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### MANOVA - I

MANOVA (multivariate analysis of variance) is a statistical procedure that allows you to determine if a set of categorical predictor variables can explain the variability in a set of continuous response variables. It is also possible to include continuous predictor variables either as covariates or as true independent variables in the design (so that you can test for the effect of interactions).

MANOVA is related to within-subject ANOVA in that both of these analyses examine multiple measurements from each case (i.e., participant) in your data set. Whether you should perform a MANOVA or a within-subject ANOVA depends on the relationship between the measurements. If the different measurements reflect observations at different levels of a theoretical factor, then
you should perform a within-subject ANOVA. For example, you might look at a person’s heart rate over successive days, such that the different measurements represent different levels of a "time" factor. If the measurements instead reflect different dependent variables, then you should perform a MANOVA. For example, using MANOVA you could simultaneously test whether a treatment program affects participants. responses on a depression scale, their GPA, and their performance on a reaction-time task.

The primary purpose of MANOVA is to show that an independent variable (manipulated either within- or between-subjects) has an overall effect on a collection of continuous dependent variables. If you have a large number of dependent variables, you can perform a MANOVA to see if there is any effect of your independent variables, taking into account the number of different dependent variables you are examining.

If one had multiple dependent variables, he/she could perform an ANOVA on each to examine the effect of the independent variable. However, if one were concerned that performing these multiple tests would increase the Type I error rate, a MANOVA would be useful alternative, as it is a single test of the independent variable’s influence on the collection of dependent variables.
In other words, MANOVA can act as protection against an inflation of your Type I error rate from performing a large number of analyses investigating the same hypothesis. If there is a significant effect of the independent variable in the MANOVA, one could then follow up that MANOVA with univariate ANOVAs (ANOVAs with a single dependent variable). Simulations performed by Hummel and Sligo (1971, Psychological Bulletin) and Rencher and Scott (1990,  Communications in Statistics: Simulation and Computation) have demonstrated that the overall experimentwide error rate when you follow up a significant multivariate test with univariate analyses is almost always below the established alpha. If the univariate tests are performed without consideration of ways to protect against alpha inflation, however, there is a significant increase in the experimentwide error rate. The most common alternative to a multivariate test would be the application of a Bonferroni correction, where the experimenter divides the alpha for each individual test by the number of tests. However, simulations demonstrate that this method leads to an overall experiment-wide error rate that is substantially below the established alpha. Many researchers feel that the use of MANOVA is the best alternative since it provides good protection against alpha inflation and is more powerful than applying a Bonferroni correction. However, it must be noted that this method does not guarantee that the experiment-wide error rate will not exceed the established alpha. Most of the time the experiment-wide error rate will be below alpha, but it will occasionally exceed it slightly.

Performing a MANOVA is not the same thing as looking for an effect on the average of your dependent variables. Therefore, it is also different from looking for a main effect of a between-subjects variable within a repeated measures analysis. One common misconception is that you cannot use MANOVA if the effect of your independent variable on the dependents varies in terms of direction, because the effects will cancel each other out. In truth, the dependent variables are never combined together in this way. MANOVA separately considers the effect of your independent variables on your dependents. It actually produces a matrix of results, which separately contains the influence of your independents on each of your dependent variables (Actually, what MANOVA does is determine the effect of your independent variables on the principle components that can be calculated from your dependent variables. However, thinking of it the way described in the text is a little simpler and is basically accurate, since the components represent the dimensions of variability found in your dependent variables.)
The multivariate test of an independent variable does not require that it affect each dependent variable in the same way. What is important is just the extent to which your independent variables create differences in each of your dependent variables. For example, reverse coding one of the dependent variables would have absolutely no influence on a MANOVA.