Reading a Multiple-Decrement Table
A multiple-decrement table records how many lives enter an interval and how many leave it through each of several competing causes. The central task is to keep three quantities distinct: the number at risk at the start, the exits during the interval, and the number remaining at the end. A row can look plausible while failing its accounting identities, so both the arithmetic within a row and the continuity between rows should be checked.
For this tutorial, consider an age-based table with one-year intervals. The row for age \(x\) describes lives present at exact age \(x\) and their exits over \([x,x+1)\). The same principles apply to other interval lengths, provided the row’s interval and cohort basis are clearly stated.
Here, “lives exposed” means lives present and at risk at the interval’s start; it does not mean person-years of exposure. A table that reports person-years or a central exposure measure is using a different quantity and should not be checked as though that value were \(l_x\). Likewise, exits must refer to the same interval and population as the stated starting lives.
For example, suppose the causes are death and withdrawal. If a person can leave through only one of these causes during the interval, the cause-specific counts partition the total exits. A person who remains in force at the interval’s end is counted among survivors, not among exits. If the causes listed are not exhaustive, then the total of the listed cause counts need not equal all exits; an “other” category or an explicit residual is needed to complete the accounting.
The Two Basic Accounting Checks
A complete row for an exhaustive set of mutually exclusive causes must satisfy two identities. The first checks how exits are divided among causes. The second checks that every starting life either exits or remains.
Together, these imply \(l_{x+1}=l_x-\sum_j d_x^{(j)}\). These are count identities, not assumptions about how exits are distributed within the interval. They do not require a mortality law, a force of decrement, or an interpolation assumption.
If the row is internally complete and its counts are observed for a closed cohort, the corresponding empirical proportions can also be checked: the total exit proportion is \(d_x/l_x\), and the cause-specific proportions are \(d_x^{(j)}/l_x\). These proportions are not additional counts. A probability model or a table of probabilities may use related notation, but a count identity should first be checked in the count columns.
Why the Identities Hold
The checks follow directly from classifying each life present at the start of the interval. Under the stated assumptions, each such life has exactly one of two outcomes by the interval’s end: exit through exactly one listed cause, or remain in the cohort. Therefore, the starting group is partitioned into cause-specific exit groups and the survivor group.
Because the exit causes are mutually exclusive and exhaustive for exits, every exit is counted once and only once. Adding the cause-specific counts gives \(d_x=\sum_j d_x^{(j)}\).
Every one of the \(l_x\) starting lives either belongs to one of the exit groups, whose combined size is \(d_x\), or remains, with count \(l_{x+1}\). Thus \(l_x=d_x+l_{x+1}\).
If either equality fails, at least one entry, definition, interval, or cause-coverage assumption needs investigation. The arithmetic alone does not establish which entry is wrong.
The last point matters in practice. An inconsistent row signals a problem, but it does not generally identify a unique correction. A value may have been transcribed incorrectly, a cause may be omitted, or the entries may refer to different bases. A correction is justified only when additional information tells us which entries are reliable.
Worked Examples
Worked Example: Checking a Complete Two-Cause Row
A hypothetical closed cohort has 240 lives exposed at age 58. During the year, 11 exit through cause A and 7 through cause B. The table reports 222 survivors at age 59. Assuming the two causes are mutually exclusive and exhaustive, check the row and calculate the observed total and cause-specific exit proportions. Name a common mistake the checks prevent.
Method. First add the cause-specific exits, then compare their sum with the reported survivors and starting lives. Only after the count identities agree, divide by the starting lives to obtain proportions.
Cause-count check. The total exits implied by the two cause columns are \(11+7=18\).
Survivor check. Starting lives minus these exits gives \(240-18=222\), exactly the reported survivor count. Equivalently, \(18+222=240\). Both within-row identities hold.
Proportions. The observed total exit proportion is \(18/240=0.075\). The proportions exiting through causes A and B are \(11/240=0.045833\ldots\) and \(7/240=0.029166\ldots\), respectively. Rounded only at the final step to four decimal places, these are 0.0750, 0.0458, and 0.0292. Their cause-specific proportions sum to \(0.0750\) before rounding.
Check. The count totals reconcile, and the rounded cause proportions also sum to the rounded total proportion: \(0.0458+0.0292=0.0750\). The common mistake avoided is dividing a cause-specific exit count by the survivors, \(222\), rather than by the lives exposed at the start, \(240\).
Worked Example: Diagnosing a Row That Does Not Reconcile
A hypothetical decrement table for a closed cohort lists 500 lives exposed at age 64, 13 exits from cause A, 8 exits from cause B, 20 total exits, and 480 survivors at age 65. Assume the two listed causes are intended to be mutually exclusive and exhaustive. Identify every failed check and explain whether the information determines a unique correction.
Method. Check the cause sum and the starting-lives partition independently. Do not choose a column to alter unless the problem states that it is reliable.
Cause-sum check. The cause-specific counts sum to \(13+8=21\), but the reported total is 20. The first identity fails by one exit.
Starting-lives check. The reported total exits and survivors sum to \(20+480=500\), so the second identity holds using those two columns. But the cause counts and survivor count imply \(21+480=501\), one more than the 500 starting lives.
Correction assessment. The row is inconsistent, but the stated information does not identify a unique erroneous entry. For instance, if the cause-specific counts are reliable and the causes are exhaustive, total exits must be 21 and survivors must be \(500-21=479\). Alternatively, if total exits and survivors are reliable, then the cause counts must be revised to sum to 20; one possible revision is 13 and 7, but other revisions are possible. The table does not tell us which interpretation is correct.
