Families, link functions and parameters

Families

The list of built-in families can be obtained by calling family_vals. This shows the name of each family, and the value that can be used to obtain it when calling functions.

family_vals$val

The parameterizations for these families are based on generalized linear models. Essentially, we have that \[\begin{align*} \mathbb{E} [Y \mid X=x] = \mu(x) = g^{-1}(\eta(x)) \end{align*}\] where \(\eta(x) = x^T\beta\) for a vector of real parameters \(\beta\).

Note that the beta and lognormal families are not actually exponential dispersion families, so they do not share all the nice properties of full GLM distributions. Here we include a table of the different models used.

fams <- family_vals[-1,]
fams$val[5] <- "0/5"
fams$canon <- c("identity", "identity", "inverse", NA, "logistic", NA, "log-linear", "log-linear")
fams$default <- c("-", "-", "log", "logistic", "-", "exponential", "-", "-")