Posterior Null Probability and Local FDR

Why a Posterior Null Probability Is Useful

A genomic analysis may produce thousands of test statistics, but the evidence against the null need not be equally strong for every test. A single summary of the overall discovery list cannot describe how plausible the null is for one particular observed statistic. The local false discovery rate addresses that pointwise question: given a test statistic at a particular value, what is the probability that its hypothesis belongs to the null component of a two-group model?

This is a model-based probability, not a direct label attached to a test. Its value depends on the proportion of null hypotheses and on how the null and alternative components distribute test statistics. A small local false discovery rate indicates that, under the specified model, the observed statistic is more characteristic of the alternative component than of the null component. It does not by itself establish a biological effect or a causal explanation.

Definition in a Two-Group Model

Let \(X\) denote a test statistic for a randomly selected hypothesis from a collection of tests. Let \(H=0\) indicate that the hypothesis is null and \(H=1\) indicate that it is nonnull. Suppose the two-group model has null proportion \(\pi_0=P(H=0)\) and alternative proportion \(\pi_1=1-\pi_0\). Conditional on the group, \(X\) has density \(f_0(x)\) under the null and density \(f_1(x)\) under the alternative. The marginal density for a test statistic drawn from the full collection is \[ f(x)=\pi_0 f_0(x)+\pi_1 f_1(x). \]

Definition: At an observed statistic \(x\) for which \(f(x)>0\), the local false discovery rate is the posterior probability that the tested hypothesis is null: \[ \operatorname{lfdr}(x)=P(H=0\mid X=x) =\frac{\pi_0 f_0(x)}{\pi_0 f_0(x)+\pi_1 f_1(x)}. \] Here \(\pi_0 f_0(x)\) is the null component's contribution to the marginal density at \(x\), and the denominator is the total marginal density there.

For a continuous statistic, the event \(X=x\) has probability zero. The notation \(P(H=0\mid X=x)\) means the posterior probability specified by the model's conditional distribution at that value, computed from the component densities. Densities have units inverse to those of \(X\); their weighted ratio is dimensionless. For a discrete statistic, the corresponding formula uses component probability masses in place of densities.

The model requires \(0\leq\pi_0\leq1\), with \(\pi_1=1-\pi_0\), and \(f_0\) and \(f_1\) must be valid densities that each integrate to one. The statistic must have a comparable interpretation across the tests being pooled, and the chosen null density must represent the null distribution for those tests. The formula is a marginal, test-specific calculation; independence among all tests is not required for that calculation. Dependence can nevertheless matter for estimating the mixture components or assessing uncertainty in them.

How the Density Contributions Determine the Result

The local false discovery rate depends on both the null proportion and the relative density at the observed value. The null proportion alone is not the answer: an observed statistic may be much more common under the alternative than under the null, or vice versa. A useful way to see this relationship is to divide numerator and denominator by \(f_0(x)\), when \(f_0(x)>0\), and define the alternative-to-null density ratio \(r(x)=f_1(x)/f_0(x)\). Then

\operatorname{lfdr}(x)=\frac{\pi_0}{\pi_0+\pi_1r(x)}.

For \(0<\pi_0<1\), the derivative of this expression with respect to \(r\) is

\frac{d}{dr}\left(\frac{\pi_0}{\pi_0+\pi_1r}\right) = -\frac{\pi_0\pi_1}{(\pi_0+\pi_1r)^2}<0.

Thus, holding \(\pi_0\) fixed, a larger alternative-to-null density ratio gives a smaller local false discovery rate. This comparison is about densities at the observed value, not about which component has the larger total probability. If \(f_0(x)=0\) and \(f(x)>0\), the formula gives local false discovery rate zero. If \(f_1(x)=0\) and \(f(x)>0\), it gives one. If \(f(x)=0\), the model assigns no density to the observation, so the formula is undefined and the model should be examined.

Worked Example: A Statistic in the Upper Tail

Consider a hypothetical collection of genomic tests with \(\pi_0=0.90\), so \(\pi_1=0.10\). Suppose the null statistic has the standard normal density, giving \(f_0(3)=0.004432\), and a normalized alternative density has \(f_1(3)=0.08000\). Calculate the local false discovery rate at \(x=3\).

The null component contributes \(\pi_0 f_0(3)=0.90(0.004432)=0.0039888\). The alternative component contributes \(\pi_1 f_1(3)=0.10(0.08000)=0.008000\). Therefore the marginal density at 3 is \(0.0039888+0.008000=0.0119888\), and

\operatorname{lfdr}(3) = \frac{0.0039888}{0.0039888+0.008000} = \frac{0.0039888}{0.0119888} \approx 0.3327.

Under this model, the posterior probability that the hypothesis associated with a statistic of 3 is null is about 0.3327, or 33.27%. This is not the probability that the null is true before observing the statistic; that model-wide proportion is \(\pi_0=0.90\). The observed value has more density contribution from the alternative component than from the null component.

Worked Example: A Statistic Near the Null Center

For a second hypothetical model, let \(X\) have null distribution \(N(0,1)\) and alternative distribution \(N(0,4)\), where the second parameter is the variance. Let \(\pi_0=0.70\), and consider \(x=0\). The component densities at zero are \(f_0(0)=0.398942\) and \(f_1(0)=0.199471\). Find the local false discovery rate.

The null contribution is \(0.70(0.398942)=0.2792594\). The alternative contribution is \(0.30(0.199471)=0.0598413\). Their sum is the marginal density \(0.3391007\). Hence

\operatorname{lfdr}(0) = \frac{0.2792594}{0.2792594+0.0598413} = \frac{0.2792594}{0.3391007} \approx 0.8235.

