Indirect methods, pioneered by Brass and Coale (1968), estimate child mortality from information on aggregate numbers of children ever born and children still alive (or dead) reported by women classified by age group (or alternatively grouped by time since first birth, or marital duration).
Such information is described as a summary birth history (SBH). The amount of detail collected varies, from just two questions (number of children ever born and number of children still alive) to the most detailed SBH asking separately about boys and girls, and enquiring separately about surviving children living at home versus those living elsewhere described in the introduction to child mortality analysis [1] as part of the DHS full birth history sequence of questions. The proportion dead of children born to women by age (or time since first birth, or duration of marriage) reflects the level of child mortality, but is also affected by other things, primarily the age pattern of childbearing and the age pattern of child mortality. Young mothers generally have young children, who have been exposed to the risk of death for short, recent periods; the proportion dead for such mothers thus reflects child mortality risks to an early age. Older mothers, in contrast, have a mix of young and older children exposed to the risk of dying for longer periods on average further in the past. Through models of fertility and child mortality, the proportions dead are converted into probabilities of dying by exact ages of childhood, _{n}q_{0}. The older the women, the greater the value of n.
If mortality has changed over time, the estimated probabilities of dying reflect the mortality rates that have prevailed at a range of ages and dates. Fortunately, a `time location’ method has been developed that estimates how many years previously each proportion dead approximates period probabilities of dying. These intervals increase with the age of respondents. Thus, if the probabilities of dying estimated from the reports of different age groups of woman are translated into a common index of mortality, these statistics will refer to different dates and can be used to infer the broad trend in mortality over time.
Data quality assessment for an SBH involves a subset of analyses described under the section on direct estimation of child mortality [3] for the analysis of a full birth history (FBH). Since an SBH contains no information on dates of individual births, assessment is limited to analysis of aggregates, tabulated by age group of mother (or duration of marriage or time since first birth, if such information is available); singleyear tabulations can also be revealing if there is substantial age heaping. Once again, assessments are carried out of internal plausibility within the data set itself, and of external consistency with other data sets for the population. Of internal assessments, a first check should be of the average number of children ever born [4] (CEB) in each group of women. (Note that the appropriate denominator for these calculations is all women, not the number of mothers or number of evermarried women). Unless fertility has been rising, the average number of CEB should increase with fiveyear group. A second check should be of the average number of children dead (CD) by each fiveyear group. Unless child mortality or fertility has been increasing, the average number of CD should also increase with age. If information is available by sex of children ever born, sex ratios of births should be calculated. As with the FBH, any tendency for such ratios to deviate far from 100 to 106 males per 100 females, or to rise with age (or marital duration or time since first birth) should be interpreted as warning signs unless the population is known to practice sexselective abortion. Among external checks, the cohort comparisons of CEB and CD described for the FBH in the section on direct estimation of child mortality [3] are often revealing.
For each fiveyear group (x, x + 5) of women, the proportions of children dead are calculated by dividing the number of children dead by the number ever born.
For each fiveyear group of women, divide the reported number of children born _{5}CEB_{x} by the number of women _{5}N_{x} in the group. Note that if the classifying variable being used is age, the denominator should be all women, regardless of marital status or childbearing history.
The age pattern of child mortality has an important bearing on the translation of a proportion dead into a standard _{n}q_{0}, and on the translation of that _{n}q_{0} into a common index such as U5MR. In a population for which fairly recent FBH data exist, the model can be selected on the basis of the FBH by plotting estimates of _{4}q_{1} against those of _{1}q_{0} on a graph showing the corresponding relationships in CoaleDemeny and United Nations model life tables. If no appropriate FBH estimates exist, the selection of a model can be based on child mortality patterns in a neighbouring population. A perfect fit of the data to any model is unlikely. The analyst should select the model that best represents the range of observations available. The same model family should be used in steps 4, 5 and 6. Steps 4 and 5 have different variants, depending on whether the data being analyzed are classified by age of mother (stream a), time since first birth of mother (stream b), or duration of marriage of mother (stream c).
This substep is only required for analysis using one of the United Nations model life tables for developing Countries [6]. It is calculated from agespecific fertility rates _{5}f_{x} as follows:
The (x+2) term in the numerator represents the midpoint of the age group x to x+5 at the time when the births occurred. This assumes that agespecific fertility rates are calculated from information on births in the year before the survey classified by age of woman at the time of the survey (see the section on assessment of recent fertility data [7] for more details). If agespecific rates are calculated from registered births by age of mother at the birth, the term would be (x+2.5). Note that it is only the relative pattern of fertility by age that determines the mean, so there is no need to adjust level by, for example, a relational Gompertz model [8], before calculating it.
