Diagnosing Inconsistent Decrement Tables

Why Audit a Decrement Table?

A multiple-decrement table describes exits from a group by cause. Its entries are linked: cause-specific probabilities determine the total probability of exit, while cause-specific exit counts and the starting number of lives determine the number remaining. Associated single-decrement columns add another layer: they must agree with the multiple-decrement probabilities under the stated conversion assumption. A table can therefore look plausible entry by entry while still being internally inconsistent.

The aim of an audit is not to force every observed count to equal a model expectation. It is to check which relationships must hold for the type of table being used, and then identify the entry or assumption responsible for a failure. This diagnostic approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: For an interval beginning at age or duration \(x\), let \(l_x\) be the number in the decrement group at the start, \(d_x^{(j)}\) the number exiting from cause \(j\), and \(l_{x+1}\) the number remaining at the end. Let \(q_x^{(j)}\) be the probability that a member of the starting group exits from cause \(j\) during the interval. The total exit probability is \(q_x=\sum_j q_x^{(j)}\), and the probability of remaining in the group is \(p_x=1-q_x\). An associated single-decrement probability, written \(q_x^{\prime(j)}\), is the probability of exit from cause \(j\) in a hypothetical model in which that cause acts alone.

A Four-Part Audit

Begin by identifying what the entries represent. A probability table, a table of observed counts, and a table of expected counts do not have identical requirements. Then check the basic relationships in order: probability bounds, cause totals, survivor recurrence, and any stated conversion between multiple-decrement and associated single-decrement columns.

Check the probability range.
Each \(q_x^{(j)}\) must be between zero and one. Their sum \(q_x\) must also be between zero and one, so \(p_x=1-q_x\) is nonnegative.

Check cause and total exits.
If counts are reported, add the cause-specific counts and compare their sum with the reported total exits. For a complete cohort observed over the interval, the total exits equal \(l_x-l_{x+1}\).

Check the survivor recurrence.
For a count table, \(l_{x+1}=l_x-\sum_j d_x^{(j)}\). For a probability-based table, the expected number remaining is \(l_xp_x\), provided \(l_x\) is the starting count to which those probabilities apply.

Check associated columns on their stated basis.
Use the specified conversion assumption, such as uniform distribution of decrements, and the relevant number of causes. Do not compare associated and multiple-decrement probabilities as if they were generally equal.

The count recurrence is an accounting identity for a cohort with no other way to enter or leave the group during the interval. If there are entrants, transfers, censoring or incomplete observation, those features must be accounted for before applying that identity. Likewise, probabilities describe a model or experience rate; they are not automatically the realized fractions in a finite cohort.

Checking Two Associated Causes Under UDD

A conversion check is meaningful only after its assumption is stated. For two causes under the uniform distribution of decrements (UDD) associated model, the isolated decrement probability for each cause is spread uniformly over the interval. The two associated decrement times are treated as independent. The multiple-decrement probability for a cause is then its probability of acting at a time when the other cause has not already removed the member.

q_x^{(1)}=q_x^{\prime(1)}\left(1-\frac{q_x^{\prime(2)}}{2}\right), \qquad q_x^{(2)}=q_x^{\prime(2)}\left(1-\frac{q_x^{\prime(1)}}{2}\right).

Here is a short justification. At fractional time \(t\), between zero and one, the probability of having avoided cause 2 in its associated single-decrement model is \(1-tq_x^{\prime(2)}\). The probability density of cause 1’s associated decrement time is \(q_x^{\prime(1)}\) per unit time. Thus cause 1 is the first decrement with probability equal to the integral of the cause 1 density times the probability of avoiding cause 2:

q_x^{(1)} =\int_0^1 q_x^{\prime(1)}(1-tq_x^{\prime(2)})\,dt =q_x^{\prime(1)}\left(1-\frac{q_x^{\prime(2)}}{2}\right).

Interchanging the cause labels gives the second relation. This two-cause formula should not be applied unchanged to a model with three or more causes: the probability of avoiding all competing causes then involves their joint survival over the interval. Nor should the formula be used if the table specifies a different conversion basis.

