Fixed-effect and DerSimonian-Laird random-effects pooling, with what the truncation does
Source:R/meta_pooling.R
morie_meta_random_effects.RdPools k site or study estimates with known variances. The
DerSimonian-Laird between-site variance is a truncation,
\(\hat\tau^2 = \max(0, (Q - (k-1))/c)\), non-negative and zero exactly
when \(Q \le k - 1\) (Research.P13.tauDL_nonneg,
tauDL_eq_zero_iff). For any \(\tau^2 \ge 0\) the random-effects
variance of the pooled estimate is at least the fixed-effect variance, with
equality only at \(\tau^2 = 0\) (re_var_ge, re_var_eq_iff):
the random-effects interval is never narrower, and the two methods agree
in a finite sample only when the estimate is truncated. Because
\(E[\max(X, 0)] > E[X]\) whenever \(X > 0\) has positive probability
(truncation_bias, pos_part_pos), the estimator has positive
expectation under homogeneity, where \(E[Q] = k - 1\)
(dl_biased_under_homogeneity); see
morie_meta_dl_bias for the size of that bias.
Value
A list with k, fixed and random (each with
estimate, variance, se, ci), Q,
df (\(k-1\)), c, tau2 (DerSimonian-Laird),
truncated (whether \(Q \le k-1\) forced \(\hat\tau^2 = 0\)),
variance_ratio (random over fixed, at least 1), weights
(fixed and random, normalised) and theorems.
Examples
est <- c(-0.25, -0.10, -0.40, 0.05, -0.30)
v <- c(0.010, 0.020, 0.015, 0.030, 0.012)
m <- morie_meta_random_effects(est, v)
c(fixed = m$fixed$estimate, random = m$random$estimate, tau2 = m$tau2, ratio = m$variance_ratio)
#> fixed random tau2 ratio
#> -0.240000000 -0.230962714 0.006967742 1.506184974