Combining Polygenic Scores With Clinical Risk Models

Why Combine Clinical Predictors and a Polygenic Score?

A polygenic score summarizes genetic information, while a clinical risk model may use predictors such as age, family history, or measured risk factors. Using both can provide a more individualized estimate than either source alone. But an absolute risk is not obtained simply by placing two risk estimates side by side or multiplying their marginal odds ratios. The predictors need to be represented together in a model whose coefficients describe their joint relationship with the outcome.

This tutorial develops a logistic model for a binary outcome defined over a specified horizon. For example, \(Y=1\) might mean that a person develops a disease within five years, and \(Y=0\) might mean that the person does not. The horizon and outcome definition must be consistent across the data used to fit the model and the person for whom risk is predicted. Modeling event times and risks over time requires additional methods.

Definition: Let \(Y\) indicate whether the disease outcome occurs within the specified horizon, let \(Z\) be a polygenic score, and let \(C=(C_1,\ldots,C_k)\) denote clinical predictors measured before that outcome. A joint absolute-risk model specifies \(P(Y=1\mid Z,C)\). A polygenic score is often standardized as \(Z=(S-\bar{S})/s_S\), where \(S\) is the raw score and \(\bar{S}\) and \(s_S\) are the reference mean and standard deviation used for standardization.

The Logistic Model and Its Interpretation

For a binary outcome, a common model uses the logit link, which maps a probability between 0 and 1 to the log of its odds. With one score and no interaction, the model is

\operatorname{logit}\{P(Y=1\mid Z,C)\} = \log\left(\frac{P(Y=1\mid Z,C)}{1-P(Y=1\mid Z,C)}\right) = \beta_0+\beta_Z Z+\sum_{j=1}^{k}\beta_j C_j.

The probability is recovered from the linear predictor \(\eta\) using the inverse-logit function:

P(Y=1\mid Z,C)=\frac{\exp(\eta)}{1+\exp(\eta)} =\frac{1}{1+\exp(-\eta)}.

Here \(\beta_Z\) is the change in conditional log odds associated with a one-unit increase in \(Z\), holding the modeled clinical predictors fixed. If \(Z\) is measured in standard-deviation units, a one-unit increase is a one-standard-deviation increase in the score. The corresponding conditional odds ratio is \(\exp(\beta_Z)\). This is an odds ratio, not a risk ratio: its size does not directly describe the change in probability.

The intercept \(\beta_0\) gives the log odds for the reference profile, where \(Z=0\) and each clinical predictor is at its coded reference value. Centering continuous predictors can make this profile easier to interpret. A predicted probability is specific to the values entered for that person; it is not automatically the population-average risk.

Worked Example: Adding a Standardized Score to Clinical Predictors

Suppose a hypothetical model for disease within five years has been fitted using a compatible target population and outcome definition. It has intercept \(-3.20\), coefficient \(0.80\) for a binary clinical indicator \(C\) (1 means the clinical risk factor is present), and coefficient \(0.55\) for standardized score \(Z\). The fitted model is \(\eta=-3.20+0.80C+0.55Z\). Estimate risk for a person with \(C=1\) and \(Z=1.2\).

First calculate the linear predictor:

\eta=-3.20+(0.80)(1)+(0.55)(1.2) =-3.20+0.80+0.66=-1.74.

Convert it to a probability:

\hat{P}(Y=1\mid C=1,Z=1.2) = \frac{\exp(-1.74)}{1+\exp(-1.74)} = \frac{1}{1+\exp(1.74)} \approx 0.149.

The estimated five-year risk for this hypothetical profile is about 14.9%. The score coefficient corresponds to a conditional odds ratio of \(\exp(0.55)\approx1.73\) per standard deviation, holding \(C\) fixed. It does not mean that risk rises by 73 percentage points or by 73%.

When the Score Effect May Depend on Clinical Risk

A model without an interaction assumes that the score has the same effect on the log-odds scale at every value of the clinical predictors. That assumption may be unsuitable when, for example, the score association differs between clinical groups. An interaction allows the score effect to vary:

\operatorname{logit}\{P(Y=1\mid Z,C)\} = \beta_0+\beta_Z Z+\beta_C C+\beta_{ZC}(ZC).

