Choosing a Basis for an Associated Table

Why the Within-Year Basis Matters

An associated single-decrement table describes the probability of exit from one cause when the other causes are removed. Annual multiple-decrement data tell us how many exits occurred from each cause, but not exactly when those exits occurred within the year. To construct an associated table, a within-year assumption is therefore needed. Uniform distribution of decrements (UDD) and constant force are two distinct choices.

The basis is not a preference made to obtain a convenient answer. It must match the assumption specified for the calculation. The same annual multiple-decrement probabilities can lead to different associated probabilities under UDD and constant force because the two assumptions describe different patterns of exits within the year.

Definition: Let \(q_x^{(j)}\) be the observed probability that a life aged \(x\) exits during the year from cause \(j\), in a model with multiple causes. Let \(q_x^{\prime(j)}\) be the one-year probability of exit from cause \(j\) in the associated single-decrement model, in which all other causes are removed. The within-year assumption specifies how to relate the observed probabilities to the associated probabilities.

The calculations below use one year of exposure and a fixed set of causes. They assume that the stated within-year model applies over that year and that the probabilities are valid for the model. Neither conversion should be used merely because it is familiar: first identify what the problem says about exits within the year.

UDD: Average Exposure to a Cause

Under UDD, decrement exits are distributed uniformly over the year in the model used to relate the multiple-decrement and associated tables. For each cause \(j\), the standard relation is

q_x^{(j)} = q_x^{\prime(j)} \left(1-\frac{1}{2}\sum_{k\ne j}q_x^{\prime(k)}\right).

The observed probability for cause \(j\) is its associated probability adjusted for the average removal of lives by the other causes during the year. With two causes, write \(a=q_x^{\prime(1)}\) and \(b=q_x^{\prime(2)}\). The equations become \(q_x^{(1)}=a(1-b/2)\) and \(q_x^{(2)}=b(1-a/2)\). In general, recovering associated probabilities from observed probabilities requires solving these equations together; simply dividing each observed probability by a single common adjustment is not the relation.

Why the UDD relation holds: For cause \(j\), the associated table has probability \(q_x^{\prime(j)}\) of exit over the year if other causes are absent. Under UDD, the other causes collectively remove a fraction \(\sum_{k\ne j}q_x^{\prime(k)}\), distributed uniformly through the year. Their average removal over the year is therefore half that fraction. The average share of the original cohort still available with respect to those other causes is \(1-\frac12\sum_{k\ne j}q_x^{\prime(k)}\). Multiplying the associated probability by that average share gives the observed probability. This relation applies under the stated UDD model, not under constant force.

Worked Example: Recovering Two UDD Associated Probabilities

A hypothetical table records annual probabilities \(q_x^{(1)}=0.16\) and \(q_x^{(2)}=0.36\). The specified within-year assumption is UDD. Find both associated probabilities and verify the result.

Method. Let \(a=q_x^{\prime(1)}\) and \(b=q_x^{\prime(2)}\). Apply the two-cause UDD equations:

0.16=a(1-b/2), \qquad 0.36=b(1-a/2).

Subtract the second equation from the first. The quadratic terms cancel:

0.16-0.36 = a-\frac{ab}{2}-b+\frac{ab}{2} = a-b, \qquad\text{so}\qquad b=a+0.20.

Substitute \(b=a+0.20\) into the first equation:

0.16=a\left(1-\frac{a+0.20}{2}\right) =a(0.90-0.50a),

so \(0.50a^2-0.90a+0.16=0\). The discriminant is \(0.90^2-4(0.50)(0.16)=0.49\), whose square root is \(0.70\). Thus \(a=(0.90\pm0.70)/1.00\), giving \(a=0.20\) or \(a=1.60\). A probability cannot exceed 1, so \(a=0.20\), and \(b=0.40\).

Check. The reconstructed probabilities are \(0.20(1-0.40/2)=0.20(0.80)=0.16\) and \(0.40(1-0.20/2)=0.40(0.90)=0.36\), exactly the supplied values. The invalid quadratic root is rejected because it is not a probability.

