profile-methods: Profile method for merMod objects

Description

Methods for profile() of [ng]lmer fitted models.

The log() method and the more flexible logProf() utility transform a lmer profile into one where logarithms of standard deviations are used, while varianceProf converts from the standard-deviation to the variance scale; see Details.

Usage

# S3 method for merMod
profile(fitted, which = NULL, alphamax = 0.01,
	maxpts = 100, delta = NULL,
    delta.cutoff = 1/8, verbose = 0, devtol = 1e-09,
    devmatchtol = 1e-5,
    maxmult = 10, startmethod = "prev", optimizer = NULL,
    control=NULL, signames = TRUE,
    parallel = c("no", "multicore", "snow"),
    ncpus = getOption("profile.ncpus", 1L), cl = NULL,
    prof.scale = c("sdcor","varcov"),
    ...)
# S3 method for thpr
as.data.frame (x, ...)
# S3 method for thpr
log(x, base = exp(1))
logProf(x, base = exp(1), ranef = TRUE,
           sigIni = if(ranef) "sig" else "sigma")
varianceProf(x, ranef = TRUE)

Value

profile(<merMod>) returns an object of S3 class

"thpr",

which is data.frame-like. Notable methods for such a profile object

confint(), which returns the confidence intervals based on the profile, and three plotting methods (which require the lattice package),

xyplot, densityplot, and

splom.

In addition, the

log() (see above) and as.data.frame()

methods can transform "thpr" objects in useful ways.

Arguments

fitted

a fitted model, e.g., the result of lmer(..).

which

NULL value, integer or character vector indicating which parameters to profile: default (NULL) is all parameters. For integer, i.e., indexing, the parameters are ordered as follows:

(1): random effects (theta) parameters; these are ordered as in getME(.,"theta"), i.e., as the lower triangle of a matrix with standard deviations on the diagonal and correlations off the diagonal.

(2)

residual standard deviation (or scale parameter for GLMMs where appropriate).

(3)

fixed effect (beta) parameters.

Alternatively, which may be a character, containing "beta_" or "theta_" denoting the fixed or random effects parameters, respectively, or also containing parameter names, such as ".sigma" or "(Intercept)".

alphamax

a number in \((0,1)\), such that 1 - alphamax is the maximum alpha value for likelihood ratio confidence regions; used to establish the range of values to be profiled.

maxpts

maximum number of points (in each direction, for each parameter) to evaluate in attempting to construct the profile.

delta

stepping scale for deciding on next point to profile. The code uses the local derivative of the profile at the current step to establish a change in the focal parameter that will lead to a step of delta on the square-root-deviance scale. If NULL, the delta.cutoff parameter will be used to determine the stepping scale.

delta.cutoff

stepping scale (see delta) expressed as a fraction of the target maximum value of the profile on the square-root-deviance scale. Thus a delta.cutoff setting of 1/n will lead to a profile with approximately 2*n calculated points for each parameter (i.e., n points in each direction, below and above the estimate for each parameter).

verbose

level of output from internal calculations.

devtol

tolerance for fitted deviances less than baseline (supposedly minimum) deviance.

devmatchtol

tolerance for match between original deviance computation and value returned from auxiliary deviance function

maxmult

maximum multiplier of the original step size allowed, defaults to 10.

startmethod

method for picking starting conditions for optimization (STUB).

optimizer

(character or function) optimizer to use (see lmer for details); default is to use the optimizer from the original model fit.

control

a list of options controlling the profiling (see lmerControl): default is to use the control settings from the original model fit.

signames

logical indicating if abbreviated names of the form .sigNN should be used; otherwise, names are more meaningful (but longer) of the form (sd|cor)_(effects)|(group). Note that some code for profile transformations (e.g., log() or varianceProf) depends on signames==TRUE.

...

potential further arguments for various methods.

an object of class thpr (i.e., output of profile)

base

the base of the logarithm. Defaults to natural logarithms.

ranef

logical indicating if the sigmas of the random effects should be log() transformed as well. If false, only \(\sigma\) (standard deviation of errors) is transformed.

sigIni

character string specifying the initial part of the sigma parameters to be log transformed.

parallel

The type of parallel operation to be used (if any). If missing, the default is taken from the option "profile.parallel" (and if that is not set, "no").

ncpus

integer: number of processes to be used in parallel operation: typically one would choose this to be the number of available CPUs.

