Defining the Polygenic Risk Score Estimand

From a Genetic Score to a Prediction Target

A polygenic risk score (PRS) compresses information from many genetic variants into one number. That number can be useful for prediction, but it is not, by itself, a probability of disease, a diagnosis, or a universal measure of “genetic risk.” Its meaning depends on what outcome is being predicted, for whom, and over what period. Those choices define the prediction target—the quantity a study intends its model to predict.

A clear analysis therefore separates two things: the rule that calculates a score from genotypes, and the prediction target against which a score-based model will be interpreted or evaluated. The same score can be used in different prediction tasks, but those tasks need not have the same predictions, interpretation, or performance.

Definition: For individual $i$ and variant $j$, let $G_{ij}$ be the genetic dosage coded as the number of copies of a designated allele, commonly from 0 to 2, and let $w_j$ be a fixed weight for that variant. A polygenic risk score is the weighted sum $$ S_i=\sum_{j=1}^{M} w_jG_{ij}, $$ where $M$ is the number of variants included. The score is a function of the genotype data and weights. It is not automatically a probability, and it does not by itself specify an outcome, target population, or time horizon.

The weights and dosages must use compatible allele definitions for the sum to have its intended meaning. Here, the weights are treated as already specified and fixed; the definition does not prescribe how they were estimated or which variants should be included. If weights represent changes in log odds per dosage unit, the score is on a log-odds contribution scale. If weights use a different scale, the score inherits that scale. Without knowing the weights’ scale and origin, the numerical value alone has no universal interpretation.

Define the Outcome, Population, and Horizon

For a binary prediction task, define an outcome $Y_i(\tau)$ for each eligible individual. For example, $Y_i(\tau)=1$ can mean that the individual experiences a specified event within $\tau$ years after a defined starting point; $Y_i(\tau)=0$ can mean that the event is not observed within that period under the study’s outcome definition. The event, starting point, and handling of incomplete follow-up must be specified. “Disease risk” without those details is not a complete outcome definition.

The target population is the population to which the prediction is intended to apply, with its eligibility criteria and relevant setting stated. A prediction task also specifies the information available when the prediction is made. For example, a model using age and clinical measurements at enrollment has a different information set from one using only genotype. Let $C_i$ denote any additional predictors included alongside the score.

Definition: For a specified target population $Q$, outcome $Y(\tau)$, score $S$, and additional predictors $C$, a conditional prediction target for a binary outcome is $$ m_Q(s,c;\tau)=P_Q\{Y(\tau)=1\mid S=s,\ C=c\}. $$ It is the event probability among individuals in population $Q$ with score $s$ and predictors $c$, for the defined outcome and horizon. A population-average prediction for a specified group is obtained by averaging the relevant conditional predictions over that group’s predictor distribution in $Q$.

A prediction rule, written $f(s,c)$, is the procedure that produces a prediction from the score and any other predictors. It might output a probability, a predicted quantitative outcome, or a ranking; the appropriate interpretation depends on the task. Even when $f$ outputs probabilities, it represents the target $m_Q$ only to the extent that the rule is suitable and calibrated for that outcome and population. A score alone does not establish that correspondence.

Prediction can also be evaluated using a specified loss function $L$. For example, the target-population mean squared prediction error for a quantitative outcome is $$ R_Q(f)=E_Q\left[\{Y-f(S,C)\}^2\right]. $$ This is a population-specific performance target, not the individual’s outcome. For a binary outcome, different loss functions answer different performance questions. The outcome, population, and prediction horizon must therefore be fixed before a performance result has a precise interpretation.

Worked Example: Calculating a Score Is Not Calculating a Risk

Suppose a hypothetical score uses four variants. Dosages count copies of the designated allele, and the fixed weights are on a log-odds contribution scale.