For binary \(C\), the conditional log-odds change for a one-unit increase in \(Z\) is \(\beta_Z\) when \(C=0\), and \(\beta_Z+\beta_{ZC}\) when \(C=1\). The corresponding odds ratios are \(\exp(\beta_Z)\) and \(\exp(\beta_Z+\beta_{ZC})\). The interaction is defined on the log-odds scale; it does not imply that the difference in absolute risk between score groups is constant.

Worked Example: A Score-by-Clinical-Group Interaction

Consider a hypothetical fitted model with \(\beta_0=-3.00\), \(\beta_Z=0.40\), \(\beta_C=0.70\), and \(\beta_{ZC}=-0.25\). Estimate risk for a person with \(C=1\) and \(Z=1.5\), and compare the score odds ratios across clinical groups.

For this person, the interaction product is \(ZC=(1.5)(1)=1.5\). Therefore,

\eta=-3.00+(0.40)(1.5)+(0.70)(1)+(-0.25)(1.5) =-3.00+0.60+0.70-0.375=-2.075.

The estimated risk is

\hat{P}(Y=1\mid Z=1.5,C=1) = \frac{1}{1+\exp(2.075)} \approx 0.112.

The estimate is about 11.2% for the specified horizon. When \(C=0\), the score odds ratio per standard deviation is \(\exp(0.40)\approx1.49\). When \(C=1\), it is \(\exp(0.40-0.25)=\exp(0.15)\approx1.16\). The negative interaction means the score association is weaker on the log-odds scale in the \(C=1\) group in this hypothetical model. It does not mean that the score lowers risk in that group: its estimated coefficient there is still positive.

An interaction should be included when it is scientifically motivated or supported by an appropriate modeling strategy, not merely because many combinations can be tested. It uses additional parameters and can make estimates less precise, especially when some combinations of score and clinical status are rare. The fitted model must still be assessed for whether its predicted risks are suitable for the intended population and use.

A Short Derivation: Why the Score Coefficient Gives a Conditional Odds Ratio

In the model without an interaction, hold all clinical predictors fixed and compare two score values, \(Z\) and \(Z+1\). Their linear predictors differ by

\eta(Z+1,C)-\eta(Z,C)=\beta_Z.

Because the logit of the probability is the log odds, exponentiating the difference in logits gives the ratio of the two odds:

\frac{\operatorname{odds}(Y=1\mid Z+1,C)} {\operatorname{odds}(Y=1\mid Z,C)} = \exp\{\eta(Z+1,C)-\eta(Z,C)\} = \exp(\beta_Z).

Thus the odds ratio is conditional on the modeled clinical predictors. With an interaction, the difference in linear predictors also includes the change in the interaction term, so the conditional odds ratio depends on \(C\). This follows from the specified model; it does not establish that changing a person's score would cause a change in disease risk.

Avoiding Double-Counting

Clinical predictors and a polygenic score can be correlated without making it wrong to use both. For example, family history may reflect inherited factors, but it can also capture shared environment and genetic variation not represented by the score. The concern is not correlation by itself. The concern is combining quantities as if their effects were independent when they are not, or counting the same information twice through an unexamined construction.

A joint model estimates the association of each predictor with the outcome conditional on the other predictors included in that model. By contrast, a marginal score odds ratio and a marginal clinical odds ratio generally do not equal the corresponding conditional effects in a joint model. Multiplying those marginal odds ratios can therefore yield an incorrect joint prediction. Likewise, if a clinical risk score already incorporates a genetic predictor or a close substitute for the same information, appending that predictor without refitting or otherwise accounting for the overlap can duplicate its contribution.

Worked Example: Why Marginal Odds Ratios Cannot Simply Be Multiplied

Suppose a hypothetical clinical model gives a 4% risk for its reference profile. Its reference odds are \(0.04/(1-0.04)\approx0.04167\). Separately reported marginal odds ratios are 2.0 for a clinical indicator \(C\) and 1.8 per score unit. A different hypothetical joint model, fitted with both predictors, has coefficients \(\beta_C=0.60\) and \(\beta_Z=0.35\) and an intercept chosen to give 4% risk at \(C=0,Z=0\). Compare the risk for \(C=1,Z=1\) using the joint model with the result of naively multiplying the marginal odds ratios.

The joint model's intercept is the reference log odds:

\beta_0=\log\left(\frac{0.04}{0.96}\right)\approx-3.178.

