Conditional Likelihood for Genome-Wide Significant Loci

Why Condition the Likelihood?

A genome-wide significant estimate is not an ordinary draw from the sampling distribution: it is reported because it crossed a threshold. Even if an estimator is unbiased before selection, estimates that pass a two-sided threshold tend to have larger absolute values than the underlying effects. A likelihood that treats the reported set as though it had been selected at random ignores this filtering and can preserve winner’s-curse inflation.

Conditional likelihood addresses this by modeling the estimate’s distribution given that it passed the reporting rule. It does not change the sampling model for the estimate. Instead, it divides the ordinary likelihood by the probability of selection under each candidate effect. The selection probability depends on that effect, so it cannot be dropped when estimating the effect.

Model and Selection Event

Consider one genetic variant with true effect \(\beta\), measured in the chosen effect units, and an estimate \(\hat{\beta}\) with known standard error \(s\). Use the normal approximation \(\hat{\beta}\mid\beta\sim N(\beta,s^2)\). The standard error is treated as fixed. Define \(Z=\hat{\beta}/s\) and \(\delta=\beta/s\). Suppose the reporting rule is two-sided significance, \(|Z|>c\), where \(c\) is the prespecified critical value. For a standard normal test at the conventional genome-wide threshold of \(5\times10^{-8}\), \(c\) is about 5.45. A smaller value such as \(c=2\) below is used only to make hand calculations easier; it represents the same type of selection rule, not the genome-wide threshold itself.

The formula for \(Q\) follows because \(Z\mid\beta\sim N(\delta,1)\): the lower-tail event has probability \(\Phi(-c-\delta)\), and the upper-tail event has probability \(1-\Phi(c-\delta)=\Phi(\delta-c)\). The two tails are disjoint. The conditional density is therefore \[ f(y\mid S,\beta)=\frac{\frac{1}{s}\phi((y-\beta)/s)}{Q(\beta)} \] on the selected region, with \(\phi\) the standard normal density. The numerator is the ordinary sampling density; the denominator renormalizes it over values that could have been reported.

This model assumes the estimate is approximately normal with the stated standard error, the threshold and two-sided rule are known, and selection depends on this statistic alone. For several estimates, the product likelihood below also assumes they are independent conditional on their effects and that each is selected by its own specified rule. Correlated variants, selection of only the most significant result, or additional filters require the corresponding joint selection event rather than an independent per-variant denominator.

Deriving the Conditional Score

For one selected observation, write the log likelihood in standardized units \(z=y/s\): \[ \ell_c(\delta;z)=\log\phi(z-\delta)-\log Q(\delta)-\log s, \] where \(Q(\delta)=\Phi(-c-\delta)+\Phi(\delta-c)\). Since \(\phi'(u)=-u\phi(u)\), differentiation gives \[ Q'(\delta)=-\phi(c+\delta)+\phi(\delta-c). \] Thus the score with respect to \(\delta\) is \[ U_c(\delta)=z-\delta-\frac{\phi(\delta-c)-\phi(\delta+c)}{Q(\delta)}. \] The maximum-likelihood estimate solves \(U_c(\delta)=0\), with \(\hat{\beta}_c=s\hat{\delta}_c\). In general the root is found numerically.

The correction term is the derivative of the log selection probability. It is not an arbitrary adjustment: it is exactly the part of the score that would be omitted by treating selected estimates as an unselected normal sample. When \(\delta=0\), symmetry gives \(Q'(0)=0\). Away from zero, the term adjusts the score for the changing probability of entering the reported set.

Several Selected Estimates

Suppose \(m\) independent association estimates \(y_1,\ldots,y_m\) are reported, with known standard errors \(s_1,\ldots,s_m\). If each has its own effect \(\beta_i\) and its own threshold \(c_i\), the conditional likelihood is \[ L_c(\beta_1,\ldots,\beta_m)=\prod_{i=1}^{m}\frac{f(y_i\mid\beta_i)}{Q_i(\beta_i)}, \] where \(Q_i(\beta_i)=P_{\beta_i}(|\hat{\beta}_i/s_i|>c_i)\). If a scientific model instead specifies a shared parameter, substitute that model’s effects into each factor and maximize the resulting likelihood. The number of reported hits alone is not included in this conditional likelihood: it conditions on which estimates were selected, rather than modeling the count selected from all tested variants.

Checking the Model and Understanding Its Limits

The conditional likelihood is useful only when its conditioning event matches how results entered the data set. Before interpreting an adjusted estimate, check that the reported set was selected by the stated threshold, on the stated test statistic and effect scale. In a two-sided analysis, both tails must be represented even if the observed hit is positive. If selection included replication, ranking, or additional filters, conditioning only on \(|Z|>c\) does not account for those steps.

Also check whether the normal approximation and standard errors are adequate, and whether dependence matters. Linkage disequilibrium can make nearby variant statistics correlated. If the analysis reports one lead variant from a correlated region, the event may be that it is the region’s most significant result, not merely that its own statistic crosses a cutoff. In that setting, independent per-variant denominators do not describe the selection process. A fitted model should also be checked for a finite, stable maximum; extreme thresholds, sparse information, or a misspecified selection rule can make estimates unstable or highly dependent on assumptions.

A common mistake is to use the ordinary normal likelihood for the selected estimate and argue that significance filtering is irrelevant because the standard error is already in the model. The standard error describes the spread of estimates before selection; it does not remove the fact that only estimates in the selected region were observed. The correction is to include \(Q(\beta)\) in the likelihood and let it vary with the candidate effect.

Another mistake is to interpret a lower conditional maximum-likelihood estimate as proof that the original result was false. It is an estimate under a specified sampling and selection model, not a test of whether the locus is associated. Uncertainty should be assessed from the conditional likelihood, and conclusions remain dependent on the validity of the assumed selection process.

The earlier material on truncated-normal correction focuses on correcting the selected estimate through its conditional distribution. Conditional likelihood instead retains the full conditional density and estimates parameters by maximizing it. Neither approach repairs an incorrect threshold, unmodeled dependence, or additional unrecorded selection.