To obtain this output:

1. File, Open, point to gss 93 subset.sav.
2. Statistics, General Linear Model GLM - Multivariate.
3. In the MANOVA dialog box, select blues, blugras, and jazz as the "dependents", and select agecat4 and race as the "factors." Enter educ as the "Covariate".
4. Click on Options and check all the options, then Continue.
5. When the output comes up, go to the "Descriptives" table. Click on the table to select it. Right-click
6. Click on Plots and let agecat4 be the "Horizontal Axis", then click Add, Continue.

   Comments in blue are by the instructor and are not part of SPSS output.

   Notes

   Output Created		19 Mar 98 07:47:58
   Comments
   Input	Data	Y:\PC\spss95\GSS93 subset.sav
   	Filter	<none>
   	Weight	<none>
   	Split File	<none>
   	N of Rows in Working Data File	1500
   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 blues blugrass jazz BY agecat4 race WITH educ /METHOD = SSTYPE(3) /INTERCEPT = INCLUDE /PLOT = PROFILE( agecat4 ) /PRINT = DESCRIPTIVE ETASQ PARAMETER HOMOGENEITY RSSCP TEST(SSCP) /PLOT = SPREADLEVEL RESIDUALS /CRITERIA = ALPHA(.05) /DESIGN .
   Resources	Elapsed Time	0:00:05.71

   
   

   
   
   
   First come the usual descriptive statistics for each cell in the factor design table.

   Between-Subjects Factors

   		Value Label	N
   Age Categories	1.00	18-29	218
   	2.00	30-39	309
   	3.00	40-49	280
   	4.00	50+	483
   Racew of Respondent	1	white	1117
   	2	black	118
   	3	other	55

   
   

   Descriptive Statistics

   	Age Categories	Racew of Respondent	Mean	Std. Deviation	N
   Blues or R & B Music	18-29	white	2.57	1.12	180
   		black	1.71	.55	24
   		other	2.21	1.12	14
   		Total	2.45	1.10	218
   	30-39	white	2.38	.96	270
   		black	2.23	1.03	26
   		other	2.23	.73	13
   		Total	2.36	.96	309
   	40-49	white	2.46	.95	235
   		black	1.86	.99	29
   		other	2.56	1.09	16
   		Total	2.41	.98	280
   	50+	white	2.66	1.00	432
   		black	1.97	1.11	39
   		other	3.33	1.07	12
   		Total	2.63	1.03	483
   	Total	white	2.54	1.01	1117
   		black	1.95	.98	118
   		other	2.56	1.08	55
   		Total	2.49	1.02	1290
   Bluegrass Music	18-29	white	2.89	1.06	180
   		black	3.58	.97	24
   		other	3.36	1.08	14
   		Total	3.00	1.08	218
   	30-39	white	2.61	.88	270
   		black	3.38	1.10	26
   		other	3.54	1.05	13
   		Total	2.71	.95	309
   	40-49	white	2.54	1.00	235
   		black	3.17	.93	29
   		other	2.75	.77	16
   		Total	2.62	1.00	280
   	50+	white	2.44	1.00	432
   		black	3.08	.98	39
   		other	2.92	.79	12
   		Total	2.51	1.01	483
   	Total	white	2.58	1.00	1117
   		black	3.27	1.00	118
   		other	3.13	.96	55
   		Total	2.66	1.02	1290
   Jazz Music	18-29	white	2.58	1.14	180
   		black	2.04	1.08	24
   		other	2.64	1.15	14
   		Total	2.52	1.14	218
   	30-39	white	2.46	.97	270
   		black	2.00	1.17	26
   		other	2.08	.95	13
   		Total	2.40	1.00	309
   	40-49	white	2.60	1.02	235
   		black	1.76	.99	29
   		other	2.25	1.29	16
   		Total	2.49	1.06	280
   	50+	white	2.91	1.10	432
   		black	2.13	1.08	39
   		other	3.00	1.13	12
   		Total	2.84	1.12	483
   	Total	white	2.68	1.08	1117
   		black	1.99	1.07	118
   		other	2.47	1.17	55
   		Total	2.61	1.10	1290

   
   
   
   
   MANOVA and MANCOVA assume that for each group (each cell in the factor design matrix) the covariance matrix is similar. Box's M tests this assumption. We want M not to be significant in order to conclude there is insufficient evidence that the covariance matrices differ. Here M is significant, so we have violated an assumption. That is, the various music groups differ in their covariance matrices. Note, however, that the F test is quite robust even when there are departures from this assumption.

