Why Compare Associated Bases?
A multiple-decrement dataset records exits by cause while several causes operate together. An associated single-decrement probability answers a different question: what annual probability would apply to one cause in a single-decrement table associated with those data under a specified assumption about decrements within the year? Because within-year exposure is represented differently under different assumptions, the associated probability can depend on the chosen basis.
This tutorial compares the Uniform Distribution of Decrements (UDD) and constant-force bases using the same hypothetical dataset. The observed counts will not change; only the assumption used to convert them will change. Keep that distinction clear: an associated probability is not the observed proportion exiting from that cause in the multiple-decrement population.
The Two Conversion Rules
Under UDD, decrements are treated as spread uniformly through the year. For a particular cause, the associated single-decrement probability is calculated using the observed probabilities for the competing causes:
Under constant force, the total force is constant during the year, and each cause’s force is allocated in proportion to its observed share of total exits. The associated single-decrement probability is:
If \(q=0\), every observed cause probability is zero, so there is no nonzero cause-specific probability to convert. For positive \(q\), the constant-force formula can also be computed by first finding the total force \(-\ln(1-q)\), multiplying by \(q_j/q\), and converting that cause-specific force to an annual probability.
Worked Example: Compare the Bases for One Cause
Worked Example: Compare the Bases for One Cause
A hypothetical group of 1,000 policyholders is exposed for one year to three causes of exit. There are 200 exits from cause A, 100 from cause B, and 100 from cause C. Calculate the associated single-decrement probability for cause A under UDD and constant force. Give probabilities to four decimal places.
Method. First convert the counts to observed probabilities. Then apply each basis separately, using the same cause A data. This avoids treating the observed cause A proportion as if it were already an associated probability.
Step 1: Observed probabilities. The initial exposure is \(l=1{,}000\), so
Step 2: UDD. The competing causes for A are B and C. Their observed probabilities sum to \(0.10+0.10=0.20\), giving
Step 3: Constant force. Cause A’s share of total exits is \(q_A/q=0.20/0.40=0.50\). Therefore,
Answer and check. To four decimal places, the associated probabilities are \(0.2222\) under UDD and \(0.2254\) under constant force. Both exceed the observed multiple-decrement probability \(q_A=0.20\), as expected when the competing causes are removed from the associated single-decrement table. The two answers differ because the bases represent within-year exposure differently.
Common mistake avoided: Using \(0.20\) as the answer simply reports the observed proportion exiting from A while all three causes operate. It does not perform either associated-table conversion.
Why the Formulas Have These Forms
The UDD expression reflects the effect of competing exits on exposure during the year. With \(l\) lives initially exposed and \(d_k\) exits from each competing cause \(k\), UDD places those exits uniformly through the year. On average, half of the exits from each competing cause occur before the end of the year. The associated exposure for cause \(j\) is therefore adjusted by half of each competing cause’s exits:
This is the UDD conversion identity: the numerator is the observed number exiting from cause \(j\), while the denominator adjusts the exposure for competing exits under the uniform-within-year assumption.
For constant force, let \(\mu\) denote the total force during the year and \(\mu_j\) the force for cause \(j\). Constant force gives total one-year survival \(1-q=e^{-\mu}\), so \(\mu=-\ln(1-q)\). The cause’s share of the total force is taken to equal its share \(q_j/q\) of observed exits. Thus \(\mu_j=(q_j/q)\mu\). In a single-decrement table with constant force \(\mu_j\), the probability of decrement is \(1-e^{-\mu_j}\). Substituting gives:
The derivation also explains why the constant-force calculation uses the total survival probability \(1-q\) and the cause’s share of total exits. It does not use the UDD exposure adjustment.
Worked Example: Compare the Other Causes
Worked Example: Compare Causes B and C
Use the same hypothetical dataset: 1,000 exposed lives, with 200 exits from A, 100 from B, and 100 from C. Calculate the associated probabilities for B and C under both bases, and compare the results for all three causes.
Method. For UDD, use the sum of the observed probabilities for causes other than the cause being calculated. For constant force, use that cause’s share of the total \(q=0.40\). Since B and C have equal observed probabilities, their associated results must be equal under either basis.
UDD for cause B. The other probabilities are \(q_A=0.20\) and \(q_C=0.10\):
For cause C, the other probabilities are \(0.20\) and \(0.10\) as well, so \(q'_{C,\mathrm{UDD}}=0.117647\ldots\).
