An intuitive method for analyzing longitudinal data is grounded in the concept that each individual within the population possesses a unique subject-specific mean response profile over time, characterized by a particular functional form.

To formally introduce the representation of longitudinal data, let y_{ij} denote the response of subject i, at time t_{ij}. Different subjects tend to have different intercepts and slopes for regression. Therefore, a plausible model is considered as

    \[y_{ij} = \tilde{\beta}_{i0}+\tilde{\beta}_{i1}t_{ij}+\varepsilon_{ij},\]

where the error terms \varepsilon_{ij} are assumed from N(0, \sigma^2). It is customary to assume that the distribution of the regression coefficients in the population is a bivariate normal distribution with mean vector \beta=(\beta_0, \beta_1)^T and the variance-covariance matrix D. We can reformulate the model as

    \[y_{ij} = (\beta_0+b_{i0})+(\beta_1+b_{i1})t_{ij}+\varepsilon_{ij}\]

where b_i=(b_{i0},b_{i1})^T are called random effect, having a bivariate normal distribution with mean zero and covariance matrix D. \beta_0 and \beta_1 describe the average longitudinal evolution in the population (i.e., averaged over the subjects) and are called fixed effects.