyses if there is any difference in the median values of three or more independent samples. The data values are ranked in an increasing order, and the rank sums calculated followed by calculation of the test statistic.

Jonckheere test

In contrast to Kruskal–Wallis test, in Jonckheere test, there is an a priori ordering that gives it a more statistical power than the Kruskal–Wallis test.[ 14 ]

Friedman test

The Friedman test is a non-parametric test for testing the difference between several related samples. The Friedman test is an alternative for repeated measures ANOVAs which is used when the same parameter has been measured under different conditions on the same subjects.[ 13 ]

Tests to analyse the categorical data

Chi-square test, Fischer's exact test and McNemar's test are used to analyse the categorical or nominal variables. The Chi-square test compares the frequencies and tests whether the observed data differ significantly from that of the expected data if there were no differences between groups (i.e., the null hypothesis). It is calculated by the sum of the squared difference between observed ( O ) and the expected ( E ) data (or the deviation, d ) divided by the expected data by the following formula:

A Yates correction factor is used when the sample size is small. Fischer's exact test is used to determine if there are non-random associations between two categorical variables. It does not assume random sampling, and instead of referring a calculated statistic to a sampling distribution, it calculates an exact probability. McNemar's test is used for paired nominal data. It is applied to 2 × 2 table with paired-dependent samples. It is used to determine whether the row and column frequencies are equal (that is, whether there is ‘marginal homogeneity’). The null hypothesis is that the paired proportions are equal. The Mantel-Haenszel Chi-square test is a multivariate test as it analyses multiple grouping variables. It stratifies according to the nominated confounding variables and identifies any that affects the primary outcome variable. If the outcome variable is dichotomous, then logistic regression is used.

SOFTWARES AVAILABLE FOR STATISTICS, SAMPLE SIZE CALCULATION AND POWER ANALYSIS

Numerous statistical software systems are available currently. The commonly used software systems are Statistical Package for the Social Sciences (SPSS – manufactured by IBM corporation), Statistical Analysis System ((SAS – developed by SAS Institute North Carolina, United States of America), R (designed by Ross Ihaka and Robert Gentleman from R core team), Minitab (developed by Minitab Inc), Stata (developed by StataCorp) and the MS Excel (developed by Microsoft).

There are a number of web resources which are related to statistical power analyses. A few are:

• StatPages.net – provides links to a number of online power calculators
• G-Power – provides a downloadable power analysis program that runs under DOS
• Power analysis for ANOVA designs an interactive site that calculates power or sample size needed to attain a given power for one effect in a factorial ANOVA design
• SPSS makes a program called SamplePower. It gives an output of a complete report on the computer screen which can be cut and paste into another document.

It is important that a researcher knows the concepts of the basic statistical methods used for conduct of a research study. This will help to conduct an appropriately well-designed study leading to valid and reliable results. Inappropriate use of statistical techniques may lead to faulty conclusions, inducing errors and undermining the significance of the article. Bad statistics may lead to bad research, and bad research may lead to unethical practice. Hence, an adequate knowledge of statistics and the appropriate use of statistical tests are important. An appropriate knowledge about the basic statistical methods will go a long way in improving the research designs and producing quality medical research which can be utilised for formulating the evidence-based guidelines.

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Statistical Methods in Theses: Guidelines and Explanations

Signed August 2018 Naseem Al-Aidroos, PhD, Christopher Fiacconi, PhD Deborah Powell, PhD, Harvey Marmurek, PhD, Ian Newby-Clark, PhD, Jeffrey Spence, PhD, David Stanley, PhD, Lana Trick, PhD

Version:  2.00

This document is an organizational aid, and workbook, for students. We encourage students to take this document to meetings with their advisor and committee. This guide should enhance a committee’s ability to assess key areas of a student’s work. 

In recent years a number of well-known and apparently well-established findings have  failed to replicate , resulting in what is commonly referred to as the replication crisis. The APA Publication Manual 6 th Edition notes that “The essence of the scientific method involves observations that can be repeated and verified by others.” (p. 12). However, a systematic investigation of the replicability of psychology findings published in  Science  revealed that over half of psychology findings do not replicate (see a related commentary in  Nature ). Even more disturbing, a  Bayesian reanalysis of the reproducibility project  showed that 64% of studies had sample sizes so small that strong evidence for or against the null or alternative hypotheses did not exist. Indeed, Morey and Lakens (2016) concluded that most of psychology is statistically unfalsifiable due to small sample sizes and correspondingly low power (see  article ). Our discipline’s reputation is suffering. News of the replication crisis has reached the popular press (e.g.,  The Atlantic ,   The Economist ,   Slate , Last Week Tonight ).

