# Lesson 7: Multiple-Decrement Life Tables with Observational Data

## 7.0 Overview

In Lesson 6, we learned to construct single decrement life tables with observational (micro) data. In this session we will learn how to extend the method to construct multiple decrement life tables.

Recall that in a multiple decrement life table individuals leave a well-defined cohort through multiple attrition factors. For example, a group of married women can end their marriage by divorce or widowhood. A single decrement life table does not distinguish between divorce and widowhood, but combines them to form a single attrition factor, "marital disruption." In an ordinary life table, we considered discontinuation of pill use without taking into account the reason for discontinuation. In a multiple decrement life table, we will differentiate various reasons for the discontinuation of pill use.

Download a zip file containing data for Lesson 4 in Excel and CSV formats.

## 7.1 Examples of Data

Data 1: Study of Marital Disruption

Table 7.1 shows tabulations of marital disruption by cause from a demographic survey of 17,045 ever-married women. This data set is the same as the data set 1 we used in constructing the single decrement life tables except that the marital disruption is now separated into divorced or widowed.

Duration of Marriage in Completed Years  NumberDivorced  NumberWidowed  Number ofMaritalDisruptions  Number Still Marriedat Survey  Total Table 7.1: Marital Disruption by Cause and Duration of Marriage 0 140 1 141 88 229 1 211 3 214 222 436 2 272 2 274 523 797 3 256 8 264 405 669 4 232 12 244 452 696 5 193 10 203 555 758 6 193 13 206 539 745 7 174 12 186 543 729 8 147 19 166 465 631 9 142 18 160 435 595 10 122 18 140 441 581 11 96 20 116 437 553 12+ 981 113 1094 8532 9626 Total 3159 249 3408 13637 17045

Table 7.1 shows that among 17,045 ever-married women interviewed, 141 ended their first marriage in the first year, 140 of them due to divorce and one due to widowhood. As before, 88 women who were married less than one year at the time of survey and still in intact marriages are considered "censored."

Essentials of a Multiple Decrement Life Table

Recall that a multiple decrement life table includes a number of summary measures. Following is a brief review of each of quantities, including the computational formulae and examples.

1. Crude probability is the probability of an event due to a specific cause in the presence of other competing events.  denotes the crude probability of the occurrence of an event of type  at duration x in the presence of other competing causes.

In the data above, there are two competing causes of marital disruption: divorce and widowhood. Qxd denotes the crude probability that a marriage will end at duration at x due to divorce for those in intact marriages at this duration when both divorce and widowhood are possible. Similarly one can define crude probability of widowhood and denote it as Qxw.

2. Total probability is the probability of occurrence of an event regardless of the type of event. In the above example, total probability, or qx, is the probability of a marital disruption due either to divorce or widowhood. It follows that:

qx = Qxd + Qxw

When more than two types of events occur, one can extend the above formula to calculate total probability as:

qx = Qx1 + Qx2 + ... + Qxk, where k denotes the number of possible types of events.

Remark

A slight change in the notation will be used when referring to interval data. In this case the probability of occurrence of an event of type Rs in an interval [x, x + n] will be denoted as . Then the probability of an event of any type in the interval is the total probability, denoted as (nqx) and calculated in relation: (nqx) = Qx1 + Qx2 + ... +Qxk

Survival Function

A survival function is the probability of no event of any type by duration x. As before, this function is denoted asS(x) and calculated using the Kaplan Meier estimate as:

In order to interpret the multiple decrement life easily use another quantity lx = 100,000 x S(x) . Because S(0) = 1,l0 = 100,000 and one can interpret l0 as the radix of the life table.

Occurrence of Events of Particular Type at a Specified Duration x

In the life table population with the radix of 100,000, there will be some events of a particular type at each duration. We denote  as the number of occurrences of event type  at duration x. The number of events of a specific type  is obtained as:

Note that the total number of events of all types at duration x will be:

dx = dx1 + dx2 + .... + dx2.

Cumulative Number of Events of a Particular Type by a Specified Duration

Denote as the cumulative number of events of type  up to duration x.

