Question

Attempts: Average: /15 Д 4. Determining main effects and interactions in a two-way ANOVA А. Аа 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 post deployment life, including the acquisition of college survival skills. Post-traumatic 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 (community service, physical activity, Journaling, 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. 1-2 Second Factor: Program Component Community Service Physical Activity Journaling Meditation X11 - 51.5 X12 48.5 R13 = 49 49 834 - 55 2 . 821 - 49 8,2 = 49.0 823 = 49 8 24 = 48 X31 - 51 X x2 = 50 33 = 50 24 = 47.5 Xp, 50.50 Rp2 = 49.17 Xp3 = 49.33 84 = 50.17 Totals Xyz - 51.00 Xv2 - 48.75 Ryg = 49.63 3-4 First Factor: Years in Military >4 Totals The researcher performs an analysis of variance (ANOVA) to test the hypothesis that the populations defined by the treatment combinations are equal. Some of the results are presented in the following ANOVA table. Enter the values for degrees of freedom that are missing from the df column in the ANOVA table.

Answer 1

Level of Factor A, a = 3

Level of Factor B, b = 4

df(A) = a-1 = 2

df(B) = b-1 = 3

df(AB) = (a-1)*(b-1) = 6

df(total) = 119

df(error) = 119-2-3-6 = 108

SSA = 102.916

SSB = 37.2930

SSAB = 302.0810

SSE = 1447.5017

SS(total) = 1566.5017

MSA = SSA/df(A) = 51.458

MSB = SSB/df(B) = 12.431

MSAB = SSAB/df(AB) = 50.34683

MSE = SSE/df(error) = 13.4028

For for Factor A:

F = MSA/MSE = 3.8393

p-value = F.DIST.RT(3.8393, 2, 108) = 0.0245

Critical value Fc = F.INV.RT(0.05,2,108) = 3.08

For for Factor B:

F = MSB/MSE = 0.9275

p-value = F.DIST.RT(0.9275, 3, 108) = 0.4301

Critical value Fc = F.INV.RT(0.05,3,108) = 2.69

For for Interaction:

F = MSAB/MSE = 3.7564

p-value = F.DIST.RT(3.7564, 6, 108) = 0.0020

Critical value Fc = F.INV.RT(0.05,6,108) = 2.18

Source of Variation	SS	df	MS	F	P-value	F crit
Years in Military(Y)	102.9160	2	51.4580	3.8393	0.0245	3.08
Program Component(P)	37.2930	3	12.4310	0.92749	0.4301	2.69
YP	302.0810	6	50.3468	3.75644	0.0020	2.18
Error	1447.5017	108	13.4028
Total	1566.5017	119

The critical value of F for factor A (years in military) is 3.08. Therefore, the main effect due to factor A is significant because calculated value of F = 3.84 > Fc = 3.08.

The critical F value for factor B (program component) is 2.69. Therefore, the main effect due to factor B is insignifcant because calculated value of F = 0.93 < Fc = 2.69.

The critical F value for interaction of factor A and B is 2.18. Therefore, the main effect due to interaction of factor A and B is significant because calculated value of F = 3.76 > Fc = 2.18.

Conclude:

Veteran with different length of service had different PTSD symptoms scores. By itself, the program that veterans took was not associated with differences in severity; however, length of service resulted in different degrees of PTSD symptoms depending on program.