Roever 2020 - Pediatric Liver Transplant Worked Example

Summary

Six controlled studies of interleukin-2 receptor antagonists (IL-2RA) after pediatric liver transplantation (Crins et al. 2014) are meta-analysed with bayesmeta. With $\mu \sim N(0,4^2)$ and $\tau \sim \text{half-normal}(0.5)$ the posterior median log-OR is $-1.57$ (95% interval $[-2.23,-0.93]$), posterior probability of a non-beneficial effect $6\times 10^{-5}$, and an implied median risk difference of $-0.33$ at 50% control risk. A grid-based Python reproduction matches the printed summary to three or more decimals.

Overview

Question. Does IL-2RA prevent acute rejection (AR) events after pediatric liver transplantation? Treatment strategies and responses differ between adults and children, so a pediatric-specific systematic review of six controlled studies was performed.

Main Content

Data

Generic 2x2 table: event yes/no for treatment ($a,b$, $n_1=a+b$) and control ($c,d$, $n_2$). First study (Heffron 2003): IL-2RA 14 events, 47 none (61); control 15 events, 5 none (20). Odds ratio $\frac{14/47}{15/5}=0.0993$; $y_1=\log (0.0993)=-2.31$, ${\sigma }_1=\sqrt{1/14+1/47+1/15+1/5}=0.60$ (the paper writes the SE formula over the 2x2 counts).

$i$	Study	IL-2RA events/total	Control events/total	$y_i$ (log-OR)	${\sigma }_i$	Randomised
1	Heffron 2003	14/61	15/20	-2.31	0.60	yes
2	Gibelli 2004	16/28	19/28	-0.46	0.56	no
3	Schuller 2005	3/18	8/12	-2.30	0.88	no
4	Ganschow 2005	9/54	29/54	-1.76	0.46	no
5	Spada 2006	4/36	11/36	-1.26	0.64	yes
6	Gras 2008	0/50	3/34	-2.42	1.53	no

Study 6 has a zero cell, so a continuity correction (0.5 added to each cell) was applied (Sweeting et al. 2004). Effect sizes via metafor::escalc(measure="OR", ai, n1i, ci, n2i, ...) return yi and vi = squared SE; bayesmeta() needs SEs, hence sigma=sqrt(vi) (Viechtbauer 2010 - Effect Size Measures and escalc). A bayesmeta() call can also take the escalc data frame directly.

The analysis

bayesmeta(y, sigma, labels, mu.prior.mean=0, mu.prior.sd=4, tau.prior=function(t) dhalfnormal(t, scale=0.5)). The effect prior is the vague unit-information prior; the half-normal(0.5) puts about 95% of prior mass on $\tau \le 0.98$ and about 5% on “fairly extreme” values.

	$\tau$	$\mu$
ML joint	0.3258	-1.5783
ML marginal	0.4644	-1.5780
MAP joint	0.0869	-1.5594
MAP marginal	0.2453	-1.5691

Marginal posterior summary:

	$\tau$	$\mu$	${\theta }_{k+1}$ (prediction)
mode	0.2453	-1.5691	-1.5633
median	0.3445	-1.5735	-1.5702
mean	0.3811	-1.5764	-1.5764
sd	0.2594	0.3295	0.5672
95% lower	0.0000	-2.2312	-2.7661
95% upper	0.8607	-0.9264	-0.4073

(shortest credible intervals). Forest plot (Fig. 2): raw estimates with 95% CI plus shrinkage intervals, “mean” $-1.57$ $[-2.23,-0.93]$ and “prediction” $-1.57$ $[-2.77,-0.41]$. Fig. 3 (plot(ma01)): top-left forest, top-right joint posterior of $(\tau ,\mu )$ with conditional mean $\hat{\mu }(\tau )$ and 95% band, ML cross and MAP cross; bottom marginal densities of $\mu$ and $\tau$.

