Computes the variance partition coefficient on the response (probability) scale for a binomial MAIHDA model, using the simulation method of Goldstein, Browne & Rasbash (2002). Stratum random effects \(u \sim N(0, \sigma^2_u)\) are simulated and converted to predicted probabilities \(p = g^{-1}(\eta + u)\) (with \(\eta\) the fixed-part linear predictor); the VPC is then the between-stratum variance of \(p\) as a share of the total (between + the binomial within-stratum variance \(\overline{p(1-p)}\)).
Unlike the latent-scale VPC (fixed level-1 variance \(\pi^2/3\) for the logit), the response-scale VPC depends on the overall outcome prevalence, so report it as a complement to – not a replacement for – the latent-scale value.
Arguments
- model
A binomial
maihda_model(lme4 engine) fromfit_maihda.- n_sim
Number of Monte-Carlo draws of the stratum random effect (>= 100). Default 10000.
- seed
Optional integer seed for reproducibility.
Value
An object of class maihda_vpc_response: a list with
estimate, scale = "response", method = "simulation",
n_sim, var_between (the latent-scale between-stratum variance),
var_other (the summed latent-scale variance of any non-stratum random
intercepts, 0 when there are none), lp_fixed (the mean fixed-part
linear predictor), inner_method ("gauss-hermite" when non-stratum
effects are integrated out, else "none") and inner_nodes (the
number of Gauss-Hermite quadrature nodes used for that inner integral, NA
when there are no non-stratum effects).
Details
The fixed part \(\eta\) is collapsed to a single value – the mean linear predictor \(\bar\eta\) over the analytic sample – before the random effect is simulated around it. The result is therefore a VPC evaluated at the mean covariate profile (a conditional-at-mean estimate), not one marginalised over the empirical covariate distribution. For the canonical strata-only (null) model \(\eta\) is constant (the intercept), so the two coincide and the value is exact. For an adjusted model (one with covariates) they can differ, because the inverse link is nonlinear and \(g^{-1}(\bar\eta) \neq \overline{g^{-1}(\eta)}\): read the response-scale VPC from the null model, or interpret an adjusted value as conditional on the average covariate profile rather than as a covariate-averaged (marginal) VPC.
The method is binomial-link agnostic: it maps the simulated stratum effects through whichever inverse link the model uses (logit, probit, cloglog, ...), so a non-logit binomial fit is computed on its own scale rather than rejected. Only the family is required to be binomial.
When the model carries random intercepts beyond the stratum (a
contextual (1 | school) or an explicit (1 | site)), the
simulation integrates over them: the reported estimate is the
stratum share \(Var(E[p \mid u_{stratum}])\) of the total
response-scale variance, where the total includes the variation the other
random effects induce in \(p\) plus the binomial within-variance.
Simulating the stratum effect alone would overstate the stratum share. The
inner integral over the combined non-stratum effect is evaluated by
deterministic Gauss-Hermite quadrature, not an inner Monte-Carlo
sample, so it contributes no simulation error of its own and increasing
n_sim converges the whole estimate (the outer stratum draw is then the
only Monte-Carlo dimension). The number of quadrature nodes used is reported in
inner_nodes.
References
Goldstein, H., Browne, W., & Rasbash, J. (2002). Partitioning variation in multilevel models. Understanding Statistics, 1(4), 223-231.
Examples
# \donttest{
strata <- make_strata(maihda_health_data, vars = c("Gender", "Race"))
d <- maihda_health_data
d$stratum <- strata$data$stratum
m <- fit_maihda(Obese ~ (1 | stratum), data = d, family = "binomial")
#> Binary outcome 'Obese' recoded to 0/1: 'No' = 0 (reference), 'Yes' = 1 (modeled event). Set the factor levels (or supply a 0/1 outcome) to control which level is the event.
maihda_vpc_response(m, seed = 1)
#> Response-scale VPC (simulation method)
#> VPC: 0.0370
#> 10000 simulated stratum effects; between-stratum variance 0.1700 (latent scale).
#>
# }