   Box's Test of Equality of Covariance Matrices(a)

   Box's M	102.028
   F	1.469
   df1	66
   df2	27295
   Sig.	.008
   Tests the null hypothesis that the observed covariance matrices of the dependent variables are equal across groups.
   a Design: Intercept+EDUC+AGECAT4+RACE+AGECAT4 * RACE

   
   

   
   

   In a repeated measures design (not the present example), the univariate ANOVA tables will not be interpreted properly unless the variance/covariance matrix of the dependent variables is circular in form (see Huynh and Mandeville, 1979). When there is a violation of this assumption, a common option then is to focus on the multivariate (simultaneous) approach to gauging effects. We want the test not to be significant in order to conclude there is insufficient evidence that conclude this assumption is violated. Here the assumption is clearly violated.

   
   

   Bartlett's Test of Sphericity(a)

   Likelihood Ratio	.000
   Approx. Chi-Square	502.060
   df	5
   Sig.	.000
   Tests the null hypothesis that the residual covariance matrix is proportional to an identity matrix.
   a Design: Intercept+EDUC+AGECAT4+RACE+AGECAT4 * RACE

   
   

   
   The "multivariate tests" section simultaneously tests each factor effect on the dependent groups. This is the most important table in the SPSS output. Each factor (agecat4 and race in this example) and each covariate (educ in this example) has a main effect, as does the intercept. Interactions among the factors (here agecat4*race) are also assessed. SPSS offers four alternative multivariate significance tests. Hotelling's Trace is commonly used for two dependent groups, and Wilks' Lambda if there are more than two groups, as there are in this example. The significance of the F tests show if that effect is significant. Here all effects are significant except the interaction effect. Eta-squared is the proportion of the total variability in the dependent variable accounted for by the variation in the independent variable. Note that the covariate serves as a control. Thus, for the table below, race accounts for about 6% of the variability in the music variables, after controlling for education. Significance, of course, is the chance of making a Type I error (thinking you have something when you don't), whereas power (the last column, below) is the chance of making a Type II error (thinking you don't have something when you do). One wants the power level to be high (ex., above .90).

   
   

   Multivariate Tests(d)

   Effect		Value	F	Hypothesis df	Error df	Sig.	Eta Squared	Noncent. Parameter	Observed Power(a)
   Intercept	Pillai's Trace	.424	313.472(b)	3.000	1275.000	.000	.424	940.417	1.000
   	Wilks' Lambda	.576	313.472(b)	3.000	1275.000	.000	.424	940.417	1.000
   	Hotelling's Trace	.738	313.472(b)	3.000	1275.000	.000	.424	940.417	1.000
   	Roy's Largest Root	.738	313.472(b)	3.000	1275.000	.000	.424	940.417	1.000
   EDUC	Pillai's Trace	.074	33.810(b)	3.000	1275.000	.000	.074	101.429	1.000
   	Wilks' Lambda	.926	33.810(b)	3.000	1275.000	.000	.074	101.429	1.000
   	Hotelling's Trace	.080	33.810(b)	3.000	1275.000	.000	.074	101.429	1.000
   	Roy's Largest Root	.080	33.810(b)	3.000	1275.000	.000	.074	101.429	1.000
   AGECAT4	Pillai's Trace	.023	3.225	9.000	3831.000	.001	.008	29.025	.983
   	Wilks' Lambda	.978	3.235	9.000	3103.166	.001	.008	23.602	.949
   	Hotelling's Trace	.023	3.242	9.000	3821.000	.001	.008	29.174	.984
   	Roy's Largest Root	.019	7.883(c)	3.000	1277.000	.000	.018	23.648	.990
   RACE	Pillai's Trace	.117	26.539	6.000	2552.000	.000	.059	159.237	1.000
   	Wilks' Lambda	.883	27.260(b)	6.000	2550.000	.000	.060	163.563	1.000
   	Hotelling's Trace	.132	27.982	6.000	2548.000	.000	.062	167.889	1.000
   	Roy's Largest Root	.127	53.981(c)	3.000	1276.000	.000	.113	161.943	1.000
   AGECAT4 * RACE	Pillai's Trace	.022	1.548	18.000	3831.000	.065	.007	27.858	.933
   	Wilks' Lambda	.978	1.549	18.000	3606.730	.065	.007	26.279	.914
   	Hotelling's Trace	.022	1.550	18.000	3821.000	.064	.007	27.896	.933
   	Roy's Largest Root	.015	3.184(c)	6.000	1277.000	.004	.015	19.105	.927
   a Computed using alpha = .05
   b Exact statistic
   c The statistic is an upper bound on F that yields a lower bound on the significance level.
   d Design: Intercept+EDUC+AGECAT4+RACE+AGECAT4 * RACE

