Calculate the expected rank of random coefficients that account for uncertainty.

Description

expectedRank calculates the expected rank and the percentile expected rank of any random term in a merMod object. A simple ranking of the estimated random effects (as produced by ranef) is not satisfactory because it ignores any amount of uncertainty.

Usage

expectedRank(merMod, groupFctr = NULL, term = NULL)

Arguments

Details

Inspired by Lingsma et al. (2010, see also Laird and Louis 1989), expectedRank sums the probability that each level of the grouping factor is greater than every other level of the grouping factor, similar to a two-sample t-test.

The formula for the expected rank is:

ExpectedRank_i = 1 + ∑ φ((θ_i - θ_k) / √(var(θ_i)+var(θ_k))

where φ is the standard normal distribution function, θ is the estimated random effect and var(θ) is the posterior variance of the estimated random effect. We add one to the sum so that the minimum rank is one instead of zero so that in the case where there is no overlap between the variances of the random effects (or if the variances are zero), the expected rank equals the actual rank. The ranks are ordered such that the winners have ranks that are greater than the losers.

The formula for the percentile expected rank is:

100 * (ExpectedRank_i - 0.5) / N_grps

where N_grps is the number of grouping factor levels. The percentile expected rank can be interpreted as the fraction of levels that score at or below the given level.

NOTE: expectedRank will only work under conditions that lme4::ranef will work. One current example of when this is not the case is for models when there are multiple terms specified per factor (e.g. uncorrelated random coefficients for the same term, e.g. lmer(Reaction ~ Days + (1 | Subject) + (0 + Days | Subject), data = sleepstudy))

Value

A data.frame with the following five columns:

Column 1

The original grouping factor

Column 2

The estimated random effects (from lme4::ranef(, condVar=TRUE)); name taken from term.

Column 3

The posterior variance of the estimate random effect (from lme4::ranef(, condVar=TRUE)); named "term"_var.

ER

The expected rank.

pctER

The percentile expected rank.

References

Laird NM and Louis TA. Empirical Bayes Ranking Methods. Journal of Education Statistics. 1989;14(1)29-46. Available at http://www.jstor.org/stable/1164724.

Lingsma HF, Steyerberg EW, Eijkemans MJC, et al. Comparing and ranking hospitals based on outcome: results from The Netherlands Stroke Survey. QJM: An International Journal of Medicine. 2010;103(2):99-108. doi:10.1093/qjmed/hcp169

Examples

#For a one-level random intercept model
require(lme4)
m1 <- lmer(Reaction ~ Days + (1 | Subject), sleepstudy)
(m1.er <- expectedRank(m1))

#For a one-level random intercept model with multiple random terms
require(lme4)
m2 <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy)
#ranked by the random slope on Days
(m2.er1 <- expectedRank(m2, term="Days"))
#ranked by the random intercept
(m2.er2 <- expectedRank(m2, term="int"))

## Not run: 
#For a two-level model with random intercepts
require(lme4)
m3 <- lmer(y ~ service * dept + (1|s) + (1|d), InstEval)
#Ranked by the random intercept on 's'
(m3.er1 <- expectedRank(m3, groupFctr="s", term="Intercept"))

## End(Not run)

Results