Average Intensity Over an Exposure Interval

Introduction

In a competing decrement model, a person contributes exposure only while still in the state from which decrements can occur. Someone who exits early contributes less person-time than someone who remains throughout the interval. This matters when relating the number of exits from a particular cause to the intensity of that cause: the relevant denominator is time at risk, not simply the initial cohort size multiplied by the interval length.

This tutorial develops that relationship for constant cause-specific intensities. It explains the expected-exits-per-exposure interpretation, shows how to calculate exposure when all decrement causes remove people from the risk set, and identifies the conditions under which the relationship applies.

Definition: For cause $j$, the cause-specific intensity at time $t$ is the instantaneous rate of exit from the active state due to cause $j$, conditional on being in that state immediately before $t$. If the intensity is constant, write it as $\mu_j$. With a finite set of competing causes, the total intensity is $\mu=\sum_j\mu_j$.

For a cohort, let $Y(t)$ denote the number still exposed to the decrements at time $t$, and let $N_j(T)$ denote the number of exits from cause $j$ during the interval from time 0 to time $T$. The cohort's person-time exposure over that interval is the accumulated time spent at risk:

E(T)=\int_0^T Y(t)\,dt.

If people are counted individually, this is the sum of their times at risk in the interval. Its unit is person-years when time is measured in years. The initial cohort multiplied by $T$ is only the exposure that would accrue if nobody left the risk state before $T$.

Constant Intensities: The Expected-Exit Relationship

Assume a fixed cohort of $n_0$ members starts in the active state at time 0. Each member is subject to the same constant cause-specific intensities throughout the interval, and exit from any cause ends that member's exposure. The total intensity is therefore the constant $\mu=\sum_j\mu_j$. Under these assumptions, the probability of remaining in the active state to time $t$ is $e^{-\mu t}$, so the expected number still at risk is $n_0e^{-\mu t}$.

The expected exits from cause $j$ over a short interval of length $dt$ are approximately the number at risk multiplied by $\mu_jdt$. Integrating those expected exits across the interval gives the expected total. Integrating the expected number at risk gives expected person-time exposure. These calculations yield:

E[N_j(T)] =\int_0^T n_0e^{-\mu t}\mu_j\,dt =\mu_j n_0\frac{1-e^{-\mu T}}{\mu}, \qquad E[E(T)] =\int_0^T n_0e^{-\mu t}\,dt =n_0\frac{1-e^{-\mu T}}{\mu}.

Thus the ratio of expected cause-$j$ exits to expected exposure is $\mu_j$. More generally, if the risk set is random but a constant intensity $\mu_j$ applies to every person while at risk, the expected cause-$j$ exit count equals $\mu_j$ times expected exposure. This is the key interpretation of an intensity as an expected rate per unit of time at risk.

Theorem: Under a homogeneous constant-intensity competing decrement model, with all relevant exits ending exposure, $$E[N_j(T)]=\mu_j E[E(T)].$$ Equivalently, $\mu_j=E[N_j(T)]/E[E(T)]$ when expected exposure is positive. This is a relationship between expectations; it does not assert that every observed cohort has exactly $N_j(T)=\mu_jE(T)$.

Why Exposure Gives the Correct Denominator

The relationship can be justified by dividing time into small intervals. Conditional on being at risk at the start of a short interval of length $dt$, the probability of an exit from cause $j$ in that interval is $\mu_jdt+o(dt)$, where $o(dt)$ is a term that becomes negligible compared with $dt$. If $Y(t)$ people are at risk, the expected cause-$j$ exits in that interval are $Y(t)\mu_jdt+o(dt)$. Summing over intervals and taking the limit gives the integral of the cause-specific intensity multiplied by the risk set.

E[N_j(T)] =\int_0^T \mu_j E[Y(t)]\,dt =\mu_j\int_0^T E[Y(t)]\,dt =\mu_j E[E(T)].

