The NESTED Procedure

Analysis of Covariation

When you specify more than one dependent variable, the NESTED procedure produces a descriptive analysis of the covariance between each pair of dependent variables in addition to a separate analysis of variance for each variable. The analysis of covariation is computed under the basic random-effects model for each pair of dependent variables:

\begin{eqnarray*} y_{i_1 i_2 \cdots i_ n r} & = & \mu + \alpha _{i_1} + \beta _{i_1 i_2} + \cdots + \epsilon _{i_1 i_2 \cdots i_ n r} \\[0.10in] y_{i_1 i_2 \cdots i_ n r}^{\prime } & = & \mu ^{\prime } + \alpha _{i_1}^{\prime } + \beta _{i_1 i_2}^{\prime } + \cdots + \epsilon _{i_1 i_2 \cdots i_ n r}^{\prime } \\ \end{eqnarray*}

where the notation is the same as that used in the preceding general random-effects model.

There is an additional assumption that all the random effects in the two models are mutually uncorrelated except for corresponding effects, for which

\begin{eqnarray*} \mbox{Corr}(\alpha _{i_1} , \alpha _{i_1}^{\prime }) & = & \rho _{\alpha } \\[0.05in] \mbox{Corr}(\beta _{i_1 i_2}, \beta _{i_1 i_2}^{\prime }) & = & \rho _{\beta } \\[0.05in]& \vdots & \\[0.05in] \mbox{Corr}(\epsilon _{i_1 i_2 \cdots i_ n r}, \epsilon _{i_1 i_2 \cdots i_ n r}^{\prime }) & = & \rho _{\epsilon } \\ \end{eqnarray*}