Then

Cumulative Probability of Occurrence of an Event of Specified Type by Duration

Denote  as the cumulative probability that an event of particular type will occur by duration x. By definition:

Computational Formula (Exact event times)

Step 1:

Like in the construction of the ordinary life table, first calculate the number in the risk set at specified durations. Let Nx denote the number in the risk set at duration x. Use the observed events and censoring at duration x to calculate the risk set. Like before, denote the number of events and censoring at duration x as dx and Cxrespectively. Then Nx+1 = Nx - dx - Cx

Example

In Table 7.1 No. At duration zero, 141 individuals experienced marital disruption and 88 individuals were censored. Therefore, the number in the risk set at duration one is: N1 = 17045 - 141 - 88 = 16816

Column 2 in Table 7.2 shows the risk set at all durations.

Duration of Marriage in Completed Years Risk Set Qxd Qxw qx S(x) Table 7.2: Computation of Crude Probabilities 0 17,045 0.008214 0.00006 0.00827 1.00000 1 16,902 0.012548 0.00018 0.01273 0.99173 2 16,680 0.016606 0.00012 0.01673 0.97911 3 16,394 0.016428 0.00051 0.01694 0.96273 4 16,120 0.015556 0.00080 0.01636 0.94642 5 15,863 0.013574 0.00070 0.01428 0.93093 6 15,648 0.014339 0.00097 0.01530 0.91764 7 15,423 0.013685 0.00094 0.01463 0.90360 8 15,219 0.012264 0.00159 0.01385 0.89038 9 15,035 0.012506 0.00159 0.01409 0.87805 10 14,855 0.011338 0.00167 0.01301 0.86568 11 14,602 0.009431 0.00196 0.01140 0.85441 12+ 14,049 0.84468

Step 2: Computation of the crude probabilities

The notation  denotes the number of occurrences of events of type  at duration x. Using this notation, the crude probability is calculated as:

 Example Tables 7.1 and 7.2 show N0 = 17045, 40 marital disruptions due to divorce and one due to widowhood. With this data we can calculate the crude probability of marital disruption due to divorce (Qod) and due to widowhood (Qow) as follows: Similarly crude probabilities at duration one are calculated as: The crude probabilities at all durations are shown in Table 7.2.

Step 3: Total probability

By definition the total probability is the sum of all crude probabilities at a specified duration. Thus:

 Example In Table 7.2 there are only two competing events for marital disruption (divorce and widowhood). Table 7.2 shows: q0 = Q0d + Q0w = .008214 + 0.00006 = 0.00827 q1 = Q1d + Q1w = 0.012548 + .00018 + 0.01273 The total probabilities for all durations are shown in Table 7.2.

Step 4: Computation of survival probabilities

We will use the Kaplan-Meier formula to compute the survival probabilities (probability of nonoccurence of any event by duration x).

Recall that the survival probability is calculated by the formula:

Remember that S(0) = 1.

 Example From Table 7.2 S(0) = 1; S(1) = 1 — .00827 = 0.99173. The survival functions at all durations are given in Table 7.2.

Step 5: Number of events of specific type

For computational ease we will create a life table lx S(x) column by multiplying the survival function S(x) by 100,000:

lx = 100000×S(x)

Then the number of occurrences of events of a specific type  in the life table population is obtained by multiplying the lx column by the corresponding crude probability:

Duration of Marriagein Completed Years lx dxd dxw dx F(x, d) F(x, w) F(x) Table 7.3: Multiple Decrement Life Table for Marital Disruption 0 100,000 821 6 827 0 0 0 1 99,173 1,244 18 1,262 821 6 827 2 97,911 1,626 12 1,638 2,066 24 2,089 3 96,273 1,582 49 1631 3692 36 3727 4 94,642 1472 76 1548 5273 85 5358 5 93,093 1264 65 1329 6745 161 6906 6 91,764 1316 89 1404 8009 227 8235 7 90,360 1237 85 1322 9325 315 9640 8 89,038 1092 141 1233 10,561 401 10,962 9 87,805 1098 139 1237 11,653 542 12,195 10 86,568 982 145 1126 12,751 681 13,432 11 85,441 806 168 974 13,733 826 14,558 12+ 84,468 14,539 994 15,532 Example Remember that S(0) = 1 and therefore l0 = 100,000. In Table 7.2 the value of the survival function at duration one is 0.99173. Therefore lx = 100,000 x S(1) = 100000 x 0.99173 = 99173 The lx values for all durations are presented in column 2 of Table 7.3. We will use the lx column in Table 7.3 and the crude probability calculated in Table 7.2 to obtain the number of divorces and widowhoods at specified marital duration in the life table. For example, crude probability of divorce at duration zero is 0.008214 and the crude probability of widowhood at this duration is 0.0006. Therefore, number of divorces at duration zero in the life table population is: D0d = 100,000 x 0.008214 = 821 Similarly the number of widowhoods at duration zero in this life table population is: D0w = 100,000 x 0.00006 = 6 The number in still-intact marriages at duration one is l1 = 99173. The respective crude probabilities of divorce and widowhood at this duration are 0.01255 and 0.00018 Therefore, the number of divorces and widowhoods at duration 1 is obtained as follows: D1d = 0.01255 = 1244 D1w = 99173 x 0.00018 = 18. Number of divorces and widowhoods at all durations are presented in Table 7.3.

