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Factorial (2 × 3) MANOVA

This study investigates whether there are differences in the outcomes of three different treatments for anxiety. The treatment conditions that are compared are treatment with medication, treatment with psychotherapy, and placebo (inactive pills). In addition, we want to see if gender of the client moderates the effects of the treatment. Every participant had been diagnosed with similar types and severity of anxiety disorders when they entered the study. Each participant was randomly assigned only to one of the three treatment conditions. After 12-weeks of treatment (or placebo), all participants completed two questionnaires to measure worry and general emotion. The higher the scores on these measures, the higher the anxiety level of the participant.

Directions: The factorial MANOVA will combine what you have learned previously about 1. Post hoc tests when you have more than two groups on an IV (one-way ANOVA), 2. Main effects and interactions (factorial ANOVA), and 3. Working with multivariate analyses of multiple DVs (one-way MANOVA).

Using the SPSS data file for Module 6 (located in Topic Materials), answer the following questions:

1. What are the independent variables in this study? What are the dependent variables?

Independent variables: TREATMENT, GENDER

Dependent variables: EMOTION, WORRY

2. Why is a factorial MANOVA appropriate to use for this research design?

There are two reasons why MANOVA is better for this research design:

(1) MANOVA is a more powerful statistical technique because it is better able to detect differences when such difference do exist, when compared to a series of ANOVAS.

(2) MANOVA provides a way to control inflated type I error.

3. Did you find any errors that the researcher made when setting up the SPSS data file (don't forget to check the variable view)? If so, what did you find? How did you correct it? HINT: Yes, there are coding errors for Measures.

Treatment and Gender variables should be nominal, not ordinal.

The Emotion and Worry variables are not labeled.

4. Perform Initial Data Screening. What did you find regarding missing values, univariate outliers, multivariate outliers, normality? HINT: Revisit instructions from last module's readings on how to compute Mahalanobis distance and then analyze for multivariate outliers.

MISING VALUES: There are no missing values.

Univariate Statistics
	N	Mean	Std. Deviation	Missing	No. of Extremesa
				Count	Percent	Low	High
Treatment	100	2.00	.829	0	.0	0	0
Gender	100	.50	.503	0	.0	0	0
Emotion	100	27.64	13.977	0	.0	0	0
Worry	100	34.88	8.985	0	.0	1	0
a. Number of cases outside the range (Q1 - 1.5IQR, Q3 + 1.5IQR).

OUTLIERS . From the box plot, there is one outlier for WORRY. Observation No. 96, value = 8.

Residuals Statisticsa
	Minimum	Maximum	Mean	Std. Deviation	N
Predicted Value	-.44	1.19	.50	.432	100
Std. Predicted Value	-2.168	1.591	.000	1.000	100
Standard Error of Predicted Value	.028	.092	.051	.011	100
Adjusted Predicted Value	-.46	1.20	.50	.433	100
Residual	-.673	.504	.000	.257	100
Std. Residual	-2.582	1.934	.000	.985	100
Stud. Residual	-2.619	1.975	-.001	1.005	100
Deleted Residual	-.692	.526	.000	.268	100
Stud. Deleted Residual	-2.703	2.006	-.002	1.016	100
Mahal. Distance	.116	11.387	2.970	1.859	100
Cook's Distance	.000	.132	.011	.018	100
Centered Leverage Value	.001	.115	.030	.019	100
a. Dependent Variable: Gender

Outlier Statisticsa
	Case Number	Statistic
Mahal. Distance	1	96	11.387
	2	39	8.173
	3	71	7.761
	4	7	7.586
	5	57	7.431
	6	3	6.373
	7	1	6.049
	8	22	5.564
	9	95	5.400
	10	68	5.382
a. Dependent Variable: Gender

outlier

Tests of Normality
	Kolmogorov-Smirnova	Shapiro-Wilk
	Statistic	df	Sig.	Statistic	df	Sig.
Emotion	.095	100	.028	.969	100	.017
Worry	.085	100	.069	.986	100	.366
a. Lilliefors Significance Correction

