A simulated cross-sectional dataset in which many intersectional strata hold
only a handful of individuals, built with a known intersectional interaction.
With strata this small the split of the between-stratum variance into additive
and interaction parts is weakly identified. An lme4 fit of the crossed-dimensions
decomposition can be singular; a brms fit of it (engine = "brms") gives
the split a posterior median and a credible interval, both of which depend on
the prior on the standard deviations.
Format
A data frame with 240 rows and 6 variables:
- gender
Strata dimension (Women/Men).
- ethnicity
Strata dimension (White/Black/Asian).
- education
Strata dimension (Low/High).
- age_group
Strata dimension (Young/Mid/Older).
- y
A continuous (Gaussian) outcome with residual SD 1. The 36 stratum effects have a variance of 0.31 (a VPC of 0.24), of which 41% is the intersectional interaction.
- event
A binary outcome (No/Yes), ~46% "Yes". On the latent scale its stratum effects have a variance of 1.32 (a VPC of 0.29), again 41% interaction.
attr(maihda_sparse_data, "truth") holds, for each outcome, the intercept
and the 36 stratum effects with their additive, interaction and between-stratum
variances, the interaction share and the VPC.
Details
Four dimensions form 36 intersectional strata with deliberately skewed sizes
(median 6 individuals, 12 of 36 cells below 5, two singletons). The stratum
effects were drawn with the interaction aimed at 40% of the between-stratum
variance; the 36 effects drawn carry 41% on both outcomes (on the latent scale
for event), by the mean-square split of the section Realised strata of
maihda. The interaction share a crossed-dimensions fit reports is a
ratio of variance parameters, and the four dimensions' variances, each estimated
from two or three levels, add up to more than the spread of the main effects
among the 36 strata: fitted to these stratum effects with 1,000 individuals in
every stratum, lme4 puts the share at about 28% on y and 35% on
event, the gap between the two being that of REML and maximum likelihood
(the two sets of effects differ only in scale). In
vignette("bayesian_sparse_maihda") both lme4 fits are singular through a
dimension's variance, not the interaction's, and put the share at about 23% on
y and 3% on event. The brms posterior medians are below 41% as
well; under the priors the vignette sets on the standard deviations, their
credible intervals run from near zero to about 41% or more.
Note
A purely illustrative dataset. The dimension labels are arbitrary and the interaction is constructed, not estimated from any real population – its only purpose is to make the sparse-cell behaviour of the ML and Bayesian estimators visible against a known answer.
Examples
data(maihda_sparse_data)
attr(maihda_sparse_data, "truth")$gaussian$interaction_share # 0.41
#> [1] 0.4056439
# lme4: the interaction share as a point estimate, from a fit that is singular here:
# m_lme4 <- maihda(y ~ 1 + (1 | gender:ethnicity:education:age_group),
# data = maihda_sparse_data, decomposition = "crossed-dimensions")
#
# brms: a posterior median and a credible interval, both resting on the prior:
# m_brms <- maihda(y ~ 1 + (1 | gender:ethnicity:education:age_group),
# data = maihda_sparse_data, decomposition = "crossed-dimensions",
# engine = "brms",
# prior = brms::set_prior("normal(0, 0.5)", class = "sd"))
# See vignette("bayesian_sparse_maihda").