An increasing number of psychologists have responded by promoting new research standards that involve open science and the elimination of  Questionable Research Practices . The open science perspective is made manifest in the  Transparency and Openness Promotion (TOP) guidelines  for journal publications. These guidelines were adopted some time ago by the  Association for Psychological Science . More recently, the guidelines were adopted by American Psychological Association journals ( see details ) and journals published by Elsevier ( see details ). It appears likely that, in the very near future, most journals in psychology will be using an open science approach. We strongly advise readers to take a moment to inspect the  TOP Guidelines Summary Table . 

A key aspect of open science and the TOP guidelines is the sharing of data associated with published research (with respect to medical research, see point #35 in the  World Medical Association Declaration of Helsinki ). This practice is viewed widely as highly important. Indeed, open science is recommended by  all G7 science ministers . All Tri-Agency grants must include a data-management plan that includes plans for sharing: “ research data resulting from agency funding should normally be preserved in a publicly accessible, secure and curated repository or other platform for discovery and reuse by others.”  Moreover, a 2017 editorial published in the  New England Journal of Medicine announced that the  International Committee of Medical Journal Editors believes there is  “an ethical obligation to responsibly share data.”  As of this writing,  60% of highly ranked psychology journals require or encourage data sharing .

The increasing importance of demonstrating that findings are replicable is reflected in calls to make replication a requirement for the promotion of faculty (see details in  Nature ) and experts in open science are now refereeing applications for tenure and promotion (see details at the  Center for Open Science  and  this article ). Most dramatically, in one instance, a paper resulting from a dissertation was retracted due to misleading findings attributable to Questionable Research Practices. Subsequent to the retraction, the Ohio State University’s Board of Trustees unanimously revoked the PhD of the graduate student who wrote the dissertation ( see details ). Thus, the academic environment is changing and it is important to work toward using new best practices in lieu of older practices—many of which are synonymous with Questionable Research Practices. Doing so should help you avoid later career regrets and subsequent  public mea culpas . One way to achieve your research objectives in this new academic environment is  to incorporate replications into your research . Replications are becoming more common and there are even websites dedicated to helping students conduct replications (e.g.,  Psychology Science Accelerator ) and indexing the success of replications (e.g., Curate Science ). You might even consider conducting a replication for your thesis (subject to committee approval).

As early-career researchers, it is important to be aware of the changing academic environment. Senior principal investigators may be  reluctant to engage in open science  (see this student perspective in a  blog post  and  podcast ) and research on resistance to data sharing indicates that one of the barriers to sharing data is that researchers do not feel that they have knowledge of  how to share data online . This document is an educational aid and resource to provide students with introductory knowledge of how to participate in open science and online data sharing to start their education on these subjects. 

Guidelines and Explanations

In light of the changes in psychology, faculty members who teach statistics/methods have reviewed the literature and generated this guide for graduate students. The guide is intended to enhance the quality of student theses by facilitating their engagement in open and transparent research practices and by helping them avoid Questionable Research Practices, many of which are now deemed unethical and covered in the ethics section of textbooks.

This document is an informational tool.

How to Start

In order to follow best practices, some first steps need to be followed. Here is a list of things to do:

• Get an Open Science account. Registration at  osf.io  is easy!
• If conducting confirmatory hypothesis testing for your thesis, pre-register your hypotheses (see Section 1-Hypothesizing). The Open Science Foundation website has helpful  tutorials  and  guides  to get you going.
• Also, pre-register your data analysis plan. Pre-registration typically includes how and when you will stop collecting data, how you will deal with violations of statistical assumptions and points of influence (“outliers”), the specific measures you will use, and the analyses you will use to test each hypothesis, possibly including the analysis script. Again, there is a lot of help available for this. 

Exploratory and Confirmatory Research Are Both of Value, But Do Not Confuse the Two

We note that this document largely concerns confirmatory research (i.e., testing hypotheses). We by no means intend to devalue exploratory research. Indeed, it is one of the primary ways that hypotheses are generated for (possible) confirmation. Instead, we emphasize that it is important that you clearly indicate what of your research is exploratory and what is confirmatory. Be clear in your writing and in your preregistration plan. You should explicitly indicate which of your analyses are exploratory and which are confirmatory. Please note also that if you are engaged in exploratory research, then Null Hypothesis Significance Testing (NHST) should probably be avoided (see rationale in  Gigerenzer  (2004) and  Wagenmakers et al., (2012) ). 

