Random Intercept

Intuition

Random Intercept

• The intercept in a simple regression model represents the average value of \(y\) when \(x\) is 0.
• In a simple regression model, there is one intercept for all individuals in the study population.
• However, when individuals are grouped, such as in classes, schools, or firms, there is the potential for separate intercepts for each group.

More visual

Random Intercept

Random Intercept

Random Intercept

Equation Level 1

The equation for a level 1 regression model in multilevel modeling is:

\[ y_{ij}=\beta_{0j}+\beta_{1j}x+\varepsilon_ij \]

where \(ij\) declare the individual \(i\) and group \(j\).

Random Intercept for Group

Random intercept for each group:

\[ y_{ij}=\beta_{0j}+\varepsilon_{ij} \]

Random Intercept Equation

Random intercept:

\[ \beta_{0j}=\gamma_{00}+U_{0j} \] \(\gamma_{00}\) represents the average intercept value across all clusters, while \(U_{0j}\) represents the group-specific effect on the intercept. When the clusters are assumed to be random, \(U_{0j}\) can be described as the residual effect on \(y_{ij}\).

Full Composite Model

If both random intercept equations are substituted into the regression model equation, we obtain the full or composite model.

\[ y_{ij}=\gamma_{00}+U_{0j}+\beta_1x+\varepsilon \]