EPS 625 INTERMEDIATE STATISTICS TWO-WAY ANOVA IN-CLASS EXAMPLE (FLEXIBILITY)

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1 EPS 625 INTERMEDIATE STATISTICS TO-AY ANOVA IN-CLASS EXAMPLE (FLEXIBILITY) A researcher conducts a study to evaluate the effects of the length of an exercise program on the flexibility of female and male students. A sample of 48 students (24 Females and 24 Males) was randomly selected to participate in the current study. The students who participated in the study were randomly assigned to one of three treatment conditions in which they participated in an exercise program for one week, two weeks, or three weeks. The dependent measure for this study is the score on a flexibility measure, with higher scores indicating higher levels of flexibility. 1. hat would the null hypotheses for the two main effects and the interaction be for this study? Show/write the appropriate symbols or the expression in words. Gender Main Effect H 0 : µ 1. µ 2. There is no difference among the gender row means. Length of Program Main Effect H 0 : µ. 1 µ. 2 µ. 3 There is no difference among the length of program column means. Gender by Length of Program Interaction H 0 : all (µ jk µ j. µ. k + µ) 0 or H 0 : all αβ effects 0 There is no difference in the gender by length of program (JK) cell means that cannot be explained by the differences among the gender (row) means, the length of program (column) means, or both. In other words, there is no interaction between the two independent variables (gender and length of program). 2. hat would the alternative hypotheses for the two main effects and the interaction be for this study? Show/write the appropriate symbols or the expression in words. Gender Main Effect H a : µ 1. µ 2. The gender row means differ (are not equal). Length of Program Main Effect H a : µ. i µ. k for some i, k At least one pair or combination of means (length of program) significantly differs (are not equal). At least one mean (length of program) differs significantly from the other two means.

2 Gender by Length of Program Interaction H a : all (µ jk µ j. µ. k + µ) 0 or H 0 : all αβ effects 0 There are differences among the cell population means that cannot be attributed to the main effects. In other words, there is an interaction between the two independent variables (gender and length of program). 3. Determine if the underlying assumptions of the two-way ANOVA are met for these data. a. as the assumption of independence met for these data? Indicate how you made this determination. YES the assumption of independence was met. The participants for the study were randomly selected and represent two independent groups, i.e., either female or male. The students were then randomly assigned to one of three exercise program lengths (i.e., one week, two weeks, or three weeks). b. as the assumption of normality met for these data? Indicate how you made this determination (using an alpha level of.001 for the Shapiro-ilks test). YES the assumption of normality was met. Looking at the standardized skewness compared to a critical value of +3.29, we find none of the levels in either independent variable to be significantly skewed Gender: Female Male Length of Program: One week Two weeks Three weeks Also, looking at the Sig. (p) value in the Shapiro-ilks test for each level of both independent variables, we see that none of them were significant. Gender: Female, p (.103) > (.001) Male, p (.416) > (.001) Length of Program: One week, p (.786) > (.001) Two weeks, p (.024) > (.001) Three weeks, p (.449) > (.001) PAGE 2

3 c. as the assumption of homogeneity of variance met for these data? Indicate how you made this determination. Use an a priori alpha level.05. YES the assumption of homogeneity of variance was met. The Levene s test resulted in an F(5, 42).669, p.649. Comparing the significance value of.649 to the a priori alpha level (α.05) we see that p (.649) > α (.05), therefore the null hypothesis of no difference is retained, which indicates the assumption of homogeneity of variance is met. 4. Determine which (if any) effects were significant. Use an a priori alpha level.05. a. as the J (row) main effect significant? Indicate how you made this determination. YES, the J (Gender) Main Effect is significant, p (.000) < (.05) F(1, 42) 22.37, p <.001 b. as the K (column) main effect significant? Indicate how you made this determination. YES, the K (Length of Program) Main Effect is significant, p (.000) < (.05) F(2, 42) 36.03, p <.001 c. as the JK (interaction) significant? Indicate how you made this determination. YES, the JK Interaction is significant, p (.000) < (.05) F(2, 42) 11.84, p < hat proportion of variance in the student s average flexibility score is attributed to the interaction of the student s gender and the length of the exercise program? You will need to calculate this value by hand you may show your work on the SPSS output. ω 2 SS JK ( J 1)( K 1) MS SS + MS T (2 1)(3 1) (1)(2) ω (2) ω or 13.45% ( 14%) Approximately 14% of the total variance in the dependent variable (DV Flexibility Score) can be attributed to the interaction of the two independent variables (IV 1 Gender and IV 2 Length of Exercise Program). PAGE 3

