Phenotype Definition and Scale in PRS Evaluation

Why Phenotype Definition Is Part of PRS Evaluation

A polygenic risk score (PRS) is evaluated against a phenotype, not against an abstract notion of genetic risk. The definition of that phenotype determines who counts as a case, which time point is relevant, and what a one-unit change in a quantitative outcome means. Two analyses can use the same score and participants yet report different effect estimates or prediction metrics because they define or code the outcome differently.

This matters when comparing results across studies and when deciding whether an estimate applies to a target population. A change from one unit to another may alter a coefficient’s numerical value without changing the underlying predictions. A change in diagnostic threshold or case ascertainment can instead change the target itself. Case-control sampling can also alter some observed metrics simply because the sampled case fraction differs from the fraction in the population where the PRS would be used.

Binary Phenotypes: Coding and Effect Direction

For a binary phenotype, define \(Y=1\) as the outcome called a case and \(Y=0\) as the reference outcome called a control. For a score \(S\), a common association model is logistic regression:

Here, \(\exp(\beta)\) is the odds ratio for a one-unit increase in \(S\), under the model. If \(S\) is standardized, one unit is one standard deviation of the score in the population used to standardize it. The coefficient is not a change in probability: the probability change depends on the starting probability as well as on \(\beta\). Interpretation also depends on which outcome was coded as 1 and how the score was scaled.

Changing a binary outcome’s labels from 0/1 to 1/2 only adds 1 to every outcome value; it does not change the underlying case definition. But reversing which group is treated as the event reverses the direction of a logistic coefficient. A more consequential change is to alter the case rule, such as using a stricter diagnostic threshold, a different age window, or self-report instead of a clinical record. That creates a different phenotype, not merely a different label.

Ascertainment and the Target Population

Many genomic evaluations use case-control samples in which investigators select participants according to case status. The case fraction in such a sample is set by study design and need not equal disease prevalence in the population where prediction is intended. This difference has important consequences for interpreting metrics and, under specific assumptions, a more limited consequence for logistic regression coefficients.

Suppose selection into the sample depends on \(Y\) but, within each outcome group, does not depend on \(S\). Write \(q_1=P(\text{sampled}\mid Y=1)\) and \(q_0=P(\text{sampled}\mid Y=0)\), with both probabilities positive. The sampled odds at a given score value are the population odds multiplied by \(q_1/q_0\). Thus:

Taking logarithms shows that outcome-dependent sampling adds \(\log(q_1/q_0)\) to the logistic intercept while leaving the slope \(\beta\) unchanged, provided the logistic model is appropriate and sampling is independent of the score within outcome groups. The sampled intercept does not generally represent the population risk. If selection also depends on the score within cases or controls, or the model is misspecified, this simple slope-preservation result need not hold.

This example illustrates why evaluation should identify the population to which a metric refers. Some conditional quantities may be estimable in an outcome-sampled design under suitable assumptions, but quantities that average over cases and controls depend on the case fraction. Correcting a metric for prevalence requires a defensible target prevalence and an appropriate sampling model; it does not make an unclear phenotype definition adequate.

Quantitative Phenotypes: Units and Transformations

For a quantitative outcome, a basic model relates the measured phenotype \(Y\) to the score \(S\): \[ Y=\alpha+\beta S+\varepsilon. \] The slope \(\beta\) is the expected difference in the measured outcome per one-unit increase in \(S\), under the model. A score measured in standard-deviation units makes the PRS increment explicit; the outcome coefficient still has the units of \(Y\). Interpreting the slope requires the phenotype’s unit, measurement method, and assessment time.

A linear conversion of units changes the coefficient and prediction error in predictable ways. If \(Y^*=a+cY\), where \(c\ne0\), then the fitted intercept becomes \(a+c\alpha\), the slope becomes \(c\beta\), and residuals are multiplied by \(c\). Consequently, mean squared error is multiplied by \(c^2\). With an intercept in the regression, \(R^2\), the fraction of observed outcome variation explained by fitted values, is unchanged by this affine conversion. If \(c\) is negative, the coefficient’s sign reverses as well.

A nonlinear transformation is different. For example, taking the logarithm of a positive measurement changes distances between values, and a linear model on the transformed scale describes changes in log outcome rather than equal additive changes on the original scale. The ranking of predictions may be preserved by a strictly increasing transformation of predictions, but errors and \(R^2\) computed on different outcome scales are not directly comparable. A changed diagnostic threshold is more fundamental still: it changes which participants are cases and may change prevalence, effect estimates, and the prediction task.

Assumptions, Diagnostics, and Alternatives

The evaluation should begin with a written phenotype definition: case criteria and reference group for binary outcomes, or measurement procedure, unit, and assessment time for quantitative outcomes. Then specify the sampled population, the intended target population, the score scale, and the model. These details let readers distinguish a change of units or labels from a change in the scientific target.

• Check outcome coding. Verify which value is the event in software and that the coefficient direction matches the stated case definition. A reversed label can make a correct estimate appear to have the opposite meaning.
• Check ascertainment. Compare the sampled case fraction with the target prevalence. Record whether selection depended only on outcome status or could also depend on score, site, severity, or covariates.
• Check measurement consistency. Look for differences in assay, instrument, diagnostic criteria, follow-up duration, missingness, and unit conversion. A nominally identical phenotype may not be measured identically across groups or cohorts.
• Check metric scale and target. State whether errors are reported in original or transformed units and whether prevalence-dependent metrics describe the validation sample or a target population.

For binary outcomes, logistic regression directly models case odds and is often suitable for case-control association analysis under the sampling conditions described above. A linear probability model instead models the conditional mean of a 0/1 outcome as a linear function; its fitted values can fall outside 0 and 1, so it does not provide probabilities that are valid over all score values. For quantitative outcomes, a linear model gives effects in the measured units and can be compared with models on transformed scales, but the interpretation and error metric must then be reported on the corresponding scale.

No model or metric repairs a poorly defined phenotype. Misclassification, selective participation, and inconsistent measurement can distort an estimated relationship; the direction and magnitude of distortion depend on how those processes operate. Likewise, a result for one case threshold or assessment age should not automatically be treated as a result for another. State the limitation rather than interpreting a coefficient as a causal effect or as a universal measure of PRS performance.