Roever 2020 - Priors for the Effect and Heterogeneity

Summary

The effect $\mu$ gets a uniform (improper) or normal prior; the heterogeneity $\tau$ gets one of four classes: proper informative (half-normal, half-$t$, half-Cauchy, log-normal, exponential), proper “non-informative” depending on the ${\sigma }_i$ (uniform shrinkage, DuMouchel, Berger-Deely conventional), improper $\sigma$-dependent (Jeffreys/Tibshirani, Berger-Deely) and improper scale-invariant (${\tau }^a$). For small $k$ an informative prior is recommended; the improper uniform prior is “conservative” in a provable sense (Appendix A.2).

Overview

Prior independence is assumed, $p(\mu ,\tau )=p(\mu )p(\tau )$, for computational convenience (dependent priors: Senn 2007; Pullenayegum 2011). Improper priors may give improper posteriors.

Main Content

Prior for the effect $\mu$

Uniform on $\mathbb{R}$ (non-informative location prior), or normal $N({\mu }_p,{\sigma }_p^2)$ (conditionally conjugate), the two cases supported in the package. Uniform is the limit ${\sigma }_p\to \infty$.

Unit information prior (Kass and Wasserman 1995; App. A.1)

A vague prior whose variance equals that of a single observational unit. For a log-OR, set the 2x2 allocation $a=b=c=d=N/4$ (neutral effect, equal allocation) and $N=1$: $\text{SE}=\sqrt{1/a+1/b+1/c+1/d}=4$, so $\mu \sim N(0,4^2)$; 95% of log-ORs fall in $\pm 7.84$, i.e. ORs from about $1/2500$ to $2500$. With event probability $p$ ($a=c=pN/2$, $b=d=(1-p)N/2$): $\text{SD}=2/\sqrt{p(1-p)}$, never below 4. For a log-RR, $a=b=c=d=N/4$ gives SD 2 and with $p$: $2\sqrt{(1-p)/p}$ (larger for $p<1/2$).

Priors for the heterogeneity $\tau$

Proper informative. Because $k$ is usually tiny, information about $\tau$ from data is weak, so informative priors help.

• For log-OR-type endpoints, $\tau$ above $\log (10)\approx 2.3$ is implausible (a tenfold OR spread). Spiegelhalter et al. (2004, Sec. 5.7): $\tau \in [0.1,0.5]$ “reasonable”, $[0.5,1.0]$ “fairly high”, $>1.0$ “fairly extreme”. Same logic for log-RR, log-HR, log-variance.
• Maximum-entropy prior for a given $E[\tau ]$: exponential with rate $1/E[\tau ]$. Half-normal with scale 0.5 or 1.0 recommended and shown to work well (Friede et al. 2017a,b). Log-normal, or heavy-tailed half-Student-$t$, half-Cauchy, Lomax (discount the prior when data conflict: O’Hagan and Pericchi 2012; Schmidli et al. 2014; Gelman 2006; Polson and Scott 2012).
• Inverse-Gamma is generally not recommended (Gelman 2006; Polson and Scott 2012).
• Empirical priors from the Cochrane Database: Rhodes et al. (2015, mean-difference endpoints) and Turner et al. (2015, log-OR endpoints; log-normal/log-Student-$t$), available as RhodesEtAlPrior() and TurnerEtAlPrior().

Proper “non-informative”. Scale-invariant but dependent on the ${\sigma }_i$ (logically awkward: Senn 2007). With the harmonic-mean squared standard error

$$s_0^2=\frac{k}{{\sum }_{i=1}^k{\sigma }_i^{-2}}\text{(Eq. 4)}$$

Uniform shrinkage, DuMouchel and Berger-Deely conventional priors

• Uniform shrinkage (Daniels 1999): uniform on the average shrinkage $S(\tau )=s_0^2/(s_0^2+{\tau }^2)$, equivalently uniform in $1-S={\tau }^2/(s_0^2+{\tau }^2)$, a quantity much like $I^2$ of Higgins-Thompson (Viechtbauer 2010 - tau2 Estimators, Q-Test and Heterogeneity Statistics). Density (Eq. 5): $p(\tau )=\frac{2\tau s_0^2}{(s_0^2+{\tau }^2{)}^2}$, median $s_0$.
• DuMouchel (DuMouchel and Normand 2000), Eq. 6: $p(\tau )=\frac{s_0}{(s_0+\tau {)}^2}$, log-logistic, mode 0, median $s_0$.
• Conventional (Berger and Deely 1988), Eq. 7: $p(\tau )\propto {\left({\prod }_{i=1}^k\frac{\tau }{({\sigma }_i^2+{\tau }^2{)}^{3/2}}\right)}^{1/k}$; proper, intended for testing/model selection.

