pb_modcomp {pbkrtest}

Model comparison using parametric bootstrap methods.

Description

Model comparison of nested models using parametric bootstrap methods. Implemented for some commonly applied model types.

Usage

PBmodcomp(
  largeModel,
  smallModel,
  nsim = 1000,
  ref = NULL,
  seed = NULL,
  cl = NULL,
  details = 0
)

## S3 method for class 'merMod'
PBmodcomp(
  largeModel,
  smallModel,
  nsim = 1000,
  ref = NULL,
  seed = NULL,
  cl = NULL,
  details = 0
)

## S3 method for class 'lm'
PBmodcomp(
  largeModel,
  smallModel,
  nsim = 1000,
  ref = NULL,
  seed = NULL,
  cl = NULL,
  details = 0
)

seqPBmodcomp(largeModel, smallModel, h = 20, nsim = 1000, cl = 1)

Arguments

largeModel

An lmer model

smallModel

An lmer model or a restriction matrix

nsim

The number of simulations to form the reference distribution.

ref

Vector containing samples from the reference distribution. If NULL, this vector will be generated using PBrefdist().

seed

A seed that will be passed to the simulation of new datasets.

cl

A vector identifying a cluster; used for calculating the reference distribution using several cores. See examples below.

details

The amount of output produced. Mainly relevant for debugging purposes.

h

For sequential computing for bootstrap p-values: The number of extreme cases needed to generate before the sampling process stops.

Details

The model object must be fitted with maximum likelihood (i.e. with REML=FALSE). If the object is fitted with restricted maximum likelihood (i.e. with REML=TRUE) then the model is refitted with REML=FALSE before the p-values are calculated. Put differently, the user needs not worry about this issue.

Under the fitted hypothesis (i.e. under the fitted small model) nsim samples of the likelihood ratio test statistic (LRT) are generated.

Then p-values are calculated as follows:

LRT: Assuming that LRT has a chi-square distribution.

PBtest: The fraction of simulated LRT-values that are larger or equal to the observed LRT value.

Bartlett: A Bartlett correction is of LRT is calculated from the mean of the simulated LRT-values

Gamma: The reference distribution of LRT is assumed to be a gamma distribution with mean and variance determined as the sample mean and sample variance of the simulated LRT-values.

F: The LRT divided by the number of degrees of freedom is assumed to be F-distributed, where the denominator degrees of freedom are determined by matching the first moment of the reference distribution.

Note

It can happen that some values of the LRT statistic in the reference distribution are negative. When this happens one will see that the number of used samples (those where the LRT is positive) are reported (this number is smaller than the requested number of samples).

In theory one can not have a negative value of the LRT statistic but in practice on can: We speculate that the reason is as follows: We simulate data under the small model and fit both the small and the large model to the simulated data. Therefore the large model represents - by definition - an over fit; the model has superfluous parameters in it. Therefore the fit of the two models will for some simulated datasets be very similar resulting in similar values of the log-likelihood. There is no guarantee that the the log-likelihood for the large model in practice always will be larger than for the small (convergence problems and other numerical issues can play a role here).

To look further into the problem, one can use the PBrefdist() function for simulating the reference distribution (this reference distribution can be provided as input to PBmodcomp()). Inspection sometimes reveals that while many values are negative, they are numerically very small. In this case one may try to replace the negative values by a small positive value and then invoke PBmodcomp() to get some idea about how strong influence there is on the resulting p-values. (The p-values get smaller this way compared to the case when only the originally positive values are used).

Author(s)

Søren Højsgaard [e-mail]

References

Ulrich Halekoh, Søren Højsgaard (2014)., A Kenward-Roger Approximation and Parametric Bootstrap Methods for Tests in Linear Mixed Models - The R Package pbkrtest., Journal of Statistical Software, 58(10), 1-30., https://www.jstatsoft.org/v59/i09/

See Also

KRmodcomp, PBrefdist

Examples

## Not run:  
(fm0 <- lmer(Reaction ~ (Days|Subject), sleepstudy))
(fm1 <- lmer(Reaction ~ Days + (Days|Subject), sleepstudy))
(fm2 <- lmer(Reaction ~ Days + I(Days^2) + (Days|Subject), sleepstudy))

NSIM <- 50 ## Simulations in parametric bootstrap; default is 1000.

