MANOVA is an extension of the univariate ANOVA as it involves more than one Dependent Variable (DV). The following are assumptions for using MANOVA:

Transcription

1 MULTIVARIATE ANALYSIS OF VARIANCE MANOVA is an extension of the univariate ANOVA as it involves more than one Dependent Variable (DV). The following are assumptions for using MANOVA: 1. Cell sizes : o It is necessary to have more subjects in each cell than the number of dependent variable. Some have suggested that it should be > 30 per cell. o Equal cell size is ideas but not essential. 2. Multicollinearity: o The correlations between the dependent variables should not be high. EXAMPLE: A researcher conducted a study among 364 students who completed a personality test comprising: o Self-esteem o Optimism o Hope Hypothesis: Is there a significant difference between and fe students on the three dependent variables (self-esteem, optimism and hope)? 1) Descriptive statistics is shown in the table below: Between-Subjects Factors Value Label N fe

2 Descriptive Statistics fe fe fe Mean Std. Deviation N ) The above table shows the number of subjects in each cell, the mean and standard deviation obtained for the 3 dependent variables (hope, self-esteem and optimism). o In your study, you must report the means, standard deviations and n size. Box's Test of Equality of Covariance Matrices a Box's M F df1 df2 Sig Tests the null hypothesis that the observed c ovariance matric es of the dependent variables are equal acros s groups. a. Design: Intercept+ 3) The Box Test above tests the homogeneity of variancecovariance matrices. o You set the alpha level at which is recommended. In other words, any time you get alpha level > 0.001, you can assume homogeneity of variance. o From the above table, the alpha is which is >0.001 and so you can assume homogeneity of variance.

3 Levene's Test of Equality of Error Variances a F df1 df2 Sig Tests the null hypothes is that the error variance of the dependent variable is equal across groups. a. Design: Intercept+ 4) To further confirm that you have not violated the homogeneity of variance assumption, the SPSS output gives you the Levene s Test as shown above. o The Levene s Test for homogeneity gives you the analysis for each of the dependent measure. o All the three dependent variables do not violate the assumption at an alpha level of o However, at the 0.05 level, optimism does violate the assumption. Multivariate Tests b Effect Intercept Pillai's Trace Wilks' Lambda Hotelling's Trace Roy's Largest Root Pillai's Trace a. Exact statistic Wilks' Lambda Hotelling's Trace Roy's Largest Root b. Design: Intercept+ Value F Hypothesis df Error df Sig a a a a a a a a ) The table above gives you the multivariate tests of significance. Remember your hypothesis! o You notice there a number of multivariate statistics given (Pillai, Wilks, Hotelling s and Roy s Largest Root). o Pillai s Trace is considered to be most acceptable because it is the most robust statistic against violations

4 of assumptions. i.e. you select this statistic because it can withstand the violation of assumptions which often happens in educational research. o Conclusion: There is a significant difference because the Pillai s Trace is which is < p In other words, there is a difference between the 3 dependent variables (self-esteem, optimism and hope); but which one? o Since there is a significant multivariate effect then, you have to examine the univariate effects (i.e ANOVA for according Gender (fortunately SPSS does it for you!). Source Corrected Model Intercept Error Corrected Dependent Variable a. R Squared =.012 (Adjusted R Squared =.010) b. R Squared =.009 (Adjusted R Squared =.007) c. R Squared =.000 (Adjusted R Squared = -.002) Tests of Between-Subjects Effects Type III Sum of Squares df Mean Square F Sig a b c ) What do you notice about the above table? Look at the Corrected Model which should read. o Hope is the only variable that differs significantly across gender when the alpha is set at 0.05.

5 Dependent Variable fe fe fe 95% Confidence Interval Mean Std. Error Lower Bound Upper Bound ) An examination of the above table reveals that Fes (29.56) have significantly higher scores than those of s (28.64); and this different is significant. Note: o In this example, your IV is gender which has only 2 factors. o In cases where your IV is more than 2 such as high, middle and low, than you have to do POS HOC analysis (Tukey Test) to determine which of the 3 are significantly different. o It is important to remember that MANOVA is straightforward when only one IV is involved. It becomes more intricate when there are more than one IV.