Midterm Examination Key

Winter 2005

1. A marital researcher is interested in comparing the effects of two different treatments for distressed couples. She randomly assigns distressed couples to one of two therapies. Couples assigned to the first "talking" therapy are encouraged to share their feelings and to discuss the issues confronting them. Couples assigned to the second "doing" therapy are encouraged to do pleasant things with each other and not to focus on their problems. Every month, the couples are brought into the laboratory and their interactions during a problem-solving task are coded. Higher scores indicate more positive affect. The researcher predicts that couples engaged in the "doing" therapy will show more positive affect at the conclusion of the 4 months and that their relationships will improve at a faster rate than couples engaged in the "talking" therapy.

a. Write out the structural model for these data; use S=subject couples, T=therapy, M=month. Treat time as fixed.

S^R(T^F)XM^F

b. Write out the expected mean square table. Be sure to identify the correct error term for each effect. Calculate the degrees of freedom for each effect. Identify the correct degrees of freedom for each F ratio. If a quasi-F ratio is required, specify the terms that would contribute to the numerator and denominator of the ratio and write out the formulas for the degrees of freedom specifying the MS that would be used.

Source df E(MS) Error Line

1. T t-1=1 nms²_T+ms²_S(T)2

2. S(T) t(n-1)= 28 ms²_S(T)

3. M m-1=3 ms²_M+ s²_S(T)M 5

4. MT (m-1)(t-1)=3 ms²_MT+ s²_S(T)M 5

5. S(T)M (n-1)t(m-1)=84 s²_S(T)M

c. Conduct a standard analysis that corresponds to your E(MS) table.

First, determine whether an uncorrected univariate repeated measures analysis is appropriate.

From “Mauchly’s Test of Sphericity”, we have no reason to believe that sphericity has been violated, so we can proceed with the unadjusted analysis.

There is significant change in observed positive affect over time (F(3,84)=10.326, MSe=9.804, p<.001) and this change is different in the two therapy groups (F(3,84)=2.637, p=.055).

d. Test the hypothesis that at the conclusion of the 4 months of therapy that the couples who had the “doing” therapy will show more positive affect than the couples who had the “talking” therapy.

This hypothesis can be tested in a variety of ways. The easiest is to do a t-test:

After 4 months, couples who participated in the “Doing” therapy were coded as displaying more positive affect than couples who participated in the “Talking” therapy (t(28)=3.042, p=.005).

e. Test the hypothesis that the rate of improvement for couples engaged in the “doing” therapy was greater than the rate of improvement for couples engaged in the “talking” therapy.

This hypothesis asks for a comparison of the way the positive affect displayed by the couples in each group changes over time. In particular, it asks for a comparison of the “rate of improvement”. “Rate” implies a simple linear comparison (but see below), so we want to compare the difference in linear trends in positive affect over time.

Members of the therapy groups demonstrate different linear trends in positive affect (F(1,28)=5.480, MSe=12.287, p=.027). The rate of improvement is greater for the “Doing” group than the “Talking” group (see Figure 1).

f. Plot the means for the therapy X time interaction.

f. Interpret your results.

Examination of the plot suggests that although the general linear trend followed by the “Talk” group is shallower than that of the “Do” group, there is actually very little difference between these groups for the first 3 months of the therapy. Only during the 4^th month is there a substantial difference. This is confirmed by comparing the means for the groups at each time period.

One could have done this analysis by simple effects contrasts using a pooled between/within error term to obtain more power (but this can be a dangerous procedure and some authors warn against it). In this case, there is no doubt about the results. Doing the simple effects analysis “the other way” -- examining trends for each type of therapy – reveals that the “Do” group demonstrates a linear trend (F(1,84)=32.04, p<.001) as does the “Talk” group (F(1,84)=3.82, p=.054). No other effects are significant.

So, we could interpret the results as follows:

Couples in both groups demonstrated increased positive affect over time (F(1,28)=23.137, MSe=12.287, p<.001). Couples who received the “Doing” therapy showed a greater rate of improvement (F(1,28)=5.480, MSe=12.287, p=.027) compared to couples who participated in the “Talking” therapy (see Figure 1). However, the type of therapy only had demonstrably different effects after 4 months (F(1,28)=9.255, p=.005).

2. A researcher has been hired to test the relative effectiveness of two different strategies for coping with the deleterious effects of situational anxiety on task performance. The first strategy (Cognitive) requires that the subject cognitively reinterpret the stressful situation and thereby alter the experienced affect. The second strategy (Emotional) requires that the subject induce a positive emotional response that is incompatible with the anxiety. To test these strategies, the researcher recruits 90 subjects and randomly assigns them to one of three conditions: No intervention (control), Cognitive strategy, and Emotional strategy. The subjects assigned to each condition are then trained in the use of the appropriate strategies in small groups of 6 participants each (the control subjects receive a “placebo” training program). Each subject is then run individually through the experimental procedure. In this procedure, the subject is asked to perform a cognitively demanding task under stress and under a no-stress condition. Higher numbers indicate better performance. Negative values are possible. The researcher expects that no training will be able to completely erase the effects of stress. Hence she hypothesizes that performance will degrade under stress for all groups. She also hypothesizes that both amelioration procedures will produce better performance under stress than the control training and that the cognitive procedure will be superior to both the control training and the emotional procedure.

a. Write out the structural model for this design; use S=subjects, G=group, T=training, and C=condition

S^R(G^R(T^F))XC^F

1. T ngcσ²_T + ncσ²_G(T) + cσ²_S(G(T))2 t-1 2

2. G(T) ncσ²_G(T) + cσ²_S(G(T)) 3 t(g-1) 3(4)=12

3. S(G(T)) cσ²_S(G(T)gt(n-1) 15(5)=75

4. C ngtσ²_C + nσ²_G(T)C + σ²_S(G(T))C 6 c-1 1

5. TC ngσ²_TC + nσ²_G(T)C + σ²_S(G(T))C 6 (c-1)(t -1) 2

6. G(T)C nσ²_G(T)C + σ²_S(G(T))C 7 t(g-1)(c-1) 12

7. S(G(T))C σ²_S(G(T))C (n-1)gt(c-1) 75

b. Conduct a standard analysis that corresponds to your E(MS) table.