EMOTION is not normal, p = 0.028 < 0.05

WORRY is normal, p = 0.069 > 0.05

5. Perform a factorial MANOVA on the data. Before interpreting the multivariate results of the MANOVA, check outcomes that test other assumptions for this statistic: equality of covariance matrices (see Box's Test) and sufficient correlation among the DVs (see Bartlett's Test of Sphericity). Also check the results of the Levene's Test of Equality of Error Variances to evaluate that assumption for the univariate ANOVAs that are run and show in the Tests of Between-Subjects Effects output. What have you found about whether the data meet these additional assumptions for the MANOVA and follow-up ANOVAs? Explain. HINTS: • Once in the Options box, remember to check box for "Residual SSCP matrix" to get results for the Bartlett's test. • Also, remember to ask for post hoc tests for Treatment because there are more than two conditions. Profile plots also help with visualizing interactions.

Equality of covariance matrices (Box's Test)

Box's Test of Equality of Covariance Matricesa
Box's M	12.642
F	.983
df1	12
df2	7143.492
Sig.	.463
Tests the null hypothesis that the observed covariance matrices of the dependent variables are equal across groups.
a. Design: Intercept + Treatment + Gender + Treatment * Gender

Sufficient correlation among the DVs (Bartlett's Test of Sphericity)

Bartlett's Test of Sphericitya
Likelihood Ratio	.000
Approx. Chi-Square	51.982
df	2
Sig.	.000
Tests the null hypothesis that the residual covariance matrix is proportional to an identity matrix.
Levene's Test of Equality of Error Variances
Levene's Test of Equality of Error Variancesa
	F	df1	df2	Sig.
Emotion	1.733	5	94	.135
Worry	1.383	5	94	.238
Tests the null hypothesis that the error variance of the dependent variable is equal across groups.
a. Design: Intercept + Treatment + Gender + Treatment * Gender

Based on the foregoing output, the data meets these additional assumptions for the MANOVA and follow-up ANOVAs. Significance levels are > 0.05. for equality of variances, and correlation among DV.

6. What are the outcomes of the multivariate tests (main effects and interaction)? Report either the Pillai's Trace or Wilks's Lambda for each result, as well as the associated F-value and its statistical significance. Use the following format for notation to report each result: Pillai's Trace OR Wilks' lambda = ____; F(df, df) = ____, p = ____. HINTS: • Use Pillai's trace if there are problems with heterogeneity of variance-covariance matrices for the DVs. Otherwise, Wilks' lambda is fine. • Eta squared cannot be calculated from the information provided in the multivariate tests results.

The outcomes are:

Pillai’s trace for Treatment:

Pillai's Trace OR Wilks' lambda = 0.39 ; F(df, df) = 9.263, p = 0.000

Pillai's Trace

.329

9.263

4.000

188.000

.000

Pillai’s Trace for Gender:

Pillai's Trace OR Wilks' lambda = 0.523 ; F(df, df) = 50.919, p = 0.000

Pillai's Trace

.523

50.919b

2.000

93.000

.000

7. Given the results of the multivariate tests, would you now move on to interpret the results of the Tests of Between-Subjects Tests? If yes, what are the results and what do they mean? (Report each of the results using the format of F(df, df) = _____, p = _____ , 2 = _____.)

