Choosing Risk Thresholds With Benefit and Harm

From Predicted Risk to a Treatment Decision

A polygenic score can help estimate a person’s absolute risk over a specified period. That estimate does not, by itself, say whether treatment is worthwhile. A decision also depends on what the intervention can prevent and what burdens or harms it causes. A person at higher risk may have more to gain from an effective intervention, but risk alone does not establish the balance of benefit and harm.

The decision rule developed here compares the expected utility of intervening with the expected utility of not intervening. The risk used in the calculation is the person’s untreated absolute risk for a defined event over a defined time horizon. The treatment effect and the consequences of both the event and the intervention must refer to that same decision context.

Definition: A risk threshold is the untreated absolute risk at which the expected utility of intervention equals the expected utility of no intervention, under specified assumptions about treatment effectiveness and benefits and harms. Under a rule that intervenes above the threshold, treatment is favored when the estimated risk exceeds it.

A Threshold With a Fixed Expected Treatment Burden

Let \(p\) be the probability of the clinical event over the chosen horizon if a person does not receive the intervention. Let \(e\) be the proportional reduction in event risk caused by treatment, so that treatment risk is \((1-e)p\). This risk model assumes the relative reduction is constant over the relevant risk range and horizon. Let \(L\) be the utility loss associated with the event, and let \(C\) be the expected utility burden of treatment, expressed on the same utility scale. The burden might summarize side effects, inconvenience, or treatment cost if these have been converted to utility units.

Under these assumptions, the expected event-related loss without treatment is \(pL\). With treatment it is \((1-e)pL\), and treatment also incurs burden \(C\). The expected utility gain from treating rather than not treating is therefore the reduction in expected event loss minus the treatment burden.

\Delta U = pL - (1-e)pL - C = peL - C

Treatment is favored when \(\Delta U>0\). Setting \(\Delta U=0\) gives the threshold. This comparison uses incremental utility: losses shared by both choices cancel, so the decision depends on the event risk reduced by treatment and the burden added by treatment.

peL > C \quad\Longleftrightarrow\quad p > p_t = \frac{C}{eL}

Here \(p_t\) is the threshold risk. A larger treatment burden raises it; a more effective treatment or a more consequential event lowers it. If \(e=0\), treatment does not reduce the event risk, so it cannot be justified by event prevention when \(C>0\). If \(C/(eL)\) is at least 1, no attainable risk strictly exceeds the threshold under this model. These conclusions are conditional on the quantities being appropriate for the person, outcome, and time horizon under consideration.

Worked Example: Deriving a Threshold From Expected Utility

Suppose, in a hypothetical decision, an event has a utility loss of 80 points, treatment reduces event risk by 25%, and treatment has an expected burden of 2 utility points. Assume the risk reduction applies over the same five-year horizon as the event risk, and that the burden is measured on the same scale as the event loss. Find the threshold.

Here \(L=80\), \(e=0.25\), and \(C=2\). The threshold is \(p_t=C/(eL)\):

p_t = \frac{2}{0.25 \times 80} = \frac{2}{20} = 0.10

The threshold is a five-year untreated event risk of 10%. At exactly 10%, the expected event loss avoided is \(0.10 \times 0.25 \times 80=2\) utility points, equal to the treatment burden. At a risk above 10%, expected benefit exceeds expected burden; below 10%, it does not. This calculation assumes the stated treatment effect and burden are applicable to the person being considered.

When Harm Depends on Whether the Event Occurs

A treatment burden need not be a fixed expected cost for every treated person. For example, consider a simplified model in which the intervention provides benefit \(B\) if it prevents the event, while a treatment harm with disutility \(H\) occurs only in people who would otherwise remain event-free. Suppose treatment prevents the event with probability \(e\) among those who would have the event, and the harm probability and disutility are represented by \(H\) in the event-free group. In this model, the expected utility change is \(peB-(1-p)H\). The second term is weighted by \(1-p\) because that harm is assigned to the event-free group, rather than charged to every treated person.

\Delta U = peB - (1-p)H

Solving \(\Delta U>0\) gives a different threshold from the fixed-burden rule:

peB > (1-p)H \quad\Longleftrightarrow\quad p(eB+H) > H \quad\Longleftrightarrow\quad p > \frac{H}{eB+H}

The two threshold formulas are not contradictory; they describe different models of treatment burden. A fixed expected burden \(C\) gives \(C/(eL)\), whereas a harm weighted by the probability of remaining event-free gives \(H/(eB+H)\). Choosing the formula requires specifying how benefits and harms arise. In particular, one should not use a harm term weighted by \(1-p\) if the stated burden is already an expected cost incurred regardless of event status.

Worked Example: A Harm Occurring in the Event-Free Group

In a hypothetical five-year decision, the event would cause 60 utility points of loss, treatment prevents 40% of events, and a treatment harm has a disutility of 3 points in the event-free group. Assume the harm description is intended to apply only to that group and the treatment effect applies over the same horizon. Calculate the threshold and decide at an untreated risk of 8%.

Use \(B=60\), \(e=0.40\), and \(H=3\). The threshold is:

p_t = \frac{H}{eB+H} = \frac{3}{0.40 \times 60+3} = \frac{3}{27} = 0.1111

The threshold is approximately 11.1%. At \(p=0.08\), the expected benefit is \(0.08 \times 0.40 \times 60=1.92\) utility points. The expected harm is \((1-0.08)\times 3=2.76\) utility points. The difference is \(1.92-2.76=-0.84\) points, so this model favors no intervention at an 8% risk. At the threshold, the two expected quantities are equal.

