UDD Conversion With Three or More Causes

Why Three or More Causes Require a Simultaneous Solution

In a multiple-decrement model, an individual can leave during the year because of any one of several causes. The observed probability of exit from a particular cause is therefore affected by the other causes: an individual who exits first from one cause cannot later exit from another. An associated single-decrement table asks a different, hypothetical question: what would the probability of exit from one cause be if that cause acted alone?

Under the uniform distribution of decrements (UDD) assumption, converting observed probabilities to associated probabilities requires solving a system. With three or more causes, each associated probability depends on the associated probabilities for all the other causes. Substituting observed probabilities for those unknowns is generally not a valid conversion.

Definition: Consider one year and $m$ competing causes, with $m\geq 3$. Let $q_j$ be the observed probability that an individual exits from cause $j$ during the year, and let $q_j'$ be the associated single-decrement probability for cause $j$. Under the UDD competing-decrement equations, $$ q_j=q_j'\left(1-\frac{1}{2}\sum_{\substack{k=1\\k\ne j}}^m q_k'\right),\qquad j=1,\ldots,m. $$ Each $q_j'$ is unknown in this system and is defined for its own single-decrement table; it is not the observed probability $q_j$.

The factor multiplying $q_j'$ reflects the effect of the other causes during the year. If their associated probabilities are larger, more individuals are removed by competing causes before they can exit from cause $j$. The equation is simultaneous because the adjustment for cause $j$ uses the associated probabilities for the other causes, not their observed probabilities.

This conversion is specific to the UDD assumption and a one-year interval. It does not assert that different causes are independent, and it should not be carried over to a different within-year assumption without deriving the corresponding equations.

Why the UDD Equation Has This Form

Under UDD, each cause's decrements are treated as uniformly distributed through the year for the conversion. In the UDD competing-decrement relationship, cause $j$'s associated decrement amount is adjusted for the competing decrements that can occur before it. For another cause $k$, the average fraction of its associated decrement amount occurring before a uniformly located cause-$j$ decrement is one half. Thus the total competing adjustment for cause $j$ is one half the sum of the other associated probabilities.

Subtracting that average competing amount from one gives the multiplier on $q_j'$. Multiplying by the hypothetical single-decrement probability for cause $j$ gives the observed probability:

q_j=q_j'\left[1-\frac{1}{2}\left(q_1'+\cdots+q_{j-1}'+q_{j+1}'+\cdots+q_m'\right)\right].

The bracket excludes $q_j'$ itself: it adjusts cause $j$ for exits from the other causes. That exclusion is important when setting up each row of the system.

A Useful Reduction: Solve for the Total Associated Probability

For more than two causes, it is helpful to replace the several coupled equations with one equation in the total associated probability. Define $S=\sum_{k=1}^m q_k'$. Then the sum of associated probabilities for causes other than $j$ is $S-q_j'$, so the UDD equation becomes

q_j=q_j'\left(1-\frac{S-q_j'}{2}\right).

For a fixed candidate value of $S$, rearrange this expression into a quadratic in $q_j'$:

$$ (q_j')^2+(2-S)q_j'-2q_j=0. $$

The nonnegative root is

q_j'=\frac{-(2-S)+\sqrt{(2-S)^2+8q_j}}{2}.

Each observed probability therefore gives an associated probability as a function of the same $S$. A solution must satisfy the consistency condition that these associated probabilities sum back to $S$:

S=\sum_{j=1}^m \frac{-(2-S)+\sqrt{(2-S)^2+8q_j}}{2}.

This is a one-variable equation that can be solved algebraically in special cases or numerically in general. After finding a candidate, substitute the resulting associated probabilities into every original UDD equation. The substitution check is essential: it confirms that the individual equations hold, not just that a computational procedure returned a number.

Record the observed probabilities.
Label the observed cause probabilities $q_1,\ldots,q_m$, and confirm that they refer to the same year and population.

Form the UDD equations.
For each cause, exclude that cause from the sum of associated probabilities in the adjustment factor.