Remark

The sum of the number of divorces and widowhoods at a specified marital duration gives the total number of marital disruptions at that duration. For example at duration zero there were 821 divorces and 6 widowhoods with a total of 827 marital disruptions. The total marital disruption at specified marital durations are shown in Column 5 of Table 7.3.

Step 6: Cumulative number of events of specified types

So far we have calculated the number of events of various types at specified times. Now we calculate the cumulative number of events of various types . By simply summing the number of events at each time point:

 Example In Table 7.3 the number of divorces at duration zero is 821. The number of divorces at marital duration one is 1244. Therefore the cumulative number of divorces at the end of year one (beginning of year 2) is 821 + 1244 = 2065. Continuing the summation yields the cumulative number of divorces in this life table population at the beginning of each marital duration. Table 7.3 shows the cumulative number of divorces, widowhoods, and total marital disruptions at all marital durations. From the table one can see that, at the beginning of year 12, there were 14539 divorces, 894 widowhoods and 15532 marital disruptions in this life table population.

Step 7: Cumulative probability of an event of specified type

In a life table with radix l0, the total number of events of a particular type occurring by time x is . Therefore the probability that an individual will experience event  by duration x is

 Example In Table 7.3, the cumulative number of divorces by marital duration 12 is 14539. Therefore the probability that a marriage will end due to divorce by year 12 is calculated as . The probability that a marriage will end due to widowhood by year 12 is  The cumulative probabilities for all durations are shown in Table 7.4.
Durationof Marriage inCompleted Years CQxd CQxw Table 7.4: Cumulative Probabilities of Divorce and Widowhood 0 0 0 1 0.00821 0.00006 2 0.02066 0.00024 3 0.03692 0.00036 4 0.05273 0.00085 5 0.06745 0.00161 6 0.08009 0.00227 7 0.09325 0.00315 8 0.10561 0.00401 9 0.11653 0.00542 10 0.12751 0.00681 11 0.13733 0.00826 12+ 0.14539 0.00994

Remark

So far we have learned to construct a multiple decrement life table with exact-time-to event data. The principal quantities we learned to compute and interpret are crude probabilities and the event type-specific cumulative probabilities.

The most often used data in the multiple decrement life tables are the cumulative probabilities. One may also want to look at the average time to an event of specific type. Median is the preferred measure. The median time for a specific event is defined as the duration at which the cumulative probability reaches one half (0.5). In the above example the cumulative probabilities of both divorce and widowhood are less than 0.5 at the maximum point of observation (12 years).

The next example extends the construction of multiple decrement life tables to grouped data.

Multiple Decrement Life Tables with Grouped Data

Following are some notations, computational formulas, and examples to illustrate the method. Most of the concepts have been introduced in the previous sections. Therefore a brief introduction is provided here.

Notations
n ~ the length of the class interval. The length need not be the same for each class interval.

~ Number of events of type  in the interval (x, x + n)

~ Total number of events in the interval (x, x + n)

(nCx) ~ Number censored in the interval (x, x + n)

Nx ~ Number in the risk set (exposed to the risk of an event) at the beginning of the interval (x, x + n)

By definition Nx + n = Nx — (nDx) — (nCx)

~ Adjusted number in the risk set.

We assume that censoring is uniform in the interval and calculate the adjusted risk set. The adjusted risk set at the beginning of each interval is computed by taking away half of the censored observations from the interval from the number who have not experienced any event at the beginning if each interval. Thus,

~ Crude probability of an event of type  in the interval (x, x + n)

(nqx) ~ Probability of an event (regardless of type) in the interval (x, x+ n)

By definition:

(nqx) = (nQx1) + (nQx2) + ... + (nQxk)

Computational Formula

By definition:

The survival function (the proportion of individuals experiencing no event until the beginning of the interval) is calculated by the Kaplan-Meier estimate as:

Recall that a quantity lx = 100,000 * S(x) and l0 = 100,000.