Querying the posterior ($dposterior, $pposterior, $qposterior, $rposterior, $post.interval)

Query	Result
$P(\mu >0\mid y)$ (non-beneficial effect)	$6.187\times 10^{-5}$
$P(\tau >1\mid y)$ (“fairly extreme”)	0.02097
99% upper limit of $\tau$	1.1092
99% interval for $\tau$, shortest	[0, 1.1092]
99% interval for $\tau$, central	[0.003548, 1.2054] (the 0.005 and 0.995 quantiles)

Monte Carlo (rposterior(n, tau.sample=FALSE) is much faster if only $\mu$ is needed): assume control AR risk $p_c=0.5$; $\text{logit}{p}_t=\text{logit}{p}_c+{\mu }^{\ast }$; risk difference $p_t-p_c$ has median $-0.328$ and 95% interval $[-0.403,-0.215]$ from 10,000 draws.

Shrinkage (ma01$theta[,1:2]):

	Heffron 2003	Gibelli 2004
$y$, $\sigma$	-2.3097, 0.5995	-0.4595, 0.5564
posterior median	-1.7411	-1.2822
posterior mean, sd	-1.7779, 0.4229	-1.2340, 0.4489
95% interval	[-2.6561, -0.9920]	[-2.0455, -0.3089]

Prior variations (Sec. 3.4)

• Prior predictive: see the table in Roever 2020 - Priors for the Effect and Heterogeneity (97.5% quantiles 1.092, 2.185, 5.051 for half-normal(0.5), half-normal(1.0), half-Cauchy(0.5)).
• Empirical prior (Turner et al. 2015): TurnerEtAlPrior(outcome="surgical", comparator1="pharmacological", comparator2="placebo / control"), log-normal with quantiles (2.5%, 50%, 97.5%) = (0.0623, 0.3430, 1.8873). Result “very similar” to the half-normal analysis (Fig. 1 shows the two densities).
• Non-informative: omit mu.prior.* for the uniform effect prior; tau.prior="Jeffreys" for Jeffreys (default is "uniform").

Examples

Reproducing the printed posterior without bayesmeta

A fine $\tau$ grid with Eq. 22 and Eq. 21 (Roever 2020 - Marginal Likelihood and Conditional Posteriors) gives: $\tau$ posterior mean 0.3811, sd 0.2594, median 0.3445, $P(\tau >1)=0.02097$, 99% quantile 1.1092; $\mu$ mean $-1.5764$, sd 0.3294, median $-1.5735$, $P(\mu >0)=6.1\times 10^{-5}$; study-1 shrinkage mean $-1.778$, median $-1.742$; REML $\hat{\tau }=0.4670$, $\hat{\mu }=-1.5915$, SE 0.3341. All agree with the paper. The $y_i,{\sigma }_i$ come from the counts with the 0.5 correction in study 6.

import numpy as np
from scipy import stats
a=np.array([14,16,3,9,4,0.]); n1=np.array([61,28,18,54,36,50.]); c=np.array([15,19,8,29,11,3.]); n2=np.array([20,28,12,54,36,34.])
b, d = n1-a, n2-c
z = (a==0)|(b==0)|(c==0)|(d==0)                              # continuity correction for studies with a zero cell
a, b, c, d = [np.where(z, x+.5, x) for x in (a, b, c, d)]
y = np.log(a*d/(b*c)); s = np.sqrt(1/a+1/b+1/c+1/d)
# then call posterior_grid(y, s) from the Marginal Likelihood note, with tau ~ half-normal(0.5), mu ~ N(0, 4^2)

Connections

• Marketing: the workflow (compute effect and SE per study, state an informative $\tau$ prior from the plausible spread of effects, report the predictive interval and the probability of a harmful or null effect, then run prior sensitivity) is a template for pooling a handful of lift or geo experiments. "$P(\mu >0\mid y)$" is a direct decision quantity; the risk-difference translation (via an assumed baseline) is the analogue of converting a log-lift into incremental conversions at a baseline conversion rate.
• Frequentist re-analysis of a comparable problem: Viechtbauer 2010 - BCG Vaccine Worked Example.

See Also

• Roever 2020 - Posterior Predictive Checks and Frequentist Connection (the two-randomised-study subset analysis)
• Roever 2020 - Bayesian Random-Effects Meta-Analysis with bayesmeta - Overview