qq.gam {mgcv}  R Documentation 
QQ plots for gam model residuals
Description
Takes a fitted gam
object produced by gam()
and produces
QQ plots of its residuals (conditional on the fitted model
coefficients and scale parameter). If the model distributional
assumptions are met then usually these plots should be close to a
straight line (although discrete data can yield marked random
departures from this line).
Usage
qq.gam(object, rep=0, level=.9,s.rep=10,
type=c("deviance","pearson","response"),
pch=".", rl.col=2, rep.col="gray80", ...)
Arguments
object 
a fitted 
rep 
How many replicate datasets to generate to simulate quantiles
of the residual distribution. 
level 
If simulation is used for the quantiles, then reference intervals can be provided for the QQplot, this specifies the level. 0 or less for no intervals, 1 or more to simply plot the QQ plot for each replicate generated. 
s.rep 
how many times to randomize uniform quantiles to data under direct computation. 
type 
what sort of residuals should be plotted? See

pch 
plot character to use. 19 is good. 
rl.col 
color for the reference line on the plot. 
rep.col 
color for reference bands or replicate reference plots. 
... 
extra graphics parameters to pass to plotting functions. 
Details
QQplots of the the model residuals can be produced in one of two ways. The cheapest method generates reference quantiles by
associating a quantile of the uniform distribution with each datum, and feeding these uniform quantiles into the quantile function associated with each datum. The resulting quantiles are then used in place of each datum to generate approximate quantiles of residuals.
The residual quantiles are averaged over s.rep
randomizations of the uniform quantiles to data.
The second method is to use direct simulatation. For each replicate, data are simulated from the fitted model, and the corresponding residuals computed. This is repeated rep
times.
Quantiles are readily obtained from the empirical distribution of residuals so obtained. From this method reference bands are also computable.
Even if rep
is set to zero, the routine will attempt to simulate quantiles if no quantile function is available for the family. If no random deviate generating function family is available (e.g. for the quasi families), then a normal QQplot is produced. The routine conditions on the fitted model coefficents and the scale parameter estimate.
The plots are very similar to those proposed in Ben and Yohai (2004), but are substantially cheaper to produce (the interpretation of residuals for binary data in Ben and Yohai is not recommended).
Note that plots for raw residuals from fits to binary data contain almost no useful information about model fit. Whether the residual is negative or positive is decided by whether the response is zero or one. The magnitude of the residual, given its sign, is determined entirely by the fitted values. In consequence only the most gross violations of the model are detectable from QQplots of residuals for binary data. To really check distributional assumptions from residuals for binary data you have to be able to group the data somehow. Binomial models other than binary are ok.
Author(s)
Simon N. Wood simon.wood@rproject.org
References
N.H. Augustin, EA Sauleaub, S.N. Wood (2012) On quantile quantile plots for generalized linear models Computational Statistics & Data Analysis. 56(8), 24042409.
M.G. Ben and V.J. Yohai (2004) JCGS 13(1), 3647.
https://www.maths.ed.ac.uk/~swood34/
See Also
Examples
library(mgcv)
## simulate binomial data...
set.seed(0)
n.samp < 400
dat < gamSim(1,n=n.samp,dist="binary",scale=.33)
p < binomial()$linkinv(dat$f) ## binomial p
n < sample(c(1,3),n.samp,replace=TRUE) ## binomial n
dat$y < rbinom(n,n,p)
dat$n < n
lr.fit < gam(y/n~s(x0)+s(x1)+s(x2)+s(x3)
,family=binomial,data=dat,weights=n,method="REML")
par(mfrow=c(2,2))
## normal QQplot of deviance residuals
qqnorm(residuals(lr.fit),pch=19,cex=.3)
## Quick QQplot of deviance residuals
qq.gam(lr.fit,pch=19,cex=.3)
## Simulation based QQplot with reference bands
qq.gam(lr.fit,rep=100,level=.9)
## Simulation based QQplot, Pearson resids, all
## simulated reference plots shown...
qq.gam(lr.fit,rep=100,level=1,type="pearson",pch=19,cex=.2)
## Now fit the wrong model and check....
pif < gam(y~s(x0)+s(x1)+s(x2)+s(x3)
,family=poisson,data=dat,method="REML")
par(mfrow=c(2,2))
qqnorm(residuals(pif),pch=19,cex=.3)
qq.gam(pif,pch=19,cex=.3)
qq.gam(pif,rep=100,level=.9)
qq.gam(pif,rep=100,level=1,type="pearson",pch=19,cex=.2)
## Example of binary data model violation so gross that you see a problem
## on the QQ plot...
y < c(rep(1,10),rep(0,20),rep(1,40),rep(0,10),rep(1,40),rep(0,40))
x < 1:160
b < glm(y~x,family=binomial)
par(mfrow=c(2,2))
## Note that the next two are not necessarily similar under gross
## model violation...
qq.gam(b)
qq.gam(b,rep=50,level=1)
## and a much better plot for detecting the problem
plot(x,residuals(b),pch=19,cex=.3)
plot(x,y);lines(x,fitted(b))
## alternative model
b < gam(y~s(x,k=5),family=binomial,method="ML")
qq.gam(b)
qq.gam(b,rep=50,level=1)
plot(x,residuals(b),pch=19,cex=.3)
plot(b,residuals=TRUE,pch=19,cex=.3)