Following Douglas Bates' advice for those required to produce p-values for fixed-effects terms in a mixed-effects model, I wrote a function to perform a likelihood ratio test for each term in an lmer() object. Bates has championed the notion that calculating the p-values for fixed effects is not trivial. That's because with unbalanced, multilevel data, the denominator degrees of freedom used to penalize certainty are unknown (i.e., we're uncertain about how uncertain we should be). As the author of lme4, the foremost mixed-effects modeling package in 
, he has practiced what he preaches by declining to approximate denominator degrees of freedom as SAS does. Bates contends that alternative inferential approaches make p-values unnecessary. I agree with that position, focusing instead on information criteria and effect sizes. However, as a program evaluator, I recognize that some stakeholders find p-values useful for understanding findings.
A likelihood ratio test can be used to test if the sample size is large. According to Fitzmaurice, Laird, and Ware, twice the difference between the maximized log-likelihoods of two nested models,
, represents the degree to which the reduced model is inadequate. When the sample size is large,
with degrees of freedom equal to the difference in parameters between the full and reduced models,
. The p-value is
.
What if the sample size is not large? A p-value based on will be too liberal (i.e., the type I error rate will exceed the nominal p-value). More conservatively, we might say that
with
numerator and
denominator degrees of freedom. According to Snijders and Bosker, the effective sample size lies somewhere between
total micro-observations (i.e., at level one) and
clusters randomly sampled in earlier stages (i.e., at higher levels). Formally, the effective sample size is
, where
observations are nested within each cluster and intraclass correlation is
. Even if
is large,
(and statistical power) could be quite small if
is small and
is large:
. Unbalanced designs, modeling three or more levels, and cross-level interactions add to our uncertainty about the denominator degrees of freedom.
The function I wrote chews up the lmer() model call and concatenates the frame and model matrix slots, after which it iteratively fits (via maximum likelihood instead of restricted ML) models reduced by each fixed effect and compares them to the full model, yielding a vector of p-values based on . As the example shows, the function can handle shortcut formulas whereby lower order terms are implied by an interaction term. The function doesn't currently handle weights,
glmer() objects, or on-the-fly transformations of the dependent variable [e.g., log(dep.var) ~ ...]. The accuracy of resulting p-values depends on large sample properties, as discussed above, so I don't recommend using the function with small samples. I'm working on another function that will calculate p-values based on the effective sample size estimated from intraclass correlation. I will post that function in a future entry. I'm sure the following function could be improved, but I wanted to go ahead share it with other applied researchers whose audience likes p-values. Please let me know if you see ways to make it better.
p.values.lmer <- function(x) {
summary.model <- summary(x)
data.lmer <- data.frame(model.matrix(x))
names(data.lmer) <- names(fixef(x))
names(data.lmer) <- gsub(pattern=":", x=names(data.lmer), replacement=".", fixed=T)
names(data.lmer) <- ifelse(names(data.lmer)=="(Intercept)", "Intercept", names(data.lmer))
string.call <- strsplit(x=as.character(x@call), split=" + (", fixed=T)
var.dep <- unlist(strsplit(x=unlist(string.call)[2], " ~ ", fixed=T))[1]
vars.fixef <- names(data.lmer)
formula.ranef <- paste("+ (", string.call[[2]][-1], sep="")
formula.ranef <- paste(formula.ranef, collapse=" ")
formula.full <- as.formula(paste(var.dep, "~ -1 +", paste(vars.fixef, collapse=" + "),
formula.ranef))
data.ranef <- data.frame(x@frame[,
which(names(x@frame) %in% names(ranef(x)))])
names(data.ranef) <- names(ranef(x))
data.lmer <- data.frame(x@frame[, 1], data.lmer, data.ranef)
names(data.lmer)[1] <- var.dep
out.full <- lmer(formula.full, data=data.lmer, REML=F)
p.value.LRT <- vector(length=length(vars.fixef))
for(i in 1:length(vars.fixef)) {
formula.reduced <- as.formula(paste(var.dep, "~ -1 +", paste(vars.fixef[-i],
collapse=" + "), formula.ranef))
out.reduced <- lmer(formula.reduced, data=data.lmer, REML=F)
print(paste("Reduced by:", vars.fixef[i]))
print(out.LRT <- data.frame(anova(out.full, out.reduced)))
p.value.LRT[i] <- round(out.LRT[2, 7], 3)
}
summary.model@coefs <- cbind(summary.model@coefs, p.value.LRT)
summary.model@methTitle <- c("\n", summary.model@methTitle,
"\n(p-values from comparing nested models fit by maximum likelihood)")
print(summary.model)
}
library(lme4)
library(SASmixed)
lmer.out <- lmer(strength ~ Program * Time + (Time|Subj), data=Weights)
p.values.lmer(lmer.out)
Yields:
Linear mixed model fit by REML
(p-values from comparing nested models fit by maximum likelihood)
Formula: strength ~ Program * Time + (Time | Subj)
Data: Weights
AIC BIC logLik deviance REMLdev
1343 1383 -661.7 1313 1323
Random effects:
Groups Name Variance Std.Dev. Corr
Subj (Intercept) 9.038486 3.00641
Time 0.031086 0.17631 -0.118
Residual 0.632957 0.79559
Number of obs: 399, groups: Subj, 57
Fixed effects:
Estimate Std. Error t value p.value.LRT
(Intercept) 79.99018 0.68578 116.64000 0.000
ProgramRI 0.07009 1.02867 0.07000 0.944
ProgramWI 1.11526 0.95822 1.16000 0.235
Time -0.02411 0.04286 -0.56000 0.564
ProgramRI:Time 0.12902 0.06429 2.01000 0.043
ProgramWI:Time 0.18397 0.05989 3.07000 0.002
Correlation of Fixed Effects:
(Intr) PrgrRI PrgrWI Time PrRI:T
ProgramRI -0.667
ProgramWI -0.716 0.477
Time -0.174 0.116 0.125
ProgrmRI:Tm 0.116 -0.174 -0.083 -0.667
ProgrmWI:Tm 0.125 -0.083 -0.174 -0.716 0.477
This is very useful. However, I would recommend that readers also consider using a fit index, such as the AIC or BIC, or in the case of comparing nested models using the anova() function. I know that lots of folks rely on p-values for decision making, I am quite skeptical of them as you know especially when compared against some arbitrary 0.05 cutpoint. In fact, I would argue using alpha = 0.05 to base decisions is bad practice indeed. Instead, I recommend examining and exploring alternative hypotheses via model fitting as I feel they more accurately capture the inherent uncertainty underlying statistics. Or at a minimum just presenting the p-values and letting the readers decide. However, I do understand as an applied researcher and someone who works for the government, that interested parties are less interested in the underlying academics and more about whether such and such effect is 'significant' or not.
Cheers,
Chris
This works well for singular random effects.
Can the function be modified to handle nested random effects? At the moment, when I try it with nested random effects, I receive the following error:
Error in names(data.ranef) 'names' attribute [6] must be the same length as the vector [2]
Thank you very much for sharing this; very handy and useful!