This document is structured around the stages of thesis work:  hypothesizing, design, data collection, analyses, and reporting – consistent with the headings used by Wicherts et al. (2016). We also list the Questionable Research Practices associated with each stage and provide suggestions for avoiding them. We strongly advise going through all of these sections during thesis/dissertation proposal meetings because a priori decisions need to be made prior to data collection (including analysis decisions). 

To help to ensure that the student has informed the committee about key decisions at each stage, there are check boxes at the end of each section.

How to Use This Document in a Proposal Meeting

• Print off a copy of this document and take it to the proposal meeting.
• During the meeting, use the document to seek assistance from faculty to address potential problems.
• Revisit responses to issues raised by this document (especially the Analysis and Reporting Stages) when you are seeking approval to proceed to defense.

Consultation and Help Line

Note that the Center for Open Science now has a help line (for individual researchers and labs) you can call for help with open science issues. They also have training workshops. Please see their  website  for details.

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Thesis life: 7 ways to tackle statistics in your thesis.

By Pranav Kulkarni

Thesis is an integral part of your Masters’ study in Wageningen University and Research. It is the most exciting, independent and technical part of the study. More often than not, most departments in WU expect students to complete a short term independent project or a part of big on-going project for their thesis assignment.

Source : www.coursera.org

This assignment involves proposing a research question, tackling it with help of some observations or experiments, analyzing these observations or results and then stating them by drawing some conclusions.

Since it is an immitigable part of your thesis, you can neither run from statistics nor cry for help.

The penultimate part of this process involves analysis of results which is very crucial for coherence of your thesis assignment.This analysis usually involve use of statistical tools to help draw inferences. Most students who don’t pursue statistics in their curriculum are scared by this prospect. Since it is an immitigable part of your thesis, you can neither run from statistics nor cry for help. But in order to not get intimidated by statistics and its “greco-latin” language, there are a few ways in which you can make your journey through thesis life a pleasant experience.

Make statistics your friend

The best way to end your fear of statistics and all its paraphernalia is to befriend it. Try to learn all that you can about the techniques that you will be using, why they were invented, how they were invented and who did this deed. Personifying the story of statistical techniques makes them digestible and easy to use. Each new method in statistics comes with a unique story and loads of nerdy anecdotes.

If you cannot make friends with statistics, at least make a truce

If you cannot still bring yourself about to be interested in the life and times of statistics, the best way to not hate statistics is to make an agreement with yourself. You must realise that although important, this is only part of your thesis. The better part of your thesis is something you trained for and learned. So, don’t bother to fuss about statistics and make you all nervous. Do your job, enjoy thesis to the fullest and complete the statistical section as soon as possible. At the end, you would have forgotten all about your worries and fears of statistics.

Visualize your data

The best way to understand the results and observations from your study/ experiments, is to visualize your data. See different trends, patterns, or lack thereof to understand what you are supposed to do. Moreover, graphics and illustrations can be used directly in your report. These techniques will also help you decide on which statistical analyses you must perform to answer your research question. Blind decisions about statistics can often influence your study and make it very confusing or worse, make it completely wrong!

Simplify with flowcharts and planning

Similar to graphical visualizations, making flowcharts and planning various steps of your study can prove beneficial to make statistical decisions. Human brain can analyse pictorial information faster than literal information. So, it is always easier to understand your exact goal when you can make decisions based on flowchart or any logical flow-plans.

Source: www.imindq.com

Find examples on internet

Although statistics is a giant maze of complicated terminologies, the internet holds the key to this particular maze. You can find tons of examples on the web. These may be similar to what you intend to do or be different applications of the similar tools that you wish to engage. Especially, in case of Statistical programming languages like R, SAS, Python, PERL, VBA, etc. there is a vast database of example codes, clarifications and direct training examples available on the internet. Various forums are also available for specialized statistical methodologies where different experts and students discuss the issues regarding their own projects.

Comparative studies

Much unlike blindly searching the internet for examples and taking word of advice from online faceless people, you can systematically learn which quantitative tests to perform by rigorously studying literature of relevant research. Since you came up with a certain problem to tackle in your field of study, chances are, someone else also came up with this issue or something quite similar. You can find solutions to many such problems by scouring the internet for research papers which address the issue. Nevertheless, you should be cautious. It is easy to get lost and disheartened when you find many heavy statistical studies with lots of maths and derivations with huge cryptic symbolical text.