The number of events of a particular type  in the life table population is calculated as:

The cumulative number of events of type  until the beginning of the interval (x, x + n) is equal to:

The cumulative probability of an event of particular type by the beginning of an interval is:

Example: Analysis of Oral Contraceptive Use Data

Earlier we constructed an ordinary life table to examine the duration of oral contraceptive use regardless of the reason for termination. In the multiple decrement life table we will differentiate the reasons for termination. Three reasons for discontinuation are identified in the data: Planning pregnancy, medical reasons, and others. The duration of use is tabulated in interval form. The data are reproduced in Table 7.5.

Durationof Pill Use  PlanningPregnancy  MedicalReasons  Others  Terminations  ContinuingUsers  Nx Table 7.5: Length of Use of Oral Contraceptives by Termination Reason 1-3 15 64 30 109 32 732 716.0 4-6 10 33 12 55 31 591 575.5 7-9 12 19 13 44 24 505 493.0 10-12 6 8 17 31 24 437 425.0 13-15 7 15 4 26 27 382 368.5 16-18 3 4 4 11 19 329 319.5 19-21 2 3 3 8 26 299 286.0 22-24 2 9 3 14 23 265 253.5 25-27 6 7 2 15 29 228 213.5 28-30 2 1 5 8 13 184 177.5 31-33 3 5 3 11 21 163 152.5 34-36 2 1 2 5 19 131 121.5 37-39 2 1 1 4 16 107 99.0 40-42 3 2 5 10 9 87 82.5 43-45 2 2 1 5 10 68 63.0 46-48 1 0 0 1 13 53 46.5 49+ 2 1 0 3 36 39

Step 1: Computation of Adjusted Risk Set

In Table 2.2 in the previous chapter we have already calculated the risk set and adjusted risk set for all the intervals considered. These numbers are reproduced in columns 7 and 8 of Table 7.5.

There were 732 women in the sample. Thirty two women were censored in the interval 1-3. Therefore the adjusted risk set is 732 - 0.5 * 32 = 716. One hundred nine women terminated the use of pill in the same interval. Therefore the number in the risk set at the beginning of the interval 4-6 is 732 - 109 - 32 = 591. There were 31 censored cases in the interval 4-6. Therefore the adjusted number in the risk set at the beginning of interval 4 is591 - 0.5 * 31 = 575.5. The risk set for other durations is shown in Table 7.5.

Step 2: Computation of Crude Probabilities

Recall that crude probabilities are calculated by dividing the number of events of a particular type by the adjusted number in the risk set. In the interval 1-3 there were 15 terminations due to planning pregnancy, 64 terminations due to medical reasons, and 109 terminations due to other reasons. The adjusted number in the risk set at the beginning of interval 1-3 is 716. Therefore the crude probabilities of terminations due to planning pregnancy, medical reasons, and other reasons are respectively

Table 7.6 shows the crude probabilities for all durations. The total probability of occurrence of an event in the interval (nqx) is the sum of all crude probabilities.

Durationof Pill Use  Qxpreg  QxMed  QxOther  qx  S(x) Table 7.6: Crude Probabilities of Pill Discontinuation 1-3 0.02095 0.08939 0.04190 0.152235 1.00000 4-6 0.01738 0.05734 0.02085 0.09557 0.84777 7-9 0.02434 0.03854 0.02637 0.08925 0.76675 10-12 0.01412 0.01882 0.04000 0.07294 0.69831 13-15 0.01900 0.04071 0.01085 0.07056 0.64738 16-18 0.00939 0.01252 0.01252 0.03443 0.60170 19-21 0.00699 0.01049 0.01049 0.02797 0.58099 22-24 0.00789 0.03550 0.01183 0.05523 0.56473 25-27 0.02810 0.03279 0.00937 0.07026 0.53355 28-30 0.01127 0.00563 0.02817 0.04507 0.49606 31-33 0.01967 0.03279 0.01967 0.07213 0.47370 34-36 0.01646 0.00823 0.01646 0.04115 0.43953 37-39 0.02020 0.01010 0.01010 0.04040 0.42145 40-42 0.03636 0.02424 0.06061 0.12121 0.40442 43-45 0.03175 0.03175 0.01587 0.07937 0.35540 46-48 0.02151 0.00000 0 0.02151 0.32719 49+ 0.32015

Step 3: Calculation of the Survival Probabilities

Using the Kaplan-Meier formula yields the survival probabilities:

S(1) = 1 [Note that in the tabulated data the interval starts at one rather than zero.]

