Lesson 7: MultipleDecrement 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 welldefined 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 evermarried 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.
Table 7.1: Marital Disruption by Cause and Duration of Marriage  
Duration of Marriage in Completed Years  Number Divorced  Number Widowed  Number of Marital Disruptions  Number Still Married at Survey  Total 

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 evermarried 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.
 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. Q_{xd} 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 Q_{xw}.

Total probability is the probability of occurrence of an event regardless of the type of event. In the above example, total probability, or q_{x}, is the probability of a marital disruption due either to divorce or widowhood. It follows that:
q_{x} = Q_{xd} + Q_{xw}
When more than two types of events occur, one can extend the above formula to calculate total probability as:
q_{x} = Q_{x1} + Q_{x2} + ... + Q_{xk}, 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 (_{n}q_{x}) and calculated in relation: (_{n}q_{x}) = Q_{x1} + Q_{x2} + ... +Q_{xk}
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 l_{x} = 100,000 x S(x) . Because S(0) = 1,l_{0} = 100,000 and one can interpret l_{0} 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:
d_{x} = d_{x1} + d_{x2} + .... + d_{x2}.
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 N_{x} 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 d_{x} and C_{x}respectively. Then N_{x+1} = N_{x}  d_{x}  C_{x}
Example In Table 7.1 N_{o}. At duration zero, 141 individuals experienced marital disruption and 88 individuals were censored. Therefore, the number in the risk set at duration one is: N_{1} = 17045  141  88 = 16816 Column 2 in Table 7.2 shows the risk set at all durations.

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 N_{0} = 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 (Q_{od}) and due to widowhood (Q_{ow}) 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:
q_{1} = Q_{1d} + Q_{1w} = 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 KaplanMeier 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 l_{x} S(x) column by multiplying the survival function S(x) by 100,000:
l_{x} = 100000×S(x)
Then the number of occurrences of events of a specific type in the life table population is obtained by multiplying the l_{x} column by the corresponding crude probability:
Table 7.3: Multiple Decrement Life Table for Marital Disruption


Duration of Marriage in Completed Years  l_{x}  d_{xd}  d_{xw}  d_{x}  F(x, d)  F(x, w)  F(x) 

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 l_{0} = 100,000. In Table 7.2 the value of the survival function at duration one is 0.99173. Therefore l_{x} = 100,000 x S(1) = 100000 x 0.99173 = 99173 The l_{x} values for all durations are presented in column 2 of Table 7.3. We will use the l_{x} 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:
Similarly the number of widowhoods at duration zero in this life table population is:
The number in stillintact marriages at duration one is l_{1} = 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:
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 l_{0}, 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. 
Table 7.4: Cumulative Probabilities of Divorce and Widowhood


Duration of Marriage in Completed Years  CQ_{xd}  CQ_{xw} 

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 exacttimeto event data. The principal quantities we learned to compute and interpret are crude probabilities and the event typespecific 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)
(_{n}C_{x}) ~ Number censored in the interval (x, x + n)
N_{x} ~ Number in the risk set (exposed to the risk of an event) at the beginning of the interval (x, x + n)
By definition N_{x + n} = N_{x} — (_{n}D_{x}) — (_{n}C_{x})
~ 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)
(_{n}q_{x}) ~ Probability of an event (regardless of type) in the interval (x, x+ n)
By definition:
(_{n}q_{x}) = (_{n}Q_{x1}) + (_{n}Q_{x2}) + ... + (_{n}Q_{xk})
Computational Formula
By definition:
The survival function (the proportion of individuals experiencing no event until the beginning of the interval) is calculated by the KaplanMeier estimate as:
Recall that a quantity l_{x} = 100,000 * S(x) and l_{0} = 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.
Table 7.5: Length of Use of Oral Contraceptives by Termination Reason


Duration of Pill Use  Planning Pregnancy  Medical Reasons  Others  Terminations  Continuing Users  N_{x}  