When all else fails, talk to an expert

All the steps above are meant to help you independently tackle whatever hurdles you encounter over the course of your thesis. But, when you cannot tackle them yourself it is always prudent and most efficient to ask for help. Talking to students from your thesis ring who have done something similar is one way of help. Another is to make an appointment with your supervisor and take specific questions to him/ her. If that is not possible, you can contact some other teaching staff or researchers from your research group. Try not to waste their as well as you time by making a list of specific problems that you will like to discuss. I think most are happy to help in any way possible.

Talking to students from your thesis ring who have done something similar is one way of help.

Sometimes, with the help of your supervisor, you can make an appointment with someone from the “Biometris” which is the WU’s statistics department. These people are the real deal; chances are, these people can solve all your problems without any difficulty. Always remember, you are in the process of learning, nobody expects you to be an expert in everything. Ask for help when there seems to be no hope.

Apart from these seven ways to make your statistical journey pleasant, you should always engage in reading, watching, listening to stuff relevant to your thesis topic and talking about it to those who are interested. Most questions have solutions in the ether realm of communication. So, best of luck and break a leg!!!

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There are 4 comments.

A perfect approach in a very crisp and clear manner! The sequence suggested is absolutely perfect and will help the students very much. I particularly liked the idea of visualisation!

You are write! I get totally stuck with learning and understanding statistics for my Dissertation!

Statistics is a technical subject that requires extra effort. With the highlighted tips you already highlighted i expect it will offer the much needed help with statistics analysis in my course.

this is so much relevant to me! Don’t forget one more point: try to enrol specific online statistics course (in my case, I’m too late to join any statistic course). The hardest part for me actually to choose what type of statistical test to choose among many options

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Statistical Treatment Of Data

Statistical treatment of data is essential in order to make use of the data in the right form. Raw data collection is only one aspect of any experiment; the organization of data is equally important so that appropriate conclusions can be drawn. This is what statistical treatment of data is all about.

This article is a part of the guide:

• Statistics Tutorial
• Branches of Statistics
• Statistical Analysis
• Discrete Variables

Browse Full Outline

• 1 Statistics Tutorial
• 2.1 What is Statistics?
• 2.2 Learn Statistics
• 3 Probability
• 4 Branches of Statistics
• 5 Descriptive Statistics
• 6 Parameters
• 7.1 Data Treatment
• 7.2 Raw Data
• 7.3 Outliers
• 7.4 Data Output
• 8 Statistical Analysis
• 9 Measurement Scales
• 10 Variables and Statistics
• 11 Discrete Variables

There are many techniques involved in statistics that treat data in the required manner. Statistical treatment of data is essential in all experiments, whether social, scientific or any other form. Statistical treatment of data greatly depends on the kind of experiment and the desired result from the experiment.

For example, in a survey regarding the election of a Mayor, parameters like age, gender, occupation, etc. would be important in influencing the person's decision to vote for a particular candidate. Therefore the data needs to be treated in these reference frames.

An important aspect of statistical treatment of data is the handling of errors. All experiments invariably produce errors and noise. Both systematic and random errors need to be taken into consideration.

Depending on the type of experiment being performed, Type-I and Type-II errors also need to be handled. These are the cases of false positives and false negatives that are important to understand and eliminate in order to make sense from the result of the experiment.

Treatment of Data and Distribution

Trying to classify data into commonly known patterns is a tremendous help and is intricately related to statistical treatment of data. This is because distributions such as the normal probability distribution occur very commonly in nature that they are the underlying distributions in most medical, social and physical experiments.

Therefore if a given sample size is known to be normally distributed, then the statistical treatment of data is made easy for the researcher as he would already have a lot of back up theory in this aspect. Care should always be taken, however, not to assume all data to be normally distributed, and should always be confirmed with appropriate testing.

Statistical treatment of data also involves describing the data. The best way to do this is through the measures of central tendencies like mean , median and mode . These help the researcher explain in short how the data are concentrated. Range, uncertainty and standard deviation help to understand the distribution of the data. Therefore two distributions with the same mean can have wildly different standard deviation, which shows how well the data points are concentrated around the mean.

Statistical treatment of data is an important aspect of all experimentation today and a thorough understanding is necessary to conduct the right experiments with the right inferences from the data obtained.

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Siddharth Kalla (Apr 10, 2009). Statistical Treatment Of Data. Retrieved Jun 01, 2024 from Explorable.com: https://explorable.com/statistical-treatment-of-data

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The Beginner's Guide to Statistical Analysis | 5 Steps & Examples

Statistical analysis means investigating trends, patterns, and relationships using quantitative data . It is an important research tool used by scientists, governments, businesses, and other organisations.

To draw valid conclusions, statistical analysis requires careful planning from the very start of the research process . You need to specify your hypotheses and make decisions about your research design, sample size, and sampling procedure.