The second equality uses the fact that $\mu_j$ is constant and therefore can be taken outside the integral. The final equality follows from the definition of exposure: $E(T)=\int_0^T Y(t)\,dt$. If the intensity changes with time, it cannot generally be taken outside the integral; exits then depend on intensity weighted by the amount of exposure at each time. The constant-intensity simplification should not be used for a changing intensity without modification.

Worked Example: Expected Exits From a Two-Cause Cohort

A hypothetical cohort of 1,000 members is observed for two years. The constant annual intensity of exit from cause A is $\mu_A=0.02$, and the constant annual intensity from cause B is $\mu_B=0.03$. Every exit ends exposure. Find expected person-time exposure and expected exits from each cause. Time is measured in years; no interest rate applies to these count and exposure calculations.

Step 1: Find the total intensity. Both causes remove members from the risk state, so $\mu=0.02+0.03=0.05$ per year.

Step 2: Calculate expected exposure. Under constant total intensity, expected exposure over $T=2$ years is $n_0(1-e^{-\mu T})/\mu$. Substituting the data gives $1{,}000(1-e^{-0.05(2)})/0.05$. Since $e^{-0.1}=0.9048374$, this is $1{,}000(1-0.9048374)/0.05=1{,}903.25$ person-years, rounded to two decimals.

Step 3: Calculate expected exits by cause. Multiply expected exposure by each constant cause-specific intensity: $E[N_A]=0.02(1{,}903.25)=38.065$, or 38.07 expected exits, and $E[N_B]=0.03(1{,}903.25)=57.0975$, or 57.10 expected exits, each rounded to two decimals.

Check. The expected total number of exits is $38.065+57.0975=95.1625$. The model gives $1{,}000(1-e^{-0.1})=95.1626$ expected exits, apart from rounding. The cause-specific counts are therefore consistent with the total. The method also avoids treating all 2,000 possible person-years as actual exposure when some members exit before the end.

Combining Exposures Across Groups

Suppose several groups are subject to the same cause-specific intensity, or the purpose is to summarize their rates over a common analysis period. Expected exits and exposures add across groups. Consequently, the pooled rate is total cause-specific exits divided by total exposure, not an unweighted average of the group rates. Groups with more person-time contribute more to the pooled result.

For observed data, the quotient $N_j/E$ is commonly used as an estimate of a constant intensity. In a finite cohort, observed exits fluctuate: the quotient need not equal the underlying intensity exactly. Under the model, however, expected exits are proportional to expected exposure. If group intensities genuinely differ, a pooled quotient describes an exposure-weighted average of those intensities rather than necessarily the intensity in any one group.

Worked Example: Pooling Cause-Specific Rates Correctly

A hypothetical study records two independent groups under the same cause-specific decrement classification. Group 1 contributes 250 person-years of exposure and has 5 exits from cause C. Group 2 contributes 750 person-years and has 30 exits from cause C. Treat the observed counts as data for estimating a common constant intensity. Find each group's rate and the pooled rate. All exposures are in years.

Step 1: Calculate the group rates. For group 1, the rate is $5/250=0.02$ exits per person-year. For group 2, it is $30/750=0.04$ exits per person-year.

Step 2: Pool counts and exposure. Total exits are $5+30=35$, and total exposure is $250+750=1{,}000$ person-years. The pooled rate is therefore $35/1{,}000=0.035$ exits per person-year.

Step 3: Check the weighting. The exposure weights are $250/1{,}000=0.25$ and $750/1{,}000=0.75$. The weighted average is $0.25(0.02)+0.75(0.04)=0.005+0.03=0.035$, matching the pooled quotient.

The unweighted average of the two group rates would be $(0.02+0.04)/2=0.03$, which is not the pooled rate because the groups contributed different amounts of exposure. The method here is to add exits and exposure separately before dividing; it avoids giving a small group the same weight as a much larger one.

