Statistics: Effects plots: some properties in multilevel analysis

From MathWiki

This page explores issues in using effects plots for multilevel data.

Interpretations of effects plot in unilevel data

The construction of confidence bands in effects plots rests on the duality between beta space in which lives the confidence ellipsoid for estimated coefficients and data space in which we plot the regression surface or sections of it.

For a linear model with

\mathbf{Y}=\mathbf{X\beta }+\mathbf{\varepsilon },\quad \mathbf{\varepsilon }\sim N_{n}(\mathbf{0},\sigma ^{2}\mathbf{I}),\ \mathbf{X}\operatorname{ is }n\times p,\ \mathbf{\beta }\operatorname{ is }p\times 1
\mathbf{Y}=\mathbf{X\beta }+\mathbf{\varepsilon }