   
   

   
   MANOVA and MANCOVA assume that each dependent variable will have similar variances for all groups (all cells in the factor design matrix), Levene's test tests this assumption. If the Levene statistic is significant at the .05 level or better, the researcher rejects the null hypothesis that the groups have equal variances. The Levene test is robust in the face of departures from normality. Note, however, that failure to meet the assumption of homogeneity of variances is not fatal to ANOVA, which is relatively robust, particularly when groups are of equal sample size. For the example below, the homogeneity of variances assumption is met for bluegrass and jazz but not for blues.

   
   

   Levene's Test of Equality of Error Variances(a)

   	F	df1	df2	Sig.
   Blues or R & B Music	2.263	11	1278	.010
   Bluegrass Music	1.591	11	1278	.095
   Jazz Music	.758	11	1278	.682
   Tests the null hypothesis that the error variance of the dependent variable is equal across groups.
   a Design: Intercept+EDUC+AGECAT4+RACE+AGECAT4 * RACE

   
   

   
   This section of output gives the univariate ANOVA effects for factor and interaction (and in MANCOVA each covariate). The significance of F and eta-squared have the same interpretation as in the multivariate analysis above. For instance, all univariate effects for race are significant, but for the agecat4 age categories factor, bluegrass is significant but not blues of jazz. The interaction of agecat4 and race is significant for blues. The "corrected model" effect reflects the variation in the dependent attributed to other effects (except the intercept) in the model, after corrected by the mean.

   
   

   Tests of Between-Subjects Effects

   Source	Dependent Variable	Type III Sum of Squares	df	Mean Square	F	Sig.	Eta Squared	Noncent. Parameter	Observed Power(a)
   Corrected Model	Blues or R & B Music	100.255(b)	12	8.355	8.604	.000	.075	103.250	1.000
   	Bluegrass Music	110.156(c)	12	9.180	9.545	.000	.082	114.537	1.000
   	Jazz Music	178.513(d)	12	14.876	13.836	.000	.115	166.031	1.000
   Intercept	Blues or R & B Music	510.050	1	510.050	525.284	.000	.291	525.284	1.000
   	Bluegrass Music	378.699	1	378.699	393.759	.000	.236	393.759	1.000
   	Jazz Music	653.179	1	653.179	607.505	.000	.322	607.505	1.000
   EDUC	Blues or R & B Music	33.602	1	33.602	34.605	.000	.026	34.605	1.000
   	Bluegrass Music	9.397	1	9.397	9.770	.002	.008	9.770	.878
   	Jazz Music	79.143	1	79.143	73.608	.000	.055	73.608	1.000
   AGECAT4	Blues or R & B Music	5.999	3	2.000	2.059	.104	.005	6.178	.530
   	Bluegrass Music	12.658	3	4.219	4.387	.004	.010	13.162	.874
   	Jazz Music	4.216	3	1.405	1.307	.271	.003	3.921	.351
   RACE	Blues or R & B Music	42.212	2	21.106	21.736	.000	.033	43.472	1.000
   	Bluegrass Music	62.846	2	31.423	32.673	.000	.049	65.345	1.000
   	Jazz Music	57.719	2	28.859	26.841	.000	.040	53.683	1.000
   AGECAT4 * RACE	Blues or R & B Music	14.805	6	2.468	2.541	.019	.012	15.248	.848
   	Bluegrass Music	3.959	6	.660	.686	.661	.003	4.117	.276
   	Jazz Music	5.551	6	.925	.860	.523	.004	5.163	.345
   Error	Blues or R & B Music	1239.965	1277	.971
   	Bluegrass Music	1228.158	1277	.962
   	Jazz Music	1373.009	1277	1.075
   Total	Blues or R & B Music	9308.000	1290
   	Bluegrass Music	10485.000	1290
   	Jazz Music	10324.000	1290
   Corrected Total	Blues or R & B Music	1340.220	1289
   	Bluegrass Music	1338.314	1289
   	Jazz Music	1551.522	1289
   a Computed using alpha = .05
   b R Squared = .075 (Adjusted R Squared = .066)
   c R Squared = .082 (Adjusted R Squared = .074)
   d R Squared = .115 (Adjusted R Squared = .107)