Once a model life table family j has been identified, appropriate parameters a(x,j), b(x,j) and c(x,j) (and d(x,j) if a UN model life table [9] is being used) are substituted into the following equation:
Note that d(x,j) is zero unless using the UN Model Life Tables.
Table 1: Values of a(x,j), b(x,j), c(x,j) and optional d(x,j) for estimating probabilities of dying by exact ages of childhood from proportions dead of children ever born classified by age of mother


Age group of mother and value of n in _{n}q_{0}  
Family j 
Coefficient 
1519 
2024 
2529 
3034 
3539 
4044 
4549 
Value of n in _{n}q_{0} 
1 
2 
3 
5 
10 
15 
20  
Princeton 'North' 
a(x,j) 
1.1119 
1.2390 
1.1884 
1.2046 
1.2586 
1.2240 
1.1772 
b(x,j) 
2.9287 
0.6865 
0.0421 
0.3037 
0.4236 
0.4222 
0.3486  
c(x,j) 
0.8507 
0.2745 
0.5156 
0.5656 
0.5898 
0.5456 
0.4624  
Princeton 'South' 
a(x,j) 
1.0819 
1.2846 
1.2223 
1.1905 
1.1911 
1.1564 
1.1307 
b(x,j) 
3.0005 
0.6181 
0.0851 
0.2631 
0.3152 
0.3017 
0.2596  
c(x,j) 
0.8689 
0.3024 
0.4704 
0.4487 
0.4291 
0.3958 
0.3538  
Princeton 'East' 
a(x,j) 
1.1461 
1.2231 
1.1593 
1.1404 
1.1540 
1.1336 
1.1201 
b(x,j) 
2.2536 
0.4301 
0.0581 
0.1991 
0.2511 
0.2556 
0.2362  
c(x,j) 
0.6259 
0.2245 
0.3479 
0.3487 
0.3506 
0.3428 
0.3268  
Princeton 'West' 
a(x,j) 
1.1415 
1.2563 
1.1851 
1.1720 
1.1865 
1.1746 
1.1639 
b(x,j) 
2.7070 
0.5381 
0.0633 
0.2341 
0.3080 
0.3314 
0.3190  
c(x,j) 
0.7663 
0.2637 
0.4177 
0.4272 
0.4452 
0.4537 
0.4435  
United Nations 'Latin America' 
a(x,j) 
0.6892 
1.3625 
1.0877 
0.7500 
0.5605 
0.5024 
0.5326 
b(x,j) 
1.6937 
0.3778 
0.0197 
0.0532 
0.0222 
0.0028 
0.0052  
c(x,j) 
0.6464 
0.2892 
0.2986 
0.1106 
0.0170 
0.0048 
0.0256  
d(x,j) 
0.0106 
0.0041 
0.0024 
0.0115 
0.0171 
0.0180 
0.0168  
United Nations 'Chilean' 
a(x,j) 
0.8274 
1.3129 
1.0632 
0.8236 
0.6895 
0.6098 
0.5615 
b(x,j) 
1.5854 
0.2457 
0.0196 
0.0293 
0.0068 
0.0014 
0.0040  
c(x,j) 
0.5949 
0.2329 
0.1996 
0.0684 
0.0032 
0.0166 
0.0073  
d(x,j) 
0.0097 
0.0031 
0.0021 
0.0081 
0.0119 
0.0141 
0.0159  
United Nations 'South Asian' 
a(x,j) 
0.6749 
1.3716 
1.0899 
0.7694 
0.6156 
0.6077 
0.6952 
b(x,j) 
1.7580 
0.3652 
0.0299 
0.0548 
0.0231 
0.0040 
0.0018  
c(x,j) 
0.6805 
0.2966 
0.2887 
0.0934 
0.0298 
0.0573 
0.0306  
d(x,j) 
0.0109 
0.0041 
0.0024 
0.0108 
0.0149 
0.0141 
0.0109  
United Nations 'Far Eastern' 
a(x,j) 
0.7194 
1.2671 
1.0668 
0.7833 
0.5765 
0.4115 
0.3071 
b(x,j) 
1.3143 
0.2996 
0.0017 
0.0307 
0.0068 
0.0014 
0.0111  
c(x,j) 
0.5432 
0.2105 
0.2424 
0.1103 
0.0202 
0.0083 
0.0129  
d(x,j) 
0.0093 
0.0029 
0.0019 
0.0098 
0.0165 
0.0213 
0.0251  
United Nations 'General' 
a(x,j) 
0.7210 
1.3115 
1.0768 
0.7682 
0.5769 
0.4845 
0.4760 
b(x,j) 
1.4686 
0.3360 
0.0109 
0.0439 
0.0176 
0.0034 
0.0071  
c(x,j) 
0.5746 
0.2475 
0.2695 
0.1090 
0.0038 
0.0036 
0.0246  
d(x,j) 
0.0095 
0.0034 
0.0021 
0.0105 
0.0165 
0.0187 
0.0189  
Sources: Princeton models: UN Population Division (1983); UN Models: UN Population Division (1991) 
For each age group (x,x + 5), _{n}q_{0} is estimated by multiplying the righthand side of the equation by the empiricallyobserved _{5}PD_{x}.
