# Calculating Analysis of Variance (ANOVA) and Post Hoc Analyses

Calculating Analysis of Variance (ANOVA) and Post Hoc Analyses Following ANOVA

Analysis of variance (ANOVA) is a statistical procedure that compares data between two or more groups or conditions to investigate the presence of differences between those groups on some continuous dependent variable (see Exercise 18). In this exercise, we will focus on the one-way ANOVA, which involves testing one independent variable and one dependent variable (as opposed to other types of ANOVAs, such as factorial ANOVAs that incorporate multiple independent variables).

Why ANOVA and not a t-test? Remember that a t-test is formulated to compare two sets of data or two groups at one time (see Exercise 23 for guidance on selecting appropriate statistics). Thus, data generated from a clinical trial that involves four experimental groups, Treatment 1, Treatment 2, Treatments 1 and 2 combined, and a Control, would require 6 t-tests. Consequently, the chance of making a Type I error (alpha error) increases substantially (or is inflated) because so many computations are being performed. Specifically, the chance of making a Type I error is the number of comparisons multiplied by the alpha level. Thus, ANOVA is the recommended statistical technique for examining differences between more than two groups (Zar, 2010).

ANOVA is a procedure that culminates in a statistic called the F statistic. It is this value that is compared against an F distribution (see Appendix C) in order to determine whether the groups significantly differ from one another on the dependent variable. The formulas for ANOVA actually compute two estimates of variance: One estimate represents differences between the groups/conditions, and the other estimate represents differences among (within) the data.

Research Designs Appropriate for the One-Way ANOVA

Research designs that may utilize the one-way ANOVA include the randomized experimental, quasi-experimental, and comparative designs (Gliner, Morgan, & Leech, 2009). The independent variable (the “grouping” variable for the ANOVA) may be active or attributional. An active independent variable refers to an intervention, treatment, or program. An attributional independent variable refers to a characteristic of the participant, such as gender, diagnosis, or ethnicity. The ANOVA can compare two groups or more. In the case of a two-group design, the researcher can either select an independent samples t-test or a one-way ANOVA to answer the research question. The results will always yield the same conclusion, regardless of which test is computed; however, when examining differences between more than two groups, the one-way ANOVA is the preferred statistical test.

Example 1: A researcher conducts a randomized experimental study wherein she randomizes participants to receive a high-dosage weight loss pill, a low-dosage weight loss pill, or a placebo. She assesses the number of pounds lost from baseline to post-treatment 378for the three groups. Her research question is: “Is there a difference between the three groups in weight lost?” The independent variables are the treatment conditions (high-dose weight loss pill, low-dose weight loss pill, and placebo) and the dependent variable is number of pounds lost over the treatment span.

Null hypothesis: There is no difference in weight lost among the high-dose weight loss pill, low-dose weight loss pill, and placebo groups in a population of overweight adults.

Example 2: A nurse researcher working in dermatology conducts a retrospective comparative study wherein she conducts a chart review of patients and divides them into three groups: psoriasis, psoriatric symptoms, or control. The dependent variable is health status and the independent variable is disease group (psoriasis, psoriatic symptoms, and control). Her research question is: “Is there a difference between the three groups in levels of health status?”

Null hypothesis: There is no difference between the three groups in health status.