11.1 CONDUCTING A ONE-WAY BETWEEN-GROUPS ANOVA

Irwin and colleagues (2004) are among a growing number of behavioral health researchers who are interested in adherence to medical regimens. These researchers studied adherence to an exercise regimen over one year in postmenopausal women, who are at increased risk for medical problems that may be reduced by exercise. Among the many factors that the research team examined was attendance at a monthly group education program that taught tactics to change exercise behavior; the researchers kept attendance and divided participants into three categories based on the number of sessions they attended. (Note: The researchers could have kept the data as numbers of sessions, a scale variable, rather than dividing them into categories based on numbers of sessions, an ordinal variable.)

Here is an abbreviated version of this study with fictional data points; the means of these data points, however, are the actual means of the study.

< 5 sessions: 155, 120, 130

5–8 sessions: 199, 160, 184

9–12 sessions: 230, 214, 195, 209

In this study, the independent variable was attendance, with three levels: <5 sessions, 5–8 sessions, and 9–12 sessions. The dependent variable was number of minutes of exercise per week. So we have one ordinal independent variable with three between-groups levels and one scale dependent variable. How can we conduct a one-way between-groups ANOVA?

Population 1: Postmenopausal women who attended fewer than 5 sessions of a group exercise-education program. Population 2: Postmenopausal women who attended 5–8 sessions of a group exercise-education program. Population 3: Postmenopausal women who attended 9–12 sessions of a group exercise-education program.

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The comparison distribution will be an F distribution. The hypothesis test will be a one-way between-groups ANOVA. The data were not selected randomly, so we must generalize only with caution. We do not know if the underlying population distributions are normal, but the sample data do not indicate severe skew. To see if we meet the homoscedasticity assumption, we will check to see if the largest variance is no greater than twice the smallest variance. From the calculations below, we see that the largest variance, 387, is not more than twice the smallest, 208.67, so we have met the homoscedasticity assumption. (The following information is taken from the calculation of SS_within.)

Null hypothesis: Postmenopausal women in different categories of attendance at a group exercise-education program exercise the same average number of minutes per week—H₀: μ₁ = μ₂ = μ₃. Research hypothesis: Postmenopausal women in different categories of attendance at a group exercise-education program do not exercise the same average number of minutes per week—H₁ is that at least one μ is different from another μ.

df_between = N_groups − 1 = 3 − 1 = 2

df₁ = 3 − 1 = 2; df₂ = 3 − 1 = 2; df₃ = 4 − 1 = 3

df_within = 2 + 2 + 3 = 7

The comparison distribution will be the F distribution with 2 and 7 degrees of freedom.

The critical F statistic based on a p level of 0.05 is 4.74.

df_total = 2 + 7 = 9 or df_total = 10 − 1 = 9

SS_total = Σ(X − GM)² = 12,222.40

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SS_within = ∑ (X − M)² = 2050.00

SS_between = ∑(M − GM)² = 10,172.40

The F statistic, 17.37, is beyond the cutoff of 4.74. We can reject the null hypothesis. It appears that postmenopausal women in different categories of attendance at a group exercise-education program do exercise a different average number of minutes per week. However, the results from this ANOVA do not tell us where specific differences lie. The ANOVA tells us only that there is at least one difference between means. We must calculate a post hoc test to determine exactly which pairs of means are different.