Baseline Incidence as the Anchor for Risk

Why Absolute Risk Needs a Baseline

A genetic effect can describe how risk differs between people, but a relative effect alone does not say how many people will experience an event. For example, the same relative increase has different absolute consequences when the starting event rate is low than when it is high. An absolute-risk estimate therefore needs an appropriate baseline: the incidence or hazard for a clearly specified population and reference profile.

That baseline must match the person and the prediction being made. Incidence varies with characteristics such as age and sex, and it can also differ across defined target populations and settings. A population-wide rate that averages across many profiles may be a poor anchor for a particular individual. The key distinction is between a rate observed across a group and a conditional risk for a person with specified characteristics.

Definition: An incidence rate is the number of new events divided by the total person-time at risk, usually reported with units such as events per 1,000 person-years. A cumulative risk is the probability that an event occurs over a stated interval among people at risk at the interval’s start. A baseline hazard is the instantaneous event rate among people still event-free, for a specified reference profile and target population. It is not necessarily the hazard of an “average” person.

From Incidence Rate to Risk

A rate and a probability are different quantities. An incidence rate has units of inverse time; a risk is dimensionless and lies between 0 and 1. For a short interval and a small rate, multiplying the rate by the interval can be a useful approximation to risk, but the exact conversion under a constant hazard is \(1-\exp(-\lambda t)\), where \(\lambda\) is the hazard per unit time and \(t\) is the interval length.

This conversion depends on assumptions. The hazard must be constant over the interval, and the calculation treats the event of interest as the only event that can remove someone from being at risk. In a real application, the hazard may vary with age or other predictors, and competing events may matter. The simple conversion is useful for illustrating the distinction, not a reason to extend a one-year rate unchanged over many years.

Constant-hazard conversion: If a person is event-free at the start of an interval and has constant hazard \(\lambda\) during an interval of length \(t\), then the probability of an event during that interval is \(1-\exp(-\lambda t)\). The units must agree: for a hazard in events per person-year, \(t\) is measured in years.

The conversion follows from the definition of a constant hazard. Let \(S(t)\) be the probability of remaining event-free through time \(t\), conditional on being event-free at time zero. Over a small interval of length \(dt\), the conditional probability of an event is approximately \(\lambda\,dt\). Thus the probability of remaining event-free changes at rate \(-\lambda S(t)\). Solving this differential equation gives the survival probability and hence the event risk:

\frac{dS(t)}{dt}=-\lambda S(t), \qquad S(0)=1

S(t)=\exp(-\lambda t), \qquad P(\text{event by }t)=1-S(t)=1-\exp(-\lambda t).

For a very small \(\lambda t\), this probability is close to \(\lambda t\), because \(1-\exp(-x)\) is approximately \(x\) when \(x\) is near zero. It is still important to label which quantity is being reported: an incidence rate per person-year is not itself a one-year probability.

Worked Example: Converting a Hypothetical Incidence Rate

Suppose a hypothetical registry follows 2,000 event-free adults from a specified age and sex group for a total of 7,000 person-years. It records 28 new events. Estimate the incidence rate and, under a constant-hazard assumption, calculate the one-year risk for a person represented by this group.

Estimate the rate. The rate is the event count divided by person-time:

\hat{\lambda}=\frac{28\text{ events}}{7{,}000\text{ person-years}} =0.00400\text{ events per person-year} =4.00\text{ events per 1,000 person-years}.

Convert to one-year risk. Assume this rate is the person’s constant hazard for one year and that no competing event removes the person from risk during that year. Then:

1-\exp(-0.00400\times 1) =1-\exp(-0.00400) \approx 0.003992.

The estimated one-year risk is about 0.399%, or 3.99 events per 1,000 people with this profile. The registry’s 4.00 events per 1,000 person-years is a rate; the 0.399% figure is a one-year probability under the stated assumptions. The two numbers are close because the hazard is small, but they are not the same quantity.

What Makes a Baseline Appropriate?

A baseline is meaningful only after its reference has been specified. For a genetic-risk prediction, the baseline might be the hazard for people in a reference score group, conditional on a particular age, sex, and target population. A relative genetic effect then describes how the hazard changes compared with that reference. For example, under a proportional-hazards model, a score-associated hazard ratio multiplies the baseline hazard. This use requires that the relative effect is appropriate for the target population and that the model’s reference profile matches the baseline being used.

In this setting, a useful notation is \(\lambda_0(a\mid s,p)\), the baseline hazard at age \(a\) for a specified reference profile, sex group \(s\), and target population \(p\). If a genetic predictor has relative hazard \(r\) compared with that reference, the model specifies \(\lambda(a)=r\lambda_0(a\mid s,p)\), subject to its assumptions. The baseline should not be substituted with an unrelated marginal rate simply because both numbers describe the same disease.

Model statement: For an interval where the baseline hazard is constant, suppose the individual’s hazard is \(\lambda=r\lambda_0\), where \(\lambda_0\) is the hazard for a defined reference profile in the target population and \(r\) is the individual’s relative hazard versus that reference. The corresponding interval risk is \(1-\exp(-r\lambda_0t)\), provided the hazard is constant over the interval and the model and target population are appropriate.

Worked Example: Anchoring a Relative Genetic Effect

Consider a hypothetical model for one-year risk among women aged 55–64 in a defined target population. Its reference-score group has a baseline hazard of 4.00 events per 1,000 person-years. For a person whose score corresponds to a hazard ratio of 1.80 relative to that reference, calculate the model-based one-year risk. Assume the relative effect applies to this population and age group, the hazard is constant for one year, and there are no competing events in this calculation.

First express the baseline hazard per person-year and multiply by the relative hazard:

\lambda_0=\frac{4.00}{1{,}000}=0.00400,\qquad \lambda=1.80(0.00400)=0.00720\text{ per year}.

Then convert the individual hazard to one-year risk:

1-\exp(-0.00720\times1) =1-\exp(-0.00720) \approx0.007174.

The model-based one-year risk is about 0.717%, or 7.17 per 1,000 people with this specified profile. The hazard ratio of 1.80 is a relative comparison with the reference group; it is not an 80-percentage-point increase in risk. The absolute estimate depends on the 4.00-per-1,000-person-year baseline as well as on the relative effect.

If the baseline instead came from a different age group or target population, the calculation could give a numerically precise but inappropriate risk. The relative effect does not make a mismatched baseline suitable.

Population Rates Are Not Individual Conditional Risks

A crude population incidence rate combines the experience of all included people, with the age and other profile distribution of that population. It answers a group-level question: how many events occurred per unit of person-time in that mixture? An individual’s conditional risk asks a different question: given that this person is event-free now and has specified characteristics, what is the probability of an event over the stated interval?

If age-specific rates differ, changing the age composition changes the crude rate even when every age-specific rate stays the same. The same issue can arise across sex groups or defined target populations. Therefore, the rate for a broad population should not automatically be used as the baseline for someone in a particular stratum. The relevant baseline should correspond to the person’s profile and to the population for which the prediction is intended.

Worked Example: Why a Crude Rate Can Mislead

Suppose a hypothetical population has two age strata. The rate is 2.00 events per 1,000 person-years for ages 50–59 and 12.0 events per 1,000 person-years for ages 60–69. In the overall population, 90% of person-time comes from the younger stratum and 10% from the older stratum. A 62-year-old has a genetic score associated with a relative hazard of 1.50 compared with the reference score. For illustration, assume the relative effect applies in both strata and hazards are constant over one year, with no competing events.

The crude population rate is the person-time-weighted average:

(0.90)(2.00)+(0.10)(12.0) =1.80+1.20 =3.00\text{ events per 1,000 person-years}.

That 3.00 rate describes the overall mixture. It is not the age-specific baseline for the 62-year-old. For the older stratum, the baseline hazard is \(12.0/1{,}000=0.0120\) per year. Applying the relative hazard gives:

\lambda=1.50(0.0120)=0.0180\text{ per year}, \qquad 1-\exp(-0.0180)\approx0.01784.

The model-based one-year risk for this profile is about 1.784%, or 17.8 per 1,000. If one incorrectly used the crude rate as the baseline, the estimated hazard would instead be \(1.50(0.00300)=0.00450\) per year, producing a one-year risk of \(1-\exp(-0.00450)\approx0.004490\), or 4.49 per 1,000. The discrepancy arises because the crude rate averages over a population with much more younger person-time than older person-time.

Checks Before Using a Baseline

Before interpreting an absolute-risk estimate, check what the baseline actually represents. A rate from a registry is not automatically a conditional hazard for a model’s reference profile. Differences in age distribution, sex composition, calendar period, geography, event definition, and eligibility can all make two baseline figures refer to different populations. The rate and relative effect must also use compatible definitions of the event and follow-up.

A baseline hazard can vary over age. If so, the constant-hazard conversion applies only to an interval over which constancy is a reasonable assumption. For a hazard that changes during the interval, cumulative risk depends on the hazard across the interval rather than on a single rate multiplied by the interval length. Estimating risk across changing hazards requires methods beyond the simple one-interval calculations here.

The target population also matters. “Population-specific” should refer to a clearly described population relevant to the prediction, not to an assumption that a broad label alone determines an individual’s risk. Use a baseline supported by data that represent the intended setting and the person’s relevant covariates. A baseline mismatch is not fixed by presenting more decimal places.

Common Mistake

Wrong reasoning: “The population incidence is 3 per 1,000 person-years, so every person in that population has a one-year risk of 3 per 1,000.”
Correction: The incidence rate summarizes events per person-time across the population’s mix of profiles. It is not automatically an individual probability or a conditional baseline hazard. Select a baseline that matches the person’s age, sex, target population, and reference profile; then use an appropriate model and interval assumptions to obtain an individual risk.
Wrong reasoning: “A relative hazard of 1.80 means the absolute risk rises by 80 percentage points.”
Correction: A relative hazard multiplies a hazard under the stated model. The corresponding absolute risk also depends on the baseline hazard and the interval. A large relative effect can still correspond to a small absolute-risk difference when the baseline is low.

Key takeaway: Absolute genetic risk is anchored, not inferred from a relative effect alone. Match the baseline incidence or hazard to the individual’s relevant profile and target population, distinguish rates from probabilities, and state the assumptions used to turn a rate into an interval risk.

Check Your Understanding

Try these questions before moving on.

In one sentence, distinguish an incidence rate from a cumulative risk, including the units of each.
A constant hazard is 0.010 per year. Write the expression for the one-year risk and explain the assumption needed to use it.
Why might a crude rate from a population with many younger people be a poor baseline for an older individual?
In a model where an individual’s hazard is 1.40 times a reference hazard, what additional information is needed to calculate an absolute risk?
Explain why a baseline hazard for a reference score group should not automatically be replaced by a marginal rate from a different target population.