The results of Test of Between Subjects Test are:

Tests of Between-Subjects Effects
Source	Dependent Variable	Type III Sum of Squares	df	Mean Square	F	Sig.	Partial Eta Squared
Corrected Model	Emotion	12652.320a	5	2530.464	35.562	.000	.654
	Worry	3911.493b	5	782.299	18.020	.000	.489
Intercept	Emotion	34815.635	1	34815.635	489.285	.000	.839
	Worry	51685.732	1	51685.732	1190.545	.000	.927
Treatment	Emotion	1786.729	2	893.364	12.555	.000	.211
	Worry	1951.834	2	975.917	22.480	.000	.324
Gender	Emotion	5538.603	1	5538.603	77.837	.000	.453
	Worry	114.424	1	114.424	2.636	.108	.027
Treatment * Gender	Emotion	806.236	2	403.118	5.665	.005	.108
	Worry	208.084	2	104.042	2.397	.097	.049
Error	Emotion	6688.673	94	71.156
	Worry	4080.870	94	43.414
Total	Emotion	95711.199	100
	Worry	129621.437	100
Corrected Total	Emotion	19340.992	99
	Worry	7992.364	99
a. R Squared = .654 (Adjusted R Squared = .636)
b. R Squared = .489 (Adjusted R Squared = .462)

The results are: Gender has no significant effect. Treatment has significant effect.

8. Because one IV has more than two conditions, you would need to do post hoc tests if the overall F-value was statistically significant. If so, what results did you find? Post Hoc Tests

Treatment Multiple Comparisons: Meds v. Placebo & Psychotherapy v. Meds are significant.
Tukey HSD
Dependent Variable	(I) Treatment	(J) Treatment	Mean Difference (I-J)	Std. Error	Sig.	95% Confidence Interval
						Lower Bound	Upper Bound
Emotion	Meds	Psychotherapy	-12.71*	2.078	.000	-17.66	-7.76
		Placebo	2.68	2.046	.393	-2.19	7.55
	Psychotherapy	Meds	12.71*	2.078	.000	7.76	17.66
		Placebo	15.39*	2.078	.000	10.44	20.34
	Placebo	Meds	-2.68	2.046	.393	-7.55	2.19
		Psychotherapy	-15.39*	2.078	.000	-20.34	-10.44
Worry	Meds	Psychotherapy	-12.62*	1.623	.000	-16.48	-8.75
		Placebo	-12.28*	1.598	.000	-16.08	-8.47
	Psychotherapy	Meds	12.62*	1.623	.000	8.75	16.48
		Placebo	.34	1.623	.976	-3.52	4.21
	Placebo	Meds	12.28*	1.598	.000	8.47	16.08
		Psychotherapy	-.34	1.623	.976	-4.21	3.52
Based on observed means. The error term is Mean Square(Error) = 43.414.
*. The mean difference is significant at the .05 level.

9. If you had a significant interaction effect, what follow-up tests do you need to perform to understand how gender moderates the effects of treatment? What are the results?

We would follow up with post hoc tests for Gender.

The result is: Gender has insignificant role in the prognosis of the patients.

10. Citing the results of your statistical analyses, what is the conclusion you can draw (and support) regarding research question that was posed in this research (see problem statement)?

Conclusion is that Meds have a significant role in reducing the anxiety of the patients. The next best treatment is Psychotherapy.

Two-Way Factorial MANOVA Using SPSS: Output

General Linear Model

Notes
Output Created	01-JUL-2014 08:11:10
Comments
Input	Data
	Active Dataset	DataSet1
	Filter	<none>
	Weight	<none>
	Split File	<none>
	N of Rows in Working Data File	100
Missing Value Handling	Definition of Missing	User-defined missing values are treated as missing.
	Cases Used	Statistics are based on all cases with valid data for all variables in the model.
Syntax	GLM Emotion Worry BY Treatment Gender /METHOD=SSTYPE(3) /INTERCEPT=INCLUDE /POSTHOC=Treatment(TUKEY) /PRINT=DESCRIPTIVE ETASQ RSSCP HOMOGENEITY /PLOT=RESIDUALS /CRITERIA=ALPHA(.05) /DESIGN= Treatment Gender Treatment*Gender.
Resources	Processor Time	00:00:01.73
	Elapsed Time	00:00:01.13