4 6. Assuming that the interaction is significant briefly describe the plot for the interaction illustrating the comparison of female and male students at each of the three program length levels. The lines should be the students gender. Since the lines do not cross at any of the data points the interaction is considered to be ordinal. The female student s average flexibility scores stay consistently higher than the male student s average flexibility scores at each of the three exercise program lengths. 7. Do the Gender Simple Main Effects analysis to test for the statistical significance of the gender differences within each of the length of programs. Determine if these effects are significant. Don t forget to control for Type I error. Report your findings. No. of α α Contrasts Females (M 24.63) vs. Males (M 21.88) within One week Mean Difference 2.750, Not Significant, p (.160) > (.0167) Females (M 27.38) vs. Males (M 27.13) within Two weeks Mean Difference.250, Not Significant, p (.897) > (.0167) Females (M 41.00) vs. Males (M 28.25) within Three weeks Mean Difference , Significant, p (.000) < (.0167) 8. Calculate and report the Effect Size (by hand) for each of the significant pairwise differences found from the above analyses. You may show your work on the SPSS output. Using the formula: ES X i X MS k here MS , therefore MS Females (M 41.00) vs. Males (M 28.25) within Three weeks Mean Difference ES PAGE 4

5 9. Do the Length of Program Simple Main Effects analysis to test for the statistical significance of the length of program differences within each of the gender levels. Determine if these effects are significant. Don t forget to control for Type I error. Report your findings. No. of α α Contrasts Length of Program within Females Significant, p (.000) < (.025) Length of Program within Males Significant, p (.004) < (.025) 10. Conduct pairwise comparisons for each of the significant simple main effects tested above. Don t forget to control for Type I error. Report your findings. Length of Program Simple Main Effect within Females Type I Error α α No.of Contrasts One week (M 24.63) vs. Two weeks (M 27.38) Mean Difference Not Significant, p (.160) > (.0083) One week (M 24.63) vs. Three weeks (M 41.00) Mean Difference Significant, p (.000) < (.0083) Two weeks (M 27.38) vs. Three weeks (M 41.00) Mean Difference Significant, p (.000) < (.0083) Length of Program Simple Main Effect within Males Type I Error α α No.of Contrasts One week (M 21.88) vs. Two weeks (M 27.13) Mean Difference Not Significant, p (.009) > (.0083) One week (M 21.88) vs. Three weeks (M 28.25) Mean Difference Significant, p (.002) < (.0083) PAGE 5

6 Two weeks (M 27.13) vs. Three weeks (M 28.25) Mean Difference Not Significant, p (.562) > (.0083) 11. Calculate and report the Effect Size (by hand) for each of the significant pairwise differences found from the above analyses. You may show your work on the SPSS output. Using the formula: ES X i X MS k here MS , therefore MS Length of Program Simple Main Effect within Females One week (M 24.63) vs. Three weeks (M 41.00) Mean Difference ES Two weeks (M 27.38) vs. Three weeks (M 41.00) Mean Difference ES Length of Program Simple Main Effect within Males One week (M 21.88) vs. Three weeks (M 28.25) Mean Difference ES rite a separate results section for the above findings including tables for the means and standard deviations and the ANOVA summary information. Round all values to two decimal places except the probability (p) values, which should be left at three decimal places. Also include the Figure showing the (HINT) significant interaction. See Understanding the Two-way ANOVA handout for Results for this study. The following is the break-down of the results section identified by the steps in the analysis. PAGE 6

