Provides a compact numeric study of random effects, including: estimated covariance matrix, correlation matrix, per-term standard deviations, empirical mean/SD of posterior modes, shrinkage ratio, and a normality check by Shapiro-Wilk (when applicable).
Details
icc is the intraclass correlation of \(\mathrm{logit}(Y)\) implied
by the fitted model: for two observations of the same group with the
covariates of observation \(i\),
$$\mathrm{ICC}_i = \frac{\mathrm{Var}_b[\psi(a_i) - \psi(b_i)]}
{\mathrm{Var}_b[\psi(a_i) - \psi(b_i)] + E_b[\psi_1(a_i) + \psi_1(b_i)]},$$
where \(a_i(b), b_i(b)\) are the beta shapes with random part
\(b \sim N(0, x_{r,i}^\top D x_{r,i})\), and \(\psi\),
\(\psi_1\) are the digamma and trigamma functions
(\(E[\mathrm{logit}\,Y] = \psi(a) - \psi(b)\),
\(\mathrm{Var}[\mathrm{logit}\,Y] = \psi_1(a) + \psi_1(b)\)). The
expectations over \(b\) use 40-point Gauss-Hermite quadrature and the
reported value is the mean of \(\mathrm{ICC}_i\) over the observations.
The level-1 variance is that of the beta, so the value depends on the
precision; with the logit link (only) and a large precision it approaches
\(\sigma_b^2 / (\sigma_b^2 + \psi_1(a) + \psi_1(b))\). It replaces the
logistic-latent formula \(\sigma_b^2 / (\sigma_b^2 + \pi^2/3)\), which
does not describe a beta response.
The moments of \(\mathrm{logit}(Y)\) over \(b\) can be infinite: with a
probit link when \(\sigma_b^2 \ge 1/2\), with a cloglog link for every
\(\sigma_b > 0\), and numerically with any link when \(\sigma_b\) is
very large. The value would then be set by the clamp of the mean
(\(10^{-5}\)), so icc is NA, with a warning, whenever
\(E_b[\psi_1(a) + \psi_1(b)]\) changes by more than 10% when that clamp
is tightened to \(10^{-8}\).
References
Lopes, J. E. (2023). Modelos de regressao beta para dados de escala. Master's dissertation, Universidade Federal do Parana, Curitiba. URI: https://hdl.handle.net/1884/86624.
Ferrari, S. L. P., and Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31(7), 799–815. doi:10.1080/0266476042000214501
Examples
# \donttest{
dat <- data.frame(
y = c(
0, 5, 20, 50, 75, 90, 100, 30, 60, 45,
10, 40, 55, 70, 85, 25, 35, 65, 80, 15
),
x1 = rep(c(1, 2), 10),
id = factor(rep(1:4, each = 5))
)
prep <- brs_prep(dat, ncuts = 100)
#> brs_prep: n = 20 | exact = 0, left = 1, right = 1, interval = 18
fit <- brsmm(y ~ x1, random = ~ 1 | id, data = prep)
rs <- brsmm_re_study(fit)
print(rs)
#>
#> Random-effects study
#> Groups: 4
#>
#> Random-effects (VarCorr):
#> Name Std.Dev.
#> (Intercept) 0.5339
#>
#> ICC (logit(Y) scale, beta level-1 variance): 0.1807
#>
#> Summary by term (SD_model = model SD; shrinkage = Var(modes)/Var(model)):
#> term sd_model mean_mode sd_mode shrinkage_ratio shapiro_p
#> (Intercept) 0.5339 0.0012 0.446 0.6976 0.824
rs$summary
#> term sd_model mean_mode sd_mode shrinkage_ratio shapiro_p
#> 1 (Intercept) 0.5339184 0.001150958 0.4459553 0.6976425 0.8240296
# }