Common mistake avoided. The observed probabilities are not the associated probabilities: setting \(a=0.16\) and \(b=0.36\) would not satisfy the UDD equations.

Constant Force: Cause-Specific Forces

Under constant force, each cause has a constant force of decrement during the year. Write that force as \(\mu_j\), and let \(\mu=\sum_j\mu_j\) be the total force. The probability of no exit from any cause over one year is \(e^{-\mu}\), so the total observed exit probability is \(q=1-e^{-\mu}\). Among those exits, cause \(j\) receives the fraction \(\mu_j/\mu\). In the associated single-decrement model, only force \(\mu_j\) remains, giving \(q_x^{\prime(j)}=1-e^{-\mu_j}\).

Consequently, when the observed cause probabilities are given, first sum them to obtain \(q=\sum_jq_x^{(j)}\). For \(q>0\), the constant-force associated probability for cause \(j\) is

q_x^{\prime(j)} = 1-\exp\left( -\frac{q_x^{(j)}}{q}\big[-\ln(1-q)\big] \right) = 1-(1-q)^{q_x^{(j)}/q}.

This follows because the force share is \(\mu_j/\mu=q_x^{(j)}/q\), while \(\mu=-\ln(1-q)\). If \(q=0\), all observed cause probabilities are zero; there is no positive total force to apportion, and the associated probabilities are zero. The constant-force formula is not the UDD relation written in a different form: it represents a different pattern of within-year exits.

Worked Example: Applying Constant Force to the Same Observed Data

Use the observed probabilities \(q_x^{(1)}=0.16\) and \(q_x^{(2)}=0.36\), but now suppose the within-year assumption is constant force. Calculate both associated probabilities to four decimal places.

Method. The total observed probability is \(q=0.16+0.36=0.52\), so the probability of no exit is \(1-q=0.48\). The total force is

\mu=-\ln(0.48)=0.733969\ldots

The cause shares are \(0.16/0.52=4/13\) and \(0.36/0.52=9/13\). The cause-specific forces are therefore

\mu_1=\frac{4}{13}(0.733969\ldots)=0.225837\ldots, \qquad \mu_2=\frac{9}{13}(0.733969\ldots)=0.508132\ldots.

In the associated single-decrement models, remove the other cause and retain the relevant force:

q_x^{\prime(1)}=1-e^{-0.225837\ldots}=0.2022, \qquad q_x^{\prime(2)}=1-e^{-0.508132\ldots}=0.3984.

Check. The retained forces add to \(0.225837\ldots+0.508132\ldots=0.733969\ldots\), the total force. Their shares of that total are \(4/13\) and \(9/13\); multiplying those shares by \(1-e^{-\mu}=0.52\) recovers \(0.16\) and \(0.36\). The associated probabilities are reported to four decimal places.

Common mistake avoided. Under constant force, cause shares apply to the total force, not directly as associated probabilities. The associated probability is \(1-e^{-\mu_j}\), not \(\mu_j\) and not the observed \(q_x^{(j)}\).

Comparing the Bases on a Common Scale

A useful comparison starts with the same associated probabilities and asks what multiple-decrement probabilities each basis implies. This distinguishes a change in assumption from a change in the data. Under UDD, use the average-exposure relation. Under constant force, first convert each associated probability to its force, then combine the forces.

Worked Example: How the Basis Changes Observed Cause Probabilities

A hypothetical associated table has \(q_x^{\prime(1)}=0.20\) and \(q_x^{\prime(2)}=0.40\). Calculate the implied observed cause probabilities under UDD and under constant force. Report probabilities to four decimal places where needed.

UDD calculation. For cause 1, the other associated probability is \(0.40\); for cause 2, it is \(0.20\). Thus

q_x^{(1)}=0.20(1-0.40/2)=0.20(0.80)=0.16, \qquad q_x^{(2)}=0.40(1-0.20/2)=0.40(0.90)=0.36.

Constant-force calculation. The associated probabilities imply forces

\mu_1=-\ln(1-0.20)=-\ln(0.80)=0.223144\ldots, \qquad \mu_2=-\ln(1-0.40)=-\ln(0.60)=0.510826\ldots.