Common Mistake: Using Initial Cohort-Time as Exposure

A frequent error is to divide expected exits by $n_0T$, even when competing decrements occur during the interval. That denominator assumes everyone remains at risk for the full period. If exits occur before the interval ends, actual cohort exposure is lower. Under the constant-intensity assumptions, using $n_0T$ will generally make the calculated rate too small because it overstates exposure.

Worked Example: Diagnosing an Overstated Denominator

Use the hypothetical cohort in the first example: $n_0=1{,}000$, cause-A intensity $0.02$ per year, cause-B intensity $0.03$ per year, and a two-year interval. A candidate divides expected cause-A exits by $1{,}000(2)=2{,}000$ person-years. Identify the error and calculate the correct average intensity.

Step 1: Identify the candidate's denominator. The value $2{,}000$ is the exposure that would accrue if all 1,000 members remained at risk for both years. But the combined intensity is $0.05$ per year, so some members leave during the interval.

Step 2: Use expected exposure instead. The expected exposure is $1{,}000(1-e^{-0.1})/0.05=1{,}903.25$ person-years, as calculated earlier. Expected cause-A exits are $0.02(1{,}903.25)=38.065$.

Step 3: Divide by the correct exposure. The expected-exits-per-expected-exposure ratio is $38.065/1{,}903.25=0.02$ exits per person-year, the stated cause-A intensity. The candidate's denominator instead gives $38.065/2{,}000=0.0190325$, which is too small. The error is not in the exit count; it is in counting time after a member has already left the risk state.

Scope and Interpretation

The calculation depends on matching the exposure definition to the intensity. Exposure must include only time during which a member is at risk of the cause being counted, and the intensity must be expressed in the same time unit. If intensities are annual, exposure should be measured in years; using months without conversion changes the numerical rate by a factor of 12. All competing causes that remove members from the active state must be included when calculating the total intensity and expected risk set.

The relation also concerns expected counts, not a deterministic equality for each realization. A hypothetical cohort can produce more or fewer exits than its expected number. Likewise, a realized count divided by realized exposure is an observed rate, which may differ from the constant underlying intensity. These distinctions are important when interpreting cohort calculations or estimating rates from observed data.

This exposure-based view is useful whenever cohorts have different follow-up durations, or when exits occur before a study interval ends. It provides a common denominator that reflects actual time at risk and makes pooled rates interpretable. The essential checks are to confirm the cause, the risk state, the time unit, and whether the requested quantity is an expectation or an observed rate.

Key takeaway: With a constant cause-specific intensity $\mu_j$, expected exits from cause $j$ equal $\mu_j$ multiplied by expected person-time exposure. In competing decrement settings, calculate exposure only while members remain at risk; do not automatically use initial cohort size times interval length.

Check Your Understanding

For each problem, use the stated constant-intensity assumptions and keep exposure and intensity in consistent time units.

A hypothetical cohort of 400 is followed for two years. Two constant annual intensities are $0.01$ and $0.03$, and either exit ends exposure. Find expected exposure and expected exits from the first cause. (Answer: 768.84 person-years; 7.69 expected exits.)
A cause produces 12 expected exits over 600 expected person-years of exposure under a constant-intensity model. Find its intensity and state its unit. (Answer: $0.0200$ per person-year.)
Two groups have exposures of 250 and 750 person-years and observed cause-specific exit counts of 5 and 30. Find the pooled rate and explain why the unweighted average of group rates is not the pooled rate. (Answer: $0.0350$ exits per person-year; the unweighted average is $0.0300$ because it gives equal weight to unequal exposures.)
A cohort of 1,000 faces two constant annual intensities whose total is $0.10$, one of which is $0.04$. Find expected exposure over one year and expected exits from the cause with intensity $0.04$. (Answer: 951.63 person-years; 38.07 expected exits.)
Two independent groups subject to a common constant cause-specific intensity contribute 500 and 300 person-years. They record 8 and 3 exits from that cause, respectively. Find the pooled observed rate. (Answer: $11/800=0.01375$ exits per person-year.)

What You'll Learn