Visual inspection of GAMM models

Jacolien van Rij

15 March 2016

In contrast with linear regression models, in nonlinear regression models one cannot interpret the shape of the regression line from the summary. Therefore, visualization is an important tool for interpretating nonlinear regression models. Here a selection of visualization functions is summarized. Detailed information on the functions and examples are found in the help files of each function.

Before summarizing the function, we would like to highlight different concepts

Example

Loading the data:

library(itsadug)
library(mgcv)
data(simdat)
# add missing values to simdat:
simdat[sample(nrow(simdat), 15),]$Y <- NA

Model

We start with a somewhat weird model, for illustration purposes. We would like to include some factorial predictors, random smooths, and a nonlinear interaction. Therefore, the continuous predictor Condition is transformed to a categorical predictor:

simdat$Condition <- as.factor(simdat$Condition)
# Note: this is really not the best fitting model for the data:
m1 <- bam(Y ~ Group * Condition + s(Time) + s(Trial) + ti(Time, Trial) + s(Time, Subject, bs='fs', m=1, k=5), data=simdat, discrete=TRUE)

The summary shows two parts: the first summary are the parametric terms, i.e., the ‘linear’ part, and the second summary presents the nonlinear smooths.

summary(m1)
## 
## Family: gaussian 
## Link function: identity 
## 
## Formula:
## Y ~ Group * Condition + s(Time) + s(Trial) + ti(Time, Trial) + 
##     s(Time, Subject, bs = "fs", m = 1, k = 5)
## 
## Parametric coefficients:
##                         Estimate Std. Error t value Pr(>|t|)    
## (Intercept)             5.464428   0.416374  13.124  < 2e-16 ***
## GroupAdults            -1.597709   0.081295 -19.653  < 2e-16 ***
## Condition0              0.219907   0.081302   2.705  0.00684 ** 
## Condition1              0.002296   0.081294   0.028  0.97747    
## Condition2             -0.776871   0.081301  -9.555  < 2e-16 ***
## Condition3             -2.116161   0.081294 -26.031  < 2e-16 ***
## Condition4             -4.024076   0.081294 -49.500  < 2e-16 ***
## GroupAdults:Condition0  0.177183   0.114973   1.541  0.12330    
## GroupAdults:Condition1 -0.004772   0.114968  -0.042  0.96689    
## GroupAdults:Condition2 -0.350877   0.114973  -3.052  0.00228 ** 
## GroupAdults:Condition3 -1.120037   0.114968  -9.742  < 2e-16 ***
## GroupAdults:Condition4 -1.710755   0.114970 -14.880  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Approximate significance of smooth terms:
##                     edf Ref.df      F p-value    
## s(Time)           8.574   8.68  387.3  <2e-16 ***
## s(Trial)          8.820   8.99 1504.0  <2e-16 ***
## ti(Time,Trial)   12.829  14.57 2876.8  <2e-16 ***
## s(Time,Subject) 149.047 168.00 4829.6  <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## R-sq.(adj) =  0.966   Deviance explained = 96.6%
## fREML = 1.4607e+05  Scale est. = 2.7428    n = 75585

Note that although the model looks good (deviance explained of 96.62%!), there are some issues with this model, which we won’t address here.

Function gamtabs

With the function gamtabs the summary of the model can be converted to a Latex of HTML table, easy to integrate in a knitr or R Markdown report (see here):

