12.3 A General Model

The presentation of all of these models here is an attempt to integrate traditions that are typically kept separate or set against each other. In reality, they are just a few examples of the many, many possible models that exist. Each model is useful in specific situations.

The objective of all of our analyses is to deconstruct the data into meaningful and interpretable pieces. Each model does this in a different way, with different assumptions.

We can judge the models based on

1. How well they articulate and test our theory, and
   
2. How well they recover the data (evaluated as misfit to the impled moments).

Recall that the regression model may be compactly written as

\[ \mathbf{Y} = \mathbf{X}\mathbf{\beta} + \boldsymbol{e} \]

We make it into a multilevel regression model by further partitioning the predictor space into between-person and within-person components. Note: this is the very same distinction that is made in traditional presentations of ANOVA when examining between-person factors and within-person factors.

The general model becomes

\[ \underbrace{\boldsymbol{Y}_i}_{\begin{subarray}{c}\text{repeated measures}\\ \text{for persion i}\end{subarray}} = \underbrace{\boldsymbol{X}_i}_{\begin{subarray}{c}\text{known}\\ \text{covariates}\end{subarray}}\underbrace{\boldsymbol{\beta}}_{\begin{subarray}{c}\text{fixed}\\ \text{effects}\end{subarray}} + \underbrace{\boldsymbol{Z}_i}_{\begin{subarray}{c}\text{known}\\ \text{covariates}\end{subarray}}\underbrace{\boldsymbol{u}_i}_{\begin{subarray}{c}\text{random}\\ \text{effects}\end{subarray}} + \underbrace{\boldsymbol{e}_i}_{\begin{subarray}{c}\text{residuals}\end{subarray}} \] which is also called the linear mixed model.