After collecting data from your sample, you can organise and summarise the data using descriptive statistics . Then, you can use inferential statistics to formally test hypotheses and make estimates about the population. Finally, you can interpret and generalise your findings.

This article is a practical introduction to statistical analysis for students and researchers. We’ll walk you through the steps using two research examples. The first investigates a potential cause-and-effect relationship, while the second investigates a potential correlation between variables.

Table of contents

Step 1: write your hypotheses and plan your research design, step 2: collect data from a sample, step 3: summarise your data with descriptive statistics, step 4: test hypotheses or make estimates with inferential statistics, step 5: interpret your results, frequently asked questions about statistics.

To collect valid data for statistical analysis, you first need to specify your hypotheses and plan out your research design.

Writing statistical hypotheses

The goal of research is often to investigate a relationship between variables within a population . You start with a prediction, and use statistical analysis to test that prediction.

A statistical hypothesis is a formal way of writing a prediction about a population. Every research prediction is rephrased into null and alternative hypotheses that can be tested using sample data.

While the null hypothesis always predicts no effect or no relationship between variables, the alternative hypothesis states your research prediction of an effect or relationship.

• Null hypothesis: A 5-minute meditation exercise will have no effect on math test scores in teenagers.
• Alternative hypothesis: A 5-minute meditation exercise will improve math test scores in teenagers.
• Null hypothesis: Parental income and GPA have no relationship with each other in college students.
• Alternative hypothesis: Parental income and GPA are positively correlated in college students.

Planning your research design

A research design is your overall strategy for data collection and analysis. It determines the statistical tests you can use to test your hypothesis later on.

First, decide whether your research will use a descriptive, correlational, or experimental design. Experiments directly influence variables, whereas descriptive and correlational studies only measure variables.

• In an experimental design , you can assess a cause-and-effect relationship (e.g., the effect of meditation on test scores) using statistical tests of comparison or regression.
• In a correlational design , you can explore relationships between variables (e.g., parental income and GPA) without any assumption of causality using correlation coefficients and significance tests.
• In a descriptive design , you can study the characteristics of a population or phenomenon (e.g., the prevalence of anxiety in U.S. college students) using statistical tests to draw inferences from sample data.

Your research design also concerns whether you’ll compare participants at the group level or individual level, or both.

• In a between-subjects design , you compare the group-level outcomes of participants who have been exposed to different treatments (e.g., those who performed a meditation exercise vs those who didn’t).
• In a within-subjects design , you compare repeated measures from participants who have participated in all treatments of a study (e.g., scores from before and after performing a meditation exercise).
• In a mixed (factorial) design , one variable is altered between subjects and another is altered within subjects (e.g., pretest and posttest scores from participants who either did or didn’t do a meditation exercise).
• Experimental
• Correlational

First, you’ll take baseline test scores from participants. Then, your participants will undergo a 5-minute meditation exercise. Finally, you’ll record participants’ scores from a second math test.

In this experiment, the independent variable is the 5-minute meditation exercise, and the dependent variable is the math test score from before and after the intervention. Example: Correlational research design In a correlational study, you test whether there is a relationship between parental income and GPA in graduating college students. To collect your data, you will ask participants to fill in a survey and self-report their parents’ incomes and their own GPA.

Measuring variables

When planning a research design, you should operationalise your variables and decide exactly how you will measure them.

For statistical analysis, it’s important to consider the level of measurement of your variables, which tells you what kind of data they contain:

• Categorical data represents groupings. These may be nominal (e.g., gender) or ordinal (e.g. level of language ability).
• Quantitative data represents amounts. These may be on an interval scale (e.g. test score) or a ratio scale (e.g. age).

Many variables can be measured at different levels of precision. For example, age data can be quantitative (8 years old) or categorical (young). If a variable is coded numerically (e.g., level of agreement from 1–5), it doesn’t automatically mean that it’s quantitative instead of categorical.

Identifying the measurement level is important for choosing appropriate statistics and hypothesis tests. For example, you can calculate a mean score with quantitative data, but not with categorical data.

In a research study, along with measures of your variables of interest, you’ll often collect data on relevant participant characteristics.

In most cases, it’s too difficult or expensive to collect data from every member of the population you’re interested in studying. Instead, you’ll collect data from a sample.

Statistical analysis allows you to apply your findings beyond your own sample as long as you use appropriate sampling procedures . You should aim for a sample that is representative of the population.

Sampling for statistical analysis

There are two main approaches to selecting a sample.

