Roever 2020 - Normal-Normal Hierarchical Model and Likelihood

Summary

The NNHM has two unknowns: the effect $\mu$ and the heterogeneity $\tau \ge 0$. Study estimates $y_i$ are normal around study-specific ${\theta }_i$ with known standard errors ${\sigma }_i$, and the ${\theta }_i$ are exchangeable normal around $\mu$ with variance ${\tau }^2$. Integrating out ${\theta }_i$ gives $y_i\sim N(\mu ,{\sigma }_i^2+{\tau }^2)$ and the likelihood in Eq. (13); $\tau =0$ is the fixed-effect model.

Overview

The model is the same as the random-effects model of Viechtbauer 2010 - Fixed, Random and Mixed-Effects Models, with notation ${\sigma }_i$ (standard error) instead of $\sqrt{v_i}$.

Main Content

Normal-normal hierarchical model (Eq. 1-2)

Data: $k$ estimates $y_i$ of $\mu$ with known standard errors ${\sigma }_i$, $i=1,\dots ,k$.

$$y_i\mid {\theta }_i,{\sigma }_i\sim N({\theta }_i,{\sigma }_i^2),{\theta }_i\mid \mu ,\tau \sim N(\mu ,{\tau }^2),\tau \ge 0.$$

The ${\theta }_i$ (true study-specific effects) are exchangeable. Sampling error magnitude is ${\sigma }_i$.

Marginal form (Eq. 3)

Integrating out ${\theta }_i$: $y_i\mid \mu ,\tau ,{\sigma }_i\sim N(\mu ,{\sigma }_i^2+{\tau }^2)$. The unknowns are $\mu$ (primary interest) and $\tau$ (usually a nuisance parameter), though shrinkage estimates of ${\theta }_i$ can also be of interest. If $\tau =0$, ${\theta }_1=\cdots ={\theta }_k=\mu$ (fixed-effect model).

Likelihood (Eq. 13-14)

$$p(\vec{y}\mid \mu ,\tau ,\vec{\sigma })=(2\pi {)}^{-k/2}\prod _{i=1}^k\frac{1}{\sqrt{{\sigma }_i^2+{\tau }^2}}\exp \left(-\frac{1}{2}\frac{(y_i-\mu {)}^2}{{\sigma }_i^2+{\tau }^2}\right),$$ $$\log p(\vec{y}\mid \mu ,\tau ,\vec{\sigma })=-\frac{k}{2}\log (2\pi )-\frac{1}{2}\sum _{i=1}^k\left[\log ({\sigma }_i^2+{\tau }^2)+\frac{(y_i-\mu {)}^2}{{\sigma }_i^2+{\tau }^2}\right].$$

Scope: the model is applied to other data types after transformation to a real-valued scale, e.g. log odds ratios from binary data (Hedges and Olkin 1985; Hartung et al. 2008; Borenstein et al. 2009; Viechtbauer 2010; Higgins and Green 2011), see Viechtbauer 2010 - Effect Size Measures and escalc.

Examples

Marginal variance with and without heterogeneity

Take ${\sigma }_1=0.6$ and $\tau =0.5$: marginal SD of $y_1$ is $\sqrt{0.36+0.25}=0.781$ versus $0.6$ under the fixed-effect model. The extra 30% spread is what a random-effects analysis propagates into the pooled SE.

import numpy as np
def nnhm_loglik(mu, tau, y, sigma):
    v = sigma**2 + tau**2
    return -0.5*len(y)*np.log(2*np.pi) - 0.5*np.sum(np.log(v) + (y-mu)**2/v)

Connections

• Hierarchical Models: the NNHM is the simplest hierarchical model with known first-level variances (“eight schools” structure, Gelman et al. 2014 Sec. 5.5).
• Marketing measurement: replace $y_i,{\sigma }_i$ with each experiment’s lift estimate and standard error; $\tau$ is cross-experiment variability of true lift.

See Also

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