Using the model life table family j identified as being appropriate, parameters e(x,j), f(x,j) and g(x,j) are substituted into the following equation:
The location of the estimates in calendar time is readily achieved by subtracting the t(x) from the census or survey date (Table 2).
Table 2: Values of e(x,j), f(x,j) and g(x,j) for estimating the reference period t(x) for probabilities of dying by exact ages of childhood from proportions dead of children ever born classified by age of mother


Age group of mother and value of n in _{n}q_{x}  
Family j 
Coefficient 
1519 
2024 
2529 
3034 
3539 
4044 
4549 
Value of n in _{n}q_{0} 
1 
2 
3 
5 
10 
15 
20  
Princeton 'North' 
e(x,j) 
1.0921 
1.3207 
1.5996 
2.0779 
2.7705 
4.1520 
6.9650 
f(x,j) 
5.4732 
5.3751 
2.6268 
1.7908 
7.3403 
12.2448 
13.9160  
g(x,j) 
1.9672 
0.2133 
4.3701 
9.4126 
14.9352 
19.2349 
19.9542  
Princeton 'South' 
e(x,j) 
1.0900 
1.3079 
1.5173 
1.9399 
2.6157 
4.0794 
7.1796 
f(x,j) 
5.4443 
5.5568 
2.6755 
2.2739 
8.4819 
13.8308 
15.3880  
g(x,j) 
1.9721 
0.2021 
4.7471 
10.3876 
16.5153 
21.1866 
21.7892  
Princeton 'East' 
e(x,j) 
1.0959 
1.2921 
1.5021 
1.9347 
2.6197 
4.1317 
7.3657 
f(x,j) 
5.5864 
5.5897 
2.4692 
2.6419 
8.9693 
14.3550 
15.8083  
g(x,j) 
1.9949 
0.3631 
5.0927 
10.8533 
17.0981 
21.8247 
22.3005  
Princeton 'West' 
e(x,j) 
1.0970 
1.3062 
1.5305 
1.9991 
2.7632 
4.3468 
7.5242 
f(x,j) 
5.5628 
5.5677 
2.5528 
2.4261 
8.4065 
13.2436 
14.2013  
g(x,j) 
1.9956 
0.2962 
4.8962 
10.4282 
16.1787 
20.1990 
20.0162  
United Nations 'Latin America' 
e(x,j) 
1.1703 
1.6955 
1.8296 
2.1783 
2.8836 
4.4580 
6.9351 
f(x,j) 
0.5129 
4.1320 
2.9020 
2.5688 
10.3282 
17.1809 
19.3871  
g(x,j) 
0.3850 
0.1635 
3.4707 
9.0883 
15.4301 
20.4296 
23.4007  
United Nations 'Chilean' 
e(x,j) 
1.3092 
1.6897 
1.8368 
2.2036 
2.9955 
4.7734 
7.4495 
f(x,j) 
1.9474 
4.6176 
2.6370 
3.3520 
11.4013 
17.8850 
19.0513  
g(x,j) 
0.7982 
0.0173 
4.0305 
9.9233 
16.3441 
20.8883 
23.0529  
United Nations 'South Asian' 
e(x,j) 
1.1922 
1.7173 
1.8631 
2.1808 
2.7654 
4.1378 
6.4885 
f(x,j) 
0.7940 
4.3117 
2.8767 
2.7219 
10.8808 
18.6219 
22.2001  
g(x,j) 
0.5425 
0.1653 
3.5848 
9.3705 
16.2255 
22.2390 
26.4911  
United Nations 'Far Eastern' 
e(x,j) 
1.2779 
1.7471 
1.9107 
2.3172 
3.2087 
5.1141 
7.6383 
f(x,j) 
1.5714 
4.2638 
2.7285 
2.6259 
9.8891 
15.3263 
15.5739  
g(x,j) 
0.6994 
0.0752 
3.5881 
9.0238 
14.7339 
18.2507 
19.7669  
United Nations 'General' 
e(x,j) 
1.2136 
1.7025 
1.8360 
2.1882 
2.9682 
4.6526 
7.1425 
f(x,j) 
0.9740 
4.1569 
2.8632 
2.6521 
10.3053 
16.6920 
18.3021  
g(x,j) 
0.5247 
0.1232 
3.5220 
9.1961 
15.3161 
19.8534 
22.4168  