Solve the shared system.
Use the total $S$ reduction, a numerical system solver, or another valid method. Do not solve each cause using only its own observed probability.

Check the result.
Verify every associated probability is in range and that substituting all the values reproduces each observed probability.

Worked Conversions

Worked Example: Three Causes With an Exact Total

A hypothetical one-year decrement study records three observed probabilities: $q_1=0.160$, $q_2=0.255$, and $q_3=0.075$. Find the associated single-decrement probabilities under UDD. The interval is one year, the within-year decrement assumption is UDD, and no interest rate is involved in this probability conversion.

Method. Set $S=q_1'+q_2'+q_3'$ and use the quadratic root for each cause. The observed probabilities sum to $0.160+0.255+0.075=0.490$, but that is not the required associated total.

Test $S=0.600$. Then $2-S=1.400$. For cause 1,

q_1'=\frac{-1.400+\sqrt{1.400^2+8(0.160)}}{2} =\frac{-1.400+\sqrt{1.960+1.280}}{2} =\frac{-1.400+1.800}{2}=0.200.

For cause 2,

q_2'=\frac{-1.400+\sqrt{1.960+8(0.255)}}{2} =\frac{-1.400+\sqrt{4.000}}{2} =\frac{-1.400+2.000}{2}=0.300.

For cause 3,

q_3'=\frac{-1.400+\sqrt{1.960+8(0.075)}}{2} =\frac{-1.400+\sqrt{2.560}}{2} =\frac{-1.400+1.600}{2}=0.100.

Their sum is $0.200+0.300+0.100=0.600$, so the chosen total is consistent. Now check each original equation. For cause 1, the other associated probabilities sum to $0.300+0.100=0.400$, giving $0.200(1-0.400/2)=0.200(0.800)=0.160$. For cause 2, the other sum is $0.200+0.100=0.300$, giving $0.300(1-0.300/2)=0.300(0.850)=0.255$. For cause 3, the other sum is $0.200+0.300=0.500$, giving $0.100(1-0.500/2)=0.100(0.750)=0.075$. All three observed probabilities are reproduced.

Answer: $q_1'=0.200$, $q_2'=0.300$, and $q_3'=0.100$. The common mistake avoided is treating the observed total $0.490$ as the associated total.

Worked Example: Four Causes and a Shared Quadratic

A hypothetical one-year study has observed probabilities $q_1=0.08000$, $q_2=0.17000$, $q_3=0.12375$, and $q_4=0.03875$. Assume UDD for the within-year decrements. Find all four associated probabilities. No interest rate is used.

Method. Solve the quadratics using a common total. Try $S=0.500$, so $2-S=1.500$. The positive-root formula gives, for cause 1,

q_1'=\frac{-1.500+\sqrt{2.250+8(0.08000)}}{2} =\frac{-1.500+\sqrt{2.890}}{2} =\frac{-1.500+1.700}{2}=0.100.

For cause 2, the radicand is $2.250+8(0.17000)=3.610$, whose square root is $1.900$. Hence $q_2'=(-1.500+1.900)/2=0.200$. For cause 3, the radicand is $2.250+8(0.12375)=3.240$, whose square root is $1.800$. Hence $q_3'=(-1.500+1.800)/2=0.150$. For cause 4, the radicand is $2.250+8(0.03875)=2.560$, whose square root is $1.600$. Hence $q_4'=(-1.500+1.600)/2=0.050$.

The associated probabilities sum to $0.100+0.200+0.150+0.050=0.500$, agreeing with the assumed total. Check the equations using the sums of the other associated probabilities:

Cause 1: $0.100[1-(0.200+0.150+0.050)/2]=0.100(0.800)=0.08000$.
Cause 2: $0.200[1-(0.100+0.150+0.050)/2]=0.200(0.850)=0.17000$.
Cause 3: $0.150[1-(0.100+0.200+0.050)/2]=0.150(0.825)=0.12375$.
Cause 4: $0.050[1-(0.100+0.200+0.150)/2]=0.050(0.775)=0.03875$.

