Why Associated-Table Conversions Go Wrong
A multiple-decrement table records exits from several competing causes. An associated single-decrement table answers a different question: what would the decrement probability be if only one specified cause operated, under a stated basis? Confusing those two quantities can produce a plausible-looking answer that is nevertheless wrong.
The conversions depend on the model within the year. Under constant forces, causes act through cause-specific forces and the overall survival probability is exponential. Under a uniform distribution of decrements (UDD) in an associated single-decrement table, the fractional-year probability is linear in elapsed time. Those are different assumptions, so their fractional-year formulas cannot be interchanged.
This tutorial uses \(q^{(j)}\) for observed probabilities and \(q'^{(j)}\) for associated probabilities. For a constant-force model, \(\mu_j\) denotes the constant force for cause \(j\) during the year, and \(M=\sum_j\mu_j\) is the total force. All probabilities below refer to a single one-year interval unless a fractional duration is explicitly given.
Constant-Force Conversions: Keep the Quantities Separate
Assume that each cause has a constant force during the year and that an individual exits at the first decrement. While still exposed, the total force is the sum \(M\) of the cause-specific forces. The probability of surviving all causes to time \(t\) is \(e^{-Mt}\). The associated table for cause \(j\), with other causes removed but its force unchanged, has annual probability \(1-e^{-\mu_j}\).
The observed probability for cause \(j\) is not the probability of that cause operating alone. It is the probability that the individual survives long enough to exit from cause \(j\) before any competing exit. The associated probability removes that competition.
The conversion can be justified directly. At time \(t\), the probability of still being exposed is \(e^{-Mt}\). Over a short interval of length \(dt\), the conditional probability of exit from cause \(j\) is approximately \(\mu_jdt\). Integrating the resulting density over the year gives the observed probability. Removing other causes leaves a one-cause survival probability instead.
Thus the observed probability includes survival against all causes up to the time of exit; the associated probability includes survival against cause \(j\) alone. The integral also explains why simply copying an observed probability into an associated table is generally unjustified.
Worked Example: Recovering an Associated Probability Under Constant Forces
A hypothetical long-term care plan has two competing exit causes during a year. Assume constant forces \(\mu_1=0.10\) and \(\mu_2=0.20\). Find the total exit probability, both observed cause probabilities, and both associated single-decrement probabilities. State one check on the results.
Method. Add the cause-specific forces to obtain the total force, use the constant-force competing-risk formulas for observed exits, and then remove the competing cause to calculate each associated probability.
Step 1: Total exit probability. The total force is \(M=0.10+0.20=0.30\). Therefore, \(q=1-e^{-0.30}=1-0.7408182=0.2591818\).
Step 2: Observed probabilities. Cause 1 has force share \(0.10/0.30=1/3\), so \(q^{(1)}=(1/3)(0.2591818)=0.0863939\). Cause 2 has force share \(0.20/0.30=2/3\), so \(q^{(2)}=(2/3)(0.2591818)=0.1727879\). Their sum is \(0.0863939+0.1727879=0.2591818=q\), as required.
Step 3: Associated probabilities. With cause 1 alone, the probability is \(q'^{(1)}=1-e^{-0.10}=0.0951626\). With cause 2 alone, it is \(q'^{(2)}=1-e^{-0.20}=0.1812692\).
The associated probabilities are each greater than their corresponding observed probabilities because the competing cause has been removed. The common mistake avoided here is treating an observed cause probability as if it were the probability under a single-decrement table.
UDD and Constant Force Are Not Interchangeable
Once an annual associated probability has been identified, a fractional-year probability still requires an assumption about how decrements are distributed within that year. Under UDD for a single decrement, the probability of decrement by time \(t\), for \(0\le t\le1\), is \(tq'^{(j)}\). Under constant force, the annual associated probability determines the force \(-\ln(1-q'^{(j)})\), giving a different fractional-year expression.
These formulas agree at \(t=0\) and \(t=1\), but generally differ between those endpoints. They are not alternative ways to evaluate the same model: each is correct only under its stated assumption. In particular, a constant-force fractional probability is not obtained by multiplying the annual probability by \(t\).
Worked Example: Applying the Correct Fractional-Year Basis
A hypothetical associated table gives annual single-decrement probabilities \(q'^{(A)}=0.16\) and \(q'^{(B)}=0.25\). Calculate each associated probability of decrement in the first 0.4 of a year under (i) UDD and (ii) constant force. Give probabilities to four decimal places.
Method. Apply each fractional-year formula separately to the same annual associated probability. Do not mix the UDD and constant-force assumptions.
Cause A under UDD: \( {}_{0.4}q'^{(A)}=0.4(0.16)=0.0640\). Cause A under constant force: \(1-(1-0.16)^{0.4}=1-0.84^{0.4}=1-0.9326=0.0674\).
Cause B under UDD: \( {}_{0.4}q'^{(B)}=0.4(0.25)=0.1000\). Cause B under constant force: \(1-(1-0.25)^{0.4}=1-0.75^{0.4}=1-0.8913=0.1087\).
