# fix.family.link: Modify families for use in GAM fitting and checking In mgcv: Mixed GAM Computation Vehicle with Automatic Smoothness Estimation

## Description

Generalized Additive Model fitting by ‘outer’ iteration, requires extra derivatives of the variance and link functions to be added to family objects. The first 3 functions add what is needed. Model checking can be aided by adding quantile and random deviate generating functions to the family. The final two functions do this.

## Usage

 ```1 2 3 4 5``` ```fix.family.link(fam) fix.family.var(fam) fix.family.ls(fam) fix.family.qf(fam) fix.family.rd(fam) ```

## Arguments

 `fam` A `family`.

## Details

Consider the first 3 function first.

Outer iteration GAM estimation requires derivatives of the GCV, UBRE/gAIC, GACV, REML or ML score, which are obtained by finding the derivatives of the model coefficients w.r.t. the log smoothing parameters, using the implicit function theorem. The expressions for the derivatives require the second and third derivatives of the link w.r.t. the mean (and the 4th derivatives if Fisher scoring is not used). Also required are the first and second derivatives of the variance function w.r.t. the mean (plus the third derivative if Fisher scoring is not used). Finally REML or ML estimation of smoothing parameters requires the log saturated likelihood and its first two derivatives w.r.t. the scale parameter. These functions add functions evaluating these quantities to a family.

If the family already has functions `dvar`, `d2var`, `d3var`, `d2link`, `d3link`, `d4link` and for RE/ML `ls`, then these functions simply return the family unmodified: this allows non-standard links to be used with `gam` when using outer iteration (performance iteration operates with unmodified families). Note that if you only need Fisher scoring then `d4link` and `d3var` can be dummy, as they are ignored. Similalry `ls` is only needed for RE/ML.

The `dvar` function is a function of a mean vector, `mu`, and returns a vector of corresponding first derivatives of the family variance function. The `d2link` function is also a function of a vector of mean values, `mu`: it returns a vector of second derivatives of the link, evaluated at `mu`. Higher derivatives are defined similarly.

If modifying your own family, note that you can often get away with supplying only a `dvar` and `d2var`, function if your family only requires links that occur in one of the standard families.

The second two functions are useful for investigating the distribution of residuals and are used by `qq.gam`. If possible the functions add quantile (`qf`) or random deviate (`rd`) generating functions to the family. If a family already has `qf` or `rd` functions then it is left unmodified. `qf` functions are only available for some families, and for quasi families neither type of function is available.

## Value

A family object with extra component functions `dvar`, `d2var`, `d2link`, `d3link`, `d4link`, `ls`, and possibly `qf` and `rd`, depending on which functions are called. `fix.family.var` also adds a variable `scale` set to negative to indicate that family has a free scale parameter.

## Author(s)

Simon N. Wood simon.wood@r-project.org

`gam.fit3`, `qq.gam`