General Linear Model (GLM) Multivariate Analysis

AIMGLM multivariate analysis takes into account the interrelation among dependent variables and analyzes the variables simultaneously.

Checklist of RequirementsDepending on the research question and hypotheses to be tested, the experiment can include or exclude classification on independent variables.When independent variables are included, there can be two or more levels for each independent variable.There should be two or more dependent variables.

AssumptionsThe observations must be independent (i.e., responses among groups of respondents should not be correlated).The variancecovariance matrices must be equal for all treatment groups.The set of dependent variables must follow a multivariate normal distribution (i.e., any linear combination of the dependent variables must follow a normal distribution).

Example 1 GLM Multivariate Analysis: One-Sample Test

Suppose that a sports psychologist has recorded the following running times (in seconds) from five men in four different events: 50-yd dash, 100-yd dash, 200-yd dash, and 300-yd dash. The hypothesis to be tested is that the observed sample comes from a population with specified values for the means of the running events. For illustrative purposes, the standard values are taken to be 6 sec for the 50-yd dash, 12 sec for the 100-yd dash, 25 sec for the 200-yd dash, and 40 sec for the 300-yd dash. Because GLM automatically tests the hypothesis that a set of means is equal to zero, the normative values must be subtracted from the observed scores, and the hypothesis that the differences are zero is tested.

The running times for the four events are presented in the following:

PROCEDURE

SPSS Output

Results and Interpretation

The Descriptive Statistics table presents means and standard deviations for the four running events after their normative values have been subtracted. The sample exceeds (i.e., is poorer than) the norm for all four running events.The Multivariate Tests table tests the hypothesis that the four sample means do not differ from the specified set of constants. Of the three multivariate tests (Pillais, Hotellings, Wilks, and Roys), Pillais trace is the most powerful (i.e., the ability to detect differences if they exist) and the most robust (i.e., the significance level based on it is reasonably correct even when the assumptions are violated). Because the observed significance level is small (p < .05), the null hypothesis that the sample means do not differ from the specified constants is rejected, multivariate Pillai F(1,4) = 232.56, p< .05. When the null hypothesis of no difference is rejected, it is often informative to examine the univariate Tests of Between-Subjects Effects to identify which variables yielded significant differences. The univariate test results (Intercept) show that the means for all running events differ significantly from their specified standards (p< .01).

Example 2 GLM Multivariate Analysis: Two-Sample Test

Following from the previous example, suppose that the researcher also recorded running times for the four events for five women. The running times for the two samples for the four events are presented in the following. The hypothesis that men and women do not differ on the four running events will be tested.

PROCEDURECreate the four variables of FIFTY, ONEHUND, TWOHUND, and THREEHUN.From the menu bar, click Analyze, then General Linear Model, and then Multivariate. The following Multivariate window will open:

SPSS Output

Results and Interpretation

Results and InterpretationThe Estimated Marginal Means table presents the mean running times for men and women, for each of the four running events after the normative values have been subtracted. Both samples exceed the norms for all four running events. The Multivariate Tests table (Pillais, Hotellings, Wilks, and Roys) tests the hypothesis that men and women do not differ significantly on overall running time. The significance level is based on the F distribution with 4 and 5 degrees of freedom. The observed significance levels for all four multivariate tests are small (p < .001), so the null hypothesis that men and women performed similarly on the running events is rejected (e.g., multivariate Pillai F(4,5) = 41.74, p < .001).

Results and InterpretationThe Tests of Between-Subjects Effects table presents the univariate test of sex difference for the individual running events. The obtained F-values are equivalent to those obtained from the one-way analysis of variance. In the case where the comparison is between two groups, the F-values are the square of the two-sample t-values. The results indicate significant sex differences in running times for three of the four running events (FIFTY: male: M = 2.60, female: M = 7.80; ONEHUND: male: M = 7.20, female: M = 17.20; THREEHUN: male: M = 23.60, female: M = 33.60); there is no significant sex difference for the 200-yd dash.

Example 3 GLM: 2 2 Factorial Design

In Example 2, the researcher used the GLM two-sample test to test for sex differences in running times across four running events. Suppose that in addition to sex differences, the researcher was also interested in whether the subjects ethnicity (white vs. nonwhite) would make a difference to their running times. In particular, the researcher was interested in the interaction between subjects sex and ethnicity in influencing their running times. The running times for the four groups (men-white, men-nonwhite, womenwhite, women-nonwhite) for the four events are presented in the following.