The constant-force results exceed the UDD results for both causes at this duration. As an endpoint check, either model returns the given annual probability at \(t=1\): the UDD formula gives \(q'\), and the constant-force formula gives \(1-(1-q')=q'\). The common mistake is to use \(tq'\) simply because the duration is a fraction of a year, without checking whether UDD was assumed.
Total Exits Are Not a Cause-Specific Rate
A total probability \(q\) measures the chance of exiting from any cause during the year. It is not a force, and it does not identify how much of the total is due to a particular cause. Under constant forces, first convert \(q\) to the total force using \(M=-\ln(1-q)\), then allocate that force according to the observed cause probabilities. Using \(q\) directly as a cause-specific force confuses both the unit and the meaning of the quantity.
Worked Example: Correcting a Total-Exit Error
In a hypothetical disability income portfolio, the observed one-year exit probabilities are \(q^{(1)}=0.08\) for recovery and \(q^{(2)}=0.12\) for another exit cause. Assume constant cause-specific forces. Find the associated single-decrement probability for recovery. Also show the incorrect result obtained by treating the total exit probability as the recovery force.
Method. First recover the total force from the total probability, then use the observed recovery share to find its force. Finally, calculate the associated probability after removing the competing cause.
Step 1: Total probability and force. The total probability is \(q=0.08+0.12=0.20\). Hence \(M=-\ln(1-0.20)=-\ln(0.80)=0.2231436\).
Step 2: Recovery force. The observed recovery share is \(0.08/0.20=0.40\). Thus \(\mu_1=0.40(0.2231436)=0.0892574\).
Step 3: Associated probability. With recovery as the only cause, \(q'^{(1)}=1-e^{-0.0892574}=0.0854\), to four decimal places.
Incorrect calculation: Treating total probability \(0.20\) as the recovery force gives \(1-e^{-0.20}=0.1813\). This is wrong: \(0.20\) is the probability of any exit, not the recovery force. It also ignores the fact that the observed recovery probability is only \(0.08\). The corrected associated probability, \(0.0854\), is slightly above the observed recovery probability, as expected when the other cause is removed.
A Practical Conversion Check
Before finalizing a conversion, label each input and output by both its meaning and its assumption. The following sequence prevents the most common category errors.
Use observed \(q^{(j)}\) for exits while all causes compete, associated \(q'^{(j)}\) for the one-cause table, and \(q\) for all exits combined.
For constant forces, use exponential survival and additive forces. For UDD in an associated table, use linear fractional-year probabilities. Do not switch basis midway.
Under constant forces, recover \(M\) from total probability, recover each \(\mu_j\) from the observed cause share, and then calculate \(q'^{(j)}\).
Observed probabilities must sum to \(q\). Each observed cause probability cannot exceed the corresponding associated probability under the constant-force competing-risk model. Every probability must lie between zero and one.
These checks are diagnostic, not substitutes for the model. For example, an answer can satisfy the sum-to-total check and still use UDD when constant force was required. A complete solution names the basis, shows how quantities are converted, and checks the interpretation as well as the arithmetic.
Common Errors to Avoid
- Equating observed and associated probabilities. An observed cause probability includes competition from other causes. An associated probability removes the competitors on the specified basis.
- Using a fractional-year formula without stating its basis. Under UDD, use \(tq'\); under constant force, use \(1-(1-q')^t\). They usually differ for \(0<t<1\).
- Calling a probability a force. Under constant force, convert the total probability by \(M=-\ln(1-q)\). A probability is dimensionless over an interval; a force describes the instantaneous rate of exit while exposed.
- Assigning the total exit probability to one cause. Use the observed cause share to allocate the total force under the constant-force model; do not assume that all exits belong to the cause being valued.
- Forgetting the model when eliminating causes. An associated probability is defined relative to the stated basis. Do not retain a competing-cause adjustment in a single-decrement calculation.
Check Your Understanding
For each problem, state the relevant model and keep observed probabilities, associated probabilities, and forces distinct.
- A hypothetical policy has two competing constant forces, \(0.05\) and \(0.15\). Find the total exit probability, the observed probability for the first cause, and its associated single-decrement probability. (Answer: total \(0.1813\); observed first-cause probability \(0.0453\); associated probability \(0.0488\).)
- An associated single-decrement probability is \(0.30\). Find the probability in the first quarter-year under UDD and under constant force. (Answer: UDD \(0.0750\); constant force \(1-0.70^{0.25}=0.0853\).)
- Under constant forces, observed annual cause probabilities are \(0.06\) and \(0.14\). Find the total force, the first cause's force, and its associated probability. (Answer: total force \(0.2231\); first-cause force \(0.0669\); associated probability \(0.0648\).)
- Two hypothetical causes have constant forces \(0.08\) and \(0.12\). Find both observed cause probabilities and both associated probabilities. (Answer: observed probabilities \(0.0725\) and \(0.1088\); associated probabilities \(0.0769\) and \(0.1131\).)
- Under constant forces, observed probabilities are \(0.10\) for cause A and \(0.15\) for cause B. Find the total force, cause A's force, and cause A's associated probability. (Answer: total force \(0.2877\); cause A force \(0.1151\); associated probability \(0.1087\).)