M A N O V A Multivariate ANOVA V. Čekanavičius, G. Murauskas 1 Data k groups; Each respondent has m measurements; Observations are from the multivariate normal distribution. No outliers. Covariance matrices among groups are equal. V. Čekanavičius, G. Murauskas 2 1
Data X 11 X 12 Y 11 Y 12 a 1 X 13 Y 13 Z 13.. +b 1.. +c 1.. = X 1n X 21 X 22 a X 23 Y 23 Z 2 +b 2 +c 23 2 =.. X 2n Y 1n Y 21 Y 22.. Y 2n Z 11 Z 12 Z 1n Z 21 Z 22.. Z 2n f 11 f 12 f 13.. f 1n f 21 f 22 f 23.. f 2n MANOVA compares means of Canonical functions V. Čekanavičius, G. Murauskas 3 Checking assumptions In there is no way to check for the multivariate normality. Therefore, we check If all variables in each group are normal and not too correlated (there is no multicolinearity); V. Čekanavičius, G. Murauskas 4 2
Checking assumptions Equality of covariance matrices is tested by the Box test. Box test is widely criticized and it is universally assumed that covariance matrices are acceptable for MANOVA if p> 0.005 (p>=0.05 is OK, but 0.005<p<0.05 is also OK). V. Čekanavičius, G. Murauskas 5 Example File MAmanova.sav We investigate students from the regional university, from medium sized city university and from the prestigious university based in the capital. Each respondent completed test reflecting his/her motivation to study and informed about living expenses (per month in euros). Do students from various universities differ statistically significantly? V. Čekanavičius, G. Murauskas 6 3
DATA V. Čekanavičius, G. Murauskas 7 ANALYZE -> GENERAL LINEAR MODEL -> MULTIVARIATE V. Čekanavičius, G. Murauskas 8 4
Dependent variables group NEXT -> POST HOC V. Čekanavičius, G. Murauskas 9 group Bonferroni V. Čekanavičius, G. Murauskas 10 5
Options NEXT -> OPTIONS V. Čekanavičius, G. Murauskas 11 Move group Check Descriptive Check Homogeneity V. Čekanavičius, G. Murauskas 12 6
Most motivated students are from the capital s university V. Čekanavičius, G. Murauskas 13 Box's Test of Equality of Covariance Matrices a Box's M 5,374 F,817 df1 6 df2 27141,231 Sig.,556 Tests the null hypothesis that the observed covariance matrices of the dependent variables are equal across groups. a. Design: Intercept + group Box test shows that covariance matrices do not differ significantly p = 0,556 > 0,05 >0,005. V. Čekanavičius, G. Murauskas 14 7
Some mean vectors differ statistically significantly. V. Čekanavičius, G. Murauskas 15 There are 4 different MANOVA tests in. Most popular (and most reliable if data is not exactly normal or covariance matrices are not equal) are Pillai s Trace and Wilk s lambda. Some Mean vectors differ statistically significantly if p < 0.05. V. Čekanavičius, G. Murauskas 16 8
Levene s test is for testing the equality of variances for each dependent variable. V. Čekanavičius, G. Murauskas 17 Levene s test is very sensitive to normality assumption. Instead of it one can apply the rule of the thumb to descriptive statistics: variances are of acceptable equality if their standard deviations differ no more that 2 times. Therefore, we conclude that motivation can be used in MANOVA, despite its p = 0,037. V. Čekanavičius, G. Murauskas 18 9
Separate ANOVA s for motivation and financing are not statistically significant, p = 0.092 and p= 0.052. V. Čekanavičius, G. Murauskas 19 In addition, we will check for normality of each dependent variable in each group. Note that some deviations from normality are acceptable and apart from statistical tests one can use histogram, Q-Q plot or other graphical analysis. V. Čekanavičius, G. Murauskas 20 10
We will apply option Split file and sue Kolmogorov-Smirnov test. Choose: DATA -> SPLIT FILE V. Čekanavičius, G. Murauskas 21 group Check V. Čekanavičius, G. Murauskas 22 11
ANALYZE -> NONPARAMETRIC TESTS -> LEGACY DIALOGS -> 1-SAMPLE K-S V. Čekanavičius, G. Murauskas 23 Move Check V. Čekanavičius, G. Murauskas 24 12
For Group=1 Both Variables are Normal, p= 0,399 p= 0,744 V. Čekanavičius, G. Murauskas 25 For Group=2 Both Variables are Normal, p= 0,859 p= 0,996, V. Čekanavičius, G. Murauskas 26 13