Healthy-Volunteer and Survivor Bias in Polygenic Prediction

Why Volunteer and Survivor Selection Matter

A polygenic score may be evaluated among people who volunteered for a cohort or survived long enough to enroll. Those participants need not represent the population for which prediction is intended. Selection can change the range of scores observed, alter the score–phenotype relationship, or create an association that was absent in the source population. These are distinct possibilities: a narrower score distribution does not, by itself, establish that the regression slope has changed.

The key question is not simply whether participants differ from nonparticipants. It is what variables influence selection, and how are those variables related to the score and phenotype? A selection diagram makes these dependencies explicit. In the diagrams described below, \(S=1\) denotes inclusion in the evaluated sample; conditioning on \(S=1\) means analyzing only people who enrolled or survived to enrollment.

Definition: Healthy-volunteer bias is bias arising when participation depends on health, behavior, access, or other characteristics that also relate to a polygenic score or phenotype. Survivor bias is the corresponding problem when inclusion requires survival to a specified enrollment time. Either mechanism can change score distributions or score–phenotype relationships. Selection bias is not a single fixed direction of error: its effect depends on the variables and mechanisms governing selection.

Read the Selection Diagram Before Interpreting Performance

Suppose \(G\) is a polygenic score, \(Y\) is the phenotype being predicted, and \(U\) represents an environmental or health-related factor. A simple selection diagram for volunteer participation is \(G \rightarrow S \leftarrow U \rightarrow Y\). Here, both the score and \(U\) affect participation, while \(U\) also affects the phenotype. The node \(S\) is a collider: two arrows point into it. If \(G\) and \(U\) are independent in the source population, conditioning on \(S=1\) can make them associated among participants. Since \(U\) predicts \(Y\), this induced association can change the observed association between \(G\) and \(Y\).

A different diagram is \(G \rightarrow S\), with no other causes of selection related to \(Y\) after conditioning on \(G\). This represents selection that depends on the score alone. It can narrow or shift the distribution of \(G\), but under appropriate outcome-model assumptions it does not change the conditional mean of \(Y\) at a given \(G\). This contrast is important: a change in the distribution of predictors and a change in their conditional predictive relationship are not the same thing.

In practice, a health measure can sit on several paths. For example, a score may affect a health condition that influences enrollment, while an environmental factor affects both that condition and the phenotype. The resulting diagram may contain multiple paths through selection. A statement such as “the cohort is healthier” does not identify which paths are active, so it does not determine whether a score slope will be biased upward or downward.

What Score-Only Selection Does to a Linear Prediction

Consider a source population in which the phenotype follows \(Y=\alpha+\beta G+\varepsilon\). The score \(G\) and phenotype \(Y\) are measured in their stated units; \(\beta\) is the conditional change in the mean phenotype per one-unit increase in score. For the result below, assume explicitly that in the source population \(E(\varepsilon\mid G)=0\) and \(\operatorname{Var}(\varepsilon\mid G)=\sigma^2\). Also assume selection depends only on \(G\): \(P(S=1\mid G,\varepsilon)=P(S=1\mid G)\). This is stronger than merely observing particular conditional moments among participants.

Because selection is conditionally independent of the residual given \(G\), it preserves the source conditional residual moments: \(E(\varepsilon\mid G,S=1)=0\) and \(\operatorname{Var}(\varepsilon\mid G,S=1)=\sigma^2\). Thus the conditional mean relationship among participants remains \(E(Y\mid G,S=1)=\alpha+\beta G\). If \(\operatorname{Var}(G\mid S=1)>0\), the population linear regression slope among participants is \(\operatorname{Cov}(G,Y\mid S=1)/\operatorname{Var}(G\mid S=1)=\beta\). The equality follows directly from the preserved conditional mean and zero residual mean; it does not follow from moments specified only among selected people.

The correlation, however, need not be preserved. With \(\operatorname{Var}(G\mid S=1)=v\), the conditional variance of \(Y\) is \(\beta^2v+\sigma^2\), so the selected-sample correlation is \(\beta\sqrt{v}/\sqrt{\beta^2v+\sigma^2}\). Restricting score variation can therefore reduce the magnitude of correlation even when the slope remains \(\beta\). This is a prediction-distribution effect, not evidence that the per-unit conditional association has changed.

Result: Under the stated linear model, source-population residual assumptions, and selection depending only on \(G\), the selected-sample regression slope of \(Y\) on \(G\) remains \(\beta\). The score variance and score–phenotype correlation can change. Do not apply this result when selection also depends on the residual, phenotype, or other variables related to the phenotype.

Worked Example: Selection Depends on the Score Alone

Consider hypothetical source-population data generated by \(G\sim N(0,1)\) and \(Y=0.4G+\varepsilon\), where \(E(\varepsilon\mid G)=0\) and \(\operatorname{Var}(\varepsilon\mid G)=1\). Suppose participation depends only on \(G\), with probability \(P(S=1\mid G)=\exp(-G^2/2)\). This is a valid probability between zero and one, and it gives relatively low participation to scores far from zero.

The selected score density is proportional to the source density times the participation probability. Since the source density is proportional to \(\exp(-G^2/2)\), the selected density is proportional to \(\exp(-G^2)\), which is the density kernel of \(N(0,0.5)\). Thus \(\operatorname{Var}(G\mid S=1)=0.5\). The assumed score-only selection preserves the conditional residual mean and variance, so the selected regression slope is still \(0.4\) phenotype units per score unit.

In the source population, \(\operatorname{Var}(Y)=0.4^2(1)+1=1.16\), and \(\operatorname{Corr}(G,Y)=0.4/\sqrt{1.16}=0.3714\). Among participants, \(\operatorname{Var}(Y\mid S=1)=0.4^2(0.5)+1=1.08\), giving \(\operatorname{Corr}(G,Y\mid S=1)=0.4\sqrt{0.5}/\sqrt{1.08}=0.2722\). The correlation is smaller because the score has less variation, even though the slope remains \(0.4\). The assumptions have been specified for the source population, and the selection mechanism depends only on \(G\), so this calculation does not assume that selected-sample moments alone determine source-population properties.

When Recruitment Also Depends on the Phenotype

Healthy-volunteer recruitment often depends on health or function related to the phenotype under study. In that setting, selection may depend on both \(G\) and the residual part of \(Y\), or on a variable \(U\) associated with that residual. The score-only result no longer applies. The conditional mean of the phenotype at a given score can change within the selected group, so the observed regression slope may differ from the source-population slope.

Worked Example: Healthier Participants Are More Likely to Enroll

For a small hypothetical illustration, let \(G\) and \(U\) each be either 0 or 1, independently, with probability \(1/2\) for each value. Let the phenotype be \(Y=G+U\), in arbitrary phenotype units. In the source population, the regression slope of \(Y\) on \(G\) is 1: increasing \(G\) from 0 to 1 increases the conditional mean of \(Y\) by 1 unit.

Suppose recruitment probability depends on \(Y\): it is 0.9 when \(Y=0\), 0.5 when \(Y=1\), and 0.1 when \(Y=2\). Lower phenotype values are more likely to be recruited. The four equally likely source-population combinations \((G,U)\) are \((0,0),(0,1),(1,0),(1,1)\), with recruitment probabilities \(0.9,0.5,0.5,0.1\). Their probabilities among recruits are these values divided by their sum, 2: \(0.45,0.25,0.25,0.05\).

Using those selected probabilities, \(E(G\mid S=1)=0.25+0.05=0.30\), \(E(Y\mid S=1)=0.25+0.25+2(0.05)=0.60\), and \(E(GY\mid S=1)=0.25+2(0.05)=0.35\). Therefore \(\operatorname{Cov}(G,Y\mid S=1)=0.35-(0.30)(0.60)=0.17\), while \(\operatorname{Var}(G\mid S=1)=0.30-(0.30)^2=0.21\). The selected regression slope is \(0.17/0.21=0.8095\), not the source slope of 1.

This example isolates phenotype-dependent recruitment: selection is more likely at lower values of \(Y\), and the selected score–phenotype slope is attenuated. Other selection rules can increase a slope or reverse its sign. The direction is not determined by the phrase “healthy volunteers”; it must be worked out from the actual selection mechanism.