S(4) = 1 — (3q1) = 1 — 0.15223 = 0.84777

S(7) = S(4) * [1 — 3q 4] = 0.84777 * (1 — 0.09557) = 0.76675

The survival functions for all durations are shown in Table 7.6.

Step 4: Calculation of Life Table Survival Population

The radix of the life table l0 = 100,000

Then the lx column in the life table is calculated as lx = 100,000 * S(x)

In Table 7.6, S(4) = 0.76675. Therefore l4 = 100000 * 0.76675 = 76675

The lx values for all durations are shown in Table 7.7.

Durationof Pill Use lx dxPregnancy dxmed dxother Table 7.7: Number of Pill Use Terminations by Cause 1-3 100,000 2095 8939 4190 4-6 84777 1473 4861 1768 7-9 76675 1866 2955 2022 10-12 69831 986 1314 2793 13-15 64738 1230 2635 703 16-18 60170 565 753 753 19-21 58099 406 609 609 22-24 56473 446 2005 668 25-27 53355 1499 1749 500 28-30 49606 559 279 1397 31-33 47370 932 1553 932 34-36 43953 724 362 724 37-39 42145 851 426 426 40-42 40442 1471 980 2451 43-45 35540 1128 1128 564 46-48 32719 704 0 0 49+ 32015

Step 5: Cumulative Number of Events by Type

This step calculates the cumulative number of terminations of pill use by type at the end of the each interval (or to the beginning of the next interval). Forming simple sums with the number events of specific types  will complete this step.

For example, in Table 7.7, the number of terminations due to pregnancy planning in the interval 1-3 is 2095 and the corresponding number for the interval 4-6 is 1473. The cumulative number of terminations for pregnancy planning at time 7 (which is the end of the interval) [Cd7 Pregnancy] is 2095 + 1473 = 3568. Cumulative numbers for all durations by types of termination are shown in Table 7.8.

Durationof Pill Use Cdxpregnancy CdxMedical CdxOthers Cdx Table 7.8: Cumulative Number of Events at the Beginning of the Interval 1-3 0 0 0 0 4-6 2095 8939 4190 15223 7-9 3568 13800 5958 23325 10-12 5434 16755 7979 30169 13-15 6420 18069 10773 35262 16-18 7650 20704 11475 39830 19-21 8215 21458 12229 41901 22-24 8621 22067 12838 43527 25-27 9067 24072 13507 46645 28-30 10566 25821 14006 50394 31-33 11125 26101 15404 52630 34-36 12057 27654 16336 56047 37-39 12781 28016 17059 57855 40-42 13632 28442 17485 59558 43-45 15103 29422 19936 64460 46-48 16231 30550 20500 67281 49+ 16934 30550 20500 67985

Step 6: Cumulative Probability of Pill Use Terminations by Cause

The cumulative probability of pill use terminations by cause is calculated by dividing the corresponding cumulative number of terminations by 100,000 (the radix of the life table).

Table 7.8 shows that the cumulative number of terminations at the beginning of month 49 is 16934, 30551, and 20500 respectively for pregnancy planning, medical reasons and other reasons. The corresponding cumulative probabilities are:

The cumulative probability of terminations of all types is the sum of all the type specific probabilities.

CQx = 0.16934 + 0.30551 + 0.20500 = 0.68285.

Remark

The above cumulative probabilities indicate that 68.3% of the women had stopped using the pill by month 49. Terminations by type show that 17.9% stopped because they are planning pregnancy, 30.5% terminated due to medical reasons, and 20.5% dropped due to other reasons. The cumulative probabilities for all durations are shown in Table 7.9.

Duration ofPill Use CQxpregnancy CQxMedical CQxOthers CQx Table 7.9: Cumulative Probability of Pill Use Terminations by Type 1-3 0.00000 0.00000 0.00000 0.00000 4-6 0.02095 0.08939 0.04190 0.15223 7-9 0.03568 0.13800 0.05958 0.23325 10-12 0.05434 0.16755 0.07979 0.30169 13-15 0.06420 0.18069 0.10773 0.35262 16-18 0.07650 0.20704 0.11475 0.39830 19-21 0.08215 0.21458 0.12229 0.41901 22-24 0.08621 0.22067 0.12838 0.43527 25-27 0.09067 0.24072 0.13507 0.46645 28-30 0.10566 0.25821 0.14006 0.50394 31-33 0.11125 0.26101 0.15404 0.52630 34-36 0.12057 0.27654 0.16336 0.56047 37-39 0.12781 0.28016 0.17059 0.57855 40-42 0.13632 0.28442 0.17485 0.59558 43-45 0.15103 0.29422 0.19936 0.64460 46-48 0.16231 0.30550 0.20500 0.67281 49+ 0.16934 0.30550 0.20500 0.67985

