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Statistics for the Behavioral Sciences, 4/e

Michael Thorne, Mississippi State University -- Mississippi State
Martin Giesen, Mississippi State University -- Mississippi State

One-Way Analysis of Variance With Post Hoc Comparisons

Chapter Overview

The analysis of variance, or ANOVA, is a widely used test for comparing more than two groups. Two reasons for not using the two-sample t test are that multiple t tests are tedious to compute and that the more tests you do on the same data, the more likely you are to commit a Type I error (reject a true null).

The total variability in some data can be partitioned or divided into the within-groups variability and the between-groups variability. The variability within each group stems from individual differences and experimental error; the variability between groups comes from individual differences, experimental error, and the treatment effect. The ANOVA test is the ratio of a measure of variability between groups to a measure of the variability within groups. If there is no treatment effect, the computed value of F will be close to 1. However, if there is a treatment effect, the F ratio will be relatively large because of the added source of variability contributing to the between-group differences. One-way between-subjects ANOVA applies to situations in which the data from three or more independent groups are analyzed.

The first step in determining the indices of variability is to compute the sums of squares. The total sum of squares is the sum of the squared deviations of each score from the total mean. The sum of squares within each group is the sum of the squared deviations of each score in a group from its group mean, with the deviations summed across groups. Finally, the sum of squares between groups can be obtained by subtraction: SS_b = SS_tot – SS_w. Also, SS_b is the square of the deviation between each group mean and the total mean multiplied by the number of subjects in a particular group and summed over groups. It’s a good idea to compute SS_b to test the accuracy of your other computations.

After the sums of squares have been determined, appropriate degrees of freedom is computed for each. For SS_tot, or the total sum of squares, df = N – 1, where N is the total number of cases sampled. For SSb, or the sum of squares between groups, df = K – 1, where K is the number of groups. df for SS_w, or the sum of squares within groups, is N – K.

Both SS_b and SS_w are divided by their respective df to give the average or mean square. The ratio of MS_b to MS_w is called the F ratio. A relatively large value of F indicates greater variability between groups than within groups and may indicate sampling from different populations. The computed value of F is compared with values known to cut off deviant portions (5% or 1%) of the distribution of F. If the computed F exceeds critical values from Table C (see Appendix 2), the null hypothesis is rejected, and we conclude that at least one of the samples probably came from a different population. To help summarize the results, as they are computed, values are entered into the analysis of variance summary table shown here.

Summary Table for Between-Subjects ANOVA

Two tests are presented for further significance testing following a significant F ratio: the Fisher LSD and the Tukey HSD. Both tests are used to make all pairwise comparisons—comparing all groups by looking at one pair at a time. The LSD test is sometimes called a protected t test because it follows a significant F test. In the LSD test, the difference between a pair of means is significant if it is greater than LSD, which is computed with a formula; the same is true for the HSD test; that is, a difference between a pair of means is significant if the difference exceeds the computed value of HSD. A table of differences is used to summarize the results of both tests.

The one-way repeated measures ANOVA applies to situations in which the same (or matched) participants are tested on more than two occasions. The first step is to compute the sums of squares. The total and between-groups sums of squares are computed using the same procedures as in one-way between-subjects ANOVA. However, the within-groups sum of squares is divided into two parts: subjects sum of squares (SS_subj) and error sum of squares (SS_error). SS_subj is the squared deviation between the mean score for each subject and the total mean, multiplied by the number of groups and summed over subjects. SS_error is the variability remaining after removing SS_b and SS_subj from SS_tot and can be obtained by subtraction: SS_error = SS_tot – SS_b – SS_subj. Computational formulas were given for each of the sums of squares.

As in one-way between-subjects ANOVA, df_tot = N – 1, and df_b = K – 1. Subjects degrees of freedom (df_subj) equal the number of subjects minus 1 (S – 1), and error degrees of freedom (df_error) equal (K – 1)(S – 1). Both SS_b and SS_error are divided by the appropriate df to give MS_b and MS_error, respectively. The F ratio is obtained by dividing MS_b by MS_error. If the computed F is greater than or equal to the critical values from Table C (Appendix 2), the null hypothesis is rejected. With slight modifications, the LSD and HSD tests can be used for post hoc testing following a significant repeated measures ANOVA. To summarize the results, values are entered in a summary table, as shown here.

Summary Table for One-Way Repeated Measures ANOVA

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