Example
Loading the data:
library(itsadug)
library(mgcv)
data(simdat)
# select subset of data to reduce processing time:
select <- 1:18
select <- select[select %% 3 ==0]
simdat <- droplevels(simdat[simdat$Subject %in% c(sprintf("a%02d",select), sprintf("c%02d", select)),])# add start.event and Event columns:
simdat <- start_event(simdat, column="Time", event=c("Subject", "Trial"), label.event="Event")Starting model
For this simulated data set, we would like to investigate whether
children and adults react differently on Condition (for
example, stimulus onset asynchrony, or frequency or some other
continuous measure) on the measurement Y.
If we would like to employ a backward-fitting model comparison procedure we could start with a model like this:
m1 <- bam(Y ~ Group + s(Time, by=Group) + s(Condition, by=Group, k=5) + ti(Time, Condition, by=Group) + s(Time, Subject, bs='fs', m=1, k=5) + s(Event, bs='re'), data=simdat, discrete=TRUE, method="fREML")Note that to keep the model simple for illustration
purposes / time reasons, we left out other effects, such as
Trial, and random smooths over Event. instead,
we account for autocorrelation in the residuals due to the underfit of
the model by including an AR1 model. See
vignette("acf", package"itsadug") for more information.
r1 <- start_value_rho(m1)
m1Rho <- bam(Y ~ Group + s(Time, by=Group) + s(Condition, by=Group, k=5) + ti(Time, Condition, by=Group) + s(Time, Subject, bs='fs', m=1, k=5) + s(Event, bs='re'), data=simdat, method="fREML", AR.start=simdat$start.event, rho=r1)1. Model comparison
To test whether the three-way interaction between Time,
Condition and Group is significant, we can
compare the model with a model that does not include this three-way
interaction:
m2Rho <- bam(Y ~ Group + s(Time, by=Group) + s(Condition, by=Group, k=5) + ti(Time, Condition) + s(Time, Subject, bs='fs', m=1, k=5) + s(Event, bs='re'), data=simdat, method="fREML", AR.start=simdat$start.event, rho=r1)Function compareML
The function compareML compares two models on the basis
of the minimized smoothing parameter selection score specified in the
model, and performes a \(\chi^2\) test
on the difference in scores and the difference in degrees of
freedom.
# make sure that info messages are printed to the screen:
infoMessages('on')
compareML(m1Rho, m2Rho)## m1Rho: Y ~ Group + s(Time, by = Group) + s(Condition, by = Group, k = 5) +
## ti(Time, Condition, by = Group) + s(Time, Subject, bs = "fs",
## m = 1, k = 5) + s(Event, bs = "re")
##
## m2Rho: Y ~ Group + s(Time, by = Group) + s(Condition, by = Group, k = 5) +
## ti(Time, Condition) + s(Time, Subject, bs = "fs", m = 1,
## k = 5) + s(Event, bs = "re")
##
## Chi-square test of fREML scores
## -----
## Model Score Edf Difference Df p.value Sig.
## 1 m2Rho 43415.15 16
## 2 m1Rho 43393.16 19 21.992 3.000 1.520e-09 ***
##
## AIC difference: -13.06, model m1Rho has lower AIC.The following conclusions can be derived from the output:
Model
m1Rhohas a lower fREML score (lower indicates better fit).But model
m1Rhois also more complex: it uses more degrees of freedom (Edf).
Note that Edf in the model comparison
are different from the edf that are presented in the model
summary. The first are reflecting the complexity of the model (number of
model terms, complexity of model terms), and the second are reflecting
the complexity of the smooth or surface pattern (i.e., number of knots
or underlying base functions used).
- Model
m1Rhois preferred, because the difference in fREML is significant given the difference in degrees of freedom: \(\chi^2\)(3)=21.836, p < .001.
Some notes on model comparison
Model comparison procedure provides an indication for the best fitting model, but can rarely used on it’s own for determining significance.
For testing the difference in fixed effects predictors the method fREML does not provide the most reliable test. Rather use ML. However, ML takes longer to run (that is why it is not included here), and penalizes wigglyness more.
An alternative test is AIC, but when an AR1 model is included, AIC does not provide a reliable test. (Like here!)
AIC(m1Rho, m2Rho)## df AIC
## m1Rho 232.6723 86484.49
## m2Rho 244.5558 86497.552. Inspection of the model summary
Beside model comparison the model summary (e.g.,
summary(m1Rho)) could provide useful information on whether
or not a model term is significantly contributing to the model.