Constant force for cause B. Its share is \(q_B/q=0.10/0.40=0.25\):
Cause C has the same observed probability and the same share of total exits, so \(q'_{C,\mathrm{CF}}=0.119888\). The results for all causes, rounded to four decimal places, are:
| Cause | Observed \(q_j\) | Associated under UDD | Associated under constant force |
|---|---|---|---|
| A | 0.2000 | 0.2222 | 0.2254 |
| B | 0.1000 | 0.1176 | 0.1199 |
| C | 0.1000 | 0.1176 | 0.1199 |
Answer and check. For both B and C, the constant-force result is slightly higher than the UDD result. The equal B and C data produce equal associated probabilities. The differences between bases are approximately \(0.2254-0.2222=0.0032\) for A and \(0.1199-0.1176=0.0023\) for each of B and C, using the rounded table entries.
Common mistake avoided: The UDD denominator for B excludes B, not A. Using only \(q_A\) in that denominator would omit the competing exits from C and produce the wrong conversion.
Interpreting Within-Year Exposure
The two sets of associated probabilities are close in this example, but closeness is not a reason to treat the bases as interchangeable. Under UDD, the adjustment is explicit: half the observed probability from each competing cause appears in the denominator. Under constant force, the calculation represents survival exponentially through the year and allocates the total force according to each cause’s observed share.
An associated table removes the competing causes for its particular cause-specific calculation. As a result, its probability need not equal the observed multiple-decrement probability, and associated probabilities for different causes are not required to sum to the observed total exit probability. For example, in this dataset the rounded UDD associated probabilities sum to \(0.2222+0.1176+0.1176=0.4574\), not \(0.4000\). That is not a failure of the conversion: each probability describes a different associated single-decrement table.
Worked Example: Check the Associated Probabilities
For the dataset in the previous examples, calculate the sum of the unrounded associated probabilities under each basis. Then state whether either sum should be expected to equal the observed total exit probability \(0.40\).
Method. Add the cause-specific associated probabilities already calculated, retaining their unrounded values until the final display.
UDD sum. The unrounded results are \(0.222222\ldots\) for A and \(0.117647\ldots\) for each of B and C:
Constant-force sum. The unrounded results are \(0.225403\ldots\) for A and \(0.119888\ldots\) for each of B and C:
Answer and check. The sums are approximately \(0.4575\) under UDD and \(0.4652\) under constant force. Neither should be expected to equal \(0.40\), because the three associated probabilities come from three separate single-decrement constructions, each removing different competing causes. By contrast, the observed probabilities \(0.20+0.10+0.10\) do sum to \(0.40\).
Common mistake avoided: Forcing the associated probabilities to sum to the observed total would confuse separate associated tables with the original multiple-decrement table.
Common Exam Traps
- Using the target cause in the UDD adjustment. The denominator uses only causes \(k\ne j\). The target probability \(q_j\) is in the numerator.
- Using an observed cause share as a probability. Under constant force, \(q_j/q\) is an exponent, not the associated annual probability itself.
- Mixing the bases. The UDD denominator adjustment does not belong in the constant-force formula, and the constant-force exponent does not belong in the UDD formula.
- Expecting the associated probabilities to add to \(q\). Each associated probability is calculated for a separate single-decrement table; their sum is not a closure check for the original multi-cause data.
- Rounding too early. Keep extra digits in the powers and intermediate probabilities, then round the final probabilities as requested.
Check Your Understanding
For each problem, calculate the requested associated probability under the stated bases. Round probabilities to four decimal places.
- A hypothetical group has 1,200 exposed lives and records 180 exits from cause A, 120 from B, and 60 from C. Calculate the associated probability for A under UDD and constant force. (Answer: UDD 0.1622; constant force 0.1633)
- Among 500 exposed lives, 50 exit from cause X and 75 from cause Y; no other exits occur. Calculate the associated probability for Y under both bases. (Answer: UDD 0.1579; constant force 0.1585)
- The observed cause probabilities are \(0.08\), \(0.12\), and \(0.10\). Calculate the associated probability for the third cause under UDD and constant force. (Answer: UDD 0.1111; constant force 0.1121)
- There are 1,000 exposed lives, with 100 exits from A, 200 from B, and 100 from C. A candidate calculates the UDD associated probability for A as \(0.10/(1-0.40/2)\). Identify the error and give the correct UDD and constant-force probabilities for A. (Answer: The UDD denominator must exclude A’s probability: UDD 0.1176; constant force 0.1199)
- Observed probabilities are \(q_A=0.20\), \(q_B=0.10\), and \(q_C=0.10\). For cause B, calculate the difference between the constant-force and UDD associated probabilities, using unrounded results before rounding the difference. (Answer: \(0.119888\ldots-0.117647\ldots=0.0022\))