Posterior Shrinkage of Test Statistics

Why Weight a Test Statistic by Its Posterior Probability?

In a genomic analysis, a large observed test statistic can reflect a genuine departure from the null, sampling variation, or both. A two-group model represents this uncertainty by treating each test as either null or nonnull, then using the observed statistic to update the prior probability of those states. The resulting posterior probability can be used to attenuate the statistic: a result with substantial posterior null probability is pulled more strongly toward zero than a result with little posterior null probability.

This tutorial develops that probability-weighted statistic and its interpretation. It is important to name what the calculation does: it multiplies the observed statistic by the posterior probability that it is nonnull. That construction is a useful soft-shrinkage score, but it is not, in general, the posterior mean of an underlying effect. The distinction matters when reporting or using the result.

Model and Definition

Let \(Z\) be a signed test statistic, such as a standardized statistic whose null value is zero. Let \(H=0\) indicate a null test and \(H=1\) a nonnull test. The two-group model assumes prior probabilities \(P(H=0)=\pi_0\) and \(P(H=1)=\pi_1=1-\pi_0\), with component densities \(f_0(z)\) and \(f_1(z)\). Its marginal density is \(f(z)=\pi_0f_0(z)+\pi_1f_1(z)\). Given an observed value \(z\), Bayes’ rule gives the posterior null probability, or local false discovery rate, and the posterior nonnull probability.

The mixture model and its component densities must be suitable for the statistics being analyzed. In the normal example below, assume a correctly standardized null \(Z\sim N(0,1)\), a symmetric alternative density, and known mixture parameters for the calculations. In practice, parameters may be estimated; then the resulting probabilities and scores also depend on the quality and uncertainty of those estimates. The posterior probabilities are conditional on the specified model, not model-free probabilities that a scientific claim is true.

How the Posterior Weight Produces Shrinkage

The posterior nonnull probability is between zero and one. Therefore \(S(z)\) has the same sign as \(z\), unless \(z=0\), and its magnitude cannot exceed the observed magnitude. A posterior nonnull probability near zero produces strong attenuation; one near one leaves the statistic nearly unchanged. For a symmetric null and symmetric alternative, the posterior weight is the same at \(z\) and \(-z\), so the shrinkage changes magnitude but not direction.

This follows directly from the posterior probability definition: its numerator is nonnegative and no larger than its denominator. Multiplying \(z\) by a number in \([0,1]\) preserves its sign and weakly reduces its absolute value. This is a simple but useful justification for calling \(S(z)\) a shrinkage score.

A useful expression uses the likelihood ratio \(L(z)=f_1(z)/f_0(z)\), when \(f_0(z)>0\):

For fixed prior probabilities, this posterior weight increases as the likelihood ratio increases: differentiating with respect to \(L\) gives \(\pi_0\pi_1/(\pi_0+\pi_1L)^2>0\), provided both prior probabilities are positive. Thus evidence that is more compatible with the alternative than with the null receives less attenuation. This does not mean that every model makes the weight increase with \(|z|\); that pattern depends on the component densities.

What the Calculation Does—and Does Not—Estimate

The interpretation of \(S(z)\) is deliberately limited: it is the observed statistic multiplied by its posterior nonnull probability under the fitted two-group model. It is not automatically an estimate of a gene’s log fold change, treatment effect, or latent noncentrality parameter. A test statistic and an effect estimate are different quantities, and a posterior probability of the nonnull component alone does not specify the effect’s magnitude conditional on being nonnull.

In particular, do not call \(z\,r(z)\) the posterior mean effect without additional assumptions and a derivation connecting the observed statistic to a latent effect. The posterior mean of an effect generally requires a model for that effect and its sampling distribution. Deriving such a mean is a separate calculation. The weighted statistic can still be useful for displaying or summarizing evidence, but its label and units should make clear what it represents.

For a symmetric normal mixture with \(f_0=N(0,1)\) and \(f_1=N(0,\tau^2)\), where \(\tau^2>1\), the likelihood ratio is \(L(z)=\tau^{-1}\exp\{(1-\tau^{-2})z^2/2\}\). It increases with \(|z|\), so the posterior nonnull weight rises as the statistic moves farther from zero. The observed sign is preserved, and the attenuation factor is the same for positive and negative statistics of equal magnitude. This behavior relies on the assumed symmetric, broader alternative; it need not hold with asymmetric or differently shaped component densities.

Comparison, Diagnostics, and Limitations

The posterior-weighted statistic differs from several familiar summaries. The raw statistic \(z\) preserves the observed evidence without shrinkage. A hard-thresholding rule instead keeps or discards a statistic according to a cutoff, creating an abrupt change at that cutoff. Posterior weighting is continuous: values with intermediate posterior nonnull probability receive intermediate attenuation. A p-value measures tail probability under a specified null model; it is not the posterior probability of the alternative and cannot be substituted for \(r(z)\). Likewise, a tail-area false discovery rate summarizes a selected set, whereas \(r(z)\) is a pointwise posterior probability under the mixture.

Before interpreting shrunken scores, check that the inputs and model support the calculation. The null should be centered and scaled appropriately for the test statistics; the alternative density and prior proportion should be plausible for the family of tests; and the assumed sign and shape structure should be appropriate. If a symmetric model is used, compare the behavior of positive and negative statistics at similar magnitudes. If the estimated mixture assigns implausibly high posterior nonnull probabilities to central statistics, or makes nearly all weights close to zero or one without a scientific or modeling explanation, investigate the component fit and parameter estimates rather than treating the scores as automatically calibrated.

The calculation also has limits. An estimated \(\pi_0\) or component density introduces estimation uncertainty that the plug-in value \(S(z)\) does not show. Dependence among genomic tests may matter for fitting and uncertainty assessment even though the pointwise Bayes formula is defined from marginal component densities. Model misspecification can yield misleading posterior weights, and a shrunken statistic does not correct a misspecified null. Finally, multiplying by \(r(z)\) does not itself set a decision threshold, control a family-wise error rate, or guarantee a false discovery rate.

Common Mistake

A more accurate report names the model-based quantity and the scale: “Under the fitted mixture, the posterior nonnull probability at \(z=-2\) is approximately 0.3591, giving a posterior-weighted statistic of approximately \(-0.7181\).” If an effect estimate is needed, report an estimator defined for that effect and explain its model and uncertainty separately.