In biology we often wish to describe effects that are complex entities and difficult to characterize in a single measure. We may, therefore, in a single experiment measure a suite of dependent variables instead of a single variable alone and then have a better chance of discovering which factor is truly important.
When considering whether a particular treatment effects a set of dependent variables we could run multiple univariate tests and adjust the p-Value using Dunn-Sedak? What are the drawbacks of an analysis that considers each variable seperately instead of a single one that considers patterns across all of them simultaneously? Specifically, we must take into account correlations among the dependent measures when performing the significance test. A multivariate analysis can protect against Type I errors which might occur if multiple univariate analyses were performed independently. Towards this goal we first create a set of new dependent variables as linear combinations of the measured dependent variables. These artificial dependent variables are chosen to maximize group differences. A correlation between two variables indicates that the two variable in essence are measuring the same thing to some degree. In the multivariate model it is assumed both that the correlations between a set of observed variables can be explained in terms of a simpler set of derived variables. So, how do we go about deriving a set of new, better suited, hypothetical variables from the information obtained about individual correlations among them (i.e., the variance/covariance matrix)?
A single, normally distributed variable may be characterize using population parameters mean and variance. Similarly, a multivariately normal set of variables may be described in similar ways