Variant	Dosage $G_{ij}$	Weight $w_j$	Contribution $w_jG_{ij}$
1	2	0.12	0.24
2	1	-0.08	-0.08
3	0	0.05	0.00
4	2	0.03	0.06

The score is $$ S_i=(0.12)(2)+(-0.08)(1)+(0.05)(0)+(0.03)(2) =0.24-0.08+0+0.06=0.22. $$

Thus this individual’s score is 0.22 on the specified score scale. Because the weights are log-odds contributions, 0.22 is the genetic contribution to a linear predictor under the model for which those weights are appropriate. It is not a 22% risk, and it cannot be converted to an absolute risk without a prediction model that defines the outcome, horizon, and relevant baseline information.

Prediction Models Give the Score a Task-Specific Meaning

For a binary outcome, one possible model maps the score and other predictors to a conditional probability through the logistic function. For example, a model with score $S$ alone can be written

\operatorname{logit}\{P_Q(Y(\tau)=1\mid S=s)\}=\alpha+\beta s, \qquad \operatorname{logit}(p)=\log\left(\frac{p}{1-p}\right).

Here, $\alpha$ is an intercept and $\beta$ is the score coefficient for this particular outcome, population, and horizon. The score formula does not supply these quantities automatically. A model that includes additional predictors may instead use $\alpha+\beta s+\gamma^\mathsf{T}c$, where $c$ is the predictor vector and $\gamma$ is its coefficient vector.

Worked Example: Converting the Same Score to a Defined Risk

Continue with the hypothetical score $S=0.22$. Suppose a prediction model for a specified event within five years in a specified target population has linear predictor $-2.00+1.00S$. Assume, for this illustration, that the logistic model is appropriate and its coefficients are calibrated for that exact outcome, population, and horizon.

The linear predictor is $$ \eta=-2.00+(1.00)(0.22)=-1.78. $$

The model’s predicted five-year probability is $$ \hat p=\frac{1}{1+\exp(-\eta)} =\frac{1}{1+\exp(1.78)} \approx \frac{1}{1+5.929} \approx 0.1443. $$

Under the stated model and target, the predicted probability is about 14.4% for the defined event within five years. The interpretation comes from the entire prediction model and its calibration, not from the score 0.22 alone. Reusing the score with a different intercept or outcome model can give a different probability.

Population-Average Predictions Depend on the Target Population

A conditional risk is not the same as an average risk across a population. To obtain an average, average the conditional risks over the distribution of predictors in the population of interest. This follows from the law of total expectation: the population probability of the event is the expected conditional event probability, using that population’s predictor distribution. If that distribution changes, the average can change even when the conditional prediction rule does not.

Worked Example: The Same Conditional Risks, Different Population Averages

Suppose a hypothetical model predicts five-year event risks for two clinical strata at a fixed score. It predicts 10% in stratum A and 30% in stratum B. These conditional predictions are assumed to apply in both target populations below.

In target population 1, 80% of eligible people are in A and 20% are in B. Its average predicted risk is $$ (0.80)(0.10)+(0.20)(0.30)=0.08+0.06=0.14, $$ or 14%.

In target population 2, 30% are in A and 70% are in B. Its average predicted risk is $$ (0.30)(0.10)+(0.70)(0.30)=0.03+0.21=0.24, $$ or 24%.

The conditional predictions by stratum are unchanged, but the population-average prediction differs because the mix of strata differs. Therefore, a population-average risk or performance summary must name the population over which it is averaged.

Justifying the Population Dependence

The distinction between conditional and average predictions can be shown directly. Let $C$ take a finite set of stratum values $c$. The event probability in population $Q$ is the sum of the event probabilities within strata, weighted by the stratum proportions:

\begin{aligned} P_Q\{Y(\tau)=1\mid S=s\} &=\sum_c P_Q\{Y(\tau)=1,C=c\mid S=s\}\\ &=\sum_c P_Q\{Y(\tau)=1\mid S=s,C=c\}\,P_Q(C=c\mid S=s). \end{aligned}

The second equality follows from the definition of conditional probability. The conditional event probabilities may stay fixed while the weights $P_Q(C=c\mid S=s)$ change across populations. Consequently, the marginal probability can change. This is why specifying a score and a conditional prediction model is not always enough to state a population-average risk: the target population’s predictor distribution is also needed.