At \(C=1,Z=1\), the joint linear predictor is \(-3.178+0.60+0.35=-2.228\). Its odds are the reference odds multiplied by \(\exp(0.60+0.35)\):

\widehat{\operatorname{odds}} = 0.04167\exp(0.95) \approx 0.1077, \qquad \hat{P}=\frac{0.1077}{1+0.1077}\approx0.0973.

The joint-model estimate is about 9.73%. The naive calculation instead multiplies the reference odds by \(2.0\times1.8=3.6\), giving odds \(0.04167(3.6)=0.1500\) and risk \(0.1500/(1+0.1500)\approx0.1304\), or 13.0%. These are not interchangeable estimates: the marginal odds ratios need not be the conditional effects in the joint model. The example illustrates why the appropriate calculation uses coefficients estimated for the joint predictor set, rather than multiplying separately reported marginal effects.

Conditions for a Meaningful Individual Risk

The logistic formula always returns a number between zero and one, but that alone does not make it a trustworthy absolute-risk estimate. A defensible application requires careful alignment between the model and its intended use.

Define the outcome and horizon. The outcome must be the same event, measured over the same period, as the one used for model fitting. A probability from a five-year binary-outcome model is not automatically a lifetime risk.
Use a compatible score scale. Score weights, alleles, and standardization must match the fitted model. If the model uses a score measured in standard-deviation units, entering an unstandardized raw score changes the meaning of the coefficient.
Represent predictors consistently. Coding, reference categories, transformations, and interaction terms must match the model specification. A coefficient cannot be applied to a differently coded predictor without converting it appropriately.
Estimate effects jointly when predicting jointly. The coefficients should come from a model containing the score and relevant clinical predictors together, or from a justified procedure that accounts for their dependence. Separate marginal effects are not automatically suitable.
Match the model to the target population. The score and clinical predictor distributions, disease definition, and baseline risk context should be sufficiently relevant to the people receiving predictions. A model can rank people plausibly yet give inaccurate absolute risks if its risk level does not apply.
Prevent information leakage. Predictors must be available at the time the risk is meant to be estimated. Data used to construct or select the score should not inadvertently include information from the outcome assessment that the model is intended to predict.

The intercept is particularly important for absolute risk. A model may provide a useful relative association but still have an unsuitable intercept for the target population. The reference risk used to define an intercept must refer to the specified reference profile; it cannot be substituted mechanically with a population-average incidence if the two quantities describe different groups. Any necessary assessment or adjustment of absolute-risk calibration belongs to the full prediction workflow, not to the inverse-logit calculation alone.

Common Mistakes

Wrong: “The score odds ratio is 1.7, so a one-standard-deviation increase adds 70 percentage points of risk.” Correction: The coefficient describes a multiplicative change in conditional odds. The change in probability depends on the starting risk and all other predictors.
Wrong: “The clinical odds ratio and score odds ratio are separate, so multiply them to get the individual effect.” Correction: Use compatible joint-model coefficients. Marginal effects need not equal conditional effects after accounting for the other predictors.
Wrong: “Family history is correlated with the score, so they cannot both be included.” Correction: Correlation alone does not prohibit joint use. It can affect coefficient precision and interpretation; the model must estimate and represent the predictors together appropriately.
Wrong: “A score-by-clinical interaction means the score has the same risk difference in both clinical groups.” Correction: The interaction above changes the score effect on the log-odds scale. Absolute-risk differences depend on the full predicted probabilities.
Wrong: “If the model returns a probability, it must be a valid absolute risk.” Correction: A number in the zero-to-one range can still be inappropriate if the outcome horizon, predictor coding, model coefficients, or target population do not match.

Key takeaway: Combine the polygenic score and clinical predictors in a clearly specified joint model, then transform its linear predictor to a probability. Interpret score coefficients conditionally, include interactions only when justified, and check that no predictor information has been counted twice.

Check Your Understanding

Answer each question using the model definitions and assumptions above.

In a logistic model, what does \(\exp(\beta_Z)\) represent when the score is standardized and there is no score-by-clinical interaction?
For the interaction model, how does the score odds ratio differ between \(C=0\) and \(C=1\)?
Why might multiplying separately estimated marginal odds ratios give a different result from using coefficients in a joint model?
Give one reason that a clinical predictor and a polygenic score could both be useful despite being correlated.
What must match between the model and the prediction setting for a five-year risk estimate to have its intended interpretation?