Improper priors.

• Scale-invariance family (Eq. 8-9): requiring $\frac{1}{s}p(\tau /s)=f(s)p(\tau )$ gives $p(\tau )\propto {\tau }^a$, $a\in \mathbb{R}$. $a=-1$: log-uniform (improper posterior in this problem, Gelman 2006). $a=0$: uniform on $[0,\infty )$ (Gelman’s default for large $k$; needs $k\ge 3$ with an improper uniform $\mu$ prior, and no proper posterior when $k\le 2$). $a=-1/2$ (Eq. 10): $p(\tau )\propto 1/\sqrt{\tau }$, uniform in $\sqrt{\tau }$, a pole at the origin. $a=1$: uniform in ${\tau }^2$, needs $k\ge 4$, not recommended.
• Jeffreys/Tibshirani (Eq. 11): treating the orthogonal location parameter $\mu$ as fixed (the general Jeffreys $\sqrt{\det J}$ is not recommended with a location parameter),

$$p(\tau )\propto \sqrt{\sum _{i=1}^k{\left(\frac{\tau }{{\sigma }_i^2+{\tau }^2}\right)}^2}.$$

Also the Berger-Bernardo reference prior (Bodnar et al. 2016, 2017). Improper (tail $\sim 1/\tau$), locally integrable with $p(0)=0$; posterior proper for $k\ge 2$. With equal ${\sigma }_i=\sigma$ its mode is at $\tau =\sigma$; otherwise near the smallest ${\sigma }_i$, possibly multimodal. Scales with the ${\sigma }_i$.

• Berger-Deely (Eq. 12): $p(\tau )\propto {\left({\prod }_i\frac{\tau }{{\sigma }_i^2+{\tau }^2}\right)}^{1/k}$, equals Jeffreys when all ${\sigma }_i$ are equal.

Table 1: four classes of heterogeneity priors

	proper informative	proper non-informative	improper non-informative	improper scale-invariant
examples	half-normal, half-$t$, half-Cauchy, log-normal, exponential	uniform shrinkage (5), DuMouchel (6), conventional (7)	Jeffreys (11), Berger-Deely (12)	uniform in $\tau$, in $\sqrt{\tau }$, … (9)
depends on ${\sigma }_i$?	no	yes	yes	no
scale-invariant?	no	scales with ${\sigma }_i$	scales with ${\sigma }_i$	yes
$k$ restrictions	none	$k\ge 1$	$k\ge 2^{\ast }$	$k\ge 3^{\ast }$
${}^{\ast }$ less if combined with a proper effect prior.

Choosing

Proper prior wanted when marginal likelihoods or Bayes factors are needed or $k$ is small. Gelman (2006) recommends the improper uniform for $\tau$ unless $k$ is small or an informative prior is desired, in which case half-$t$ (half-Cauchy, half-normal as special/limiting cases; half-Cauchy supported by Polson and Scott 2012). Jeffreys is the default non-informative choice (proper posterior for $k\ge 2$). For a log-OR, half-normal(0.5) leaves roughly 95% of mass below $\tau \approx 0.98$ (about 5% “fairly extreme”).

