Linear Mixed Effects Models

Here we'll be considering an alternative approach for analyzing longitudinal data using linear mixed effects models. These models have some subset of the individual parameters vary randomly from one subject to another, thereby accounting for sources of natural heterogeneity in the population.

• Fixed effects are the population characteristics that are assumed to be shared by all subjects
• Random effects are characteristics, unique for each subject
• The combination of both gives rise to mixed effect models

We can write the linear mixed effects model as:
Y_i = X_i β + Z_ib_i + e_i
Where:

• β is a s*1 vector of fixed effects
• b_i is a q*1 vector of random effects (which follows  ~N(0, G) we will see below)
• e_i is an n_i*1 vector of within subject errors (~N(0, R_i)),
• X_i is an n_i*s matrix of covariates, 
• Z_i is a n_i*q matrix of covariates with q <= s
  

We usually assume R_i = σ²I_n, where I_n is an n_i*n_i identity matrix with b_i and e_i mutually independent.

Model Formulation

The simplest mixed effects model for longitudinal data is:

Where b_i (some random intercept) and e_ij (error) are independent.

The marginal mean response in the population is:

The response for the i^th subject at the j^th measurement occasion differs from the population mean response by a subject effect b_i, and a within subject measurement error e_ij.

The conditional mean response for any individual:

The residuals of profile analysis and parametric curves (ϵ_ij) and parametric curves such that ϵ_ij = b_i + e_ij

Variance and Covariance Structure

Similarly, the marginal covariance between any pair of responses:

With some linear algebra we can write the variance-covariance matrix as follows:

Where ρ is called the intra-cluster correlation coefficient. For ρ > 0, the random intercept mixed effects model is observationally equivalent to a compound symmetry model.

Random Intercept and Slope Model

As the name suggests, this adds a random effect for both the slope and intercept.

Consider the following model:

Where

So the variance of each response varies with the time it was taken:

And likewise the covariance between any pair of responses: