Expected Exposure Before Competing Exits

Why Expected Exposure Matters

In a multiple-decrement model, a member of a group can leave during an interval through one of several competing causes. Expected exposure measures the total time, in person-years, that the group is expected to remain under observation or at risk during that interval. It is not generally equal to the number of people present at the beginning or end of the interval: some members may exit partway through it.

The calculation depends on how exits are distributed within the interval. Under the uniform distribution of decrements (UDD) assumption, exits are spread uniformly through the interval. Under a constant-force assumption, the total force of exit is constant, so the expected number remaining declines exponentially. These assumptions can give different expected exposures even when they produce the same annual exit probabilities.

For one life entering a one-year interval, the corresponding quantity is the expected fraction of a year spent in the group. For a cohort of \(N_0\) lives entering together, multiply that per-life exposure by \(N_0\). If the starting number is itself an expected number of lives from a table, the same multiplication gives expected person-years.

Exposure Under UDD

Suppose the interval is one year and the mutually exclusive causes have multiple-decrement probabilities \(q_1,\ldots,q_m\). Each \(q_j\) is the probability, for a life in the group at the start of the year, of exiting through cause \(j\) during that year. Assume all exits are included, so the probability of remaining in the group for the full year is \(p=1-q\), where \(q=\sum_{j=1}^m q_j\). Under UDD, the expected proportion remaining at time \(t\), for \(0\leq t\leq1\), is \(1-tq\).

The cause-specific probabilities are added to obtain the total probability of exit; the exposure formula uses that total. For successive years, calculate exposure separately in each year using that year’s total exit probability and the number expected to be present at its start. Then add the yearly exposures.

Exposure Under Constant Competing Forces

Now suppose each cause has a constant force of exit during the year. Write \(\lambda_j\) for the constant force for cause \(j\), and let \(\lambda=\sum_{j=1}^m\lambda_j\) be the total force. The probability of remaining in the group to time \(t\) is \(e^{-\lambda t}\), and the one-year total exit probability is \(q=1-e^{-\lambda}\). Therefore, expected exposure per life entering the interval is the integral of this survival probability.

If \(q=0\), then \(\lambda=0\): no one exits during the interval, so exposure is exactly one year per entering life. The ratio \(q/[-\ln(1-q)]\) is not defined at \(q=0\), although its limit as \(q\) approaches zero is 1. State this zero-exit case separately when using the ratio form.

Under constant competing forces, the cause-specific annual probabilities are linked to the forces by \(q_j=(\lambda_j/\lambda)(1-e^{-\lambda})\) when \(\lambda>0\). Their sum is \(q\). Thus total exposure depends on the total force, or equivalently the total annual exit probability, rather than on the allocation among causes.

Why the Formulas Hold

A member who exits at time \(T\) contributes \(\min(T,1)\) years of exposure during a one-year interval. For any nonnegative time-to-exit \(T\), the identity \(\min(T,1)=\int_0^1 I(T>t)\,dt\) expresses that time as the accumulated indicator of still being present. Taking expectations gives the integral of the probability of remaining at each time. For \(N_0\) entering lives, summing their indicators gives the expected number remaining at each time, so integrating that expected number gives cohort exposure.

Under UDD, \(P(T>t)=1-tq\), and integrating gives \(\int_0^1(1-tq)\,dt=1-q/2\). Under a constant total force, \(P(T>t)=e^{-\lambda t}\), and integrating gives \((1-e^{-\lambda})/\lambda\) for \(\lambda>0\). The formulas therefore use the same underlying principle; the difference is the assumed pattern of survival within the year.

Multiple Intervals and Common Traps

For a sequence of intervals, let \(N_k\) be the expected number present at the start of interval \(k\). Calculate exposure in that interval using its own decrement probabilities and interpolation assumption, then add the results. Under UDD the one-year contribution is \(N_k(1-q_k/2)\); under constant force it is \(N_k(1-e^{-\lambda_k})/\lambda_k\) if \(\lambda_k>0\), and \(N_k\) if \(\lambda_k=0\). The starting population for the next interval must reflect survival through the preceding interval.

• Do not count a full year for everyone. Members who exit during the interval contribute only their time in the group before exit, on average less than a full year.
• Do not apply the single-cause exposure formula separately and add the results. That would count the same starting population more than once. First use the total exit probability or total force to describe whether a member remains in the group.
• Keep the model assumption consistent. UDD and constant force specify different survival curves within the year, even when their annual exit probabilities agree.
• Handle a zero total exit probability directly. Under either model, exposure is one year per entering life if there is no possibility of exit. In the constant-force ratio, do not substitute zero into a \(0/0\) expression.

Expected exposure is a person-time measure, not an exit probability or an expected count of exits. It is useful wherever calculations require the amount of time a group is at risk or subject to a rate during an interval. The key steps are to identify who is present at the interval’s start, combine competing exits into a total, and then integrate the appropriate within-interval survival pattern.