7 Results This first section sets the stage for the results section letting your reader know what statistic was used and what were the two independent variables and the dependent variable (along with their operational definitions). An indication that Table 1 will provide the means and standard deviations is also made here. A two-factor (2 3) Analysis of Variance was conducted to evaluate the effects of the length of an exercise program on the flexibility of female and male subjects. The two independent variables in this study are gender and length of exercise program (1-week, 2-weeks, and 3-weeks). The dependent variable is the score on the flexibility measure, with higher scores indicating higher levels of flexibility. The means and standard deviations for the flexibility measure as a function of the two factors are presented in Table 1. *Insert Table 1 about here This next section lets your reader know that you have tested the underlying assumptions of the two-way ANOVA (and what adjustments/corrections were made, e.g., transformation, if applicable). This section also contains the results of the two main effects and the interaction, with a general explanation of the effect (e.g., explaining what the significant interaction indicates). An indication that Table 2 will provide the two-way ANOVA results is also made here. If there is a significant interaction (as is the case in this example), an indication that Figure 1 is provided to show/illustrate that interaction is made. The applicable measures of association (e.g., ω 2 ) are also reported with a general explanation. The test for normality, examining standardized skewness and the Shapiro-ilks test indicated the data were statistically normal. The test for homogeneity of variance was not significant, Levene F(5, 42).67, p.649, indicating that this assumption underlying the application of the two-way ANOVA was met. An alpha level of.05 was used for the initial analyses. The results for the two-way ANOVA indicated a significant main effect for gender, F(1, 42) 22.37, p <.001 and a significant main effect for length of exercise program, F(2, 42) 36.03, p <.001. Additionally, the results show a significant interaction between gender and length of exercise program, F(2, 42) 11.84, p <.001 (see Table 2), indicating that any differences between the length of exercise programs were dependent upon which gender the subjects were and that any differences between females and males were dependent upon which length of exercise program they were in (see Figure 1 for a graph of this interaction). Approximately 14% (ω 2.14) of the total variance of the flexibility levels was attributed to the interaction of gender and length of exercise program. * Insert Table 2 about here *Insert Figure 1 about here PAGE 7

8 This section explains that we will be interpreting the interaction since it was significant. e break down the interaction by its Simple Main Effects, starting here with the Gender Simple Effect, which is the difference between Females and Males at each of the levels of Length of Program. e also indicate how we have protected for Type I Error for this comparison. Because these are pairwise comparisons, if a significant difference is found, we report the means, results, and the Effect Sizes. Because the interaction between gender and the length of exercise program was significant, we chose to ignore the two main effects and instead first examined the gender simple main effects, that is, the differences between females and males for each of the three lengths of exercise programs. To control for Type I error rate across the three simple effects, we set the alpha level for each at.0167 (α/3.05/3). The only significant difference between females and males was found in the 3-weeks exercise program. A review of the group means indicated that females (M 41.00) had a significantly higher level of flexibility than males (M 28.25), F(1, 42) 43.98, p <.001, ES This section continues the break-down of the significant interaction. Here we are looking at the second Simple Main Effect (Length of Program) which is the difference among the length of program at each of the gender levels. e also indicate how we have protected for Type I Error for this comparison. Results are presented for the simple effect. Because this Simple Effect is not a pairwise comparison, it needs to be broken-down further (our indication is that we are still at df 2). The follow-up tests are explained along with an indication of how we protected for Type I Error. These follow-up tests are now pairwise comparisons, so if any significant differences are found, we report the means, results, and the Effect Sizes. Note, it is not necessary to duplicate the reporting of the means when breaking down the pairwise comparisons. Additionally, we examined the length of exercise program simple main effects, that is, the differences among the three lengths of exercise programs for females and males separately. To control for Type I error across the two simple main effects, we set the alpha level for each at.025 (α/2.05/2). There was a significant difference among the three lengths of exercise programs for females, F(2, 42) 41.60, p <.001, and for males, F(2, 42) 6.26, p <.01. Followup tests were conducted to evaluate the three lengths of exercise programs pairwise differences for females. The alpha level was set at.0083 (.025/3) to control for Type I error over the three pairwise comparisons. The females in the 3-weeks exercise program (M 41.00) had significantly higher flexibility levels compared to the females in the 1-week program (M 24.63), F(1, 42) 72.54, p <.001, ES 4.26 and the females in the 2-weeks program (M 27.38), F(1, 42) 50.22, p <.001, ES Follow-up tests were also conducted to evaluate the three lengths of exercise programs pairwise differences for males. The alpha level was set at.0083 (.025/3) to control for Type I error over these three pairwise comparisons. The males in the 3-weeks exercise program (M 28.25) had significantly higher flexibility levels compared to the males in the 1-week program (M 21.88), F(1, 42) 11.00, p <.0083, ES PAGE 8

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