• Probability sampling: every member of the population has a chance of being selected for the study through random selection.
• Non-probability sampling: some members of the population are more likely than others to be selected for the study because of criteria such as convenience or voluntary self-selection.

In theory, for highly generalisable findings, you should use a probability sampling method. Random selection reduces sampling bias and ensures that data from your sample is actually typical of the population. Parametric tests can be used to make strong statistical inferences when data are collected using probability sampling.

But in practice, it’s rarely possible to gather the ideal sample. While non-probability samples are more likely to be biased, they are much easier to recruit and collect data from. Non-parametric tests are more appropriate for non-probability samples, but they result in weaker inferences about the population.

If you want to use parametric tests for non-probability samples, you have to make the case that:

• your sample is representative of the population you’re generalising your findings to.
• your sample lacks systematic bias.

Keep in mind that external validity means that you can only generalise your conclusions to others who share the characteristics of your sample. For instance, results from Western, Educated, Industrialised, Rich and Democratic samples (e.g., college students in the US) aren’t automatically applicable to all non-WEIRD populations.

If you apply parametric tests to data from non-probability samples, be sure to elaborate on the limitations of how far your results can be generalised in your discussion section .

Create an appropriate sampling procedure

Based on the resources available for your research, decide on how you’ll recruit participants.

• Will you have resources to advertise your study widely, including outside of your university setting?
• Will you have the means to recruit a diverse sample that represents a broad population?
• Do you have time to contact and follow up with members of hard-to-reach groups?

Your participants are self-selected by their schools. Although you’re using a non-probability sample, you aim for a diverse and representative sample. Example: Sampling (correlational study) Your main population of interest is male college students in the US. Using social media advertising, you recruit senior-year male college students from a smaller subpopulation: seven universities in the Boston area.

Calculate sufficient sample size

Before recruiting participants, decide on your sample size either by looking at other studies in your field or using statistics. A sample that’s too small may be unrepresentative of the sample, while a sample that’s too large will be more costly than necessary.

There are many sample size calculators online. Different formulas are used depending on whether you have subgroups or how rigorous your study should be (e.g., in clinical research). As a rule of thumb, a minimum of 30 units or more per subgroup is necessary.

To use these calculators, you have to understand and input these key components:

• Significance level (alpha): the risk of rejecting a true null hypothesis that you are willing to take, usually set at 5%.
• Statistical power : the probability of your study detecting an effect of a certain size if there is one, usually 80% or higher.
• Expected effect size : a standardised indication of how large the expected result of your study will be, usually based on other similar studies.
• Population standard deviation: an estimate of the population parameter based on a previous study or a pilot study of your own.

Once you’ve collected all of your data, you can inspect them and calculate descriptive statistics that summarise them.

Inspect your data

There are various ways to inspect your data, including the following:

• Organising data from each variable in frequency distribution tables .
• Displaying data from a key variable in a bar chart to view the distribution of responses.
• Visualising the relationship between two variables using a scatter plot .

By visualising your data in tables and graphs, you can assess whether your data follow a skewed or normal distribution and whether there are any outliers or missing data.

A normal distribution means that your data are symmetrically distributed around a center where most values lie, with the values tapering off at the tail ends.

In contrast, a skewed distribution is asymmetric and has more values on one end than the other. The shape of the distribution is important to keep in mind because only some descriptive statistics should be used with skewed distributions.

Extreme outliers can also produce misleading statistics, so you may need a systematic approach to dealing with these values.

Calculate measures of central tendency

Measures of central tendency describe where most of the values in a data set lie. Three main measures of central tendency are often reported:

• Mode : the most popular response or value in the data set.
• Median : the value in the exact middle of the data set when ordered from low to high.
• Mean : the sum of all values divided by the number of values.

However, depending on the shape of the distribution and level of measurement, only one or two of these measures may be appropriate. For example, many demographic characteristics can only be described using the mode or proportions, while a variable like reaction time may not have a mode at all.

Calculate measures of variability

Measures of variability tell you how spread out the values in a data set are. Four main measures of variability are often reported:

• Range : the highest value minus the lowest value of the data set.
• Interquartile range : the range of the middle half of the data set.
• Standard deviation : the average distance between each value in your data set and the mean.
• Variance : the square of the standard deviation.

Once again, the shape of the distribution and level of measurement should guide your choice of variability statistics. The interquartile range is the best measure for skewed distributions, while standard deviation and variance provide the best information for normal distributions.

Using your table, you should check whether the units of the descriptive statistics are comparable for pretest and posttest scores. For example, are the variance levels similar across the groups? Are there any extreme values? If there are, you may need to identify and remove extreme outliers in your data set or transform your data before performing a statistical test.