## Test for no effect of Days in fm1, i.e. test fm0 under fm1
PBmodcomp(fm1, "Days", cl=1, nsim=NSIM)
PBmodcomp(fm1, ~.-Days, cl=1, nsim=NSIM)
L1 <- cbind(0, 1) 
## PBmodcomp(fm1, L1, cl=1, nsim=NSIM) ## FIXME
PBmodcomp(fm1, fm0, cl=1, nsim=NSIM)
anova(fm1, fm0)

## Test for no effect of Days and Days-squared in fm2, i.e. test fm0 under fm2
PBmodcomp(fm2, "(Days+I(Days^2))", cl=1, nsim=NSIM)
PBmodcomp(fm2, ~. - Days - I(Days^2), cl=1, nsim=NSIM)
L2 <- rbind(c(0, 1, 0), c(0, 0, 1))
## PBmodcomp(fm2, L2, cl=1, nsim=NSIM) ## FIXME
PBmodcomp(fm2, fm0, cl=1, nsim=NSIM)
anova(fm2, fm0)


## Test for no effect of Days-squared in fm2, i.e. test fm1 under fm2
PBmodcomp(fm2, "I(Days^2)", cl=1, nsim=NSIM)
PBmodcomp(fm2, ~. - I(Days^2), cl=1, nsim=NSIM)
L3 <- rbind(c(0, 0, 1))
## PBmodcomp(fm2, L3, cl=1, nsim=NSIM) ## FIXME
PBmodcomp(fm2, fm1, cl=1, nsim=NSIM)
anova(fm2, fm1)


## Linear normal model:
sug <- lm(sugpct ~ block + sow + harvest, data=beets)
sug.h <- update(sug, .~. -harvest)
sug.s <- update(sug, .~. -sow)

PBmodcomp(sug, "harvest", nsim=NSIM, cl=1)
PBmodcomp(sug, ~. - harvest, nsim=NSIM, cl=1)
PBmodcomp(sug, sug.h, nsim=NSIM, cl=1)
anova(sug, sug.h)

## Generalized linear model
mm <- glm(ndead/ntotal ~ sex + log(dose), family=binomial, weight=ntotal, data=budworm)
mm0 <- update(mm, .~. -sex)

### Test for no effect of sex
PBmodcomp(mm, "sex", cl=1, nsim=NSIM)
PBmodcomp(mm, ~.-sex, cl=1, nsim=NSIM)
## PBmodcomp(mm, cbind(0, 1, 0), nsim=NSIM): FIXME
PBmodcomp(mm, mm0, cl=1, nsim=NSIM)
anova(mm, mm0, test="Chisq")

## End(Not run)

## Generalized linear mixed model (it takes a while to fit these)

## Not run: 
(gm1 <- glmer(cbind(incidence, size - incidence) ~ period + (1 | herd),
              data = cbpp, family = binomial))
(gm2 <- update(gm1, .~.-period))

PBmodcomp(gm1, "period", nsim=NSIM)
PBmodcomp(gm1, ~. -period, nsim=NSIM)
PBmodcomp(gm1, gm2, nsim=NSIM)
anova(gm1, gm2)

## End(Not run)


## Not run: 
## Linear mixed effects model:
sug   <- lmer(sugpct ~ block + sow + harvest + (1|block:harvest),
              data=beets, REML=FALSE)
sug.h <- update(sug, .~. -harvest)
sug.s <- update(sug, .~. -sow)

anova(sug, sug.h)
PBmodcomp(sug, sug.h, nsim=NSIM, cl=1)
PBmodcomp(sug, "harvest", nsim=NSIM, cl=1)

anova(sug, sug.s)
PBmodcomp(sug, sug.s, nsim=NSIM, cl=1)
PBmodcomp(sug, "sow", nsim=NSIM, cl=1)

## Simulate reference distribution separately:
refdist <- PBrefdist(sug, sug.h, nsim=1000, cl=1)
refdist <- PBrefdist(sug, "harvest", nsim=1000, cl=1)
refdist <- PBrefdist(sug, ~.-harvest, nsim=1000, cl=1)

## Do computations with multiple processors:
## Number of cores:

(nc <- detectCores())
## Create clusters
cl <- makeCluster(rep("localhost", nc))

## Then do:
refdist <- PBrefdist(sug, sug.h, nsim=1000, cl=cl)

## It is recommended to stop the clusters before quitting R:
stopCluster(cl)

## End(Not run)