spaMM (version 3.11.3)

LRT: Likelihood ratio test of fixed effects.

Description

LRT performs a likelihood ratio (LR) test between two model fits, the ``full'' and the ``null'' model fits. The anova method for fit objects from spaMM is an alias for LRT.

If the models differ only in their fixed effects, a standard test based on the asymptotic chi-square distribution is performed, with number of degrees of freedom determined by the function. In addition, parametric bootstrap p-values can be computed, either using the raw bootstrap distribution of the likelihood ratio, or a bootstrap estimate of the Bartlett correction of the LR statistic. This function differs from fixedLRT in its arguments (model fits for LRT, but all arguments required to fit the models for fixedLRT), and in the format of its return value. By conceptual drift, anova works as an alias for LRT.

If the two models differ in their random effects, a bootstrap test may be performed. Either the full and null model can be told apart by a simple comparison of the model formulas, or the model with the lower likelihood is assumed to be the null one (the latter is subject to numerical uncertainties when both fit are equivalent). No number of degrees of freedom is deduced, so no asymptotic test is performed.

If the two models differ in both their fixed and random components, the same bootstrap test can be performed (see Examples), but the procedure further checks that the same model is nested in the other for both components. This requires that a simple comparison of the model formulas is sufficient to assess this for random effects, and may therefore fail).

Usage

# S3 method for HLfit
anova(object, object2, ..., method="")
LRT(object, object2, boot.repl = 0, resp_testfn = NULL, 
    simuland = eval_replicate, 
    #     many further arguments can be passed to spaMM_boot via the '...'
    #     These include arguments for parallel computations, such as
    # nb_cores, fit_env,
    #     as well as other named arguments and spaMM_boot's own '...'
    ...)

Arguments

object,object2

Two models fits being compared (their order does not matter).

boot.repl

the number of bootstrap replicates.

resp_testfn

See argument resp_testfn of spaMM_boot

method

For development purposes, not documented.

simuland

a function, passed to spaMM_boot. See argument eval_replicate for default value and requirements.

…

Further arguments, passed to spaMM_boot (e.g., for parallelization).

Value

An object of class fixedLRT, actually a list with as-yet unstable format, but here with typical elements (depending on the options)

fullfit

the HLfit object for the full model;

nullfit

the HLfit object for the null model;

basicLRT

A data frame including values of the likelihood ratio chi2 statistic, its degrees of freedom, and the p-value;

and, if a bootstrap was performed:

rawBootLRT

A data frame including values of the likelihood ratio chi2 statistic, its degrees of freedom, and the raw bootstrap p-value;

BartBootLRT

A data frame including values of the Bartlett-corrected likelihood ratio chi2 statistic, its degrees of freedom, and its p-value;

bootInfo

a list with the following elements:

bootreps

A table of fitted likelihoods for bootstrap replicates;

meanbootLRT

The mean likelihood ratio chi-square statistic for bootstrap replicates;

%- item

Details

A raw bootstrap p-value can be computed from the simulated distribution as (1+sum(t >= t0))/(N+1) where t0 is the original likelihood ratio, t the vector of bootstrap replicates and N its length. See Davison & Hinkley (1997, p. 141) for discussion of the adjustments in this formula. However, a computationally more economical use of the bootstrap is to provide a Bartlett correction for the likelihood ratio test in small samples. According to this correction, the mean value \(m\) of the likelihood ratio statistic under the null hypothesis is computed (here estimated by a parametric bootstrap) and the original LR statistic is multiplied by \(n/m\) where \(n\) is the number of degrees of freedom of the test.

If random effects are tested, only the raw p-value is computed. Its null distribution may include a probability mass in 1 (the discussion in Details of get_RLRsim_args applies).

References

Bartlett, M. S. (1937) Properties of sufficiency and statistical tests. Proceedings of the Royal Society (London) A 160: 268-282.

Davison A.C., Hinkley D.V. (1997) Bootstrap methods and their applications. Cambridge Univ. Press, Cambridge, UK.

See Also

See also fixedLRT and get_RLRsim_args.

Examples

# NOT RUN {
data("wafers")
## Gamma GLMM with log link
m1 <- HLfit(y ~X1+X2+X1*X3+X2*X3+I(X2^2)+(1|batch),family=Gamma(log),
          resid.model = ~ X3+I(X3^2) ,data=wafers,method="ML")
m2 <- update(m1,formula.= ~ . -I(X2^2))
anova(m1,m2)

# Using resp_testfn argument:
# }
# NOT RUN {
set.seed(1L)
d <- data.frame(success = rbinom(10, size = 1, prob = 0.9), x = 1:10)
xx <- cbind(1,d$x)
table(d$success)
m_x <- fitme(success ~ x, data = d, family = binomial())
m_0 <- fitme(success ~ 1, data = d, family = binomial())
anova(m_x, m_0, boot.repl = 100,
      resp_testfn=function(y) {! is_separated(xx,as.numeric(y),verbose=FALSE)})
# }
# NOT RUN {
## Models differing both in fixed and random effects:
if (spaMM.getOption("example_maxtime")>11) { 
 set.seed(123)
 dat <- data.frame(g = rep(1:10, e = 10), x = (x<-rnorm(100)), 
                   y = 0.1 * x + rnorm(100))
 m <- fitme(y ~ x + (1|g), data=dat)
 m0 <- fitme(y ~ 1, data=dat) 
 (bootpval <- LRT(m,m0, boot.repl = 199L)$rawBootLRT$p_value)
 ## See help("get_RLRsim_args") for a fast and accurate test procedure
}

# }