From this table, we can see that the mean score increased after the meditation exercise, and the variances of the two scores are comparable. Next, we can perform a statistical test to find out if this improvement in test scores is statistically significant in the population. Example: Descriptive statistics (correlational study) After collecting data from 653 students, you tabulate descriptive statistics for annual parental income and GPA.

It’s important to check whether you have a broad range of data points. If you don’t, your data may be skewed towards some groups more than others (e.g., high academic achievers), and only limited inferences can be made about a relationship.

A number that describes a sample is called a statistic , while a number describing a population is called a parameter . Using inferential statistics , you can make conclusions about population parameters based on sample statistics.

Researchers often use two main methods (simultaneously) to make inferences in statistics.

• Estimation: calculating population parameters based on sample statistics.
• Hypothesis testing: a formal process for testing research predictions about the population using samples.

You can make two types of estimates of population parameters from sample statistics:

• A point estimate : a value that represents your best guess of the exact parameter.
• An interval estimate : a range of values that represent your best guess of where the parameter lies.

If your aim is to infer and report population characteristics from sample data, it’s best to use both point and interval estimates in your paper.

You can consider a sample statistic a point estimate for the population parameter when you have a representative sample (e.g., in a wide public opinion poll, the proportion of a sample that supports the current government is taken as the population proportion of government supporters).

There’s always error involved in estimation, so you should also provide a confidence interval as an interval estimate to show the variability around a point estimate.

A confidence interval uses the standard error and the z score from the standard normal distribution to convey where you’d generally expect to find the population parameter most of the time.

Hypothesis testing

Using data from a sample, you can test hypotheses about relationships between variables in the population. Hypothesis testing starts with the assumption that the null hypothesis is true in the population, and you use statistical tests to assess whether the null hypothesis can be rejected or not.

Statistical tests determine where your sample data would lie on an expected distribution of sample data if the null hypothesis were true. These tests give two main outputs:

• A test statistic tells you how much your data differs from the null hypothesis of the test.
• A p value tells you the likelihood of obtaining your results if the null hypothesis is actually true in the population.

Statistical tests come in three main varieties:

• Comparison tests assess group differences in outcomes.
• Regression tests assess cause-and-effect relationships between variables.
• Correlation tests assess relationships between variables without assuming causation.

Your choice of statistical test depends on your research questions, research design, sampling method, and data characteristics.

Parametric tests

Parametric tests make powerful inferences about the population based on sample data. But to use them, some assumptions must be met, and only some types of variables can be used. If your data violate these assumptions, you can perform appropriate data transformations or use alternative non-parametric tests instead.

A regression models the extent to which changes in a predictor variable results in changes in outcome variable(s).

• A simple linear regression includes one predictor variable and one outcome variable.
• A multiple linear regression includes two or more predictor variables and one outcome variable.

Comparison tests usually compare the means of groups. These may be the means of different groups within a sample (e.g., a treatment and control group), the means of one sample group taken at different times (e.g., pretest and posttest scores), or a sample mean and a population mean.

• A t test is for exactly 1 or 2 groups when the sample is small (30 or less).
• A z test is for exactly 1 or 2 groups when the sample is large.
• An ANOVA is for 3 or more groups.

The z and t tests have subtypes based on the number and types of samples and the hypotheses:

• If you have only one sample that you want to compare to a population mean, use a one-sample test .
• If you have paired measurements (within-subjects design), use a dependent (paired) samples test .
• If you have completely separate measurements from two unmatched groups (between-subjects design), use an independent (unpaired) samples test .
• If you expect a difference between groups in a specific direction, use a one-tailed test .
• If you don’t have any expectations for the direction of a difference between groups, use a two-tailed test .

The only parametric correlation test is Pearson’s r . The correlation coefficient ( r ) tells you the strength of a linear relationship between two quantitative variables.

However, to test whether the correlation in the sample is strong enough to be important in the population, you also need to perform a significance test of the correlation coefficient, usually a t test, to obtain a p value. This test uses your sample size to calculate how much the correlation coefficient differs from zero in the population.

You use a dependent-samples, one-tailed t test to assess whether the meditation exercise significantly improved math test scores. The test gives you:

• a t value (test statistic) of 3.00
• a p value of 0.0028

Although Pearson’s r is a test statistic, it doesn’t tell you anything about how significant the correlation is in the population. You also need to test whether this sample correlation coefficient is large enough to demonstrate a correlation in the population.