Answer: $(q_1',q_2',q_3',q_4')=(0.100,0.200,0.150,0.050)$. The check shows why one common $S$, rather than a separate adjustment for each cause, is needed.

Worked Example: Recovering One Cause From the Other Associated Values

A hypothetical three-cause UDD conversion has observed $q_3=0.070$. The associated probabilities for causes 1 and 2 have already been established as $q_1'=0.250$ and $q_2'=0.150$. Find $q_3'$, then calculate the observed probabilities for causes 1 and 2 as a consistency check. The period is one year; UDD is assumed, and interest is not relevant.

Method. Use the cause-3 equation, which adjusts for causes 1 and 2:

0.070=q_3'\left(1-\frac{0.250+0.150}{2}\right) =q_3'(1-0.200)=0.800q_3'.

Therefore $q_3'=0.070/0.800=0.0875$. The associated total is $0.250+0.150+0.0875=0.4875$. Next calculate the other observed probabilities using their own competing sums. For cause 1, the other sum is $0.150+0.0875=0.2375$, so

q_1=0.250\left(1-\frac{0.2375}{2}\right) =0.250(0.88125)=0.2203125.

For cause 2, the other sum is $0.250+0.0875=0.3375$, so

q_2=0.150\left(1-\frac{0.3375}{2}\right) =0.150(0.83125)=0.1246875.

Checking cause 3 gives $0.0875[1-(0.250+0.150)/2]=0.0875(0.800)=0.070$, as required. The observed probabilities sum to $0.2203125+0.1246875+0.070=0.415$.

Answer: $q_3'=0.0875$, with $q_1=0.2203125$ and $q_2=0.1246875$. The common mistake avoided is applying the cause-3 adjustment factor to causes 1 and 2 as well; each equation has its own excluded cause.

Common Mistake: Adjusting With Observed Probabilities

A tempting shortcut is to replace the other associated probabilities in the UDD equation with the other observed probabilities. In the first worked example, this would give the incorrect calculation

q_1'\ \text{(incorrect)} =\frac{0.160}{1-(0.255+0.075)/2} =\frac{0.160}{0.835} \approx 0.1916.

The denominator should use $q_2'+q_3'$, not $q_2+q_3$. The simultaneous solution gives $q_2'+q_3'=0.300+0.100=0.400$, so the correct calculation is $q_1'=0.160/(1-0.400/2)=0.160/0.800=0.200$. This illustrates the key dependency: even the equation for cause 1 needs the associated values for the other causes.

Key takeaway: Under UDD, write one equation per cause, with the sum of all the other associated probabilities in that cause's adjustment factor. Solve the equations together, then substitute every result into the original system. An associated probability is not obtained by adjusting with the other observed probabilities.

Check Your Understanding

For each problem, use the stated one-year UDD assumption. No interest rate is involved.

A hypothetical model has three associated probabilities $q_1'=0.100$, $q_2'=0.200$, and $q_3'=0.100$. Calculate the three observed probabilities. (Answer: $q_1=0.085,\ q_2=0.180,\ q_3=0.085$)
In a four-cause model, the associated probabilities are $0.120,\ 0.080,\ 0.200,\ 0.100$, in that order. Find the observed probability for cause 2. (Answer: $0.0632$)
For three causes, $q_1'=0.200$, $q_2'=0.100$, and the observed $q_3=0.085$. Find $q_3'$. (Answer: $0.100$)
For one cause in a four-cause model, the total associated probability is $S=0.500$ and its observed probability is $q_j=0.12375$. Find its associated probability using the nonnegative quadratic root. (Answer: $q_j'=0.150$)
A three-cause study reports observed probabilities $0.180,\ 0.085,\ 0.085$. A proposed associated solution is $(0.200,0.100,0.100)$. Check whether it satisfies all three UDD equations. (Answer: Yes; substitution gives $0.180,\ 0.085,\ 0.085$, respectively.)

What You'll Learn