Question

12. Measuring effect size for two-factor ANOVA

It is projected that approximately 580,000 veterans will take advantage of the GI Bill for the 21st Century. Boots to Books is a course for all veterans, current military members, and their family members, friends, and supporters. The goal of Boots to Books is to assist deployed, postdeployed, and veteran students in making a positive transition from military to civilian life or from deployment to postdeployment life, including the acquisition of college survival skills.

Posttraumatic stress disorder (PTSD) is quite common among combat veterans. Suppose that a researcher wants to revise the Boots to Books course in order to reduce the effects of PTSD. He recruits a group of combat veterans and collects data on the length of time they were in the military (factor A) and the potential components of the revised course (journaling, community service, physical activity, and meditation: factor B). He then assigns them one of the four components. The study will evaluate which components of the program are the most effective for reducing PTSD symptoms.

The results of the hypothetical study are summarized in the following data matrix. Each cell reports the average (M), the total (T), and the sum of squares (SS) of the symptom score (X) on the Clinician-Administered PTSD Scale (CAPS) of 10 veterans. Veterans are grouped according to the number of years they served in the military.

Factor B: Program Component

		Journaling	Community Service	Physical Activity	Meditation
		M = 36.5	M = 39	M = 42	M = 45
	1–2	T = 365	T = 390	T = 420	T = 450	TROW1ROW1 = 1,625
		SS = 252.5	SS = 840	SS = 460	SS = 350
Factor A: Years in Military							ΣX² = 156,870
		M = 44	M = 44.8	M = 47.6	M = 49.1
	3–4	T = 440	T = 448	T = 476	T = 491	TROW2ROW2 = 1,855
		SS = 840	SS = 743.6	SS = 670.4	SS = 644.9
		TCOL1COL1 = 805	TCOL2COL2 = 838	TCOL3COL3 = 896	TCOL4COL4 = 941

The researcher performs an analysis of variance (ANOVA) to test the hypothesis that the populations defined by the treatment combinations are equal. The results are presented in the following ANOVA table.

ANOVA Table

Source	SS	df	MS	F
Between treatments	1,238.6000	7
Factor A	661.2520	1	661.2520	11.20
Factor B	548.3000	3	182.7667	3.10
AXB interaction	29.0480	3	9.6827	0.16
Within treatments	4,251.4000	72	59.0472
Total	5,490.0000	79

Use the significance level α = 0.05 to complete the following conclusions.

The critical F value for factor A (years in military) is 3.97. Therefore, the main effect due to factor A is_____ .

The critical F value for factor B (program component) is 2.73. Therefore, the main effect due to factor B is____ .

The critical F value for the interaction of factors A and B is 2.73. Therefore, the effect due to the interaction of factors A and B is_____ .

Given the results of the preceding analysis, what does the researcher conclude?

-Veterans who took different components of the program had different PTSD symptom scores. The severity of PTSD symptoms was unrelated to length of service either by itself or in conjunction with the various components of the program.

-Length of service and program component were each related to the severity of PTSD symptoms. However, the effect of each factor was independent of the other.

-Neither length of service nor program component was related to the severity of PTSD symptoms when looked at alone. However, there was an interaction between the two factors such that severity differed by program component differently depending on length of service (and vice versa).

-Veterans with different lengths of service had different PTSD symptom scores. The severity of PTSD symptoms was unrelated to program component either by itself or in conjunction with length of service.

Measure the effect size for factor A (years in military), factor B (program component), and the interaction by computing (partial) eta squares. (Express the value of each eta square as a percent.)

η² for factor A (years in military) = _____  

η² for factor B (program component) = _____

η² for the interaction = ______

Answer 1

Result is significant if test statistic F > critical F value.

The critical F value for factor A (years in military) is 3.97. Therefore, the main effect due to factor A is Significant.

The critical F value for factor B (program component) is 2.73. Therefore, the main effect due to factor B is significant.

The critical F value for the interaction of factors A and B is 2.73. Therefore, the effect due to the interaction of factors A and B is not significant.

Overal; conclusion:-

-The length of service and program component were each related to the severity of PTSD symptoms. However, the effect of each factor was independent of the other.

effect size calculation:-

Partial eta square = (SSeffect)/(SSeffect + SSerror)

η² for factor A (years in military) = (661.252)/(661.252+4,251.4) = 0.1346 = 13.46%

η² for factor B (program component) = (548.3)/(548.3+4,251.4) = 0.1142 = 11.42%

η² for the interaction = (29.048)/(29.048+4,251.4) = 0.0068 = 0.68%