   
   

   Parameter Estimates

   When MANOVA or MANCOVA are computed through the GLM (general linear model) module, coefficients are computed as part of GLM's full parameterization approach. This additional optional output allows the researcher to assess the significance of each parameter coefficient. For instance, for jazz music, the parameter associated with race=1 is not significant but it is significant for race=2 (there is no coefficient for race=3 as one category is left out as the reference category, similar to dummy variables).

   		B	Std. Error	t	Sig.	95% Confidence Interval		Eta Squared	Noncent. Parameter	Observed Power(a)
   Dependent Variable	Parameter					Lower Bound	Upper Bound
   Blues or R & B Music	Intercept	3.949	.303	13.028	.000	3.354	4.543	.117	13.028	1.000
   	EDUC	-5.552E-02	.009	-5.883	.000	-7.403E-02	-3.700E-02	.026	5.883	1.000
   	[AGECAT4=1.00]	-1.013	.388	-2.609	.009	-1.774	-.251	.005	2.609	.741
   	[AGECAT4=2.00]	-.953	.395	-2.412	.016	-1.729	-.178	.005	2.412	.674
   	[AGECAT4=3.00]	-.581	.378	-1.539	.124	-1.322	.160	.002	1.539	.337
   	[AGECAT4=4.00]	0(b)	.	.	.	.	.	.	.	.
   	[RACE=1]	-.599	.289	-2.074	.038	-1.165	-3.231E-02	.003	2.074	.545
   	[RACE=2]	-1.412	.325	-4.339	.000	-2.050	-.774	.015	4.339	.991
   	[RACE=3]	0(b)	.	.	.	.	.	.	.	.
   	[AGECAT4=1.00] * [RACE=1]	.993	.397	2.498	.013	.213	1.773	.005	2.498	.704
   	[AGECAT4=1.00] * [RACE=2]	.874	.464	1.881	.060	-3.735E-02	1.785	.003	1.881	.468
   	[AGECAT4=1.00] * [RACE=3]	0(b)	.	.	.	.	.	.	.	.
   	[AGECAT4=2.00] * [RACE=1]	.756	.402	1.880	.060	-3.284E-02	1.544	.003	1.880	.468
   	[AGECAT4=2.00] * [RACE=2]	1.395	.467	2.988	.003	.479	2.311	.007	2.988	.848
   	[AGECAT4=2.00] * [RACE=3]	0(b)	.	.	.	.	.	.	.	.
   	[AGECAT4=3.00] * [RACE=1]	.457	.385	1.187	.235	-.298	1.213	.001	1.187	.220
   	[AGECAT4=3.00] * [RACE=2]	.628	.447	1.405	.160	-.249	1.506	.002	1.405	.289
   	[AGECAT4=3.00] * [RACE=3]	0(b)	.	.	.	.	.	.	.	.
   	[AGECAT4=4.00] * [RACE=1]	0(b)	.	.	.	.	.	.	.	.
   	[AGECAT4=4.00] * [RACE=2]	0(b)	.	.	.	.	.	.	.	.
   	[AGECAT4=4.00] * [RACE=3]	0(b)	.	.	.	.	.	.	.	.
   Bluegrass Music	Intercept	2.591	.302	8.591	.000	2.000	3.183	.055	8.591	1.000
   	EDUC	2.936E-02	.009	3.126	.002	1.093E-02	4.779E-02	.008	3.126	.878
   	[AGECAT4=1.00]	.384	.386	.995	.320	-.373	1.142	.001	.995	.169
   	[AGECAT4=2.00]	.543	.393	1.380	.168	-.229	1.315	.001	1.380	.281
   	[AGECAT4=3.00]	-.267	.376	-.710	.478	-1.004	.470	.000	.710	.109