gamtabs(m1, type="HTML")
A. parametric coefficients Estimate Std. Error t-value p-value
(Intercept) 5.4644 0.4164 13.1238 < 0.0001
GroupAdults -1.5977 0.0813 -19.6533 < 0.0001
Condition0 0.2199 0.0813 2.7048 0.0068
Condition1 0.0023 0.0813 0.0282 0.9775
Condition2 -0.7769 0.0813 -9.5555 < 0.0001
Condition3 -2.1162 0.0813 -26.0309 < 0.0001
Condition4 -4.0241 0.0813 -49.5002 < 0.0001
GroupAdults:Condition0 0.1772 0.1150 1.5411 0.1233
GroupAdults:Condition1 -0.0048 0.1150 -0.0415 0.9669
GroupAdults:Condition2 -0.3509 0.1150 -3.0518 0.0023
GroupAdults:Condition3 -1.1200 0.1150 -9.7422 < 0.0001
GroupAdults:Condition4 -1.7108 0.1150 -14.8800 < 0.0001
B. smooth terms edf Ref.df F-value p-value
s(Time) 8.5739 8.6799 387.3232 < 0.0001
s(Trial) 8.8204 8.9904 1504.0250 < 0.0001
ti(Time,Trial) 12.8285 14.5662 2876.8288 < 0.0001
s(Time,Subject) 149.0468 168.0000 4829.6242 < 0.0001

Partial versus Summed effects

An important distinction in regression model is the distinction between partial effects and summed effects:

Inspection of summed effects

Function plot_smooth

The function plot_smooth visualizes 1-dimensional nonlinear effects as summed effects. The summary of the first plot below shows that this plot is specifically plotting the summed effect of a certain Subject and a certain Trial for the “Adults” age group.

## Summary:
##  * Group : factor; set to the value(s): Adults. 
##  * Condition : factor; set to the value(s): 1. 
##  * Time : numeric predictor; with 30 values ranging from 0.000000 to 2000.000000. 
##  * Trial : numeric predictor; set to the value(s): 0. 
##  * Subject : factor; set to the value(s): c10.

Generally, we would like to generalize over the random effects predictors, such as Subject. We can exclude the random effects from the summed effects plots by the argument rm.ranef=TRUE.

Note the labels at the right side of the plot indicate the chosen settings.

par(mfrow=c(1,2), cex=1.1)
# including random effects:
plot_smooth(m1, view="Time", cond=list(Group="Adults"))
# excluding random effects:
plot_smooth(m1, view="Time", cond=list(Group="Adults"), rm.ranef=TRUE)

The plots below show two different ways for combining the two age groups in one plot:

par(mfrow=c(1,2), cex=1.1)
# version 1:
plot_smooth(m1, view="Time", plot_all="Group", rm.ranef=TRUE, ylim=c(-15,15))
# version 2:
plot_smooth(m1, view="Time", cond=list(Group="Adults"), rm.ranef=TRUE, rug=FALSE, col="red", ylim=c(-15,15))
plot_smooth(m1, view="Time", cond=list(Group="Children"), rm.ranef=TRUE, rug=FALSE, col="cyan", add=TRUE)

Function fvisgam

Two dimensional surfaces are used for visualizing higher order interactions, such as TimexTrial. Again, the argument rm.ranef=TRUE is used for canceling the random effects.

Note the difference in scale on the color legend:

par(mfrow=c(1,2), cex=1.1)
# including random effects:
fvisgam(m1, view=c("Time", "Trial"), cond=list(Group="Adults"))
# excluding random effects:
fvisgam(m1, view=c("Time", "Trial"), cond=list(Group="Adults"), rm.ranef=TRUE)

Setting the color range to the same scale is important when comparing surfaces:

par(mfrow=c(1,2), cex=1.1)
# group children:
fvisgam(m1, view=c("Time", "Trial"), cond=list(Group="Children"), rm.ranef=TRUE, zlim=c(-15,15), main="Children")
# group Adults:
fvisgam(m1, view=c("Time", "Trial"), cond=list(Group="Adults"), rm.ranef=TRUE, zlim=c(-15,15), main="Adults")

Function plot_parametric

The function plot parametric plots the summed effects of the parametric terms.

par(mfrow=c(1,3), cex=1.1)
# all:
plot_parametric(m1,  pred=list(Condition=levels(simdat$Condition),Group=c('Adults', 'Children')), rm.ranef=TRUE)
# children:
plot_parametric(m1,  pred=list(Condition=levels(simdat$Condition)), cond=list(Group="Children"), rm.ranef=TRUE, main="Children")
# adults:
plot_parametric(m1,  pred=list(Condition=levels(simdat$Condition)), cond=list(Group="Adults"), rm.ranef=TRUE, main="Adults")