Summary
This example demonstrated how to construct a multiple decrement life table using event-time data in grouped form. The key parameters of interest are the cumulative probabilities of events by type for different durations. Intermediate steps in obtaining these parameters are calculation of adjusted risk set and crude probabilities of type-specific probabilities.

## 7.2 Assignment

The following information was collected in a demographic survey from ever-married women with no premarital birth. Women were asked about the date of birth of their first child or date of marital disruption whichever came first. Time from marriage to the event (either birth or marital disruption) was noted. For women who were in intact first marriage at the time of survey, time from marriage to the time of survey was noted. For these women the time to event is considered as censored at the duration of marriage at the time of survey. Data are tabulated in one-year intervals in Table 7.10.

TimeSince Marriage(Years) Women withFirst Birthin the Year Women DivorcedBeforeFirst Birth Childless Womenin Intact Marriageat Survey Table 7.10 0 3871 192 577 1 6263 190 466 2 2854 150 386 3 1560 86 261 4 981 85 185 5 602 61 141 6 325 45 110 7 238 34 64 8 161 26 55 9 102 22 58 10 89 28 34 11 65 18 37 12+ 158 108 28

Assume that censoring is distributed uniformly within the one-year interval. For simplicity assume that all marital disruptions are due to divorce. With this assumption calculate:

1.  Number in the risk set at the beginning of each interval (Nx).
2.  Adjusted number in the risk set at the beginning of each interval   .
3.  Crude probability of birth and divorce in each interval   QxBirthQxDivorce.
4.  Total probability of either birth or divorces in the interval (dxBirthdxDivorce).
5.  Survival probability (probability neither birth or divorce occurring) (qx).
6.  Assume that l0 = 100,000 and complete lx column in the life table.
7.  Number of births and divorce in each interval in the life table population (dxBirthQxDivorce).
8.  Cumulative number of births and divorces in each interval (CdxBirthCdxDivorce).
9.  Cumulative probabilities of birth and divorce by the beginning of each interval (CQxBirthCQxDivorce).

Interpret the results.

SEE ANSWERS IN TABLE 7.11 AND 7.12.

TimeSince Marriage(Years) Risk Set AdjustedRisk Set S(x) qx Qxdivorce Qxbirth Table 7.11 Crude Probabilities of Birth and Divorce 0 20716 20427.5 0.1895 0.02825 0.21775 1.00000 1 16076 15843 0.39532 0.02941 0.42473 0.78225 2 9157 8964 0.31838 0.04306 0.36145 0.45001 3 5767 5636.5 0.27677 0.04631 0.32307 0.28735 4 3860 3767.5 0.26038 0.0491 0.30949 0.19452 5 2609 2538.5 0.23715 0.05554 0.29269 0.13432 6 1805 1750 0.18571 0.06286 0.24857 0.095 7 1325 1293 0.18407 0.0495 0.23357 0.07139 8 989 961.5 0.16745 0.0572 0.22465 0.05471 9 747 718 0.14206 0.08078 0.22284 0.04242 10 565 548 0.16241 0.06204 0.22445 0.03297 11 414 395.5 0.16435 0.09355 0.2579 0.02557 12+ 294 280 0.56429 0.10000 0.66429 0.01897

TimeSince Marriage(Years) lx dxBirth dxDivorce dx Cdxbirth Cdx CdxDivorce CQxbirth CQxDivorce CQx Table 7.12 Cumulative Probabilities of Birth and Divorce 0 100000 18950 2825 21775 0 0 0 1 78225 30924 2301 33225 18950 2825 21775 0 0 0 2 45001 14328 1938 16266 49874 5126 55000 0.1895 0.02825 0.21775 3 28735 7953 1331 9284 64201 7064 71265 0.49874 0.05126 0.55 4 19452 5065 955 6020 72154 8394 80548 0.64201 0.07064 0.71265 5 13432 3185 746 3931 77219 9349 86568 0.72154 0.08394 0.80548 6 9500 1764 597 2362 80405 10095 90500 0.77219 0.09349 0.86568 7 7139 1314 353 1667 82169 10693 92862 0.80405 0.10095 0.905 8 5471 916 313 1229 83483 11046 94529 0.82169 0.10693 0.92862 9 4242 603 343 945 84399 11359 95758 0.83483 0.11046 0.94529 10 3297 535 205 740 85002 11702 96704 0.84399 0.11359 0.95758 11 2557 420 239 659 85537 11906 97443 0.85002 0.11702 0.96704 12+ 1897 85958 12145 98103 0.85537 0.11906 0.97443