Worked Example: Finding a Survivor-Count Error

A hypothetical decrement table begins an interval with \(l_x=500\). It reports \(d_x^{(1)}=22\), \(d_x^{(2)}=13\), \(l_{x+1}=466\), \(q_x^{(1)}=0.044\), and \(q_x^{(2)}=0.026\). Treat the counts as a complete cohort’s exits and the probabilities as an unrounded probability-based table. Identify any inconsistency.

Method. Check the count recurrence and the probability totals separately. This avoids treating a survivor count as if it could be checked from only one cause column.

First add the reported cause-specific exits:

d_x^{(1)}+d_x^{(2)}=22+13=35.

The count recurrence therefore requires:

l_{x+1}=500-35=465.

The reported survivor count is 466, one higher than required. Equivalently, the reported starting and ending counts imply \(500-466=34\) exits, but the cause-specific counts total 35. The count entries cannot all be correct under the stated complete-cohort assumption.

Now check the probability columns:

q_x=0.044+0.026=0.070, \qquad p_x=1-0.070=0.930.

The corresponding probability-based expected number remaining is \(500(0.930)=465\). Also, the expected cause-specific exits are \(500(0.044)=22\) and \(500(0.026)=13\). The probability columns agree with the cause counts and with 465 survivors; the reported \(l_{x+1}=466\) is the inconsistent entry. Check: \(22+13+465=500\), as required. The common mistake would be to accept 466 because it is close to 465; exact table identities are not approximate unless rounding or another stated adjustment is involved.

Worked Example: Testing Associated Columns Under UDD

A hypothetical table has two causes and reports associated single-decrement probabilities \(q_x^{\prime(1)}=0.20\) and \(q_x^{\prime(2)}=0.10\). The multiple-decrement columns are listed as \(q_x^{(1)}=0.19\) and \(q_x^{(2)}=0.08\). The table states that it uses the two-cause UDD associated model. Determine whether the columns agree.

Method. Convert each associated probability separately using the UDD relation, then compare the results with the reported multiple-decrement columns.

For cause 1, the probability of avoiding cause 2 over the interval is represented in the conversion by the factor \(1-0.10/2=0.95\). Hence:

q_x^{(1)}=0.20(0.95)=0.19.

For cause 2, the corresponding factor is \(1-0.20/2=0.90\). Hence:

q_x^{(2)}=0.10(0.90)=0.09.

The listed cause 1 probability, 0.19, agrees with the conversion. The listed cause 2 probability, 0.08, does not: it should be 0.09 under the stated assumptions. The converted total exit probability is \(0.19+0.09=0.28\), leaving \(p_x=1-0.28=0.72\). Check: all converted probabilities are nonnegative and their total does not exceed one. The common mistake is to compare the associated and multiple-decrement columns directly, or to subtract half of the same cause’s associated probability rather than that of the competing cause.

Expected Counts Are Not Realized Counts

A probability-based table of expected counts and a cohort’s observed counts answer different questions. If a probability model is applied to \(l_x\) starting members, its expected cause-specific count is \(l_xq_x^{(j)}\), and its expected survivor count is \(l_xp_x\). These are expectations, not identities that every observed cohort must satisfy cause by cause. Realized counts can differ because of random variation.

Important distinction: If the table is an unrounded probability-based table of expected counts, each cause-specific exit count must be \(d_x^{(j)}=l_xq_x^{(j)}\). Observed counts need only reconcile with the observed total and survivors; they need not equal the model’s expected cause-specific counts.

Worked Example: Separating Observed Experience from Model Expectations

A hypothetical cohort has 200 members at the start of an interval. The observed exits are 10 from cause A and 7 from cause B, and 183 members remain. Separately, a probability model assigns \(q_x^{(A)}=0.06\) and \(q_x^{(B)}=0.04\). Determine whether the observed counts contradict the model merely because they differ from the expected counts.