A t test can also determine how significantly a correlation coefficient differs from zero based on sample size. Since you expect a positive correlation between parental income and GPA, you use a one-sample, one-tailed t test. The t test gives you:

• a t value of 3.08
• a p value of 0.001

The final step of statistical analysis is interpreting your results.

Statistical significance

In hypothesis testing, statistical significance is the main criterion for forming conclusions. You compare your p value to a set significance level (usually 0.05) to decide whether your results are statistically significant or non-significant.

Statistically significant results are considered unlikely to have arisen solely due to chance. There is only a very low chance of such a result occurring if the null hypothesis is true in the population.

This means that you believe the meditation intervention, rather than random factors, directly caused the increase in test scores. Example: Interpret your results (correlational study) You compare your p value of 0.001 to your significance threshold of 0.05. With a p value under this threshold, you can reject the null hypothesis. This indicates a statistically significant correlation between parental income and GPA in male college students.

Note that correlation doesn’t always mean causation, because there are often many underlying factors contributing to a complex variable like GPA. Even if one variable is related to another, this may be because of a third variable influencing both of them, or indirect links between the two variables.

Effect size

A statistically significant result doesn’t necessarily mean that there are important real life applications or clinical outcomes for a finding.

In contrast, the effect size indicates the practical significance of your results. It’s important to report effect sizes along with your inferential statistics for a complete picture of your results. You should also report interval estimates of effect sizes if you’re writing an APA style paper .

With a Cohen’s d of 0.72, there’s medium to high practical significance to your finding that the meditation exercise improved test scores. Example: Effect size (correlational study) To determine the effect size of the correlation coefficient, you compare your Pearson’s r value to Cohen’s effect size criteria.

Decision errors

Type I and Type II errors are mistakes made in research conclusions. A Type I error means rejecting the null hypothesis when it’s actually true, while a Type II error means failing to reject the null hypothesis when it’s false.

You can aim to minimise the risk of these errors by selecting an optimal significance level and ensuring high power . However, there’s a trade-off between the two errors, so a fine balance is necessary.

Frequentist versus Bayesian statistics

Traditionally, frequentist statistics emphasises null hypothesis significance testing and always starts with the assumption of a true null hypothesis.

However, Bayesian statistics has grown in popularity as an alternative approach in the last few decades. In this approach, you use previous research to continually update your hypotheses based on your expectations and observations.

Bayes factor compares the relative strength of evidence for the null versus the alternative hypothesis rather than making a conclusion about rejecting the null hypothesis or not.

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.

The research methods you use depend on the type of data you need to answer your research question .

• If you want to measure something or test a hypothesis , use quantitative methods . If you want to explore ideas, thoughts, and meanings, use qualitative methods .
• If you want to analyse a large amount of readily available data, use secondary data. If you want data specific to your purposes with control over how they are generated, collect primary data.
• If you want to establish cause-and-effect relationships between variables , use experimental methods. If you want to understand the characteristics of a research subject, use descriptive methods.

Statistical analysis is the main method for analyzing quantitative research data . It uses probabilities and models to test predictions about a population from sample data.

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Heterogeneity, Optimality, and Sensitivity in Causal Inference

 Identifying and efficiently estimating causal effects under minimal assumptions is a crucial endeavor across the sciences and society. We address several problems related to identification under violations of causal assumptions and efficient estimation of causal effects. In the first part of the thesis, we study conditional effect estimation under violations of the positivity assumption, which asserts that each subject has a non-zero probability of receiving treatment. We propose conditional effects based on incremental propensity score interventions, which are robust to violations of the positivity assumption, and develop efficient estimators for them. In the second part of the thesis, we focus on functional estimation, efficiency, and inference. We develop efficient estimators for the Expected Conditional Covariance. We use a recently proposed “double cross-fit doubly robust” (DCDR) estimator and establish that it achieves semiparametric efficiency under minimal conditions and minimax optimality in Hölder smoothness classes, and that it can be undersmoothed for slower-than-√ n inference. In the third part of the thesis we study average effect estimation under violations of the no unmeasured confounding assumption, which says that treatment is as-if randomized within covariate strata. For this purpose, we propose novel calibrated sensitivity models, which directly incorporate measured confounding into a sensitivity model, thereby bounding the error due to unmeasured confounding by measured confounding multiplied by a sensitivity parameter. We illustrate how to construct calibrated sensitivity models via several examples, demonstrate their advantages over standard sensitivity analyses and post hoc calibration/benchmarking, and establish methods for estimation and inference.

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• Dissertation
• Statistics and Data Science

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• Doctor of Philosophy (PhD)

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Thesis Statistical Treatment of Data

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