PROCEDURE

PROCEDURECreate the four variables of FIFTY, ONEHUND, TWOHUND, and THREEHUN.From the menu bar, click Analyze, then General Linear Model, and then Multivariate. The following Multivariate window will open:

SPSS Output

Results and Interpretation

Results and InterpretationThe Estimated Marginal Means table presents the means of the overall running times for the four running events as a function of subjects SEX, ETHNICITY, and SEX*ETHNICITY interaction.The Multivariate Tests table presents the multivariate tests of significance for the main effects of the between-groups variables of SEX and ETHNICITY, and the SEX*ETHNICITY interaction. For all three effects, the observed significance levels for the four multivariate tests (Pillais, Wilks, Hotellings, and Roys) are small. Thus, their associated null hypotheses (no sex difference, no ethnicity difference, and no sex*ethnicity interaction) are rejected. The results from the multivariate tests have been converted to approximate F-values and can be interpreted in the same way that F-values from oneway ANOVA are interpreted. The multivariate tests for both SEX (multivariate Pillai F(4,13) = 392.43, p < .001) and ETHNICITY (multivariate Pillai F(4,13) = 6.25, p < .01) are statistically significant, indicating that men (M = 11.1), when compared to women (M = 22.32), and whites (M = 13.18), when compared to nonwhites (M = 17.2), differed significantly in their overall running times.

Results and InterpretationThe Tests of Between-Subjects Effects can be examined for significant SEX and ETHNICITY differences for each of the four running events. For SEX, the results show significant sex differences for all four running events (p < .01) (FIFTY: men: M = 1.10, women: M = 10.30; ONEHUND: men: M = 5.70, women: M = 19.60; TWOHUND: men: M = 15.50, women: M = 23.30; THREEHUN: men: M = 22.10, women: M = 36.10). For ETHNICITY, the results show significant ethnicity difference for only the FIFTY-yd dash (white: M = 5.20, nonwhite: M = 6.20; F(1,16) = 10.00, p < .01).

Results and InterpretationGiven that the SEX*ETHNICITY interaction is significant, the Tests of Between-Subjects Effects table can be examined to see which of the four running events are significantly affected by this interaction. The univariate F-tests for the SEX*ETHNICITY interaction are identical to the F ratios generated from the SEX*ETHNICITY interaction in a two-way ANOVA. The univariate results show significant SEX*ETHNICITY interaction effect for three of the four running events (p < .01). Thus, the subjects performances on the 50-yd dash, the 100-yd dash, and the 300-yd dash are dependent on the joint effects of their sex and ethnicity.

Results and InterpretationTo interpret the SEX*ETHNICITY interactions, it would be useful to graph the means of the four running events (FIFTY, ONEHUND, TWOHUND, and THREEHUN) from the estimated marginal means table.

Results and Interpretation

Results and InterpretationUsing the 100-yd dash as an example, the SEX*ETHNICITY interaction can be interpreted as follows. The effect of subjects sex on the running times in the 100-yd dash is dependent on the ethnicity of the subjects, such that for men, nonwhites ran faster than whites; for women, the effect is opposite with whites running faster than nonwhites.

Post hoc comparisons analysis can also be used to identify which of the four groups (male-white, male-nonwhite, female-white, female-nonwhite) generated from the SEX*ETHNICITY interaction is significantly different from each other in the 100-yd dash. The following procedures will accomplish this analysis.

Data Transformation

Data Transformation

Post Hoc Comparisons

Post Hoc ComparisonsOnce the four levels (male-white, male-nonwhite, female-white, femalenonwhite) have been created, Scheff post hoc comparisons can be conducted to test for differences (simple effects) between these four levels.

SPSS Output

Results and InterpretationResults from the Scheff comparisons indicate that white females (M = 17.20) and nonwhite females (M = 22.00) ran significantly slower than white males (M = 7.20) and nonwhite males (M = 4.20). Whereas there was no significant difference in running times between white males and nonwhite males, the results show that nonwhite females ran significantly slower than white females.