Fix the prediction occasion.
State who is eligible and when predictors are measured. This establishes the population and the information available at prediction time.

Define the outcome and horizon.
Specify the event or measurement, its starting point, and the time window. For a binary event, define what counts as an observed event and how follow-up is handled.

State the score and prediction rule.
Give the dosage-weight definition and clarify whether the rule uses the score alone or also uses other predictors. Distinguish the score value from the rule’s output.

Name the estimand being reported.
Say whether it is a conditional prediction, a population-average risk, or a performance measure, and identify the population over which it is defined.

Comparisons, Diagnostics, and Limitations

A score-only predictor and a model combining the score with clinical covariates are different prediction rules. The latter may answer a more useful clinical prediction task, but its predictions should not be attributed to the PRS alone. Likewise, a score can be used for ranking individuals without being calibrated to absolute probabilities. Ranking concerns relative ordering; absolute-risk prediction requires a mapping to the specified outcome scale and target.

Several checks help reveal a mismatch between the intended estimand and the analysis:

Outcome and horizon: Confirm that the outcome definition and follow-up period used to fit or evaluate a prediction rule match the stated target. A prediction for an event by five years is not automatically a prediction for the same event by ten years.
Population: Check that eligibility, setting, and predictor distributions are relevant to the intended target. Differences in baseline event frequency or covariate mix can affect absolute predictions and population-average performance.
Information timing: Ensure that every predictor would be available at the stated prediction occasion. Using information measured after that occasion changes the task and can make evaluation misleading.
Score definition: Verify that the weights and dosage coding correspond to the intended score. A coding mismatch can change the score itself, not merely its interpretation.
Calibration: For probability predictions, compare predicted probabilities with event frequencies in an appropriate target sample. Systematic disagreement indicates that the predicted probabilities should not be read as calibrated risks for that target.

A common mistake is to describe a higher PRS as causing a higher probability of disease, or to call the score itself an individual’s probability. The score is a weighted summary of genetic dosages. A prediction model can associate that score with an outcome, but prediction does not establish a causal effect. Another mistake is to report a risk without stating the population or time horizon. The correction is to describe the score, prediction rule, outcome, eligible population, and horizon separately.

These definitions do not guarantee useful prediction. A score may have limited predictive information, the prediction rule may be poorly estimated, or the target setting may differ from the setting in which the rule was developed. A precisely defined estimand makes those limitations assessable; it does not remove them.

Key takeaway: A PRS is a weighted sum of genetic dosages. Its prediction meaning comes only after specifying the outcome, target population, time horizon, available predictors, and prediction quantity—such as a conditional probability, population-average risk, or prediction loss.

Check Your Understanding

For each question, identify the relevant prediction target or calculation. Solutions follow the questions.

A score is 0.40. Does this mean the individual has a 40% probability of the outcome? Explain.
A model predicts event risk by three years. What must be changed or re-established before interpreting its output as risk by eight years?
Two populations have identical conditional risks within each of two strata, but different proportions in the strata. Can their average risks differ? Give the reason.
In one sentence, define the information needed to interpret a reported population-average PRS-based risk.

Solutions. (1) No. The score is a weighted sum on the scale determined by its weights; a risk requires a prediction rule calibrated for a defined outcome, population, and horizon. (2) The outcome must be defined over the eight-year horizon, and a prediction rule appropriate and calibrated for that target is needed; a three-year prediction is not automatically an eight-year prediction. (3) Yes. The average is a weighted average of the conditional risks, and different stratum proportions give different weights. (4) State the outcome and horizon, eligible target population, prediction rule and predictors, and the population distribution over which the conditional predictions are averaged.