13  15  64  30  109  32  732  716.0 
46  10  33  12  55  31  591  575.5 
79  12  19  13  44  24  505  493.0 
1012  6  8  17  31  24  437  425.0 
1315  7  15  4  26  27  382  368.5 
1618  3  4  4  11  19  329  319.5 
1921  2  3  3  8  26  299  286.0 
2224  2  9  3  14  23  265  253.5 
2527  6  7  2  15  29  228  213.5 
2830  2  1  5  8  13  184  177.5 
3133  3  5  3  11  21  163  152.5 
3436  2  1  2  5  19  131  121.5 
3739  2  1  1  4  16  107  99.0 
4042  3  2  5  10  9  87  82.5 
4345  2  2  1  5  10  68  63.0 
4648  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 13. 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 46 is 732  109  32 = 591. There were 31 censored cases in the interval 46. 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 13 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 13 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 (_{n}q_{x}) is the sum of all crude probabilities.
Table 7.6: Crude Probabilities of Pill Discontinuation  
Duration of Pill Use  Q_{xpreg}  Q_{xMed}  Q_{xOther}  q_{x}  S(x)  

13  0.02095  0.08939  0.04190  0.152235  1.00000  
46  0.01738  0.05734  0.02085  0.09557  0.84777  
79  0.02434  0.03854  0.02637  0.08925  0.76675  
1012  0.01412  0.01882  0.04000  0.07294  0.69831  
1315  0.01900  0.04071  0.01085  0.07056  0.64738  
1618  0.00939  0.01252  0.01252  0.03443  0.60170  
1921  0.00699  0.01049  0.01049  0.02797  0.58099  
2224  0.00789  0.03550  0.01183  0.05523  0.56473  
2527  0.02810  0.03279  0.00937  0.07026  0.53355  
2830  0.01127  0.00563  0.02817  0.04507  0.49606  
3133  0.01967  0.03279  0.01967  0.07213  0.47370  
3436  0.01646  0.00823  0.01646  0.04115  0.43953  
3739  0.02020  0.01010  0.01010  0.04040  0.42145  
4042  0.03636  0.02424  0.06061  0.12121  0.40442  
4345  0.03175  0.03175  0.01587  0.07937  0.35540  
4648  0.02151  0.00000  0  0.02151  0.32719  
49+  0.32015 
Step 3: Calculation of the Survival Probabilities
Using the KaplanMeier 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 — (_{3}q_{1}) = 1 — 0.15223 = 0.84777
S(7) = S(4) * [1 — _{3}q _{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 l_{0} = 100,000
Then the l_{x} column in the life table is calculated as l_{x} = 100,000 * S(x)
In Table 7.6, S(4) = 0.76675. Therefore l_{4} = 100000 * 0.76675 = 76675
The l_{x} values for all durations are shown in Table 7.7.
Table 7.7: Number of Pill Use Terminations by Cause


Duration of Pill Use  l_{x}  d_{xPregnancy}  d_{xmed}  d_{xother} 

13  100,000  2095  8939  4190 
46  84777  1473  4861  1768 
79  76675  1866  2955  2022 
1012  69831  986  1314  2793 
1315  64738  1230  2635  703 
1618  60170  565  753  753 
1921  58099  406  609  609 
2224  56473  446  2005  668 
2527  53355  1499  1749  500 
2830  49606  559  279  1397 
3133  47370  932  1553  932 
3436  43953  724  362  724 
3739  42145  851  426  426 
4042  40442  1471  980  2451 
4345  35540  1128  1128  564 
4648  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 13 is 2095 and the corresponding number for the interval 46 is 1473. The cumulative number of terminations for pregnancy planning at time 7 (which is the end of the interval) [Cd_{7 Pregnancy}] is 2095 + 1473 = 3568. Cumulative numbers for all durations by types of termination are shown in Table 7.8.
Table 7.8: Cumulative Number of Events at the Beginning of the Interval


Duration of Pill Use  Cd_{xpregnancy}  Cd_{xMedical}  Cd_{xOthers}  Cd_{x} 

13  0  0  0  0 
46  2095  8939  4190  15223 
79  3568  13800  5958  23325 
1012  5434  16755  7979  30169 
1315  6420  18069  10773  35262 
1618  7650  20704  11475  39830 
1921  8215  21458  12229  41901 
2224  8621  22067  12838  43527 
2527  9067  24072  13507  46645 
2830  10566  25821  14006  50394 
3133  11125  26101  15404  52630 
3436  12057  27654  16336  56047 
3739  12781  28016  17059  57855 
4042  13632  28442  17485  59558 
4345  15103  29422  19936  64460 
4648  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.
CQ_{x} = 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.
Table 7.9: Cumulative Probability of Pill Use Terminations by Type


Duration of Pill Use  CQ_{xpregnancy}  CQ_{xMedical}  CQ_{xOthers}  CQ_{x} 

13  0.00000  0.00000  0.00000  0.00000 
46  0.02095  0.08939  0.04190  0.15223 
79  0.03568  0.13800  0.05958  0.23325 
1012  0.05434  0.16755  0.07979  0.30169 
1315  0.06420  0.18069  0.10773  0.35262 
1618  0.07650  0.20704  0.11475  0.39830 
1921  0.08215  0.21458  0.12229  0.41901 
2224  0.08621  0.22067  0.12838  0.43527 
2527  0.09067  0.24072  0.13507  0.46645 
2830  0.10566  0.25821  0.14006  0.50394 
3133  0.11125  0.26101  0.15404  0.52630 
3436  0.12057  0.27654  0.16336  0.56047 
3739  0.12781  0.28016  0.17059  0.57855 
4042  0.13632  0.28442  0.17485  0.59558 
4345  0.15103  0.29422  0.19936  0.64460 
4648  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 eventtime 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 typespecific probabilities.
7.2 Assignment
The following information was collected in a demographic survey from evermarried 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 oneyear intervals in Table 7.10.
Table 7.10  
Time Since Marriage (Years)  Women with First Birth in the Year  Women Divorced Before First Birth  Childless Women in Intact Marriage at Survey 

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 oneyear interval. For simplicity assume that all marital disruptions are due to divorce. With this assumption calculate:
 Number in the risk set at the beginning of each interval (N_{x}).
 Adjusted number in the risk set at the beginning of each interval .
 Crude probability of birth and divorce in each interval Q_{xBirth}, Q_{xDivorce}.
 Total probability of either birth or divorces in the interval (d_{xBirth}, d_{xDivorce}).
 Survival probability (probability neither birth or divorce occurring) (q_{x}).
 Assume that l_{0} = 100,000 and complete l_{x} column in the life table.
 Number of births and divorce in each interval in the life table population (d_{xBirth}, Q_{xDivorce)}.
 Cumulative number of births and divorces in each interval (Cd_{xBirth}, Cd_{xDivorce)}.
 Cumulative probabilities of birth and divorce by the beginning of each interval (CQ_{xBirth}, CQ_{xDivorce)}.
Interpret the results.
SEE ANSWERS IN TABLE 7.11 AND 7.12.
Table 7.11 Crude Probabilities of Birth and Divorce  
Time Since Marriage (Years)  Risk Set  Adjusted Risk Set  S(x)  q_{x}  Q_{xdivorce}  Q_{xbirth} 

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 
Table 7.12 Cumulative Probabilities of Birth and Divorce  
Time Since Marriage (Years)  l_{x}  d_{xBirth}  d_{xDivorce}  d_{x}  Cd_{xbirth}  Cd_{x}  Cd_{xDivorce}  CQ_{xbirth}  CQ_{xDivorce}  CQ_{x} 

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: (_{3}q_{1}) = 0.15223; (_{3}p_{1}) = 1 — (_{3}q_{1}) = 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.