   	[AGECAT4=4.00]	0(b)	.	.	.	.	.	.	.	.
   	[RACE=1]	-.512	.287	-1.782	.075	-1.075	5.176E-02	.002	1.782	.429
   	[RACE=2]	.188	.324	.581	.561	-.447	.824	.000	.581	.089
   	[RACE=3]	0(b)	.	.	.	.	.	.	.	.
   	[AGECAT4=1.00] * [RACE=1]	2.983E-02	.396	.075	.940	-.746	.806	.000	.075	.051
   	[AGECAT4=1.00] * [RACE=2]	5.502E-02	.462	.119	.905	-.852	.962	.000	.119	.052
   	[AGECAT4=1.00] * [RACE=3]	0(b)	.	.	.	.	.	.	.	.
   	[AGECAT4=2.00] * [RACE=1]	-.425	.400	-1.062	.289	-1.209	.360	.001	1.062	.186
   	[AGECAT4=2.00] * [RACE=2]	-.333	.465	-.717	.473	-1.244	.578	.000	.717	.111
   	[AGECAT4=2.00] * [RACE=3]	0(b)	.	.	.	.	.	.	.	.
   	[AGECAT4=3.00] * [RACE=1]	.325	.383	.847	.397	-.427	1.077	.001	.847	.135
   	[AGECAT4=3.00] * [RACE=2]	.278	.445	.625	.532	-.595	1.151	.000	.625	.096
   	[AGECAT4=3.00] * [RACE=3]	0(b)	.	.	.	.	.	.	.	.
   	[AGECAT4=4.00] * [RACE=1]	0(b)	.	.	.	.	.	.	.	.
   	[AGECAT4=4.00] * [RACE=2]	0(b)	.	.	.	.	.	.	.	.
   	[AGECAT4=4.00] * [RACE=3]	0(b)	.	.	.	.	.	.	.	.
   Jazz Music	Intercept	3.944	.319	12.368	.000	3.319	4.570	.107	12.368	1.000
   	EDUC	-8.520E-02	.010	-8.580	.000	-.105	-6.572E-02	.055	8.580	1.000
   	[AGECAT4=1.00]	-.194	.408	-.475	.635	-.995	.607	.000	.475	.076
   	[AGECAT4=2.00]	-.694	.416	-1.669	.095	-1.510	.122	.002	1.669	.385
   	[AGECAT4=3.00]	-.459	.397	-1.155	.248	-1.239	.321	.001	1.155	.211
   	[AGECAT4=4.00]	0(b)	.	.	.	.	.	.	.	.
   	[RACE=1]	1.318E-02	.304	.043	.965	-.583	.609	.000	.043	.050
   	[RACE=2]	-.953	.342	-2.784	.005	-1.625	-.281	.006	2.784	.794
   	[RACE=3]	0(b)	.	.	.	.	.	.	.	.
   	[AGECAT4=1.00] * [RACE=1]	-2.240E-02	.418	-.054	.957	-.843	.798	.000	.054	.050
   	[AGECAT4=1.00] * [RACE=2]	.302	.489	.619	.536	-.656	1.261	.000	.619	.095
   	[AGECAT4=1.00] * [RACE=3]	0(b)	.	.	.	.	.	.	.	.
   	[AGECAT4=2.00] * [RACE=1]	.381	.423	.901	.368	-.449	1.211	.001	.901	.147
   	[AGECAT4=2.00] * [RACE=2]	.850	.491	1.731	.084	-.114	1.814	.002	1.731	.409
   	[AGECAT4=2.00] * [RACE=3]	0(b)	.	.	.	.	.	.	.	.
   	[AGECAT4=3.00] * [RACE=1]	.271	.405	.670	.503	-.524	1.067	.000	.670	.103
   	[AGECAT4=3.00] * [RACE=2]	.334	.471	.710	.478	-.589	1.257	.000	.710	.109
   	[AGECAT4=3.00] * [RACE=3]	0(b)	.	.	.	.	.	.	.	.
   	[AGECAT4=4.00] * [RACE=1]	0(b)	.	.	.	.	.	.	.	.
   	[AGECAT4=4.00] * [RACE=2]	0(b)	.	.	.	.	.	.	.	.
   	[AGECAT4=4.00] * [RACE=3]	0(b)	.	.	.	.	.	.	.	.
   a Computed using alpha = .05
   b This parameter is set to zero because it is redundant.