Sources: Princeton models: UN Population Division (1983); UN Models: UN Population Division (1991) 
Once a model life table family j has been identified, appropriate parameters a(x,j), b(x,j) and c(x,j) are substituted into the following equation:
Table 3: Values of a(x,j), b(x,j), c(x,j) for estimating probabilities of dying by exact ages of childhood from proportions dead of children ever born classified by time since first birth of mother


Time since first birth of mother and value of n in _{n}q_{0}  
Family j 
Coefficient 
04 
59 
1014 
1519 
2024 
Value of n in _{n}q_{0} 
2 
5 
5 
5 
10  
Princeton 'North' 
a(x,j) 
1.1980 
1.2248 
1.2076 
1.2030 
1.3292 
b(x,j) 
0.1266 
0.1919 
0.0105 
0.0896 
0.1598  
c(x,j) 
0.0038 
0.0870 
0.2911 
0.4265 
0.5778  
Princeton 'South' 
a(x,j) 
1.1705 
1.3166 
1.2952 
1.2836 
1.5269 
b(x,j) 
0.1461 
0.3157 
0.0423 
0.1308 
0.2659  
c(x,j) 
0.0051 
0.0971 
0.4295 
0.6496 
0.9174  
Princeton 'East' 
a(x,j) 
1.2182 
1.2769 
1.2731 
1.2585 
1.3410 
b(x,j) 
0.1809 
0.2268 
0.0005 
0.1216 
0.1749  
c(x,j) 
0.0214 
0.1052 
0.3720 
0.5013 
0.5964  
Princeton 'West' 
a(x,j) 
1.2049 
1.2573 
1.2431 
1.2469 
1.4258 
b(x,j) 
0.1553 
0.2266 
0.0230 
0.0999 
0.1948  
c(x,j) 
0.0135 
0.0944 
0.3409 
0.5267 
0.7454 
Note: the coefficients and values of n in _{n}q_{0} have been updated by Hill from those published in Hill and Figueroa (2001).
For each age group (x,x + 5), _{n}q_{0} is estimated by multiplying the righthand side of the equation by the empiricallyobserved _{5}PD_{x }(Table 3).
Using the model life table family j identified as being appropriate, parameters e(x,j), f(x,j) and g(x,j) are substituted into the following equation:
Table 4: Values of e(x,j), f(x,j) and g(x,j) for estimating the reference period t(x) for probabilities of dying by exact ages of childhood from proportions dead of children ever born classified by time since first birth of mother


Time since first birth of mother and value of n in _{n}q_{0}  
Family j 
Coefficient 
04 
59 
1014 
1519 
2024 
Value of n in _{n}q_{0} 
2 
5 
5 
5 
10  
Princeton 'North' 
e(x,j) 
1.71 
2.16 
0.66 
1.96 
3.85 
f(x,j) 
1.07 
4.36 
3.50 
0.90 
6.42  
g(x,j) 
0.35 
0.12 
6.65 
17.66 
28.94  
Princeton 'South' 
e(x,j) 
1.68 
2.29 
1.19 
1.01 
2.68 
f(x,j) 
0.96 
3.84 
3.45 
0.18 
5.06  
g(x,j) 
0.32 
0.01 
5.41 
15.03 
25.21  
Princeton 'East' 
e(x,j) 
1.68 
2.19 
0.71 
1.96 
4.06 
f(x,j) 
0.99 
4.28 
3.63 
0.71 
6.35  
g(x,j) 
0.33 
0.02 
6.36 
17.42 
29.14  
Princeton 'West' 
e(x,j) 
1.70 
2.20 
0.86 
1.46 
2.97 
f(x,j) 
1.03 
4.20 
3.47 
0.69 
5.80  
g(x,j) 
0.34 
0.06 
6.21 
16.49 
26.65 
Note: the coefficients and values of n in _{n}q_{0} have been updated by Hill from those published in Hill and Figueroa (2001).
