Specify a Random Effects Fit in a GAM Formula

A symbolic wrapper for a factor term, to specify a random effect term in a formula argument to gam

Usage

gam.random(f, y, w, df = sum(non.zero), lambda = 0, intercept = TRUE, xeval)

random(f, df = NULL, lambda = 0, intercept = TRUE)

Arguments

f: factor variable, or expression that evaluates to a factor.
y: a response variable passed to gam.random during backfitting
w: weights
df: the target equivalent degrees of freedom, used as a smoothing parameter. The real smoothing parameter (lambda below) is found such that df=tr(S), where S is the implicit smoother matrix. Values for df should be greater than 0 and less than the number of levels of f. If both df and lambda are supplied, the latter takes precedence. Note that df is not necessarily an integer.
lambda: the non-negative penalty parameter. This is interpreted as a variance ratio in a mixed effects model - namely the ratio of the noise variance to the random-effect variance.
intercept: if intercept=TRUE (the default) then the estimated level effects are centered to average zero, otherwise they are left alone.
xeval: If this argument is present, then gam.random produces a prediction at xeval.

Value

random returns the vector f, endowed with a number of attributes. The vector itself is used in computing the means in backfitting, while the attributes are needed for the backfitting algorithms general.wam. Note that random itself does no smoothing; it simply sets things up for gam.

One important attribute is named call. For example, random(f, lambda=2) has a call component gam.random(data[["random(f, lambda = 2)"]], z, w, df = NULL, lambda = 2, intercept = TRUE). This is an expression that gets evaluated repeatedly in general.wam (the backfitting algorithm).

gam.random returns an object with components

residuals: The residuals from the smooth fit.
nl.df: the degrees of freedom
var: the pointwise variance for the fit
lambda: the value of lambda used in the fit

When gam.random is evaluated with an xeval argument, it returns a vector of predictions.

Details

This "smoother" takes a factor as input and returns a shrunken-mean fit. If lambda=0, it simply computes the mean of the response at each level of f. With lambda>0, it returns a shrunken mean, where the j'th level is shrunk by nj/(nj+lambda), with nj being the number of observations (or sum of their weights) at level j. Using such smoother(s) in gam is formally equivalent to fitting a mixed-effect model by generalized least squares.

References

Hastie, T. J. (1992) Generalized additive models. Chapter 7 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.

Hastie, T. and Tibshirani, R. (1990) Generalized Additive Models. London: Chapman and Hall.

Cantoni, E. and hastie, T. (2002) Degrees-of-freedom tests for smoothing splines, Biometrika 89(2), 251-263

Author

Written by Trevor Hastie, following closely the design in the "Generalized Additive Models" chapter (Hastie, 1992) in Chambers and Hastie (1992).

Examples


# fit a model with a linear term in Age and a random effect in the factor Level
y ~ Age + random(Level, lambda=1)
#> y ~ Age + random(Level, lambda = 1)
#> <environment: 0x590fc77b8e50>