## 7.3 Calculation of Standard Errors

Because life tables are calculated with sample data it is desirable to know the variability in the estimated quantities in the life table. This section covers computation of the standard errors of various life table quantities. Some of the elements in the calculation of the standard errors are computationally involved.

1. Variance of  (Conditional probability of surviving the interval x, x+ n)

, where  is the adjusted risk set at the beginning of the interval[x, x + n]. Standard error is the square root of the variance.

Example

In Table 7.6, for the first interval:

(3q1) = 0.15223(3p1)  = 1 — (3q1) = 1 — .15223; N1 = 716.0

Standard Error

Similarly one can compute standard errors of all other conditional probabilities. The calculated standard errors are shown in Table 7.13.

Duration of Pill Use Standard Error Standard Error Covariance Table 7.13: Standard Errors and Covariances of Event-Specific Conditional Probabilities 1-3 0.013425735 0.005352234 -0.0000248051 4-6 0.012255294 0.005446886 -0.0000273077 7-9 0.012840417 0.006940532 -0.0000449663 10-12 0.012613789 0.005722677 -0.0000307950 13-15 0.013340150 0.007111266 -0.0000479122 16-18 0.010200410 0.005395618 -0.0000283768 19-21 0.009750308 0.004927483 -0.0000237671 22-24 0.014346628 0.005556701 -0.0000294037 25-27 0.017491581 0.011310658 -0.0001223822 28-30 0.015571567 0.007922386 -0.0000606184 31-33 0.020949314 0.011245440 -0.0001196928 34-36 0.018021193 0.011543421 -0.0001299054 37-39 0.019789691 0.014139956 -0.0001958159 40-42 0.035932553 0.020609301 -0.0003873445 43-45 0.034055572 0.022088646 -0.0004639128 46-48 0.021272879 0.021272879 -0.0004525354

2. Variance of Event-Specific Conditional Probabilities (equation )

Standard error is the square root of the variance.

 Example In Table 7.6: 3Q1Pregnancy = 0.02095 Therefore  Standard Error . Similarly one can calculate the standard error of pregnancy planning specific probabilities for other intervals. The calculated probabilities are shown in Table 7.13.

3. Covariance

For the computation of the variance of the cumulative probabilities it is necessary to calculate the covariance .

Covariance .

 Example From Table 7.6:(3p1) = 0.84775; 3Q1Pregnancy= 0.02095;  = 716 Table 7.13 shows covariance for all other intervals.

4. Variance of Event-Specific Cumulative Probabilities

Before presenting a general formula for event-specific Cumulative probabilities, we first look at some specific cases.

Case 1
The cumulative probability at the end of the first interval is simply the event specific probability in the first interval. So we can denote:

Therefore

 Example 1 From Table 7.6 CQ1Pregnancy = 0.02095. Variance  = 0.000028646 or standard error

Case 2

The event specific cumulative probability at the end of the second interval is:

Therefore an approximate variance of the event-specific cumulative probability  is obtained as:
Variance of [SORRY, IMAGE 46-8.gif NOT AVAILABE]

Variance

For simplicity we rewrite the above variance with some additional notations:
B0 = 1 and B1 = p0 (Note that S(1) = p0).