Once a model life table family j has been identified, appropriate parameters a(x,j), b(x,j) and c(x,j) are substituted into the following equation:
Table 5: Values of a(x,j), b(x,j), c(x,j) for estimating probabilities of dying by exact ages of childhood from proportions dead of children ever born classified by duration of marriage of mother


Duration of marriage of mother and value of n in _{n}q_{0}  
Family j 
Coefficient 
04 
59 
1014 
1519 
2024 
2529 
Value of n in _{n}q_{0} 
2 
3 
5 
10 
15 
20  
Princeton 'North' 
a(x,j) 
1.2615 
1.1957 
1.3067 
1.4701 
1.5039 
1.4798 
b(x,j) 
0.5340 
0.4103 
0.0103 
0.1763 
0.0039 
0.2487  
c(x,j) 
0.1252 
0.0930 
0.4618 
0.7268 
0.7071 
0.5582  
Princeton 'South' 
a(x,j) 
1.3103 
1.2309 
1.2774 
1.3493 
1.3592 
1.3532 
b(x,j) 
0.5856 
0.3463 
0.0336 
0.1366 
0.0315 
0.1978  
c(x,j) 
0.1367 
0.1073 
0.3987 
0.5403 
0.4944 
0.4099  
Princeton 'East' 
a(x,j) 
1.2299 
1.1611 
1.2036 
1.2773 
1.3014 
1.3160 
b(x,j) 
0.3998 
0.2451 
0.0171 
0.1015 
0.0219 
0.1630  
c(x,j) 
0.0910 
0.0797 
0.2992 
0.4276 
0.4195 
0.3751  
Princeton 'West' 
a(x,j) 
1.2584 
1.1841 
1.2446 
1.3353 
1.3875 
1.4227 
b(x,j) 
0.4683 
0.3006 
0.0131 
0.1157 
0.0193 
0.1954  
c(x,j) 
0.1080 
0.0892 
0.3555 
0.5245 
0.5472 
0.5127 
For each age group (x,x+5), _{n}q_{0} is estimated by multiplying the righthand side of the equation by the empiricallyobserved _{5}PD_{x}.
Using the model life table family j identified as being appropriate, parameters e(x,j), f(x,j) and g(x,j) are substituted into the following equation:
The results are presented in Table 6.
Table 6: Values of e(x,j), f(x,j) and g(x,j) for estimating reference period t(x) for probabilities of dying by exact ages of childhood from proportions dead of children ever born classified by duration of marriage of mother


Duration of marriage of mother and value of n in _{n}q_{0}  
Family j 
Coefficient 
04 
59 
1014 
1519 
2024 
2529 
Value of n in _{n}q_{0} 
2 
3 
5 
10 
15 
20  
Princeton 'North' 
e(x,j) 
1.03 
1.70 
1.43 
0.08 
1.97 
2.19 
f(x,j) 
1.31 
4.21 
3.27 
1.08 
3.48 
0.61  
g(x,j) 
0.33 
0.02 
4.41 
12.93 
21.33 
23.94  
Princeton 'South' 
e(x,j) 
1.02 
1.66 
1.21 
0.65 
2.91 
3.16 
f(x,j) 
1.31 
4.51 
3.47 
1.60 
4.14 
1.21  
g(x,j) 
0.33 
0.03 
4.95 
14.68 
24.01 
26.35  
Princeton 'East' 
e(x,j) 
1.04 
1.64 
1.11 
0.86 
3.22 
3.39 
f(x,j) 
1.42 
4.70 
3.30 
1.97 
4.11 
1.67  
g(x,j) 
0.35 
0.06 
5.45 
15.52 
24.86 
26.98  
Princeton 'West' 
e(x,j) 
1.03 
1.67 
1.21 
0.54 
2.47 
2.21 
f(x,j) 
1.37 
4.59 
3.33 
1.77 
3.92 
1.31  
g(x,j) 
0.34 
0.02 
5.14 
14.64 
23.10 
24.45  
In the applications of the indirect child mortality estimation method presented here, each of the probabilities of dying by exact ages of childhood, _{n}q_{0}, is converted into a value of α, the level parameter of a system of relational logit model life tables. The α is then used to estimate the corresponding probability of dying between birth and exact age 5, _{5}q_{0}:
where the estimates of _{n}q_{0} come from Step 4 and the Y^{s}(n) values are logit transformations of the standard life table. Thus, one obtains a series of values of α corresponding to the probabilities of dying estimated from data on the different age groups of respondents. Then for each α:
To apply the relational model approach, it is necessary to choose a standard life table. In order to apply the indirect estimation procedure, it is necessary for Steps 4 and 5 to identify an appropriate model pattern, and the standard should be drawn from the same model family. The precise level of mortality within the family is less important than the family itself (the appropriate selection of which allows β to be assumed to be 1 in the relational logit model life table). We therefore recommend choosing a standard with a life expectancy of 60 years.
