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## Two-Way MANOVA

The two-way multivariate analysis of variance (two-way MANOVA) is often considered as an extension of the two-way ANOVA for situations where there are two or more dependent variables. The primary purpose of the two-way MANOVA is to understand if there is an interaction between the two independent variables on the two or more combined dependent variables.

For example, you could use a two-way MANOVA to understand whether there were differences in students’ short-term and long-term recall of facts based on lecture duration and fact type (i.e., the two dependent variables are “short-term memory recall” and “long-term memory recall”, whilst the two independent variables are “lecture duration”, which has four groups – “30 minutes”, “60 minutes”, “90 minutes” and “120 minutes” – and “fact type”, which has two groups: “quantitative (numerical) facts” and “qualitative (textual/contextual) facts”). Alternately, you could use a two-way MANOVA to understand whether there were differences in the effectiveness of male and female police officers in dealing with violent crimes and crimes of a sexual nature taking into account a citizen’s gender (i.e., the two dependent variables are “perceived effectiveness in dealing with violent crimes” and “perceived effectiveness in dealing with sexual […]

## Multiple Regression Analysis

A multiple regression is used to predict a continuous dependent variable based on multiple independent variables. As such, it extends simple linear regression, which is used when you have only one continuous independent variable. Multiple regression also allows you to determine the overall fit (variance explained) of the model and the relative contribution of each of the predictors to the total variance explained.

Note 1: The dependent variable can also be referred to as the “outcome”, “target” or “criterion” variable, whilst the independent variables can be referred to as “predictor”, “explanatory” or “regressor” variables. It does not matter which of these you use, but we will continue to use “dependent variable” and “independent variable” for consistency.

Note 2: This guide deals with “standard” multiple regression rather than a specific type of multiple regression, such as hierarchical multiple regression, stepwise regression, amongst others.

For example, you could use multiple regression to understand whether exam performance can be predicted based on revision time, test anxiety, lecture attendance, course studied and gender. Here, your continuous dependent variable would be “exam performance”, whilst you would have three continuous independent variables – “revision time”, measured in hours, “test anxiety”, measured using the TAI […]

## One-Way RM ANOVA

The one-way repeated measures analysis of variance (ANOVA) is an extension of the paired-samples t-test and is used to determine whether there are any statistically significant differences between the means of three or more levels of a within-subjects factor. The levels are related because they contain the same cases (e.g., participants) in each level. The participants are either the same individuals tested on three or more occasions on the same dependent variable or the same individuals tested under three or more different conditions on the same dependent variable. This test is also referred to as a within-subjects ANOVA or ANOVA with repeated measures.

Note: Whilst a one-way repeated measures ANOVA can be used when your within-subjects factor has just two levels, it is typically only used when the within-subjects factor has three or more levels. The reason for this is that when there are only two levels, a paired-samples t-test is more commonly used. This is why we refer to the one-way repeated measures ANOVA having three or more levels in this guide.

For example, you could use a one-way repeated measures ANOVA to understand whether there is a difference in cigarette consumption amongst heavy smokers after […]

## Correlation Analysis

Correlation Analysis

The Pearson product-moment correlation is used to determine the strength and direction of a linear relationship between two continuous variables. More specifically, the test generates a coefficient called the Pearson correlation coefficient, denoted as r (i.e., the italic lowercase letter r), and it is this coefficient that measures the strength and direction of a linear relationship between two continuous variables. Its value can range from -1 for a perfect negative linear relationship to +1 for a perfect positive linear relationship. A value of 0 (zero) indicates no relationship between two variables. This test is also known by its shorter titles, the Pearson correlation or Pearson’s correlation, which are often used interchangeably.

For example, you could use a Pearson’s correlation to determine the strength and direction of a linear relationship between salaries, measured in US dollars, and length of employment in a firm, measured in days (i.e., your two continuous variables would be “salary” and “length of employment”). You could also use a Pearson’s correlation to determine the strength and direction of a linear relationship between reaction time, measured in milliseconds, and hand grip strength, measured in kilograms (i.e., your two continuous variables would […]

## Chi-Square Test

The chi-square test can be used to test a variety of sizes of contingency tables, as well as more than one type of null and alternative hypotheses. This guide focuses on contingency tables that are greater than 2 x 2, which are often referred to as r x c contingency tables, and tests whether two variables measured at the nominal level are independent (i.e., whether there is an association between the two variables). Most commonly this test is called the chi-square test of independence, but it is also known as the chi-square test for association. Whilst it is also possible to perform the chi-square test of independence on ordinal variables, you will lose the ordered nature of the data by doing so and there will most likely be more suitable tests to run (see our Statistical Test Selector). In order to make the correct inferences from a chi-square test of independence you will need to have undertaken a naturalistic study design.