Unfortunately, SPSS won’t run the analysis that you have just designed exactly as is, but only the error terms are wrong because it assumes that all effects are fixed.

So, we’ll need to take this information and put it in the right place in the table and do a few hand calculations:

Source	Type III Sum of Squares	df	Mean Square	F	Sig.
Intercept
2872.006	1	2872.006	380.286	.000
TRAINING
28.078	2	14.039	1.859	.163
GROUP(TRAINING)
76.000	12	6.333	.839	.611
S(G(T))
566.417	75	7.552

Tests of Within-Subjects Effects

Source	Type III Sum of Squares	df	Mean Square	F	Sig.
COND
140.450	1	140.450	15.723	.002
COND * TRAINING
51.100	2	25.550	2.860	.096
COND * GROUP ( TRAINING )
107.200	12	8.933	1.160	.328
Error(COND)
577.750	75	7.703

c. Test the following hypotheses (HINT: there is more than one way to do this!)

1. Test whether there was an overall decrease in performance under stress.

Effect labeled “COND” above: F(1,12)=15.723, p=.002

2. Test whether there were differences between training conditions in the no stress condition.

There were no differences between training conditions in the no stress condition (F(2,12)=0.091, ns).

3. For each of the training conditions, test whether there was a decrease in performance under stress.

Simple effects of stress @ each training condition:

Simple Effect of stress @	MS	F	p
Cognitive	4.266667	0.47763	0.502649
Control	147.2667	16.48569	0.001581
Emotional	40.01667	4.479645	0.05588

Performance decreases under stress in the control (F(1,12)=16.49, p=.002) and emotional intervention conditions (F(1,12)=4.48, p=.056).

4. Test whether the decrease in performance under stress (from the performance observed without stress) in the control condition was greater than the decrease in performance observed in the other two conditions.

Use interaction contrast form from handout:

Contrast comparing control to experimental conditions: -1 2 –1

Contrast comparing stress to no stress: 1 –1

Interaction contrast pattern obtained by multiplying coefficients from these contrasts:

		Cognitive	Control	Emotional
		-1	2	-1
Stress	1	-1	2	-1
No stress	-1	1	-2	1

F(1,12)=4.70, p=.051

5. Test whether the decrease in performance under stress (from the performance observed without stress) in the cognitive condition was greater than the decrease in performance observed in the emotion condition.

Contrast weights:

		Cognitive	Control	Emotional
		-1	0	1
Stress	1	-1	0	1
No stress	-1	1	0	-1

F(1,12)=1.02, p=.33

6. Are you concerned about the number of tests you have conducted? If you were, what would you do?

Yes, we’ve done a lot of tests on the same data; I’d use a Bonferroni correction.

d. Interpret your results.

In general, performance decreases under stress F(1,12)=15.723, p=.002; see Table 1). However, when individuals were trained to utilize a cognitive coping strategy, the observed decrease in performance was negligible (F(1,12)=.478, n.s.). Individuals trained to use an emotional coping strategy demonstrated a nearly significant decrease in performance (F(1,12)=4.48, p=.056). But the cognitive coping strategy was not significantly better at reducing the effects of stress compared to the emotional strategy (F(1,12)=1.02, ns).

Note that the effects of group are not significant either alone or in interaction with other factors and the values of the associated F tests are less than 2. So, one could reasonably obtain more power from this design by eliminating this factor.

Simple Effect of stress @	MS	F	p
Cognitive	4.266667	0.541937	0.464
Control	147.2667	18.70528	.00004
Emotional	40.01667	5.082772	0.027

3. While waiting for you to analyze her data, the researcher in problem 2 decides to design a follow-up study to determine whether the results would generalize. So, she replicates the study in problem 2 but this time each subject works under only one condition -- stress or no-stress. This time, she recruits 180 subjects and then randomly assigns them to work under stress or no stress. Then 1/3 of each group is assigned to the control, emotional, or cognitive training conditions. The subjects in each of these conditions are further divided into 6 person groups for training.

a. Write out the structural model for this design; use S=subjects, G=group, T=training, and C=condition.

S^R(G^R(C^F X T^F))

	Source	E(MS)	df	Error Line
1	C	ntgσ²_C + nσ²_G(CT) + σ²_S(G(CT))	c-1=1	4
2	T	ncgσ²_T + nσ²_G(CT) + σ²_S(G(CT))	t-1=2	4
3	CT	ngσ²_CT + nσ²_G(CT) + σ²_S(G(CT))	(c-1)(t-1)=2	4
4	G(CT)	nσ²_G(CT) + σ²_S(G(CT))	ct(g-1)=24	5
5	S(G(CT))	σ²_S(G(CT))	ctg(n-1)=150
			179