Survival to Enrollment Can Create an Association

Survivor bias can arise even when the score and phenotype have no association in the source population. If both a score and another phenotype-related factor affect survival, survival is a common effect of those causes. Restricting analysis to survivors can then induce an association between them. The diagram \(G \rightarrow S \leftarrow U \rightarrow Y\) represents this possibility: \(G\) and \(U\) need not be associated before selection, but they can become associated after conditioning on survival \(S=1\). Because \(U\) predicts \(Y\), that induced association can appear as a score–phenotype relationship.

Worked Example: A Spurious Score Association Among Survivors

Let \(G\) and \(U\) be independent binary variables, each with probability \(1/2\) of being 1, and let \(Y=U\). There is no source-population association between \(G\) and \(Y\): \(E(Y\mid G=0)=E(Y\mid G=1)=0.5\), so the source regression slope is 0. Suppose survival to enrollment has probabilities 0.9 for \((G,U)=(0,0)\), 0.6 for \((0,1)\), 0.6 for \((1,0)\), and 0.2 for \((1,1)\). Both a higher score and \(U=1\) are associated with lower survival.

Because the four source combinations are equally likely, their probabilities among survivors are proportional to \(0.9,0.6,0.6,0.2\), whose sum is 2.3. The survivor probabilities are therefore \(9/23,6/23,6/23,2/23\). From these, \(E(G\mid S=1)=E(Y\mid S=1)=8/23\), and \(E(GY\mid S=1)=2/23\). Hence \(\operatorname{Cov}(G,Y\mid S=1)=2/23-(8/23)^2=-18/529\), while \(\operatorname{Var}(G\mid S=1)=(8/23)(15/23)=120/529\). The survivor-sample slope is \((-18/529)/(120/529)=-0.15\).

The negative slope is induced by conditioning on survival; it does not reveal a source-population score effect, since that effect was zero by construction. The numerical result follows from the stated survival probabilities and source distribution. In real cohorts, survival can depend on age, disease, treatment, environment, and other factors, so the resulting association may have a different size or direction.

Practical Interpretation and a Common Mistake

The relevant prediction target is the population in which performance will be used. A score evaluated in a volunteer cohort describes performance in that selected cohort unless additional assumptions and analyses support transport to another target. A restricted score distribution can affect correlation-based summaries even when the conditional regression slope is stable. Selection connected to phenotype or common causes can also alter slopes and calibration, or induce associations. These possibilities should not be collapsed into a single claim that volunteer studies always overestimate or always underestimate prediction.

A useful first step is to write down the selection event and its plausible causes. Then ask whether selection is determined by the score alone, by the phenotype or its residual, or by a common effect of score-related and phenotype-related variables. The distinction guides interpretation; the diagram is not proof that a particular path operates. Assumptions about the recruitment and survival process must be justified in the study context.

Wrong reasoning: “The selected participants have a narrower score distribution, so the score–phenotype slope must be smaller.” A narrower predictor range can lower correlation, but under score-only selection and the stated model the slope remains \(\beta\).
Correction: Separate distributional changes from changes in the conditional mean relationship. Check whether selection depends only on \(G\); if it depends on phenotype or related variables, derive or assess the selected relationship under that mechanism.
Wrong reasoning: “A score associated with a phenotype among survivors must also be associated in the source population.” Conditioning on survival can open a collider path.
Correction: Interpret the association as applying to the selected group unless the selection mechanism and target-population relationship have been addressed.

Key takeaway: Healthy-volunteer and survivor selection can affect polygenic prediction in different ways. Score-only selection can change score variance and correlation without changing a linear conditional slope under explicit assumptions. Selection involving phenotype-related causes can change the slope or induce an association, and its direction is mechanism-specific.

Check Your Understanding

Use the selection mechanism, rather than the label of the cohort, to reason through each question.

Under what assumptions does selection depending only on \(G\) preserve the regression slope of \(Y\) on \(G\)? State the source-population residual conditions as well as the selection condition.
Why can correlation between \(G\) and \(Y\) change under score-only selection even when the regression slope does not?
In the recruitment example, which selection probabilities correspond to each value of \(Y\), and what selected-sample slope results?
Describe the collider path in the survivor example and explain why the selected association does not establish a source-population association.
Give one reason that “healthy volunteers always overestimate prediction” is not a valid general conclusion.