R: Model comparison using parametric bootstrap methods.

pb-modcomp {pbkrtest} | R Documentation

## 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)

### Arguments

`largeModel` | A model object. Can be a linear mixed effects model or generalized linear mixed effects model (as fitted with `lmer()` and `glmer()` function in the lme4 package) or a linear normal model or a generalized linear model. The `largeModel` must be larger than `smallModel` (see below).
`smallModel` | A model of the same type as `largeModel` 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.

### 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 generetated.

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 overfit; 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 sorenh@math.aau.dk

### 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., http://www.jstatsoft.org/v59/i09/

### See Also

`KRmodcomp`, `PBrefdist`

### Examples

    data(beets, package="pbkrtest")
    head(beets)

    ## 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=50)
    anova(sug, sug.h)
    PBmodcomp(sug, sug.s, nsim=50)

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

    anova(sug, sug.h)
    PBmodcomp(sug, sug.h, nsim=50)
    anova(sug, sug.s)
    PBmodcomp(sug, sug.s, nsim=50)

    ## Generalized linear model
    counts    <- c(18,17,15,20,10,20,25,13,12)
    outcome   <- gl(3,1,9)
    treatment <- gl(3,3)
    d.AD      <- data.frame(treatment, outcome, counts)
    head(d.AD)
    glm.D93   <- glm(counts ~ outcome + treatment, family = poisson())
    glm.D93.o <- update(glm.D93, .~. -outcome)
    glm.D93.t <- update(glm.D93, .~. -treatment)

    anova(glm.D93, glm.D93.o, test="Chisq")
    PBmodcomp(glm.D93, glm.D93.o, nsim=50)
    anova(glm.D93, glm.D93.t, test="Chisq")
    PBmodcomp(glm.D93, glm.D93.t, nsim=50)

    ## 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))
    anova(gm1, gm2)
    PBmodcomp(gm1, gm2)

    ## End(Not run)

    ## Not run:
    (fmLarge <- lmer(Reaction ~ Days + (Days|Subject), sleepstudy))
    ## removing Days
    (fmSmall <- lmer(Reaction ~ 1 + (Days|Subject), sleepstudy))
    anova(fmLarge, fmSmall)
    PBmodcomp(fmLarge, fmSmall)

    ## The same test using a restriction matrix
    L <- cbind(0,1)
    PBmodcomp(fmLarge, L)

    ## Vanilla
    PBmodcomp(beet0, beet_no.harv, nsim=1000)

    ## Simulate reference distribution separately:
    refdist <- PBrefdist(beet0, beet_no.harv, nsim=1000)
    PBmodcomp(beet0, beet_no.harv, ref=refdist)

    ## Do computations with multiple processors:
    ## Number of cores:
    (nc <- detectCores())
    ## Create clusters
    cl <- makeCluster(rep("localhost", nc))

    ## Then do:
    PBmodcomp(beet0, beet_no.harv, cl=cl)

    ## Or in two steps:
    refdist <- PBrefdist(beet0, beet_no.harv, nsim=1000, cl=cl)
    PBmodcomp(beet0, beet_no.harv, ref=refdist)

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

    ## End(Not run)

[Package pbkrtest version 0.4-7 Index]