From Intensities to a Cohort Account
A competing-decrement cohort calculation links several quantities that are easy to confuse: the number still exposed to risk, the exits attributed to each cause, and the time spent in the cohort. The central discipline is to update the cohort at every change in intensity. A cause’s intensity applies only to people who are still in the cohort at that time, not to the original starting number.
The methods below assume a closed cohort with no entries. Each person still active at time \(t\) is subject to cause-specific intensities \(\mu_j(t)\), for causes \(j=1,\ldots,r\). These intensities are conditional rates per unit time. We assume that competing exits occur one at a time, so an individual does not exit simultaneously for two causes. The cohort quantities calculated are expected counts; they need not be integers.
This tutorial focuses on piecewise-constant intensities, a common setup for cohort questions. Within each interval, the total intensity is constant, so the cohort decreases exponentially. When the intensities change, the calculation starts the new interval with the cohort remaining at its beginning.
The One-Interval Calculation
Suppose an interval has length \(h\), begins with \(n\) people, and has constant cause-specific intensities \(\mu_1,\ldots,\mu_r\). Write \(\mu=\sum_j\mu_j\). The probability of remaining in the cohort for elapsed time \(t\) within this interval is \(e^{-\mu t}\), so the expected number remaining is \(ne^{-\mu t}\).
Here \(D_j\) is the expected number of exits from cause \(j\) during the interval, and \(E\) is exposure in person-time units. The interval’s total expected exits are \(n(1-e^{-\mu h})\). If \(\mu>0\), the share of those exits due to cause \(j\) is \(\mu_j/\mu\). If every intensity is zero, there are no exits, and the formula involving division by \(\mu\) is not used; exposure is then \(nh\).
The exposure formula follows by integrating the changing number at risk, rather than treating the starting cohort as exposed throughout the entire interval.
Similarly, the cause-specific exit count is \(\int_0^h ne^{-\mu t}\mu_j\,dt\). Since the intensities are constant, \(\mu_j\) can be taken outside the integral, giving \(D_j=(\mu_j/\mu)n(1-e^{-\mu h})\). This explains why cause shares within an interval equal intensity shares under the stated constant-intensity assumption.
Worked Example: One Year With Two Competing Causes
A hypothetical cohort of 200 people is followed for one year. The constant cause-specific intensities are \(0.05\) per year for cause 1 and \(0.15\) per year for cause 2. Find the remaining cohort, expected exits by cause, total exit share by cause, and exposure. Use the one-interval method and report counts and exposure to four decimal places.
Method. The total intensity is \(\mu=0.05+0.15=0.20\) per year, and the interval length is \(h=1\). The survival factor is \(e^{-0.20}=0.8187307531\).
The expected number remaining is \(200(0.8187307531)=163.7461506\), or 163.7462.
Total expected exits are \(200(1-0.8187307531)=200(0.1812692469)=36.2538494\). Cause 1 has intensity share \(0.05/0.20=0.25\), so its exits are \(36.2538494(0.25)=9.0634624\), or 9.0635. Cause 2 has share \(0.15/0.20=0.75\), so its exits are \(36.2538494(0.75)=27.1903870\), or 27.1904.
The conditional shares among those who exit during this year are therefore 25.00% for cause 1 and 75.00% for cause 2. Exposure is \(200(1-0.8187307531)/0.20=181.2692469\), or 181.2692 person-years.
Check. The exits sum to \(9.0634624+27.1903870=36.2538494\), matching total exits. Also, \(163.7461506+36.2538494=200\), matching the starting cohort. The common mistake is to multiply each intensity by 200 and one year as though the full cohort remained at risk all year; that would ignore exits occurring during the year.
Updating Across Multiple Intervals
For a piecewise-constant model, apply the one-interval formulas successively. The ending cohort in one interval becomes the starting cohort in the next. Total exits and exposure over the full period are sums of their interval-specific values. An overall cause share must be calculated from total exits by that cause divided by total exits from all causes; it is not generally the average of the interval cause shares.
Use the original cohort for the first interval and the preceding interval’s remaining cohort thereafter.
