Multilevel Model Equations

We write down the equations for the growth curve models of mathematics achievement fitted in this page. Equations for other models may be found in the class slides for random coefficient models here.

Let Y_ijk be the outcome at time k for child j in school i, with t_ijk as the corresponding time (study year).

Growth Curves

The growth curve model with random intercepts and slopes is Y_ijk = (α+a_i+a_ij) + (β+b_i+b_ij)t_ijk + e_ijk

where e_ijk ∼ N(0,σ²) is the error term, (a_ij,b_ij) ∼ N(0,Σ)} are the intercept and slope random effects at the child level, with unstructured (free) covariance matrix $$ \Sigma = \begin{bmatrix} \sigma^2_{a} & \sigma_{ab} \\ \sigma_{ab} & \sigma^2_{b} \end{bmatrix} $$
and (a_i,b_i) ∼ N(0,Φ) are the intercept and slope random effects at the school level, with unstructured covariance matrix $$ \Phi = \begin{bmatrix} \varphi^2_{a} & \varphi_{ab} \\ \varphi_{ab} & \varphi^2_{b} \end{bmatrix} $$ This model has 2 fixed parameters (intercept and slope) and 6 variances and covariances of random effects (3 at each at levels 3 and 2), plus the variance of the error term.

The parameters and the estimates from fitting the model match as follows. First the fixed effects

And next the variances and covariances of the random effects

Note that Stata reports level 3 and then level 2, and puts the constant at the end. R starts with the constant and reports level 2 and then level 3, as I have done here.

Growth Curve Predictors

Let x_2ij and x_3ij be the indicators of black and hispanic ethnicity for child j in school i and let z_i be the percent of low-income students in school i.

There are two-ways of writing the model. Let us start with level-specific models. For each child in a given occassion Y_ijk = A_ij + B_ijt_ijk + e_ijk
where the error term e_ijk is as before, and the intercept A_ij and slope B_ij are specific to each child. These follow their own models depending on child ethnicity $$ \begin{align} A_{ij} = A_i + \alpha_1 x_{1ij} + \alpha_2 x_{2ij} + a_{ij} \\ B_{ij} = B_i + \beta_1 x_{1ij} + \beta_2 x_{2ij} + b_{ij} \end{align} $$

where the intercept A_i and slope B_i are specific to each school, and follow their own models depending on the percent of low-income students $$ \begin{align} A_i = \alpha + \alpha_3 z_i + a_i \\ B_i = \beta + \beta_3 z_i + b_i \end{align} $$ Here the residuals (a_i,b_i) at level 3 and (a_ij,b_ij) at level 2 follow bivariate normal distributions as shown in the previous section.

If we substitute the level 3 and level 2 equations on the level 1 model we obtain $$ \begin{align} Y_{ijk} = & (\alpha + \alpha_1 x_{1ij} + \alpha_2 x_{2ij} + \alpha_3 z_i + a_i + a_{ij}) + \\ & ( \beta + \beta_1 x_{1ij} + \beta_2 x_{2ij} + \beta_3 z_i + b_i + b_{ij}) t_{ijk} + e_{ijk} \end{align} $$
which shows that the fixed effects include main effects of black, hispanic, low-income and year, as well as the interactions of black, hispanic and low-income with year, which enter as cross-products. Carrying out the multiplication by t_ijk and reordering a bit helps make this clear: $$ \begin{align} Y_{ijk} = & \alpha + \alpha_1 x_{1ij} + \alpha_2 x_{2ij} + \alpha_3 z_i + \beta t_{ijk} + \\ & \beta_1 x_{1ij} t_{ijk} + \beta_2 x_{2ij} t_{ijk} + \beta_3 z_i t_{ijk} + \\ & a_i + a_{ij} + b_i t_{ijk} + b_{ij} t_{ijk} + e_{ijk} \end{align} $$

The three lines here correspond to main effects, interactions, and random effects.

Here are the fixed effect parameters with their names and estimates

The names of the variances and covariances of the random effects are the same as in the previous section.