To include the summary in a R markdown or knitr report use the
function gamtabs:
gamtabs(m1Rho, type="HTML")The summary provides the following information:
There is an overall difference in Y for children and adults (parametric terms)
The F values / p-values of the ‘fixed’ effects smooth terms indicate that all these smooth terms are significantly different from 0, so each line or surface is significantly wiggly.
However, we can NOT conclude that the lines or surfaces are different from each other. This is only possible when we would use difference smooths or tensors, with ordered factors or binomial predictors. See below for an example.
For the random effects the statistics indicates whether or not these terms contribute to the model (
s(Event)) or not (s(Time,Subject)).
Using ordered factors (advanced)
It is possible to change the contrasts for grouping predictors in
mgcv so that the smooth terms represent
differences with the reference level, similar to the treatment
coding used in lmer or in the summary of parametric terms
in GAMMs. The trick is to first convert the factors to ordered
factors so that gam() and bam() won’t use
the default contrast coding.
Here’s an example:
simdat$OFGroup <- as.ordered(simdat$Group)
contrasts(simdat$OFGroup) <- "contr.treatment"
contrasts(simdat$OFGroup)## Adults
## Children 0
## Adults 1Note that in the case of using ordered factors we need to include the reference curves or surfaces as well.
m1Rho.OF <- bam(Y ~ OFGroup + s(Time) + s(Time, by=OFGroup) + s(Condition, k=5) + s(Condition, by=OFGroup, k=5) + ti(Time, Condition) + ti(Time, Condition, by=OFGroup) + s(Time, Subject, bs='fs', m=1, k=5) + s(Event, bs='re'), data=simdat, method="fREML", AR.start=simdat$start.event, rho=r1)With the ordered factors suddenly the lines
s(Time):OFGroupAdults and similar lines represent the
difference between the adults and the reference group, the
children. When the smooth term is significant, the difference smooth is
is significantly different from zero. So that means that the two groups
are different from each other:
gamtabs(m1Rho.OF, type="HTML")In summary, with continuous predictors or ordered factors we can use the summary startistics to determine the difference of smooth terms.
The function report_stats describes how one could report
the smooth terms in the text of an article:
report_stats(m1Rho.OF)3. Visual inspection of the model’s estimates
Function plot_diff
The function plot_diff allows to plot the (1
dimensional) estimated difference between two conditions. The argument
rm.ranef=TRUE indicates that random effects should be
excluded first, and the argument cond can be used to
specify values for other predictors.
The plots below visualize the difference between adults and children.
par(mfrow=c(1,2))
# PLOT 1:
plot_diff(m1Rho, view="Time", comp=list(Group=c("Adults", "Children")), cond=list(Condition=1), rm.ranef=TRUE, ylim=c(-15,15))
plot_diff(m1Rho, view="Time", comp=list(Group=c("Adults", "Children")), cond=list(Condition=4), add=TRUE, col='red')
# add legend:
legend('bottom', legend=c("Condition=1", "Condition=4"), col=c(1,2), lwd=1, cex=.75, bty='n')
# PLOT 2:
plot_diff(m1Rho, view="Condition", comp=list(Group=c("Adults", "Children")), cond=list(Time=1000), rm.ranef=TRUE, ylim=c(-15,15))
plot_diff(m1Rho, view="Condition", comp=list(Group=c("Adults", "Children")), cond=list(Time=2000), add=TRUE, col='red')
# add legend:
legend('bottom', legend=c("Time=1000", "Time=2000"), col=c(1,2), lwd=1, cex=.75, bty='n')
Function plot_diff2
The function plot_diff allows to plot the (2
dimensional) estimated difference between two conditions. The argument
rm.ranef=TRUE indicates that random effects should be
excluded first, and the argument cond can be used to
specify values for other predictors.
The plots below visualize the difference between adults and children.
par(mfrow=c(1,2), cex=1.1)
plot_diff2(m1Rho, view=c("Time", "Condition"), comp=list(Group=c("Adults", "Children")), zlim=c(-15,15), se=0, rm.ranef=TRUE)## Warning in gradientLegend(zlim, n.seg = 3, pos = 0.875, dec = dec, color =
## color, : Increase right margin to fit labels or decrease the number of decimals,
## see help(gradientLegend).# with CI:
plot_diff2(m1Rho, view=c("Time", "Condition"), comp=list(Group=c("Adults", "Children")), zlim=c(-15,15), se=1.96, rm.ranef=TRUE, show.diff = TRUE)## Warning in gradientLegend(zlim, n.seg = 3, pos = 0.875, dec = dec, color =
## color, : Increase right margin to fit labels or decrease the number of decimals,
## see help(gradientLegend).