The total force is \(0.733969\ldots\). The total observed probability is \(1-e^{-0.733969\ldots}=1-(0.80)(0.60)=0.52\). Allocate that probability according to the force shares:

q_x^{(1)}=0.52\frac{0.223144\ldots}{0.733969\ldots}=0.1581, \qquad q_x^{(2)}=0.52\frac{0.510826\ldots}{0.733969\ldots}=0.3619.

Check. Under both assumptions the cause probabilities sum to \(0.52\). Their individual allocations differ: UDD gives \(0.1600\) and \(0.3600\), whereas constant force gives \(0.1581\) and \(0.3619\). The total agrees here because the same associated survival factors, \(0.80\) and \(0.60\), give a combined survival probability of \(0.48\) under constant force; the cause shares nevertheless differ.

Common mistake avoided. Matching the total probability does not establish that the cause-specific results match. The distribution of exits across causes depends on the within-year model.

A Basis-Selection Procedure

A short sequence of decisions keeps the calculation tied to the model rather than to the algebra that looks easiest. In particular, distinguish what is given (observed multiple-decrement probabilities or associated probabilities) from what is being requested.

Read the within-year condition.
Use UDD when exits are specified as uniformly distributed over the year; use constant force when cause-specific decrement forces are specified as constant. If the condition is not stated, do not silently treat the two bases as interchangeable.

Identify the direction of conversion.
From observed probabilities under UDD, solve the associated-probability equations together. From observed probabilities under constant force, calculate the total probability, total force, and cause-force shares. If associated probabilities are given, apply the corresponding forward relation.

Check the result under the same basis.
Substitute the associated values back into the UDD equations, or reconstruct the constant-force cause probabilities from the cause-specific forces. Check that probabilities are in the interval from 0 to 1 and that cause probabilities sum to the stated total.

Interpretation and Exam Traps

An associated probability is conditional on a different decrement environment from the observed multiple-decrement probability: all competing causes have been removed. It is therefore not generally equal to the observed cause probability. The within-year basis specifies how the changed environment is represented and is part of the model, not a cosmetic convention.

A common trap is to see annual data and assume that annual probabilities alone determine a unique associated table. They do not. The data give the observed annual exits, while UDD and constant force supply different within-year descriptions. If a question specifies UDD, the UDD relation must govern even if a constant-force conversion seems shorter; if it specifies constant force, use the force allocation and exponential survival relation.

The method also matters when rounding. Keep unrounded forces and probabilities through intermediate steps, then round the requested final probabilities. Premature rounding can make a reconstruction check appear not to match the observed probabilities.

Key takeaway: Choose the basis from the stated within-year assumption before doing any conversion. UDD adjusts for average exposure to competing decrements; constant force allocates exits using cause-specific forces. Because those models differ within the year, their associated-table results can differ even when they start from the same annual data.

Check Your Understanding

For each problem, identify and use the stated basis. Probabilities in final answers are rounded to four decimal places when needed.

Under UDD, a two-cause associated table has \(q_x^{\prime(1)}=0.12\) and \(q_x^{\prime(2)}=0.30\). Find the two implied observed cause probabilities. (Answer: \(q_x^{(1)}=0.1020,\ q_x^{(2)}=0.2820\))
Observed probabilities are \(q_x^{(1)}=0.17\) and \(q_x^{(2)}=0.27\), and UDD is specified. Find the associated probabilities, choosing the valid probability solution and checking by substitution. (Answer: \(q_x^{\prime(1)}=0.2000,\ q_x^{\prime(2)}=0.3000\))
Under constant force, the two cause-specific forces are \(0.20\) and \(0.30\) per year. Calculate the two observed annual cause probabilities. (Answer: \(q_x^{(1)}=0.1574,\ q_x^{(2)}=0.2361\))
Under constant force, the associated probabilities are \(0.10\) and \(0.20\). Convert them to cause-specific forces, find the total observed probability, and allocate it to the two causes. (Answer: \(\mu_1=0.1054,\ \mu_2=0.2231,\ q_x^{(1)}=0.0898,\ q_x^{(2)}=0.1902\))

What You'll Learn