The resulting estimates of _{n}q_{0} for each age group, the corresponding estimates of _{5}q_{0}, the time location estimates, and the time trend in _{5}q_{0} must then be assessed. Plotted against time, the series of values of _{5}q_{0} will give an indication of the time trend in levels of child mortality. If data from more than one census or survey are available, estimates of _{5}q_{0} can be compared for the same time periods to evaluate the consistency and reliability of the data.
The example uses data on children ever born and children surviving by age of mother from the 2008 Census of Malawi. The method is implemented in the associated Excel workbook [10].
Table 7 shows the basic data on number of women, number of children ever born, and number of children surviving by fiveyear age group of mother from the 2008 Malawi Census. The proportion dead of children ever born _{5}PD_{x} is calculated by dividing the number of surviving children by the number of children ever born, and subtracting the result from 1:
The results are shown in the fifth and sixth columns of Table 7.
Table 7: Children ever born and children surviving, Malawi, 2008 Census
Age group x,x+4 
Number of women 
Children ever born 
Children surviving 
Mean children ever born 
Mean children surviving  Proportion dead, _{5}PD_{x} 
1519 
635,927 
180,178 
161,541 
0.2833 
0.2540  0.1034 
2024 
678,071 
1,038,556 
919,584 
1.5316 
1.3562  0.1145 
2529 
566,350 
1,613,374 
1,398,776 
2.8487 
2.4698  0.1330 
3034 
405,602 
1,697,566 
1,426,516 
4.1853 
3.5170  0.1597 
3539 
298,004 
1,553,676 
1,266,514 
5.2136 
4.2500  0.1848 
4044 
221,274 
1,335,242 
1,043,357 
6.0343 
4.7152  0.2186 
4549 
174,875 
1,128,423 
851,048 
6.4527 
4.8666  0.2458 
Although the average number of children born to women in each fiveyear age group, _{5}P_{x}, is only needed for the age groups 1519, 2024 and 2529, it is recommended to calculate these values for all age groups as part of data evaluation. The average is calculated simply by dividing children ever born by the number of women in the age group:
The results are shown in the sixth column of Table 7. The required parity ratios are then calculated as follows:
and
A single set of proportions dead by age of mother contains essentially no information about the age pattern of child mortality in a population. However, in an actual country analysis, there would invariably be some other relevant information to guide a choice. The ideal information is the age pattern of child mortality from a full birth history for the same population. The comparison is of _{1}q_{0} and _{4}q_{1}, and it is generally made graphically, superimposing observed points over curves showing the relationship at different mortality levels in each model family.
Several full birth history surveys have been conducted in Malawi. Figure 1 plots direct estimates for the 0 to 4 years before the DHS surveys of 1992, 2000 and 2004 (and also the 5 to 9 years before the 2000 survey) against model patterns. (Note that only three of the United Nations patterns are shown. This is because the General, South Asian, and Far Eastern patterns are indistinguishable in their age pattern of mortality under age 5.) All the observations show higher _{4}q_{1} relative to _{1}q_{0} than any of the model patterns. The optimal model choice in this instance would probably be the Princeton ‘North’ family, and it is this model that we use in what follows.
We are using the Princeton 'North' family of model life tables, but for illustrative purposes, we estimate the mean age of the fertility schedule. The Malawi 2008 Census included a question for women of reproductive age on how many births they had in the year before the census. Table 8 shows the basic data, and the calculation of
.
Table 8: Calculation of mean age of childbearing
Age group x,x+4 
Number of women 
Births in previous 12 months 
Agespecific fertility rates, _{5}f_{x} 
Midpoint of age group (x+2) 
_{5}f_{x}.(x+2) 
1519 
635,927 
70,737 
0.1112 
17 
1.891 
2024 
678,071 
169,406 
0.2498 
22 
5.496 
2529 
566,350 
130,331 
0.2301 
27 
6.213 
3034 
405,602 
79,232 
0.1953 
32 
6.251 
3539 
298,004 
43,747 
0.1468 
37 
5.432 
4044 
221,274 
13,956 
0.0721 
42 
3.029 
4549 
174,875 
5,599 
0.0320 
47 
1.505 
Sum 

1.0375 
29.817 
is then calculated as the sum of column (vi) divided by the sum of column (iv), = 29.817/1.0375 = 28.74.
