Local FDR in Normal Mixture Models

Local FDR at a Specific Test Statistic

In a large genomic analysis, a test statistic is observed for each feature, but the null and alternative groups are not labeled. A two-group mixture model describes the distribution of these statistics: one component represents null features, and another represents non-null features. The local false discovery rate, or local FDR, asks a pointwise question: given a feature’s observed statistic, what is the posterior probability that it belongs to the null group?

For normal mixture models, this calculation depends on the density of the observed statistic under each group. It is therefore important to evaluate the densities at the observed value, use the mixture proportions as prior probabilities, and distinguish a density from the probability of an interval. The introductory material on local FDR establishes the posterior-null interpretation; here the focus is calculating it for normal components, including symmetric and asymmetric alternatives.

A normal density with mean \(\mu\) and variance \(\sigma^2\) is written \(\phi(z;\mu,\sigma^2)\). In the common standard-null model, \(f_0(z)=\phi(z;0,1)\). The alternative density may be one normal density or a mixture of normal components. For example, with \(K\) alternative components, \(f_1(z)=\sum_{k=1}^{K} w_k\phi(z;\mu_k,\tau_k^2)\), where each \(w_k\geq 0\) and \(\sum_{k=1}^{K}w_k=1\). The alternative density is the weighted sum of its component densities, not a choice of whichever component happens to fit the observed value best.

These formulas presume that the group proportions and component distributions describe the population of test statistics being modeled. In worked calculations below, their parameters are treated as specified. If they were estimated from data, the resulting local FDR would also depend on estimation uncertainty and on how well the model describes the data. A z-statistic is dimensionless, and the normal densities here are densities per unit of z.

Bayes’ Rule and the Density-Ratio Form

The posterior formula follows directly from Bayes’ rule: the probability of being null before seeing \(z\) is \(\pi_0\), and the likelihood contribution of observing \(z\) under that group is \(f_0(z)\). The same multiplication applies to the alternative group. Dividing the null contribution by the sum of both contributions gives the posterior null probability.

When \(\pi_0>0\) and \(f_0(z)>0\), divide the numerator and denominator by \(\pi_0f_0(z)\). This gives a useful density-ratio expression:

Thus, for fixed prior proportions, a larger alternative-to-null density ratio at \(z\) means a smaller local FDR. The ratio is a comparison of densities at the observed value; it is not itself a posterior probability. The original posterior formula also handles endpoint proportions: if \(\pi_0=0\), then the local FDR is 0 wherever \(f(z)>0\); if \(\pi_1=0\), it is 1 wherever \(f(z)>0\).

For a single normal alternative \(N(\mu,\tau^2)\) and a null \(N(\mu_0,\sigma_0^2)\), the density ratio can be written explicitly:

For an alternative mixture, calculate its weighted density first and then compare that full density with \(f_0(z)\). The component weights describe how the alternative group is distributed; they are not substitutes for the overall prior probabilities \(\pi_0\) and \(\pi_1\).

Worked Calculations

Calculation Procedure and Checks

The following sequence keeps the two kinds of mixture weights distinct and makes arithmetic errors easier to find.

Some diagnostics are available even for a single calculation. The two weighted contributions must be nonnegative, and the local FDR must lie between 0 and 1. Their sum must equal the mixture density \(f(z)\). A symmetric alternative with equal component variances and equal weights at opposite means has an even density, so its \(f_1(z)\), and hence its local FDR under a symmetric null, are the same at \(z\) and \(-z\). An asymmetric alternative need not have this property.

What Symmetry Changes—and What It Does Not

Symmetry describes the shape of the alternative density; it does not imply that the alternative density equals the null density at the center. For the equal mixture of \(N(-2,1)\) and \(N(2,1)\), at \(z=0\) each alternative component has density \(\phi(0;2,1)=\phi(2;0,1)=0.053991\). Thus \(f_1(0)=0.053991\), whereas the null density is \(f_0(0)=\phi(0;0,1)=0.398942\). With \(\pi_0=0.90\) and \(\pi_1=0.10\), the local FDR is \(0.90(0.398942)/[0.90(0.398942)+0.10(0.053991)]\approx0.9852\). Near zero, this model assigns very little alternative density relative to null density, so the posterior null probability is high.

For asymmetric alternatives, sign and component weights can matter substantially. At a negative statistic, a component centered at a negative value may provide a large density contribution, but its influence is moderated by its weight within \(f_1\). At the opposite sign, the density can differ, even when the component means have equal magnitude. These changes enter through \(f_1(z)\); the Bayes formula itself does not change.

Common Mistakes and Limitations

• Using a component instead of the full alternative density. The alternative density in a mixture is the weighted sum across all its components. Calculate every component’s density at \(z\), multiply by its within-alternative weight, and then add.
• Confusing the two sets of weights. The \(\pi\) values are the overall proportions of null and alternative features. The \(w_k\) values describe the composition of the alternative group. Both are needed.
• Treating a density as a point probability. For a continuous statistic, \(f_0(z)\) and \(f_1(z)\) are densities, not probabilities that \(Z\) equals exactly \(z\). Bayes’ rule uses their relative density contributions to define the posterior group probability at \(z\).
• Assuming symmetry makes the null and alternative equally plausible at zero. Symmetry only implies matching alternative density values at \(z\) and \(-z\). It does not make \(f_1(0)=f_0(0)\).
• Reading a fitted-model result as certainty. A calculated local FDR is conditional on the specified proportions and densities. If the null model, alternative shape, or fitted parameters are inappropriate, the number may not be well calibrated for the data.

A single normal alternative is simple to evaluate and can represent a directional shift, but may fail to represent effects in both directions. A symmetric mixture can represent signals on either side of zero, while an asymmetric mixture allows unequal prevalence or strength across directions. A more flexible estimated alternative density can accommodate shapes not captured by a few normal components, but its local density estimates may be less stable, particularly where observations are sparse. None of these choices removes the need to assess whether the null and alternative models are plausible for the test statistics under study.