2. Variance of EventSpecific Conditional Probabilities (equation )
Standard error is the square root of the variance.
Example In Table 7.6: _{3}Q_{1Pregnancy} = 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. 
For the computation of the variance of the cumulative probabilities it is necessary to calculate the covariance .
Covariance .
Example From Table 7.6: Table 7.13 shows covariance for all other intervals. 
4. Variance of EventSpecific Cumulative Probabilities
Before presenting a general formula for eventspecific 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 CQ_{1Pregnancy} = 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 eventspecific cumulative probability is obtained as:
Variance of [SORRY, IMAGE 468.gif NOT AVAILABE]
Variance
For simplicity we rewrite the above variance with some additional notations:
; B_{0} = 1 and B_{1} = p_{0} (Note that S(1) = p_{0}).
Then the variance equation is expressed as:
Example 2 From Table 7.6: From Table 7.13:
Taking square root yields: = 0.00701 and = 0.03568 
Case 3
The eventspecific 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: B_{0} = 1 B_{1} = S(1) B_{2} = S(2)
Then
Example In Table 7.9: CQ_{3Pregnancy} = 0.05434 Simple calculations gives: A_{0} = 0.039394 A_{1} = 0.02063449 B_{0} = 1 B_{1} = 0.84776 B_{2} = 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 eventspecific 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
B_{1} = S(x) and B_{0} = 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 CQ_{16Pregnancy} = 0.16934
[Note: Denoting interval 13 as zero and interval 46 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 A_{0}, A_{1}, ....A_{14} and B_{0}, B_{1}, ....B_{15}. 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 A_{i} can be calculated.
Table 7.14 Standard Error Calculation of


i  Duration of Pill Use  B_{i}  S(x) Q_{xPregnancy}  P_{i}  A_{i}  

0  13  1.00000  0.0209497207  0.1483947937  0.84777  0.175042294 
1  46  0.84777  0.0147309359  0.1336638578  0.90443  0.147787800 
2  79  0.76675  0.0186631695  0.1150006882  0.91075  0.126270243 
3  1012  0.69831  0.0098585449  0.1051421434  0.92706  0.113414749 
4  1315  0.64738  0.0122975426  0.0928446008  0.92944  0.099892658 
5  1618  0.60170  0.0056497764  0.0871948243  0.96557  0.090303878 
6  1921  0.58099  0.0040628346  0.0831319897  0.97203  0.085524277 
7  2224  0.56473  0.0044554951  0.0786764946  0.94477  0.083275538 
8  2527  0.53355  0.0149942541  0.0636822406  0.92974  0.068494501 
9  2830  0.49606  0.0055894074  0.0580928332  0.95493  0.060834678 
10  3133  0.47370  0.0093187333  0.0487740999  0.92787  0.052565726 
11  3436  0.43953  0.0072351208  0.0415389791  0.95885  0.043321768 
12  3739  0.42145  0.0085140563  0.0330249228  0.95960  0.034415446 
13  4042  0.40442  0.0147060972  0.0183188256  0.87879  0.020845560 
14  4345  0.35540  0.0112824555  0.0070363701  0.92063  0.007642954 
15  4648  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.
Table 7.15: Standard Errors of EventSpecific (Pregnancy Planning)  
Interval Number  Duration of Pill Use  Standard Error  

0  13  0.0000000  
1  46  0.000028646  0.0053522 
2  79  0.000051150  0.0071519 
3  1012  0.000079329  0.0089067 
4  1315  0.000093574  0.0096734 
5  1618  0.000114376  0.0106947 
6  1921  0.000123593  0.0111172 
7  2224  0.000131095  0.0114497 
8  2527  0.000140457  0.0118515 
9  2830  0.000176920  0.0133011 
10  3133  0.000189327  0.0137596 
11  3436  0.000216772  0.0147232 
12  3739  0.000240747  0.0155160 
13  4042  0.000274396  0.0165649 
14  4345  0.000341488  0.0184794 
15  4648  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(CQ_{xMedical}) and other reasons (CQ_{xOther}) in Table 7.9.