An optional parallel or snow cluster for use if parallel = "snow". If not supplied, a cluster on the local machine is created for the duration of the profile call.

prof.scale

whether to profile on the standard deviation-correlation scale ("sdcor") or on the variance-covariance scale ("varcov")

Details

The log method and the more flexible logProf() function transform the profile into one where \(\log(\sigma)\) is used instead of \(\sigma\). By default all sigmas including the standard deviations of the random effects are transformed i.e., the methods return a profile with all of the .sigNN parameters replaced by .lsigNN. If ranef is false, only ".sigma", the standard deviation of the errors, is transformed (as it should never be zero, whereas random effect standard deviations (.sigNN) can be reasonably be zero).
The forward and backward splines for the log-transformed parameters are recalculated. Note that correlation parameters are not handled sensibly at present (i.e., they are logged rather than taking a more applicable transformation such as an arc-hyperbolic tangent, atanh(x)=\(\log((1+x)/(1-x))/2\)).

The varianceProf function works similarly, including non-sensibility for correlation parameters, by squaring all parameter values, changing the names by appending sq appropriately (e.g. .sigNN to .sigsqNN). Setting prof.scale="varcov" in the original profile() call is a more computationally intensive, but more correct, way to compute confidence intervals for covariance parameters.

Methods for function profile (package stats), here for profiling (fitted) mixed effect models.

Examples

Run this code

fm01ML <- lmer(Yield ~ 1|Batch, Dyestuff, REML = FALSE)
system.time(
  tpr  <- profile(fm01ML, optimizer="Nelder_Mead", which="beta_")
)## fast; as only *one* beta parameter is profiled over -> 0.09s (2022)

## full profiling (default which means 'all') needs longer:
system.time( tpr  <- profile(fm01ML, signames=FALSE))
## ~ 0.26s (2022) + possible warning about convergence
(confint(tpr) -> CIpr)
# too much precision (etc). but just FYI:
 trgt <- array(c(12.19854, 38.22998, 1486.451,
                 84.06305, 67.6577,  1568.548), dim = 3:2)
 stopifnot(all.equal(trgt, unname(CIpr), tol = .0001)) # had 3.1e-7

if (interactive()) {
 library("lattice")
 xyplot(tpr)
 xyplot(tpr, absVal=TRUE) # easier to see conf.int.s (and check symmetry)
 xyplot(tpr, conf = c(0.95, 0.99), # (instead of all five 50, 80,...)
        main = "95% and 99% profile() intervals")
 xyplot(logProf(tpr, ranef=FALSE),
        main = expression("lmer profile()s"~~ log(sigma)*" (only log)"))
 densityplot(tpr, main="densityplot( profile(lmer(..)) )")
 densityplot(varianceProf(tpr), main=" varianceProf( profile(lmer(..)) )")
 splom(tpr)
 splom(logProf(tpr, ranef=FALSE))
 doMore <- lme4:::testLevel() > 2 
 if(doMore) { ## not typically, for time constraint reasons
   ## Batch and residual variance only
   system.time(tpr2 <- profile(fm01ML, which=1:2)) # , optimizer="Nelder_Mead" gives warning
   print( xyplot(tpr2) )
   print( xyplot(log(tpr2)) )# log(sigma) is better
   print( xyplot(logProf(tpr2, ranef=FALSE)) )

   ## GLMM example
   gm1 <- glmer(cbind(incidence, size - incidence) ~ period + (1 | herd),
               data = cbpp, family = binomial)
   ## running ~ 10 seconds on a modern machine {-> "verbose" while you wait}:
   print( system.time(pr4 <- profile(gm1, verbose=TRUE)) )
   print( xyplot(pr4, layout=c(5,1), as.table=TRUE) )
   print( xyplot(log(pr4), absVal=TRUE) ) # log(sigma_1)
   print( splom(pr4) )
   print( system.time( # quicker: only sig01 and one fixed effect
       pr2 <- profile(gm1, which=c("theta_", "period2"))))
   print( confint(pr2) )
   ## delta..: higher underlying resolution, only for 'sigma_1':
   print( system.time(
       pr4.hr <- profile(gm1, which="theta_", delta.cutoff=1/16)))
   print( xyplot(pr4.hr) )
 }
} # only if interactive()