Between-Subjects Factors
	Value Label	N
Treatment	1	Meds	34
	2	Psychotherapy	32
	3	Placebo	34
Gender	0	Females	50
	1	Males	50

Descriptive Statistics
	Treatment	Gender	Mean	Std. Deviation	N
Emotion	Meds	Females	14.03	8.499	8
		Males	27.70	9.863	26
		Total	24.48	11.120	34
	Psychotherapy	Females	17.07	6.709	10
		Males	46.33	8.730	22
		Total	37.19	15.958	32
	Placebo	Females	20.31	7.498	32
		Males	45.59	1.653	2
		Total	21.80	9.453	34
	Total	Females	18.66	7.743	50
		Males	36.61	13.048	50
		Total	27.64	13.977	100
Worry	Meds	Females	26.61	8.305	8
		Males	26.68	7.379	26
		Total	26.66	7.475	34
	Psychotherapy	Females	33.85	6.330	10
		Males	41.75	7.260	22
		Total	39.28	7.823	32
	Placebo	Females	38.83	4.973	32
		Males	40.66	1.691	2
		Total	38.94	4.849	34
	Total	Females	35.88	7.309	50
		Males	33.87	10.374	50
		Total	34.88	8.985	100

Box's Test of Equality of Covariance Matricesa
Box's M	12.642
F	.983
df1	12
df2	7143.492
Sig.	.463
Tests the null hypothesis that the observed covariance matrices of the dependent variables are equal across groups.
a. Design: Intercept + Treatment + Gender + Treatment * Gender

Bartlett's Test of Sphericitya
Likelihood Ratio	.000
Approx. Chi-Square	51.982
df	2
Sig.	.000
Tests the null hypothesis that the residual covariance matrix is proportional to an identity matrix.
a. Design: Intercept + Treatment + Gender + Treatment * Gender

Multivariate Testsa
Effect	Value	F	Hypothesis df	Error df	Sig.	Partial Eta Squared
Intercept	Pillai's Trace	.927	589.144b	2.000	93.000	.000	.927
	Wilks' Lambda	.073	589.144b	2.000	93.000	.000	.927
	Hotelling's Trace	12.670	589.144b	2.000	93.000	.000	.927
	Roy's Largest Root	12.670	589.144b	2.000	93.000	.000	.927
Treatment	Pillai's Trace	.329	9.263	4.000	188.000	.000	.165
	Wilks' Lambda	.671	10.270b	4.000	186.000	.000	.181
	Hotelling's Trace	.490	11.276	4.000	184.000	.000	.197
	Roy's Largest Root	.490	23.016c	2.000	94.000	.000	.329
Gender	Pillai's Trace	.523	50.919b	2.000	93.000	.000	.523
	Wilks' Lambda	.477	50.919b	2.000	93.000	.000	.523
	Hotelling's Trace	1.095	50.919b	2.000	93.000	.000	.523
	Roy's Largest Root	1.095	50.919b	2.000	93.000	.000	.523
Treatment * Gender	Pillai's Trace	.118	2.953	4.000	188.000	.021	.059
	Wilks' Lambda	.883	2.988b	4.000	186.000	.020	.060
	Hotelling's Trace	.131	3.021	4.000	184.000	.019	.062
	Roy's Largest Root	.121	5.669c	2.000	94.000	.005	.108
a. Design: Intercept + Treatment + Gender + Treatment * Gender
b. Exact statistic
c. The statistic is an upper bound on F that yields a lower bound on the significance level.

