Fixed Effects

Study guide: Exam MAS-II

Fixed effects are the regression coefficients in a Linear Mixed Model that describe the population-average relationship between the predictors and the response. They are unknown constants to be estimated, not draws from a distribution.

\mathbf{y} = \underbrace{\mathbf{X}\boldsymbol{\beta}}_{\text{fixed}} + \underbrace{\mathbf{Z}\mathbf{b}}_{\text{random}} + \boldsymbol{\varepsilon}

E[\mathbf{y}] = \mathbf{X}\boldsymbol{\beta}

• A factor belongs in the fixed part when its levels are the specific levels you care about and would reappear in a replication of the study — territory, vehicle class, coverage limit
• Each fixed effect costs one parameter per level (less the baseline), so a factor with many levels is expensive and unstable — that is the case for making it a random effect instead
• Interpretation is the same as in a Generalized Linear Model: \beta_j is the change in the mean response per unit of x_j, holding the rest fixed
• Fixed effects are estimated (\hat{\boldsymbol\beta}) by generalized least squares given the variance parameters; random effects are predicted (BLUP)
• Comparing two models that differ in their fixed effects requires maximum likelihood, not REML

Fixed or Random? A claims model has three candidate grouping variables: coverage (3 levels: liability, collision, comprehensive), territory (68 rating territories), and year (5 calendar years of data). Which belong in the fixed part?

Answer Coverage — fixed. Three levels, all of interest, and each has plenty of data. Territory — random. 68 levels, most thin, and the interest is in the distribution of territory effects and a shrunk estimate for each, not in 67 separately fitted coefficients. Year — fixed if the five specific years are the object of interest (a trend), random only if they are treated as a sample of years.