# Data for Additional Computational Practice for Questions to be Graded

The second table contains the Levene’s test for equality of variances. The Levene’s test is a statistical test of the equal variances assumption. The p value is 0.488, indicating there was no significant difference among the three groups’ variances; thus, the data have met the equal variances assumption for ANOVA.

The last table contains the contents of the ANOVA summary table, which looks much like This table contains an additional value that we did not compute by hand—the exact p value, which is 0.002. Because the SPSS output indicates that we have a significant ANOVA, post hoc testing must be performed.384

Return to the ANOVA window and click “Post Hoc.” You will see a window similar to the one below. Select the “LSD” and “Tukey” options. Click “Continue” and “OK.”

The following output is added to the original output. This table contains post hoc test results for two different tests: the LSD (Least Significant Difference) test and the Tukey HSD (Honestly Significant Difference) test. The LSD test, the original post hoc test, explores all possible pairwise comparisons of means using the equivalent of multiple t-tests. However, the LSD test, in performing a set of multiple t-tests, reports inaccurate p values that have not been adjusted for multiple computations Consequently, researchers should exercise caution when choosing the LSD post hoc test following an ANOVA.

The Tukey HSD comparison test, on the other hand, is a more “conservative” test, meaning that it requires a larger difference between two groups to indicate a significant difference than some of the other post hoc tests available. By requiring a larger difference between the groups, the Tukey HSD procedure yields more accurate p values of 0.062 to reflect the multiple comparisons 385

### Post Hoc Tests

Observe the “Mean Difference” column. Any difference noted with an asterisk (*) is significant at p < 0.05. The p values of each comparison are listed in the “Sig.” column, and values below 0.05 indicate a significant difference between the pair of groups. Observe the p values for the comparison of the Bachelor’s degree group versus the Master’s degree group. The Tukey HSD test indicates no significant difference between the groups, with a p of 0.062; however, the LSD test indicates that the groups significantly differed, with a p of 0.025. This example enables you see the difference in results obtained when calculating a conservative versus a lenient post hoc test. However, it should be noted that because an a priori power analysis was not conducted, there is a possibility that these analyses are underpowered. See  for more information regarding the consequences of low statistical power.

### Final Interpretation in American Psychological Association (Apa) Format

The following interpretation is written as it might appear in a research article, formatted according to APA guidelines A one-way ANOVA performed on months to program completion revealed significant differences among the three groups, F(2,24) = 7.96, p = 0.002. Post hoc comparisons using the Tukey HSD comparison test indicated that the students in the Associate’s degree group took significantly longer to complete the program than the students in the Bachelor’s degree group (19.8 versus 13.9 months, respectively) However, there were no significant differences in program completion time between the Associate’s degree group and the Master’s degree group or between the Bachelor’s degree group and the Master’s degree group.386

### Study Questions

1. Is the dependent variable in the  example normally distributed? Provide a rationale for your answer.

2. What are the two instances that must occur to warrant post hoc testing following an ANOVA?

3. Do the data in this example meet criteria for homogeneity of variance? Provide a rationale for your answer.

4. What is the null hypothesis in the example?

5. What was the exact likelihood of obtaining an F value at least as extreme as or as close to the one that was actually observed, assuming that the null hypothesis is true?

6. Do the data meet criteria for “mutual exclusivity”? Provide a rationale for your answer.387

7. What does the numerator of the F ratio represent?

8. What does the denominator of the F ratio represent?

9. How would our final interpretation of the results have changed if we had chosen to report the LSD post hoc test instead of the Tukey HSD test?

10. Was the sample size adequate to detect differences among the three groups in this example? Provide a rationale for your answer.388

1. Yes, the data are approximately normally distributed as noted by the frequency distribution generated from SPSS, below. The Shapiro-Wilk (covered in p value for months to completion was 0.151, indicating that the frequency distribution did not significantly deviate from normality.

2. The two instances that must occur to warrant post hoc testing following an ANOVA are (1) the ANOVA was performed on data comparing more than two groups, and (2) the F value is statistically significant.

3. Yes, the data met criteria for homogeneity of variance because the Levene’s test for equality of variances yielded a p of 0.488, indicating no significant differences in variance between the groups.

4. The null hypothesis is: “There is no difference between groups (Associate’s, Bachelor’s, and Master’s degree groups) in months until completion of an RN to BSN program.”

5. The exact likelihood of obtaining an F value at least as extreme as or as close to the one that was actually observed, assuming that the null hypothesis is true, was 0.2%.

6. Yes, the data met criteria for mutual exclusivity because a student could only belong to one of the three groups of the highest degree obtained prior to enrollment (Associate, Bachelor’s, and Master’s degree).

7. The numerator represents the between groups variance or the differences between the groups/conditions being compared.

8. The denominator represents within groups variance or the extent to which there is dispersion among the dependent variables.

9. The final interpretation of the results would have changed if we had chosen to report the LSD post hoc test instead of the Tukey HSD test. The results of the LSD test indicated that the 389students in the Master’s degree group took significantly longer to complete the program than the students in the Bachelor’s degree group (p = 0.025).

10. The sample size was most likely adequate to detect differences among the three groups overall because a significant difference was found, p = 0.002. However, there was a discrepancy between the results of the LSD post hoc test and the Tukey HSD test. The difference between the Master’s degree group and the Bachelor’s degree group was significant according to the results of the LSD test but not the Tukey HSD test. Therefore, it is possible that with only 27 total students in this example, the data were underpowered for the multiple comparisons following the ANOVA.390

#### Data for Additional Computational Practice for Questions to be Graded

Using the example from study, participants were randomized to receive Supported Employment or treatment as usual. A third group, also a treatment as usual group, consisted of a nonrandomized observational group of participants. A simulated subset was selected for this example so that the computations would be small and manageable. The independent variable in this example is treatment group (Supported Employment, Treatment as Usual–Randomized, and Treatment as Usual–Observational/Not Randomized), and the dependent variable was the number of hours worked post-treatment. Supported employment refers to a type of specialized interdisciplinary vocational rehabilitation designed to help people with disabilities obtain and maintain community-based competitive employment in their chosen occupation

The null hypothesis is: “There is no difference between the treatment groups in post-treatment number of hours worked among veterans with spinal cord injuries.”

Compute the ANOVA on the data in  below.

TABLE 33-3

POST-TREATMENT HOURS WORKED BY TREATMENT GROUP