Phenotype Misclassification Under Selective Ascertainment

Why the Recorded Phenotype Matters

A polygenic score is evaluated against an outcome label, but that label may be a clinical diagnosis rather than the underlying condition itself. A person with the condition may never have been diagnosed, while a person without it may have received a diagnosis. If inclusion in a study also depends on diagnosis, the observed score–phenotype relationship can reflect both classification error and the way participants were selected.

These issues affect different quantities in different ways. Imperfect classification can change the apparent association between score and phenotype. Diagnosis-based sampling can change the proportion classified as cases, and therefore undermine direct interpretation of predicted risks. Under specific sampling assumptions, an association with the recorded diagnosis can nevertheless be preserved. Keeping the true condition, recorded phenotype, and inclusion process distinct is essential.

A misclassification mechanism is nondifferential with respect to the score when sensitivity and specificity do not vary with \(Z\), within the relevant population or selection stratum. This does not mean that diagnosis is perfect, or that its errors are unrelated to every other variable. It is a specific assumption about how classification error varies across score values. If, for example, people with different scores have different access to testing, diagnosis may be differential even when the underlying condition is measured using the same clinical criteria.

Nondifferential Misclassification and Score-Group Risks

Suppose the score has two categories, such as lower and higher score groups. In a source population, let \(p_z=P(Y=1\mid Z=z)\) be the true condition risk in group \(z\), and let \(q_z=P(D=1\mid Z=z)\) be the recorded diagnosis rate. If sensitivity \(Se\) and specificity \(Sp\) are the same in both groups, then a recorded positive can arise either from a true positive or from a false positive:

The term \(1-Sp\) is the false-positive rate. The factor \(Se+Sp-1\) is the Youden index, which describes how informative the classification is beyond chance in this binary setting. Subtracting the recorded rates for two score groups gives

Thus, if \(0<Se+Sp-1<1\), the recorded risk difference is smaller in magnitude than the true risk difference. This statement is about a risk difference under these assumptions; it does not imply that every association measure, prediction metric, or selected-sample contrast is reduced by exactly the same factor. If the error rates vary by score group, the displayed relationship no longer holds.

When the error rates are known and common across score groups, the relationship can be inverted if \(Se+Sp-1\ne0\):

This correction uses the recorded rate and the assumed sensitivity and specificity to recover a true risk. It is only as credible as those inputs and the model: the error rates must apply to the target population, classification must be represented by the stated binary mechanism, and the observed rate must refer to the population whose risk is being corrected. In particular, inserting a diagnosis rate from a case-control sample does not undo the sample's altered case fraction.

Diagnosis-Based Inclusion Changes the Sample

Let \(a_1=P(S=1\mid D=1)\) and \(a_0=P(S=1\mid D=0)\) denote inclusion probabilities for people classified as cases and controls. If inclusion depends only on \(D\), then among people in score group \(z\), the diagnosis fraction in the selected sample is

This equation shows why the selected fraction classified as cases need not estimate the diagnosis prevalence in the source population. If cases are deliberately oversampled, the selected fraction rises even if the underlying score–diagnosis relationship has not changed. The selected fraction is not, by itself, a calibrated risk.

There is an important, limited invariance. If inclusion depends on recorded diagnosis alone, with the same \(a_1\) and \(a_0\) at every score value, then the odds of diagnosis at score group \(z\) in the selected sample equal \((a_1/a_0)\) times the source-population odds. Taking the odds ratio across two score groups cancels that common factor. Therefore, the odds ratio for \(Z\) and \(D\) is preserved. It remains an association with the potentially misclassified diagnosis, not necessarily with \(Y\). The result does not apply if selection also depends on score or other variables in a way that changes selection odds across score groups.

Differential Classification Can Create or Magnify an Association

The nondifferential formula relies on common sensitivity and specificity across score groups. If those quantities vary with \(Z\), the recorded rate becomes

Now differences in recorded diagnosis can reflect differences in true risk, differences in sensitivity or specificity, or both. For example, if a high score is associated with more clinical contact, people with the condition may be more likely to receive a diagnosis in that score group. Even with identical true risks, this pattern can create an apparent score association. False-positive rates can also differ across groups.

Consequences for Calibration and Performance

Calibration concerns agreement between predicted probabilities and outcome frequencies. If a model is trained and evaluated against \(D\), good calibration means that predictions agree with the frequency of recorded diagnoses in that evaluation setting. It does not establish calibration for the true condition \(Y\). The two targets coincide only when classification is sufficiently accurate for the purpose, or when an appropriate correction based on valid error-rate information is made.

Case-control sampling adds a separate calibration problem: the selected case fraction is set partly by the sampling design, so probabilities estimated directly from the selected sample need not be source-population risks. Correcting for the sampling design and correcting for phenotype errors are distinct tasks. A correction for diagnosis-based sampling does not repair misclassification, and a sensitivity-specificity correction does not restore a population prevalence that was changed by sampling. Both need to be addressed if the goal is risk calibration for \(Y\) in a specified target population.

Discrimination metrics also require a stated target. An area under the receiver operating characteristic curve calculated using \(D\) describes ranking of people with versus without the recorded diagnosis in the evaluated sample. Classification errors can change which people are counted in each group and thus can change that metric relative to one based on \(Y\). Outcome-dependent sampling alone can preserve this ranking metric when it leaves the score distributions within each recorded outcome group unchanged; that condition does not make the recorded outcome accurate or make the metric a measure of discrimination for \(Y\).

Assumptions to Check Before Interpreting Results

A sensitivity and specificity calculation is useful only when its assumptions match how phenotypes were collected and how the evaluation sample was formed. Check, at minimum, the following:

• Target: State whether the outcome of interest is the underlying condition \(Y\) or the recorded phenotype \(D\). Do not silently interpret results for one as results for the other.
• Error rates: Establish what population, score range, and selection stratum the sensitivity and specificity refer to. Applying validation rates from a different setting requires justification.
• Differential error: Consider whether score, ancestry, healthcare access, age, or other measured factors could change the probability of diagnosis given the true condition. If rates plausibly vary across score groups, a single nondifferential correction is inadequate.
• Selection: Identify whether inclusion depends on diagnosis alone or also on score and other factors. Diagnosis-only sampling supports the odds-ratio invariance above; broader selection mechanisms need not.
• Uncertainty: Sensitivity and specificity are estimated quantities when obtained from validation data. Treating them as known can understate uncertainty in corrected risks and associations.