Lab 4 -- MANOVA on SPSS

A Multivariate Analysis of Variance (MANOVA) is often used as a multivariate analog of the ANOVA, but when there are multiple dependent variables. (Note that an ANOVA can handle multiple independent variables, but only one dependent variable at a time.) I will have much to say in lecture about the limited appropriateness of MANOVA, however you should know of its existence since it is used often. These are the SPSS instructions and an example of a simple MANOVA.

For a simple oneway MANOVA, the data set should have one independent variable
(grouping variable) and at least two dependent variables. In the example
herein I use **gender** as the IV and the GRE verbal and quantitative (**grev
**and **greq**) scores as DVs. The data set used herein (DESC.sav) is
included in this MANOVA folder.

Click Analyze/General Linear Model/Multivariate. Place the dependent variables in the box labeled such and the independent variable in the "Fixed Factors" box. Open "Options" and check the boxes as denoted in the screen shot showing all of this below.

Click "Continue" in the Options window and "OK" in the "Multivariate" window and your job will run. The output for this run is:

Value Label | N | ||
---|---|---|---|

GENDER | 1.00 | female | 193 |

2.00 | male | 45 |

GENDER | Mean | Std. Deviation | N | |
---|---|---|---|---|

GRE- VERBAL | female | 493.8860 | 90.35767 | 193 |

male | 470.8889 | 93.07432 | 45 | |

Total | 489.5378 | 91.12631 | 238 | |

GRE-QUANTITATIVE | female | 485.1813 | 95.87912 | 193 |

male | 525.7778 | 118.92490 | 45 | |

Total | 492.8571 | 101.62098 | 238 |

Box's M | 3.722 |
---|---|

F | 1.219 |

df1 | 3 |

df2 | 86913.177 |

Sig. | .301 |

Tests the null hypothesis that the observed covariance matrices of the dependent variables are equal across groups. | |

a Design: Intercept+GENDER |

Effect | Value | F | Hypothesis df | Error df | Sig. | Partial Eta Squared | Noncent. Parameter | Observed Power(a) | |
---|---|---|---|---|---|---|---|---|---|

Intercept | Pillai's Trace | .959 | 2771.204(b) | 2.000 | 235.000 | .000 | .959 | 5542.409 | 1.000 |

Wilks' Lambda | .041 | 2771.204(b) | 2.000 | 235.000 | .000 | .959 | 5542.409 | 1.000 | |

Hotelling's Trace | 23.585 | 2771.204(b) | 2.000 | 235.000 | .000 | .959 | 5542.409 | 1.000 | |

Roy's Largest Root | 23.585 | 2771.204(b) | 2.000 | 235.000 | .000 | .959 | 5542.409 | 1.000 | |

GENDER | Pillai's Trace | .054 | 6.707(b) | 2.000 | 235.000 | .001 | .054 | 13.415 | .913 |

Wilks' Lambda | .946 | 6.707(b) | 2.000 | 235.000 | .001 | .054 | 13.415 | .913 | |

Hotelling's Trace | .057 | 6.707(b) | 2.000 | 235.000 | .001 | .054 | 13.415 | .913 | |

Roy's Largest Root | .057 | 6.707(b) | 2.000 | 235.000 | .001 | .054 | 13.415 | .913 | |

a Computed using alpha = .05 | |||||||||

b Exact statistic | |||||||||

c Design: Intercept+GENDER |

F | df1 | df2 | Sig. | |
---|---|---|---|---|

GRE- VERBAL | .001 | 1 | 236 | .976 |

GRE-QUANTITATIVE | 3.443 | 1 | 236 | .065 |

Tests the null hypothesis that the error variance of the dependent variable is equal across groups. | ||||

a Design: Intercept+GENDER |

Source | Dependent Variable | Type III Sum of Squares | df | Mean Square | F | Sig. | Partial Eta Squared | Noncent. Parameter | Observed Power(a) |
---|---|---|---|---|---|---|---|---|---|

Corrected Model | GRE- VERBAL | 19299.223(b) | 1 | 19299.223 | 2.337 | .128 | .010 | 2.337 | .331 |

GRE-QUANTITATIVE | 60140.712(c) | 1 | 60140.712 | 5.945 | .015 | .025 | 5.945 | .680 | |

Intercept | GRE- VERBAL | 33966035.357 | 1 | 33966035.357 | 4113.398 | .000 | .946 | 4113.398 | 1.000 |

GRE-QUANTITATIVE | 37295811.300 | 1 | 37295811.300 | 3686.906 | .000 | .940 | 3686.906 | 1.000 | |

GENDER | GRE- VERBAL | 19299.223 | 1 | 19299.223 | 2.337 | .128 | .010 | 2.337 | .331 |

GRE-QUANTITATIVE | 60140.712 | 1 | 60140.712 | 5.945 | .015 | .025 | 5.945 | .680 | |

Error | GRE- VERBAL | 1948749.937 | 236 | 8257.415 | |||||

GRE-QUANTITATIVE | 2387316.431 | 236 | 10115.748 | ||||||

Total | GRE- VERBAL | 59004100.000 | 238 | ||||||

GRE-QUANTITATIVE | 60259600.000 | 238 | |||||||

Corrected Total | GRE- VERBAL | 1968049.160 | 237 | ||||||

GRE-QUANTITATIVE | 2447457.143 | 237 | |||||||

a Computed using alpha = .05 | |||||||||

b R Squared = .010 (Adjusted R Squared = .006) | |||||||||

c R Squared = .025 (Adjusted R Squared = .020) |

An assumption of the MANOVA is that the covariance matrices of the dependent variables are the same across groups (determined by levels of the independent variable) in the population. This is the multivariate analog of the assumption of equal variances for the ANOVA. Box's M tests that assumption. In the case at hand the p value of .301 suggests that the hypothesis of equal covariance matrices can not be rejected. So we have not violated an assumption of MANOVA, and may feel confident in continuing (at least in respect to this assumption).

The Multivariate Tests (Pillai's, Wilks', Hotelling's, and Roy's) all test the MANOVA null hypothesis -- that the mean on the composite variable is the same across groups. In the multivariate case, these tests can, in general, provide different results. In our present simple example contrasting across two groups, they are necessarily the same. Thus we find the multivariate hypothesis that the mean on the composite is the same across groups rejected. Remember that this is a test of the equality of a composite of the means (optimized to yield the maximum possible F-ratio) across groups.

Almost all MANOVA programs provide univariate tests for each of the dependent variables used in the MANOVA. This is probably done for a bad reason, as the practice has been to only pursue univariate tests if the multivariate test is significant (in an incorrect attempt to protect against a Type I error).

For this reason, we have the standard Levene's test of the assumption of
equal variances for each of our dependent variables as this is an assumption of
the ANOVA. For both **grev** and **greq**, the test produces an
nonsignificant p value, so the null hypotheses regarding equal variances can not
be rejected for either dependent variable, thus ANOVA is fine.

We can, however, consider these univariate tests if we wish (although we
should realize that they are not directly related to the multivariate test), as
long as we treat the error rate appropriately. A simple (although not
necessarily optimal) way to adjust the error rate is to use the Bonferroni
inequality, thus we test each of our two null hypotheses regarding each of our
two dependent variables at the α/2 level. For the sake of demonstration,
let α=.05, thus the adjusted error rate is .025. We see (under the Tests
of Between-Subjects Effects") that, using this modified α, the null hypothesis
regarding **greq** would be rejected (and looking at the means we see that
the males were superior), but that the null for **grev** would not be
rejected.