The posterior null probability is about 0.8235, or 82.35%. The null proportion was 0.70, but the statistic at zero raises the posterior null probability because the null density there is twice the alternative density. A statistic near the null center is not automatically null; its interpretation depends on both component densities.

Worked Example: Comparing Two Observed Statistics

Suppose a hypothetical model has null density \(f_0(z)\) from \(N(0,1)\), alternative density \(f_1(z)\) from \(N(2,1)\), and \(\pi_0=0.80\). Compare the local false discovery rates at \(z=1\) and \(z=2\). Both normal distributions have standard deviation one.

At \(z=1\), the statistic is one unit from each component mean, so \(f_0(1)=f_1(1)=0.241971\). The null and alternative contributions are \(0.80(0.241971)=0.193577\) and \(0.20(0.241971)=0.048394\). Therefore

\operatorname{lfdr}(1) = \frac{0.193577}{0.193577+0.048394} =0.8000.

At \(z=2\), \(f_0(2)=0.053991\) and \(f_1(2)=0.398942\). The weighted contributions are \(0.80(0.053991)=0.043193\) and \(0.20(0.398942)=0.079788\). Thus

\operatorname{lfdr}(2) = \frac{0.043193}{0.043193+0.079788} \approx 0.3512.

In this model, \(z=2\) has a lower posterior null probability than \(z=1\), despite the null component being the larger group overall. This ordering follows from the relative component densities at each observed value, not merely from the fact that one statistic is numerically larger. With a two-sided or otherwise differently shaped alternative, evidence need not increase monotonically with the signed statistic.

Interpretation, Diagnostics, and Alternatives

Local false discovery rate is a probability for one tested hypothesis conditional on its statistic and the specified mixture model. It is distinct from an error-rate summary for a selected set of tests. Procedures such as the Benjamini–Hochberg method operate on a collection of p-values to control a multiple-testing error rate under their stated conditions; their adjusted quantities are not automatically posterior null probabilities for individual tests. Likewise, a tail-area summary concerns a region or selected group rather than the density at one value. These quantities answer related but different questions.

Before interpreting calculated values, check whether the fitted or specified mixture is credible. The following diagnostics can expose problems:

Check density and proportion validity. Component densities should be nonnegative and integrate to one, and \(\pi_0\) must lie between zero and one. The weighted component densities should add to the marginal density.
Check the null calibration. If the null density is too narrow, too wide, or incorrectly centered, the null contribution at observed statistics can be systematically wrong. Genomic test statistics may also fail to be comparable when their sample sizes or standard errors differ substantially.
Compare the fitted mixture with the observed distribution. A mixture that does not describe the central and tail behavior of the test statistics can produce misleading local probabilities, even when the arithmetic is correct.
Investigate extreme or erratic values. A local false discovery rate outside \([0,1]\) signals an implementation or input error. Abrupt spikes or implausible patterns can indicate unstable density estimates or a poor model, although a nonmonotone pattern can be genuine when the alternative is not a simple one-directional shift.

A local false discovery rate is only as reliable as its inputs. If \(\pi_0\), \(f_0\), or \(f_1\) is estimated, uncertainty in those estimates is not represented by treating the displayed value as exact. Dependence between tests can affect estimation and uncertainty even though the pointwise mixture formula itself does not require independent tests. In addition, a single alternative density may conceal different biological mechanisms or effect directions. The resulting posterior is conditional on the model's grouping and does not identify which alternative mechanism generated the statistic.

Common Mistake

Common mistake: “The null proportion is 0.90, so every test has a 90% chance of being null,” or “a small p-value is the probability that this null hypothesis is true.” The correction is that \(\pi_0\) is a model-wide mixture proportion, while the local false discovery rate varies with the observed statistic through \(f_0(x)\) and \(f_1(x)\). A p-value is not a posterior probability. Calculate the local probability from the weighted component densities, and state that its interpretation depends on the model.

Key takeaway: The local false discovery rate at \(x\) is the null component's weighted density divided by the full mixture density at \(x\). It is a test-specific posterior probability under the two-group model, not a universal property of a p-value or a guarantee that a biological claim is true.

Check Your Understanding

Work each problem using the two-group mixture definition. Solutions follow the questions.

A model has \(\pi_0=0.60\), \(f_0(x)=0.20\), and \(f_1(x)=0.50\). Calculate the local false discovery rate at \(x\).
With \(\pi_0=0.80\), at what value of \(f_1(x)/f_0(x)\) does the local false discovery rate equal 0.50, assuming both component densities are positive?
At an observed value, \(f_0(x)=0\), \(f_1(x)>0\), and \(\pi_1>0\). What is the local false discovery rate, and what does the model imply?
Explain why two tests with the same null proportion can have different local false discovery rates at their observed statistics.

Solutions. (1) The weighted null contribution is \(0.60(0.20)=0.12\); the weighted alternative contribution is \(0.40(0.50)=0.20\). Thus \(\operatorname{lfdr}(x)=0.12/(0.12+0.20)=0.3750\). (2) Write \(r=f_1(x)/f_0(x)\). Then \(0.50=0.80/(0.80+0.20r)\), so \(0.40+0.10r=0.80\), giving \(r=4\). (3) The null contribution is zero while the alternative contribution is positive, so the local false discovery rate is zero; under this model the observation has no null density at that value. (4) Their local false discovery rates can differ because \(f_0(x)\) and \(f_1(x)\) can have different relative sizes at their respective observed values, even when \(\pi_0\) is shared.