   
   

   
   Below is the error sums-of-squares and crossproducts (SSCP) matrix for the effects noted in earlier tables. The ratio of effect sums of squares and error sums of squares is what was used in testing the significance of each effect. Thus this optional table shows the researcher more about the data on the basis of which significance was computed.

   
   

   Between-Subjects SSCP Matrix

   			Blues or R & B Music	Bluegrass Music	Jazz Music
   Hypothesis	Intercept	Blues or R & B Music	510.050	439.494	577.195
   		Bluegrass Music	439.494	378.699	497.351
   		Jazz Music	577.195	497.351	653.179
   	EDUC	Blues or R & B Music	33.602	-17.769	51.569
   		Bluegrass Music	-17.769	9.397	-27.271
   		Jazz Music	51.569	-27.271	79.143
   	AGECAT4	Blues or R & B Music	5.999	-6.196	2.125
   		Bluegrass Music	-6.196	12.658	-.156
   		Jazz Music	2.125	-.156	4.216
   	RACE	Blues or R & B Music	42.212	-43.851	48.232
   		Bluegrass Music	-43.851	62.846	-56.819
   		Jazz Music	48.232	-56.819	57.719
   	AGECAT4 * RACE	Blues or R & B Music	14.805	-.672	4.305
   		Bluegrass Music	-.672	3.959	-.034
   		Jazz Music	4.305	-.034	5.551
   Error		Blues or R & B Music	1239.965	337.327	675.337
   		Bluegrass Music	337.327	1228.158	198.951
   		Jazz Music	675.337	198.951	1373.009
   Based on Type III Sum of Squares

   
   

   Residual SSCP Matrix

   		Blues or R & B Music	Bluegrass Music	Jazz Music
   Sum-of-Squares and Cross-Products	Blues or R & B Music	1239.965	337.327	675.337
   	Bluegrass Music	337.327	1228.158	198.951
   	Jazz Music	675.337	198.951	1373.009
   Covariance	Blues or R & B Music	.971	.264	.529
   	Bluegrass Music	.264	.962	.156
   	Jazz Music	.529	.156	1.075
   Correlation	Blues or R & B Music	1.000	.273	.518
   	Bluegrass Music	.273	1.000	.153
   	Jazz Music	.518	.153	1.000
   Based on Type III Sum of Squares

   
   

   
   The spread-versus-level plots below first depict standard deviations vs. means, then variances vs. means, for each dependent variable. Each point shows the value of a factor design matrix group cell on the mean and on the standad deviation or variance. This is useful in testing the homogeneity of variances assumption, and in identifying cells which deviate substantially from the assumption.

   
   

   Spread-versus-Level Plots

   
   

   
   

   
   

   
   

   
   

   
   

   
   

   Variances versus Means

   
   

   
   

   
   

   
   

   
   

   
   

   
   

   
   

   
   

   
   For each dependent variable, a plot is produced which shows the 6 comparisons among observed, predicted, and standardized residuals. For observed by predicted, one would like to see a clear pattern, but for the plots involving standardized residuals, one would like not to see a pattern. Here the reverse is the case.

   
   

   Observed * Predicted * Std. Residual Plots

   
   

   
   

   
   

   
   

   
   

   
   

   
   

   
   

   
   
   Profile plots, below, are line plots of the predicted means of each dependent variable across levels of each factor. When two or three factors are involved, these are called interaction plots. Where a plot of observed means would show the effect being studied and the error, the profile plots of predicted means show the effect without the error. Each point in a profile plot indicates the estimated marginal mean of the dependent variable (adjusted for covariates in MANCOVA) at one level of a given factor. The profile plot shows if the estimated marginal means are increasing across levels. A second or third factor can be represented by a second or third line (not shown below), where parallel lines indicate no interaction and crossing lines indicate interaction among the factors.

   
   

   Profile Plots

   Blues or R & B Music

   
   

   
   

   
   

   Bluegrass Music

   
   

   
   

   
   

   Jazz Music