5 LMM: Model Equation

5.1 Model equation

\[y_{ij} = \beta_0 + \beta_1 \times SBP0_{i} + \beta_2 \times AGE_{i} + \beta_3 \times SEX_{3i} + \] \[\beta_4 \times GROUP_{i} + \beta_5 \times TIME_{ij} + \beta_6 \times GROUP_{i} \times TIME_{ij} + \]

\[ u_{0i} + e_{ij}\]

Here:

\(y_{ij}\) = the SBP value of the i-th patient (\(i = 1, ..., 20\)) at the j-th time point (\(j = 1, ..., 4\))

\(\beta_0\) = the intercept at the reference level of categorical variable and zero value of the covariates

\(\beta_1, \beta_2\) = the coefficient (slope) with baseline SBP (SBP0) and Age of the i-th patient, respectively

\(\beta_3\) = the effect of sex of the i-th patient; \(Sex = {Male, Female}\)

\(\beta_4\) = the effect of the allocated treatment group of i-th patient; \(Group = {Treatment, Control}\)

\(\beta_5\) = the effect of j-th time of the i-th patient

\(\beta_6\) = the interaction effect of i-th patient group at the j-th time

\(u_{0i}\) = the random intercept of the i-th patient

\(e_{ij}\) = the random error

Assumption:

\(u_{0i} \sim N(0, \sigma_P^2)\)

\(e_{ij} \sim N(0, \sigma_e^2)\)

Model equation: extension

Model intercept consists of two components: fixed part and random part:

\[Intercept: b_{0i} = \beta_0 + u_{0i}\]

5.2 Hypothesis

\[Null \space hypothesis, H_0: \beta_k = 0\]

\[Alternative \space hypothesis, H_1: \beta_k \ne 0\]