Note: If you are interested in understanding (and modelling) associations between three or more categorical variables you should consider loglinear analysis instead of the chi-square test of independence.

For example, you could use a chi-square test of independence […]

## Two-Way ANOVA

The two-way ANOVA is used to determine whether there is an interaction effect between two independent variables on a continuous dependent variable (i.e., if a two-way interaction effect exists). In many ways, the two-way ANOVA can be considered as an extension of the one-way ANOVA, which deals with just one independent variable rather than the two-way independent variables of the two-way ANOVA.

Note: It is quite common for the independent variables to be called “factors” or “between-subjects factors”, but we will continue to refer to them as independent variables in this guide. Furthermore, it is worth noting that the two-way ANOVA is also referred to as a “factorial ANOVA” or, more specifically, as a “two-way between-subjects ANOVA”.

A two-way ANOVA can be used in a number of situations. For example, consider an experiment where two drugs were being given to elderly patients to treat heart disease. One of the drugs was the current drug being used to treat heart disease and the other was an experimental drug that the researchers wanted to compare to the current drug. The researchers also wanted to understand how the drugs compared in low and high-risk elderly patients. The goal […]

## One-Way MANOVA

The one-way multivariate analysis of variance (MANOVA) is an extension of the one-way ANOVA to incorporate two or more dependent variables (i.e., the one-way ANOVA investigates just one dependent variable). Unlike the one-way ANOVA, which tests for differences in the mean values of the dependent variable between the groups of the independent variable, the one-way MANOVA tests for the ‘linear composite’ or vector of the means between the groups of the independent variable. Essentially, it combines the two or more dependent variables to form a ‘new’ dependent variable in such a way as to maximize the differences between the groups of the independent variable. It is between this new, composite variable that you test for statistically significant differences between the groups.

Also, like the one-way ANOVA, it is important to realize that the one-way MANOVA is an omnibus test statistic and cannot tell you which specific groups were significantly different from each other; it only tells you that at least two groups were different. Since you may have three, four, five or more groups in your study design, determining which of these groups differ from each other is important.

For example, you could use a one-way MANOVA […]

## One-Way ANCOVA

The analysis of covariance (ANCOVA) can be thought of as an extension of the one-way ANOVA to incorporate a covariate variable. This covariate is linearly related to the dependent variable and its inclusion into the analysis can increase the ability to detect differences between groups of an independent variable. An ANCOVA is used to determine whether there are any statistically significant differences between the adjusted population means of two or more independent (unrelated) groups.

For example, you could use a one-way ANCOVA to determine whether exam performance differed based on test anxiety levels amongst students whilst controlling for revision time (i.e., your dependent variable would be “exam performance”, measured from 0-100, your independent variable would be “test anxiety level”, which has three groups – “low-stressed students”, “moderately-stressed students” and “highly-stressed students” – and your covariate would be “revision time”, measured in hours). You want to control for revision time because you believe that the effect of test anxiety levels on exam performance will depend, to some degree, on the amount of time students spent revising.

## Assumptions of One-Way ANCOVA

In order to run a one-way ANCOVA, there are ten assumptions that need to be considered. The […]

## Three-Way RM ANOVA

The three-way repeated measures ANOVA is used to determine if there is a statistically significant interaction effect between three within-subjects factors on a continuous dependent variable (i.e., if a three-way interaction exists). As such, it extends the two-way repeated measures ANOVA, which is used to determine if such an interaction exists between just two within-subjects factors (i.e., rather than three within-subjects factors).

Note: It is quite common for “within-subjects factors” to be called “independent variables”, but we will continue to refer to them as “within-subjects factors” (or simply “factors”) in this guide. Furthermore, it is worth noting that the three-way repeated measures ANOVA is also referred to more generally as a “factorial repeated measures ANOVA” or more specifically as a “three-way within-subjects ANOVA”.

A three-way repeated measures ANOVA can be used in a number of situations. For example, you might be interested in the effect of two different types of ski goggle (i.e., blue-tinted or gold-tinted ski goggles) for improving ski performance (i.e., time to complete a ski run). However, you are concerned that the effect of the different lens colors on ski performance might be different depending on the snow condition (i.e., whether there […]