Method. First test the observed count identities, then calculate the model expectations. This avoids treating expected values as required realized outcomes.

The observed cause counts total \(10+7=17\). The observed survivor recurrence gives \(200-17=183\), exactly the reported number remaining. The observed cohort counts therefore reconcile.

Under the model, the total exit probability is \(0.06+0.04=0.10\), so \(p_x=0.90\). The expected cause A count is \(200(0.06)=12\); the expected cause B count is \(200(0.04)=8\); and the expected number remaining is \(200(0.90)=180\). These expectations sum to \(12+8+180=200\).

The observed counts, 10 and 7, differ from their expected values, 12 and 8, but that difference alone is not an inconsistency. The observed totals reconcile, and the expected counts reconcile separately. Check: the model expects 20 total exits and 180 survivors, while the cohort actually has 17 exits and 183 survivors. The common mistake is to call the observed table inconsistent solely because its counts do not equal expected counts.

Common Diagnostic Traps

Checking only the total probability. A valid sum can conceal an individual cause probability that is negative or exceeds one. Check each cause column as well as the sum.
Mixing counts and probabilities. Counts are numbers of members; probabilities are proportions of the starting group. When converting a probability to an expected count, multiply by the relevant starting count.
Using the wrong survivor relationship. For a count cohort, subtract total exits from the starting count. For a probability model, multiply the starting count by \(p_x\) to obtain the expected number remaining.
Ignoring the table basis. Associated single-decrement probabilities do not generally equal multiple-decrement probabilities. A conversion check requires the stated decrement assumption and the correct number of causes.
Forcing observed data to equal expectations. Observed cause counts need to reconcile with observed totals and survivors, but they need not equal \(l_xq_x^{(j)}\) for model probabilities.

A useful audit records each check and its result, rather than immediately replacing a suspect entry. If a recurrence fails, identify which reported entries participate in it. If a conversion fails, recompute each cause using the specified basis. Several candidate errors may remain until other information is available; the audit should distinguish what is proven inconsistent from what is merely unexplained.

Key takeaway: Audit decrement tables in layers: validate probabilities, reconcile cause totals, apply the correct survivor recurrence, and test associated columns only under their stated conversion assumption. Keep observed counts separate from probability-based expected counts.

Check Your Understanding

For each problem, identify the relevant checks before deciding whether the table is internally consistent.

A count table starts with \(l_x=360\), reports 18 exits from cause 1 and 9 from cause 2, and lists 333 survivors. Find the empirical cause probabilities and check the recurrence. (Answer: \(q_x^{(1)}=0.0500\), \(q_x^{(2)}=0.0250\); 333 survivors, so the counts reconcile.)
Under a two-cause UDD model, \(q_x^{\prime(1)}=0.30\) and \(q_x^{\prime(2)}=0.20\). Find both multiple-decrement probabilities and the survival probability. (Answer: \(q_x^{(1)}=0.2700\), \(q_x^{(2)}=0.1700\), \(p_x=0.5600\).)
A cohort of 250 has 8 observed exits from cause A, 12 from cause B and 230 survivors. A model specifies probabilities 0.04 and 0.05 for those causes. Check observed reconciliation and state the model’s expected cause counts. (Answer: observed counts reconcile; expected counts are 10 from A and 12.5 from B, so the observed counts need not equal those expectations.)
A probability-based expected-count table has \(l_x=800\), \(q_x^{(1)}=0.025\), and \(q_x^{(2)}=0.015\), but reports \(d_x^{(2)}=11\) and \(l_{x+1}=769\). Find the expected counts and identify the inconsistent reports. (Answer: expected exits are 20 and 12, with 768 survivors; the reported cause 2 count and survivor count are inconsistent.)
Under two-cause UDD, associated probabilities are 0.12 and 0.08, while the table reports multiple-decrement probabilities 0.1152 and 0.0742. Audit the columns. (Answer: the converted probabilities are 0.1152 and 0.0752; the second reported multiple-decrement probability is inconsistent.)

What You'll Learn