Inspection of partial effects

Function plot (mgcv)

par(mfrow=c(1,3), cex=1.1)
# first smooth of time:
plot(m1, select=1, shade=TRUE, scale=0, ylim=c(-15,15))
abline(h=0)
# second smooth, of trial:
plot(m1, select=2, shade=TRUE, scale=0, ylim=c(-15,15))
abline(h=0)
# third smooth, nonlienar interaction between Time and Trial:
plot(m1, select=3, scale=0, rug=FALSE)

Function pvisgam

The function pvisgam provides a more colorful version of the contour plot that visualizes the partial interaction between Time and Trial. The function can also plot two dimensional representations of higher order interactions.

Below a comparison:

par(mfrow=c(1,3), cex=1.1)
# Partial effect of nonlinear interaction between Time and Trial:
plot(m1, select=3, scale=0, rug=FALSE)
# Partial effect of nonlinear interaction between Time and Trial:
pvisgam(m1, select=3, view=c("Time", "Trial"), main="pvisgam", dec=1)
# Summed effect of nonlinear interaction between Time and Trial:
fvisgam(m1, view=c("Time", "Trial"), rm.ranef=TRUE, main="fvisgam", dec=1)

Function inspect_random

The function plot plots also random smooths, just as the function inspect_random:

par(mfrow=c(1,2), cex=1.1)
plot(m1, select=4)
abline(h=0)
inspect_random(m1, select=4)

The function inspect_random can also plot selective random effects, or apply a function over the random effects:

par(mfrow=c(1,2), cex=1.1)
# PLOT 1: selection
# ... the adults in black:
inspect_random(m1, select=4, cond=list(Subject=unique(simdat[simdat$Group=="Adults", "Subject"])), col=1, ylim=c(-15, 15), xpd=TRUE)
# ... add children in red:
inspect_random(m1, select=4, cond=list(Subject=unique(simdat[simdat$Group=="Children", "Subject"])), col=2, add=TRUE, xpd=TRUE)

# PLOT 2: averages
# ... of the adults:
inspect_random(m1, select=4, fun=mean, ylim=c(-15,15), cond=list(Subject=unique(simdat[simdat$Group=="Adults", "Subject"])), lwd=2)
# ... of the children:
inspect_random(m1, select=4, fun=mean, cond=list(Subject=unique(simdat[simdat$Group=="Children", "Subject"])), lwd=2, col=2, add=TRUE)

Note that this last plot suggest that children and adults not oly differ overall, but also differ in their pattern over time. So it may be preferred to include the grouping predictor Group in the nonlinear ‘fixed’ (non-random) effects, to capture this difference in the fixed effects terms of the model.

Other functions

Function plot_data

Inspect the data on which the model is based:

# All data, clustered by Group (very small dots):
plot_data(m1, view="Time", split_by="Group", cex=.5)

Function plot_modelfit

Inspect the model fit with plot_modelfit:

simdat$Event <- interaction(simdat$Subject, simdat$Trial)
plot_modelfit(m1, view="Time", event=simdat$Event, n=8)

Function plot_topo

The function plot_topo is a variant on the functions pvisgam and fvisgam for plotting EEG topographies. See the help file of plot_topo for more examples.

data(eeg)
# simple GAMM model:
m1 <- gam(Ampl ~ te(Time, X, Y, k=c(10,5,5), 
    d=c(1,2)), data=eeg)
# get electrode postions:
electrodes <- eeg[,c('X','Y','Electrode')]
electrodes <- as.list( electrodes[!duplicated(electrodes),] )
# topo plot, by default uses fvisgam 
# and automatically selects a timestamp (270ms):
plot_topo(m1, view=c("X", "Y"), el.pos=electrodes, dec=2)