Conservatism of the uniform prior (Appendix A.2)

Uniform prior is the stochastically largest posterior

Let the parameter domain be $[a,\infty )$, the likelihood $f(x)\ge 0$ with ${\int }_a^{\infty }f<\infty$, and the prior density $p$ monotonically decreasing with $0<p(a)<\infty$. Let $F_1$ and $F_p$ be the posterior CDFs under the improper uniform prior and under $p$. Then for all $y>a$,

$$F_p(y)=\frac{{\int }_a^{y}f(x)p(x)dx}{{\int }_a^{\infty }f(x)p(x)dx}\ge \frac{{\int }_a^{y}f(x)dx}{{\int }_a^{\infty }f(x)dx}=F_1(y)(\text{paper Eq. 30-31}),$$

so the uniform-prior posterior is stochastically larger than the posterior under any prior in this class (provided the uniform one is proper); quantiles and expectations are larger. The class includes exponential, half-normal, half-$t$, half-Cauchy, Lomax and bounded uniform priors. Proof sketch (restated). Write $F_p(y)=A/(A+B)$ with $A={\int }_a^{y}fp$ and $B={\int }_y^{\infty }fp$. Since $p$ is decreasing, $A\ge p(y){\int }_a^{y}f$ and $B\le p(y){\int }_y^{\infty }f$. Because $A/(A+B)$ increases in $A/B$, $F_p(y)\ge {\int }_a^{y}f/{\int }_a^{\infty }f$. (The paper’s printed chain bounds numerator and denominator separately via $p(a)$; the ratio argument here is the form that makes the final inequality hold.) $\square$

Interpretation: overestimating heterogeneity is “conservative” (wider intervals), though large $\tau$ also shifts $\hat{\mu }(\tau )$ so intervals need not nest (Poole and Greenland 1999); the uniform prior can nonetheless put too much weight on implausibly large $\tau$.

Prior predictive implications (Sec. 3.4)

A fixed $\tau$ implies ${\theta }_i\sim N(\mu ,{\tau }^2)$; a prior on $\tau$ implies a normal scale mixture for ${\theta }_i$ (prior predictive), evaluated by normalmixture(). For $\mu =0$ the 97.5% quantile of the prior predictive ${\theta }_i$:

$\tau$ prior	$q_{0.975}({\theta }_i)$	$\exp (q_{0.975})$
half-normal(0.5)	1.092	2.98
half-normal(1.0)	2.185	8.89
half-Cauchy(0.5)	5.051	156.1

So half-normal(0.5) places 95% of OR-scale study effects within a factor of about 3 (13 for the full 95% range) of the mean OR, while half-Cauchy(0.5) allows absurdly large heterogeneity. The empirical Turner et al. prior for “surgical / device-related success/failure”, pharmacological vs placebo, has median 0.343 and 95% range [0.0623, 1.887] (log-normal; Fig. 1 parameters $\text{log-normal}(-1.07,0.87)$).

Examples

Choosing a $\tau$ prior for log-lift in marketing experiments

For a log-relative-lift endpoint where plausible between-experiment OR-like variation is up to about 2x (a ratio of true lifts), $\tau \approx \log 2/1.96\approx 0.35$ is the 95% scale. A half-normal with scale 0.25 to 0.5 puts mass there; the first prior-predictive check is: does the implied 97.5% quantile of ${\theta }_i$ correspond to a ratio you would believe? This is the same workflow as Constructing Priors for Effect Sizes. Reproduce the check with:

import numpy as np
from scipy import stats, optimize
def prior_pred_q(cdf_tau_pdf, upper=50, q=0.975):
    tt = np.linspace(1e-6, upper, 400001); pp = cdf_tau_pdf(tt)
    F = lambda x: np.trapezoid(pp*stats.norm.cdf(x/tt), tt) - q
    return optimize.brentq(F, 1e-3, 100)
print(prior_pred_q(lambda t: stats.halfnorm.pdf(t, scale=0.5)))   # 1.091 here (paper: 1.092)
print(prior_pred_q(lambda t: stats.halfnorm.pdf(t, scale=1.0)))   # 2.182 here (paper: 2.185)

Connections

• Constructing Priors for Effect Sizes and Hierarchical Models (the half-Cauchy/half-$t$ recommendation originates in Gelman 2006).
• Sensitivity of results to these choices is demonstrated in Roever 2020 - Pediatric Liver Transplant Worked Example.

See Also

• Roever 2020 - Marginal Likelihood and Conditional Posteriors
• Roever 2020 - Bayesian Random-Effects Meta-Analysis with bayesmeta - Overview