Generalised linear mixed models

Many situations where residuals are not normally distribution - presence/absence data, count data etc. Generalised linear models and generalised mixed models relax the assumption of normally distributed residuals (to exponential-family distributed) for fixed effect and mixed models respectively.

Generalised linear models

GLM derived from simple fixed effects model by instead of making $\mu = , $g(\mu) = X \alpha$ . Can also think of link function as way of mapping data from their scale to the reals (the scale of normally distributed data).

Distributions

Binomial: parameter of interest proportion of sucesses, $f(y, n) = \frac{n! . Poisson: count data, $f(y) = \mu^y e^{-\mu}/y!$ Poisson with offset: counts per unit scale, $f(y, t) = (\mu t)^{yt} e^{-\mu

All members of exponential family of distributions: $f(y; \theta, \phi) = . $\theta$ , a function of the original parameters is known as the canonical link.

Model selection and measure of fit

Expressed by deviance $= 2(LR_full - LR_model)$ or generalised Pearson $\chi^2 = \sum (y_i - \hat{mu}_i) / V(\hat{\mu})$ , both asymptotically chi-squared. Pearson and deviance residuals defined as contribution of individual observations.

Common problem with Poisson and Binomial is over-dispersion where variance is greater than assumed distribution. Can use quasi-likelihood model where we model the variance independently of the underlying model.