Comparing the Means of Independent Groups: ANOVA, ANCOVA, MANOVA, and MANCOVA

Summary / Explanation

Testing the difference between the means of independent (unrelated) categorical groups involves parametric statistical tests.[1] Comparing the means between 2 independent groups requires a two-sample Student’s T-test (or unpaired or independent samples t-tests).[2][3] This article focuses on statistical methods for testing the difference between the means of 3 or more independent categorical groups based on a continuous dependent variable. These methods require an Analysis of Variance (ANOVA) test or one of its variations.[4][5][6]

Definitions and Descriptions

Analysis of Variance 

Analysis of Variance (ANOVA), sometimes called the F-test, compares the sample means of 1 categorical independent variable with 3 or more groups regarding 1 continuous dependent variable.[4][7] This is commonly called a one-way ANOVA. A two-way ANOVA compares 2 or more categorical independent variables and 1 dependent continuous variable.[5][6][8]

Analysis of Covariance 

Analysis of Covariance (ANCOVA) is similar to ANOVA (1 independent variable and 1 dependent variable) but in addition, adjusts or accounts for the effects of another variable (a covariate) on the mean.[4][8][9] A covariate is an independent variable that is not a main variable of interest but may affect the relationship between the dependent and independent variables.[9] A covariate is conventionally a continuous variable, but categorical variables may technically be used, although some sources consider this as not ANCOVA.[4][9]

A Multivariate Analysis of Variance/Covariance 

A Multivariate Analysis of Variance/Covariance (MANOVA/MANCOVA) compares the sample means of a set of 2 or more dependent continuous variables between 1 or more independent categorical variables, and a covariate(s) for MANCOVA.[4][8] As with all statistical tests, several assumptions are necessary for ANOVA, ANCOVA, and MANOVA/MANCOVA: normal distribution, independent observations, and homogeneity or similarity of variance across groups.[5]

Example

ANOVA, ANCOVA, and MANOVA/MANCOVA methods apply to continuous dependent variables and categorical independent variables, with or without a covariate(s).[4] 

• Continuous variables (including interval or ratio): are measured on a scale with clearly defined and equal spacing.[2][10] Examples of continuous data are blood glucose levels and AIC.

• Categorical (nominal) variables: are measured in groups with no specific order.

• A dependent variable (sometimes called the outcome): is the variable that changes due to the effects of the independent variable (sometimes called the explanatory variable) or other covariates.[8]

For example, to understand the effect of 3 different diabetic medication regimens (Drug A, Drug B, Drug C) on fasting blood glucose, fasting blood glucose is the continuous dependent variable and medication regimen is the categorical independent variable (the grouping variable). In this example, the ANOVA question is, "Do the medication regimens have the same mean fasting blood glucose?" This is an example of a one-way ANOVA because there is one grouping factor (ie, medication regimen). The summary statistic is the mean blood glucose in each of the 3 drug groups. The variability of the data points needs to be considered to decide if the groups are different beyond what might be expected by chance alone.[5] To do this, ANOVA considers the total scatter in the data in two parts - variability between the groups and what is left over (random error, within-group variance). ANOVA generates a p-value, which is the probability that the difference between groups is because of chance, and shows how likely the observed difference would have occurred purely by random chance. If the p-value for the group effect is small (eg, p<0.05), this indicates that at least some of the groups are different. If ANOVA finds that at least some of the groups are different, then groups can be compared by what is called a post-hoc analysis using multiple comparison procedures such as Fisher’s Least Significant Difference, Bonferroni Test, and Duncan’s Multiple Range Test.[6][8][11]

In the example, body weight may influence blood glucose and its effect needs to be accounted for, then this could be included in an ANCOVA model as a covariate.[9] Here, the p-value for the group effect answers the question, "Is the mean blood glucose different between medication regimens after adjusting for the effect of body weight?" The p-value for the covariate answers the question, "Does body weight affect the blood glucose beyond the effect of the medication regimen?"

MANOVA/MANCOVA is just an extension of ANOVA/ANCOVA involving 2 or more continuous dependent variables (hence the term multivariate) rather than 1.[4][8] For instance, researchers may be interested in a set (or family) of dependent variables - blood glucose and A1C - and their mean differences between medication regimens. Rather than conducting a one-way ANOVA test for each of the dependent variables separately (eg, one ANOVA for blood glucose and one ANOVA for A1C), a single MANOVA test can be used. By using a single MANOVA test instead of 2 separate ANOVA tests, the probability of finding a significant difference by chance alone is decreased, thereby decreasing the family-wise Type I error.[4]

Clinical Significance

A basic understanding of the application and implications of ANOVA, ANCOVA, and MANOVA/MANCOVA extends clinicians' knowledge beyond the simple one-sample t-test. These methods are useful for comparing means of 3 or more groups, adjusting for covariates, and studying a set of continuous dependent variables.