Each _{5}PD_{x} is then converted into an estimated _{n}q_{0 }using the appropriate coefficients from Table 3, as shown in Table 9. Thus for the age group 3034,
Table 9: Estimation of _{n}q_{0} from each _{5}PD_{x}
Age Group 
Proportion dead of CEB 
Regression coefficients for _{n}q_{0} (Princeton 'North' Model) 
_{n}q_{0}  
a(i) 
b(i) 
c(i) 

1519 
0.1034 
1.1119 
2.9287 
0.8507 
0.1063 
2024 
0.1146 
1.2390 
0.6865 
0.2745 
0.1105 
2529 
0.1330 
1.1884 
0.0421 
0.5156 
0.1222 
3034 
0.1597 
1.2046 
0.3037 
0.5656 
0.1528 
3539 
0.1848 
1.2586 
0.4236 
0.5898 
0.1885 
4044 
0.2186 
1.2240 
0.4222 
0.5456 
0.2205 
4549 
0.2458 
1.1772 
0.3486 
0.4624 
0.2441 
The time reference t(x) of each estimate before the survey or census is then obtained, using the appropriate coefficients from Table 4, as shown in Table 10. Thus for the age group 3034,
The Census was taken between 8 and 28 June 2008, so the estimated reference date can be found by subtracting t from 2008.46 (the decimal year representation of 18 June 2008). Results are shown in the last column of Table 10.
Table 10: Estimation of time reference t(x) of each estimate in years before the census, Malawi, 2008 Census
Regression coefficients for time ago Princeton 'North' model 
Time ago t 
Reference date  
Age group 
e(i) 
f(i) 
g(i) 
(2008.46t)  
1519 
1.0921 
5.4732 
1.9672 
1.05 
2007.42 
2024 
1.3207 
5.3751 
0.2123 
2.43 
2006.03 
2529 
1.5996 
2.6268 
4.3701 
4.44 
2004.03 
3034 
2.0779 
1.7908 
9.4126 
6.81 
2001.66 
3539 
2.7705 
7.3403 
14.9352 
9.44 
1999.02 
4044 
4.1520 
12.2448 
19.2349 
12.23 
1996.24 
4549 
6.9650 
13.9160 
19.9542 
15.12 
1993.35 
The final step is to convert each estimated _{n}q_{0} into an estimate of the common index
. This will make it possible to compare estimates across age groups. Each _{n}q_{0} is converted into its logit Y(n) by means of the identity Y(n)=0.5.ln(_{n}q_{0}/(1_{n}q_{0})). The value of α is then found by subtracting the standard logit Y^{s}(n), from the North jointsex model life tables with an expectation of life at birth of 60 years [12] (column (vi)) from Y(n). Each α is then used with the standard Y^{s}(5) to get the estimated
. Thus for the age group 2529, Y(3) = 0.5.ln(0.1222/(10.1222)) = 0.9857, and α = 0.9857  (1.1664) = 0.1806. Then,
Estimates of
are derived in analogous fashion using the standard logit for age 1.
Table 11: Estimating the logit life table parameter α for each estimate, and deriving a set of
and
Age group 
_{n}q_{0} 
n 
logit Y(n) 
Standard logit Y^{s}(n) 
α 
$${\text{\hspace{0.17em}}}_{1}{\widehat{q}}_{0}\text{\hspace{0.17em}}$$ 
$${\text{\hspace{0.17em}}}_{5}{\widehat{q}}_{0}\text{\hspace{0.17em}}$$ 
1519 
0.1063 
1 
1.0647 
1.3300 
0.2653 
0.1063 
0.1612 
2024 
0.1105 
2 
1.0431 
1.2273 
0.1842 
0.0918 
0.1405 
2529 
0.1222 
3 
0.9857 
1.1664 
0.1806 
0.0912 
0.1396 
3034 
0.1528 
5 
0.8566 
1.0900 
0.2334 
0.1004 
0.1528 
3539 
0.1885 
10 
0.7299 
1.0091 
0.2791 
0.1089 
0.1650 
4044 
0.2205 
15 
0.6313 
0.9664 
0.3350 
0.1203 
0.1809 
4549 
0.2441 
20 
0.5652 
0.9138 
0.3487 
0.1232 
0.1850 
Figure 2 plots each estimate of
and
against the corresponding reference date. The figure indicates a declining trend in underfive mortality over time, from around per 1000 in the early 1990s to around 140 in 2005. The apparent uptick in child mortality in 2007 is to be disregarded owing to the likely exaggeration of mortality estimated from very young mothers, as discussed above.