Add the cause-specific intensities applying during that interval.
Use the interval length and its starting cohort in each formula.
Sum exits and exposure across intervals, then verify that starting cohort equals ending cohort plus all exits.
Worked Example: Reversing Cause Shares Between Years
A hypothetical cohort of 400 is followed for two years. During year 1 the cause-specific intensities are \(0.30\) and \(0.10\) per year; during year 2 they are \(0.10\) and \(0.30\) per year. Find remaining lives after each year, exits by cause over the full two years, overall conditional cause shares among all exits, and total exposure. Report counts and exposure to four decimal places and shares to four decimal places.
Method. In each year the total intensity is \(0.40\), so the one-year survival factor is \(e^{-0.40}=0.6703200460\), and the one-year exit fraction is \(1-e^{-0.40}=0.3296799540\).
Year 1. Starting with 400, the remaining cohort is \(400e^{-0.40}=268.1280184\). Total exits are \(400(0.3296799540)=131.8719816\). The cause shares are \(0.30/0.40=0.75\) and \(0.10/0.40=0.25\), giving cause 1 exits \(131.8719816(0.75)=98.9039862\) and cause 2 exits \(131.8719816(0.25)=32.9679954\). Exposure is \(131.8719816/0.40=329.6799540\) person-years.
Year 2. The starting cohort is 268.1280184, not 400. The remaining cohort is \(268.1280184e^{-0.40}=179.7315856\). Total exits are \(268.1280184(0.3296799540)=88.3964328\). Now the cause shares are \(0.10/0.40=0.25\) and \(0.30/0.40=0.75\). Thus cause 1 exits are \(88.3964328(0.25)=22.0991082\), and cause 2 exits are \(88.3964328(0.75)=66.2973246\). Exposure is \(88.3964328/0.40=220.9910821\) person-years.
Across both years, cause 1 exits total \(98.9039862+22.0991082=121.0030944\), and cause 2 exits total \(32.9679954+66.2973246=99.2653200\). All exits total \(220.2684144\). Therefore the overall conditional shares are \(121.0030944/220.2684144=0.5493\), or 54.93%, for cause 1, and \(99.2653200/220.2684144=0.4507\), or 45.07%, for cause 2. Total exposure is \(329.6799540+220.9910821=\mathbf{550.6710}\) person-years.
Check and exam trap. The final cohort is \(400e^{-0.80}=179.7315856\), so total exits are \(400-179.7315856=220.2684144\), agreeing with the cause totals. A wrong calculation would apply year 2’s cause 1 share to the original 400, producing \(400(0.3296799540)(0.25)=32.9680\) cause 1 exits in year 2. The correction is to use the year 2 starting cohort, 268.1280, giving \(22.0991\). The later-year cause rates act only on people who remain exposed.
Three Causes and an Overall Share
With three or more causes, the same method applies: divide each interval’s total exits among causes using the intensity shares that apply in that interval, then add like causes across intervals. To find a conditional cause share for the whole follow-up period, condition on having exited at some time during that period. The denominator is all exits during the period, not the starting cohort and not the number remaining at the end.
Worked Example: Three Causes Over Two Half-Year Intervals
A hypothetical cohort of 500 is observed for one year. Each interval is half a year. In the first half-year, the intensities for causes 1, 2, and 3 are \(0.20\), \(0.10\), and \(0.10\) per year. In the second half-year they are \(0.10\), \(0.20\), and \(0.10\) per year. Find the remaining cohort, exits by cause over the year, the overall conditional cause shares, and total exposure. Report counts and exposure to four decimal places and shares to four decimal places.
Method. In both intervals the total intensity is \(0.40\) per year. For a half-year interval, the exponent is \(0.40(0.5)=0.20\), so the survival factor is \(e^{-0.20}=0.8187307531\), and the exit fraction is \(1-e^{-0.20}=0.1812692469\).