Levene's Test of Equality of Error Variancesa
	F	df1	df2	Sig.
Emotion	1.733	5	94	.135
Worry	1.383	5	94	.238
Tests the null hypothesis that the error variance of the dependent variable is equal across groups.
a. Design: Intercept + Treatment + Gender + Treatment * Gender

Tests of Between-Subjects Effects
Source	Dependent Variable	Type III Sum of Squares	df	Mean Square	F	Sig.	Partial Eta Squared
Corrected Model	Emotion	12652.320a	5	2530.464	35.562	.000	.654
	Worry	3911.493b	5	782.299	18.020	.000	.489
Intercept	Emotion	34815.635	1	34815.635	489.285	.000	.839
	Worry	51685.732	1	51685.732	1190.545	.000	.927
Treatment	Emotion	1786.729	2	893.364	12.555	.000	.211
	Worry	1951.834	2	975.917	22.480	.000	.324
Gender	Emotion	5538.603	1	5538.603	77.837	.000	.453
	Worry	114.424	1	114.424	2.636	.108	.027
Treatment * Gender	Emotion	806.236	2	403.118	5.665	.005	.108
	Worry	208.084	2	104.042	2.397	.097	.049
Error	Emotion	6688.673	94	71.156
	Worry	4080.870	94	43.414
Total	Emotion	95711.199	100
	Worry	129621.437	100
Corrected Total	Emotion	19340.992	99
	Worry	7992.364	99
a. R Squared = .654 (Adjusted R Squared = .636)
b. R Squared = .489 (Adjusted R Squared = .462)

Residual SSCP Matrix
	Emotion	Worry
Sum-of-Squares and Cross-Products	Emotion	6688.673	3273.441
	Worry	3273.441	4080.870
Covariance	Emotion	71.156	34.824
	Worry	34.824	43.414
Correlation	Emotion	1.000	.627
	Worry	.627	1.000
Based on Type III Sum of Squares

Post Hoc Tests

Treatment

Multiple Comparisons
Tukey HSD
Dependent Variable	(I) Treatment	(J) Treatment	Mean Difference (I-J)	Std. Error	Sig.	95% Confidence Interval
						Lower Bound	Upper Bound
Emotion	Meds	Psychotherapy	-12.71*	2.078	.000	-17.66	-7.76
		Placebo	2.68	2.046	.393	-2.19	7.55
	Psychotherapy	Meds	12.71*	2.078	.000	7.76	17.66
		Placebo	15.39*	2.078	.000	10.44	20.34
	Placebo	Meds	-2.68	2.046	.393	-7.55	2.19
		Psychotherapy	-15.39*	2.078	.000	-20.34	-10.44
Worry	Meds	Psychotherapy	-12.62*	1.623	.000	-16.48	-8.75
		Placebo	-12.28*	1.598	.000	-16.08	-8.47
	Psychotherapy	Meds	12.62*	1.623	.000	8.75	16.48
		Placebo	.34	1.623	.976	-3.52	4.21
	Placebo	Meds	12.28*	1.598	.000	8.47	16.08
		Psychotherapy	-.34	1.623	.976	-4.21	3.52
Based on observed means. The error term is Mean Square(Error) = 43.414.
*. The mean difference is significant at the .05 level.

Homogeneous Subsets

Emotion
Tukey HSDa,b,c
Treatment	N	Subset
		1	2
Placebo	34	21.80
Meds	34	24.48
Psychotherapy	32		37.19
Sig.		.400	1.000
Means for groups in homogeneous subsets are displayed. Based on observed means. The error term is Mean Square(Error) = 71.156.
a. Uses Harmonic Mean Sample Size = 33.306.
b. The group sizes are unequal. The harmonic mean of the group sizes is used. Type I error levels are not guaranteed.
c. Alpha = .05.

Worry
Tukey HSDa,b,c
Treatment	N	Subset
		1	2
Meds	34	26.66
Placebo	34		38.94
Psychotherapy	32		39.28
Sig.		1.000	.976
Means for groups in homogeneous subsets are displayed. Based on observed means. The error term is Mean Square(Error) = 43.414.
a. Uses Harmonic Mean Sample Size = 33.306.
b. The group sizes are unequal. The harmonic mean of the group sizes is used. Type I error levels are not guaranteed.
c. Alpha = .05.

Observed * Predicted * Std. Residual Plots