Then the variance equation is expressed as:

 Example 2 From Table 7.6:A0 = Q1Pregnancy= 0.01738; B0 = 1    B1 = B1 = P0 = 0.84776 From Table 7.13: ) = -0.000024805 Substituting in the above equation yields: Taking square root yields: = 0.00701 and = 0.03568

Case 3
The event-specific cumulative probability to the end of the third interval is calculated as:

The variance of is approximated as:

We rewrite this equation with symbols as before:

Denote: B0 = 1     B1 = S(1)     B2 = S(2)

Then

 Example In Table 7.9: CQ3Pregnancy = 0.05434 Simple calculations gives: A0 = 0.039394     A1 = 0.02063449 B0 = 1     B1 = 0.84776     B2 = 0.76674 Substituting in the equation yields: Taking square root we get

Case 4: A general formula for the event specific cumulative probabilities

A close look at equations 7.4.1 and 7.4.2 will reveal a pattern that will help in writing a general formula for the standard error of the cumulative probabilities. In general event-specific cumulative probability is denoted as . Note that in this case x denotes the beginning of the interval (x, x+ n) or exact duration x. The computation is by the formula:

Then the variance of this computed quantity can be expressed as:

where

B1 = S(x) and B0 = 1

Calculations of the required quantities can be implemented through a spreadsheet.

We illustrate here the method for the steps for calculating the standard error of the cumulative probability of pill discontinuation in month 49 due to pregnancy planning.

In Table 7.9 we calculate this number as CQ16Pregnancy = 0.16934

[Note: Denoting interval 1-3 as zero and interval 4-6 as one and so on. the interval 49 or more will be 16.]

We will use equation 7.4.3 to obtain the variance of .

In the equation x = 16.

Therefore we need to calculate A0A1, ....A14 and B0B1, ....B15. The Bs are simply the survival function S(x)with S(0) = 1. Table 7.14 shows the calculation of the A values. Note that by definition

So simply cumulating the products  and dividing by the proper survival probability the quantities Ai can be calculated.

i Durationof Pill Use Bi S(x) QxPregnancy Pi Ai Table 7.14 Standard Error Calculation of 0 1-3 1.00000 0.0209497207 0.1483947937 0.84777 0.175042294 1 4-6 0.84777 0.0147309359 0.1336638578 0.90443 0.147787800 2 7-9 0.76675 0.0186631695 0.1150006882 0.91075 0.126270243 3 10-12 0.69831 0.0098585449 0.1051421434 0.92706 0.113414749 4 13-15 0.64738 0.0122975426 0.0928446008 0.92944 0.099892658 5 16-18 0.60170 0.0056497764 0.0871948243 0.96557 0.090303878 6 19-21 0.58099 0.0040628346 0.0831319897 0.97203 0.085524277 7 22-24 0.56473 0.0044554951 0.0786764946 0.94477 0.083275538 8 25-27 0.53355 0.0149942541 0.0636822406 0.92974 0.068494501 9 28-30 0.49606 0.0055894074 0.0580928332 0.95493 0.060834678 10 31-33 0.47370 0.0093187333 0.0487740999 0.92787 0.052565726 11 34-36 0.43953 0.0072351208 0.0415389791 0.95885 0.043321768 12 37-39 0.42145 0.0085140563 0.0330249228 0.95960 0.034415446 13 40-42 0.40442 0.0147060972 0.0183188256 0.87879 0.020845560 14 43-45 0.35540 0.0112824555 0.0070363701 0.92063 0.007642954 15 46-48 0.32719 0.0070363701 16 49+

The required values of variances and covariances are given in Table 7.13.

Substituting the A and B values in Table 7.14 and the variances and covariances in Table 7.13 in the formula 7.4.3 we get the variance = 0.000406789. By taking the square root we get the standard error of  as 0.020169.

Use calculations similar to the one shown in Table 7.14 to get standard errors of the event specific cumulative probabilities for all durations. The final result is displayed in Table 7.15.

Interval Number Durationof Pill Use   Standard Error Table 7.15: Standard Errors of Event-Specific (Pregnancy Planning) 0 1-3 0.0000000 1 4-6 0.000028646 0.0053522 2 7-9 0.000051150 0.0071519 3 10-12 0.000079329 0.0089067 4 13-15 0.000093574 0.0096734 5 16-18 0.000114376 0.0106947 6 19-21 0.000123593 0.0111172 7 22-24 0.000131095 0.0114497 8 25-27 0.000140457 0.0118515 9 28-30 0.000176920 0.0133011 10 31-33 0.000189327 0.0137596 11 34-36 0.000216772 0.0147232 12 37-39 0.000240747 0.0155160 13 40-42 0.000274396 0.0165649 14 43-45 0.000341488 0.0184794 15 46-48 0.000400410 0.0200102 16 49+ 0.000447663 0.0211580

#### Exercise

Calculate the standard errors of the cumulative probabilities of pill termination due to medical reasons(CQxMedical) and other reasons (CQxOther) in Table 7.9.