Using Calibrated Polygenic Absolute Risk

For a risk-based decision, the predicted risk must be an absolute risk for the relevant outcome and horizon, not merely a relative genetic effect or a ranking of people. It should be calibrated for the population and clinical setting in which it is used. Earlier material on absolute-risk estimation and calibration establishes why baseline incidence, the time horizon, and the target population matter: a score can rank people well while still producing risks that are too high or too low.

Once a suitable risk estimate is available, the decision rule is conceptually direct: compare the person’s estimated untreated risk with the threshold derived from the intervention’s expected effects and consequences. The treatment effect used in the threshold is a causal effect of the intervention, not an association between a polygenic score and disease. Unless evidence supports treatment-effect differences by score, use the same treatment-effect assumption across score levels.

Worked Example: Applying a Threshold to Polygenic Risk

Suppose a hypothetical clinical program uses a calibrated ten-year absolute-risk model that combines clinical predictors with a polygenic score. For a particular patient, it estimates a 9% untreated risk of the target event over ten years. A hypothetical intervention is assumed to reduce that event risk by 30%, and its expected treatment burden is 1.8 utility points. The event loss is assigned 50 utility points. Assume these estimates concern the same population, outcome, and ten-year horizon, and that the treatment effect is applicable to this patient.

The fixed-burden threshold is:

p_t = \frac{C}{eL} = \frac{1.8}{0.30 \times 50} = \frac{1.8}{15} = 0.12

The threshold is a 12% ten-year untreated risk. The patient’s estimated risk, 9%, is below it. The expected event loss avoided is \(0.09 \times 0.30 \times 50=1.35\) utility points, less than the 1.8-point expected burden. Under the stated assumptions, the rule therefore does not favor intervention for this patient. This is a conditional decision calculation, not a claim that the treatment is never appropriate for other reasons.

For comparison, if another hypothetical patient with the same treatment assumptions had a calibrated ten-year untreated risk of 15%, the expected event loss avoided would be \(0.15 \times 0.30 \times 50=2.25\) utility points. Since \(2.25>1.8\), the model would favor intervention. The difference in decision follows from the different absolute risks; it does not imply that the polygenic score changes the treatment’s effectiveness.

Assumptions and Practical Checks

A numerical threshold is only as useful as the assumptions behind it. Before applying one to a polygenic absolute-risk estimate, check that the quantities are defined on a common basis.

Match the horizon and outcome. A ten-year risk cannot be combined uncritically with a treatment effect reported for a different follow-up period or a different endpoint.
Use untreated risk. The \(p\) in these formulas is the risk without the intervention. Do not compare a treatment-adjusted risk with a threshold derived for untreated risk.
Use a relevant treatment effect. The proportional reduction \(e\) must be supported for the intervention, outcome, and population. The calculation assumes it applies across the risk range considered; if that assumption is not credible, the simple formula may not apply.
Keep utility quantities comparable. Event loss and treatment burden must be expressed on the same scale, and the burden must be defined consistently as either a fixed expected burden or a harm conditional on a specified outcome.
Check risk calibration. A threshold does not repair systematic overprediction or underprediction. A miscalibrated absolute risk can place people on the wrong side of an otherwise correctly derived threshold.

A threshold can also depend on individual preferences. People may value prevention of the event differently or experience treatment burdens differently. In that setting, a single population threshold is a useful summary only if it represents the relevant preferences; it need not be the right decision boundary for every person.

Why the Threshold Matters

Risk thresholds make explicit the trade-off that is otherwise hidden in an eligibility rule. They show why the same predicted risk can support different decisions for interventions with different effectiveness, burdens, or consequences of the prevented event. They also distinguish risk estimation from decision-making: a polygenic score may change an estimate of baseline risk, while the threshold is determined by the intervention and the values assigned to its benefits and harms.

The threshold is not a universal biological boundary, nor does crossing it prove that treatment will benefit a particular person. It is the equality point under a specified decision model. Uncertainty in risk, treatment effect, and utility can matter, especially for estimates near the threshold; the calculation itself should not be mistaken for certainty about the best action.

Common mistake: “The patient’s risk is 12%, so treatment must be beneficial because the threshold is 10%.” A threshold comparison alone is not sufficient if the 12% is not calibrated untreated risk, the treatment effect refers to a different horizon, or the treatment burden was modeled inconsistently. Check those inputs first. When they are aligned and the assumptions hold, being above the threshold means expected utility favors intervention under that model—not that benefit is guaranteed or that the score proves treatment works better for this patient.

Check Your Understanding

Use the stated assumptions in each question; focus on identifying the correct decision quantities and interpretation.

If treatment reduces event risk by 20%, the event loss is 40 utility points, and the fixed expected treatment burden is 2 points, what is the untreated-risk threshold?
Why does a harm that occurs only in the event-free group lead to a different threshold formula from a fixed expected treatment burden?
A polygenic model reports a relative risk but no calibrated absolute risk for a specified horizon. What additional risk quantity is needed before applying the threshold rule?
If a patient’s risk is above a threshold derived from a treatment effect that does not apply to their population, what part of the decision calculation is unsupported?
Does a higher polygenic score, by itself, establish that treatment is more effective for people with that score? Explain.