Check. Under the first explicitly stated assumption, \(13+8=21\) and \(21+479=500\). Under the alternative, \(13+7=20\) and \(20+480=500\). Both corrected versions satisfy the identities, demonstrating why an unsupported “obvious” correction is unsafe. The common mistake avoided is treating the cause counts as automatically authoritative when the problem has not said they are.
Worked Example: Checking Continuity Across Successive Rows
A hypothetical table for the same closed cohort reports the following counts. The listed causes are exhaustive and mutually exclusive.
| Interval | Lives exposed | Cause A exits | Cause B exits | Total exits | Survivors |
|---|---|---|---|---|---|
| [70, 71) | 320 | 9 | 7 | 16 | 304 |
| [71, 72) | 303 | 6 | 5 | 11 | 292 |
Check the two rows internally and then check continuity between them. State what can and cannot be concluded about the second row.
Method. Apply both within-row identities to each interval, then compare the first row’s survivors with the next row’s exposure. For one continuous closed cohort with adjacent intervals, those two counts must agree.
First row. Cause exits sum to \(9+7=16\), matching the total exits. Total exits plus survivors are \(16+304=320\), matching the starting lives. The row reconciles.
Second row, cause sum. The two cause counts sum to \(6+5=11\), matching the reported total exits.
Second row, starting-lives check. The reported total exits and survivors sum to \(11+292=303\), matching the reported exposure of 303. The row also reconciles internally.
Across-row check. The first row ends with 304 survivors, so the next row should begin with 304 lives for the same closed cohort and adjacent intervals. Instead, it reports 303. Thus both rows are internally consistent, but the table fails the across-row continuity check by one life.
Conclusion and check. The discrepancy identifies a continuity problem, not its unique source. If the first row’s survivor count is known to be reliable, the next exposure should be 304; if the second row’s exposure is known to be reliable, then the preceding survivor count needs investigation. Without further information, neither entry can be selected as the correction. The common mistake avoided is assuming that an internally balanced row must also connect correctly to the next row.
Across-Row Checks and Their Limits
For adjacent intervals of the same closed cohort, the end count in one row is the start count in the next:
In standard age notation both quantities are written \(l_{x+1}\); comparing the reported entries is still a useful data check. This equality assumes that the rows follow the same cohort, the intervals meet without a gap or overlap, and there are no entrants or other unrecorded changes between the end of one interval and the start of the next. A table assembled from different cohorts, or one that allows new entries, need not satisfy this equality as written.
Use the checks in a fixed order:
- Confirm that each row’s exposed lives and exits refer to the stated interval and the same population basis.
- Check whether the cause categories are mutually exclusive and exhaustive. If not, determine whether an additional exit category is needed.
- Sum the cause-specific exits and compare the result with total exits.
- Add total exits and survivors and compare the result with lives exposed.
- For a continuous closed cohort, compare one row’s survivors with the next row’s exposed lives.
A table may also be presented using proportions rather than counts. In that case, verify that the stated denominator and interval are consistent before interpreting the values. Do not confuse a count of lives exposed at the start of an interval with person-years at risk over the interval, and do not assume a missing count can be recovered uniquely unless enough entries are known to be reliable.
Common Mistakes
- Adding survivors to each cause count separately. The partition identity uses total exits once: \(l_x=d_x+l_{x+1}\), where \(d_x\) is the sum over causes.
- Calling an inconsistency a unique correction. A failed check locates a contradiction, not necessarily the erroneous cell. State which entries are assumed reliable before calculating a correction.
- Checking cause columns but not survivors. A correct cause sum does not guarantee that the total exits and survivors equal the starting lives.
- Checking rows but not their connection. Two rows may each balance and still disagree across the boundary between their intervals.
- Assuming the same continuity rule for every dataset. The across-row equality requires a continuous closed cohort and adjacent intervals. Entry, transfer, or a change of cohort basis can invalidate it.
Check Your Understanding
For each problem, check the relevant count identities before deciding whether a table reconciles. Treat all scenarios as hypothetical.
- A closed cohort has 180 lives exposed during a year. Cause A records 4 exits, cause B records 9, total exits are 13, and survivors are 167. Does the row reconcile? (Answer: Yes; \(4+9=13\) and \(13+167=180\).)
- A table reports 260 lives exposed, 12 cause A exits, 6 cause B exits, 17 total exits, and 243 survivors. Identify the failed identity or identities. (Answer: The cause sum fails because \(12+6=18\ne17\); the reported total and survivors do satisfy \(17+243=260\).)
- For one closed cohort, an interval ends with 415 survivors and the next interval begins with 414 exposed lives. Both rows separately satisfy their within-row identities. What check fails, and is the correction uniquely determined? (Answer: Across-row continuity fails by one life; no unique correction is determined without knowing which entry is reliable.)
- A row has 700 exposed lives and reports 28 total exits and 672 survivors. The two listed causes are intended to be exhaustive, but their counts are 16 and 11. If those cause counts are confirmed correct, find the corrected total exits and survivors. (Answer: Total exits are 27 and survivors are \(700-27=673\).)
- A closed cohort’s first row reports 900 exposed lives, 21 total exits, and 879 survivors. The next row reports 879 exposed lives, 15 total exits, and 864 survivors, but its cause counts are 8 and 8. Check both rows and identify the issue. (Answer: The first row balances; the second row balances across exposure, exits, and survivors, and continuity holds, but its cause counts sum to 16 rather than the reported 15, so the cause-sum identity fails.)