First half-year. The cohort remaining is \(500e^{-0.20}=409.3653765\). Total exits are \(500(0.1812692469)=90.6346235\). The cause shares are \(0.20/0.40=0.50\), \(0.10/0.40=0.25\), and \(0.10/0.40=0.25\). The corresponding exits are \(45.3173117\), \(22.6586559\), and \(22.6586559\). Exposure is \(90.6346235/0.40=226.5865587\) person-years.
Second half-year. Starting with \(409.3653765\), the ending cohort is \(409.3653765e^{-0.20}=335.1600230\). Total exits are \(409.3653765(0.1812692469)=74.2053535\). The cause shares are now \(0.25\), \(0.50\), and \(0.25\), giving exits \(18.5513384\), \(37.1026768\), and \(18.5513384\). Exposure is \(74.2053535/0.40=185.5133838\) person-years.
Over the full year, cause 1 exits are \(45.3173117+18.5513384=63.8686501\); cause 2 exits are \(22.6586559+37.1026768=59.7613326\); and cause 3 exits are \(22.6586559+18.5513384=41.2099942\). Total exits are \(164.8399770\). The overall cause shares are therefore \(63.8686501/164.8399770=0.3875\), \(59.7613326/164.8399770=0.3625\), and \(41.2099942/164.8399770=0.2500\), to four decimal places. Total exposure is \(226.5865587+185.5133838=\mathbf{412.0999}\) person-years.
Check. The three exit totals sum to \(164.8399770\), and \(500-335.1600230=164.8399770\). The third-cause share is \(25.00\%\) in both intervals, so its overall share remains \(25.00\%\); the shares for causes 1 and 2 are weighted by the different numbers exiting in the two intervals.
Checks That Protect the Calculation
A complete cohort account should satisfy several checks. These are not substitutes for using the correct interval starting value, but they often expose arithmetic or bookkeeping errors.
- Cohort balance: starting cohort equals ending cohort plus the total exits, allowing for rounding.
- Cause balance: the sum of cause-specific exits equals total exits in each interval and across the full period.
- Share balance: conditional cause shares sum to 1 (or 100%) when the denominator is all exits over the same period.
- Exposure scale: exposure is in person-years when time is measured in years. It cannot exceed starting cohort multiplied by interval length in a closed cohort, because the number at risk never exceeds its starting value.
The constant-intensity cause share \(\mu_j/\mu\) applies within an interval, not automatically over a longer period with changing intensities. Once rates change, the overall share depends both on the cause shares in each interval and on how many people remain to experience those intervals. Similarly, exposure must account for the declining cohort throughout each interval; multiplying the starting number by the full interval length overstates exposure whenever exits occur.
Check Your Understanding
For each problem, calculate with the interval-specific starting cohort. Round counts and exposure to four decimal places and probabilities or shares to four decimal places unless stated otherwise.
- A cohort of 100 is exposed for one year to constant cause-specific intensities of \(0.10\) and \(0.10\) per year. Find the remaining cohort, exits by cause, and exposure. (Answer: remaining 81.8731; exits 9.0635 and 9.0635; exposure 90.6346 person-years.)
- A cohort of 300 faces constant intensities \(0.20\) and \(0.10\) per year for two years. Find the ending cohort, exits by cause, and exposure. (Answer: ending cohort 164.6435; exits 90.2377 and 45.1188; exposure 451.1884 person-years.)
- A cohort of 200 faces cause intensities \(0.10\) and \(0.30\) in year 1, then \(0.30\) and \(0.10\) in year 2. Find total exits by cause and the overall conditional share due to cause 1. (Answer: cause 1 exits 49.6327; cause 2 exits 60.5015; cause 1 share 0.4507.)
- A cohort of 250 faces three constant cause intensities of \(0.10\), \(0.20\), and \(0.10\) per year for one year. Find the remaining cohort, exits by cause, and exposure. (Answer: remaining 167.5800; exits 20.6050, 41.2099, and 20.6050; exposure 206.0499 person-years.)
- A cohort of 120 is followed through two half-year intervals. The cause intensities are \(0.20\) and \(0.10\) per year in the first interval and \(0.10\) and \(0.20\) per year in the second. Find the overall conditional share of exits due to cause 1. (Answer: 0.5000.)