Regardless of how data have been collected, or of one’s knowledge of how thoroughly interviewers were trained and supervised, careful data quality review is an essential first step of any analysis. All data sets contain errors, which can result from many sources, such as an interviewer cutting corners or an interviewee simply not knowing the correct answer to a question.
One specific aspect of summary birth history data needs to be emphasized for the data checking, and that is the failure of assumption 2, that mortality risks of children for a particular time period do not vary by age of mother. In many applications it is abundantly clear that this assumption does not hold. The risks for children of women aged 1519 (and the indirect estimate of child mortality based on CEB and CD for this age group) are frequently higher, sometimes very substantially so, than the population average. The same is true to a lesser extent for the children of mothers aged 2024. Two factors account for this distortion: the distribution of children by birth order and socioeconomic factors. First births are known to be at higher risk of dying than higherorder births, and the children born to younger women clearly consist of an aboveaverage proportion of first births. Women having children at early ages also tend to come from belowaverage socioeconomic groups, and their children are thus exposed to aboveaverage mortality risks. Consequently the estimates of mortality derived from women aged 1519 should be treated with a high degree of circumspection, or ignored altogether.
Two key characteristics of this method need to be borne in mind when interpreting results. First, there is no information about specific dates or ages in the basic data. The only timing information in the number of children ever borne by a woman is that those births occurred at some point between when she had her first birth and her age at the time of the survey. Even less information is available from the number of those children that have died about ages at death, since possible age ranges depend on time distributions of births. It is thus impossible to draw conclusions about shortterm fluctuations in child mortality from a SBH. Reports of two women of the same age (duration of marriage, time since first birth) reporting the same numbers of children ever born and died can reflect experiences of different mortality conditions. The best that a summary birth history can offer is a broad indication of an average past trend. Even this average trend needs to be interpreted with care for the recent past because of the selection biases affecting reports of women aged 1519 and, though to a lesser extent, 2024.
The second characteristic is that information is provided only for surviving women who still live in surveyed households, with an associated risk of respondent selection bias. The mortality experience of children born in a community whose mothers no longer live in the community will not be included in the measures. If such children have higher mortality than those born to mothers who do still live in the community, mortality will be underestimated. The most severe form of this bias is likely to result from substantial levels of HIV prevalence in the community, since such prevalence in the absence of widespread antiretroviral therapy will result in a strong positive correlation between survival of child and survival of mother (described in the section on the effects of HIV on methods for child mortality estimation [2]). However, some positive correlation between mother and child survival is almost certain in any population.
There are also other possible sources of respondent selection bias. For example, high inmigration rates will result in women reporting on the survival of children born and raised elsewhere, while high outmigration will remove responses about children who were born and raised in the community. Though it is impossible to know a priori the direction or magnitude of such biases, the analyst needs to keep in mind their potential effect. Nonresponse itself may actually be a smaller problem for the summary birth history than the full birth history, since information is often collected from third parties, not necessarily from the woman herself. Thus a summary birth history for a woman absent from the community on an extended trip might be collected when a full birth history for the same woman would not be since she could not be interviewed in person.
The idea that proportions of children dead among children ever born were indicators of child mortality has a long history. Questions on children ever born and children surviving were included in the 1900 Census of the United States (Preston and Haines 1991), the 1911 Census of Britain, and the 1940 Census of Brazil, among others. However, the first methodology for translating such proportions into standard life table indicators was proposed by Brass and Coale (1968).
To illustrate the basic idea, take a simple if implausible example of a population in which all women have exactly one child, born at exact age 25, that all the women survive from age 25 to age 30, and that there is no migration. In a survey, the proportion dead among children borne by women of exact age 30 will precisely measure the cohort probability of dying between birth and exact age 5, _{5}q_{0}. In another population, all women also have exactly one child, but at age 27; in this population, the proportion dead will precisely measure the cohort probability of dying by age 3, _{3}q_{0}.
These two examples illustrate a number of important points. First, age of the women is a proxy for the exposure to risk of their children. Other things being equal, the older the mother, the longer on average her children have been exposed to the risk of dying. Second, the interpretation of a proportion dead of children ever born in terms of a standard life table measure depends on the age of childbearing. Third, equivalence of a proportion dead to a life table measure requires that there be no selection effects by mortality or out migration, nor contamination effects by inmigration. Fourth, the measures obtained are for cohorts (or averages across cohorts) rather than for time periods.
Of course in real populations children are born at a range of ages of mother and are exposed to agespecific mortality risks that may change over time. Estimation methods use model age patterns of fertility and child mortality to create model proportions dead of children ever born that can then be related to underlying life table parameters. A common feature of data on survival of children classified by age group of mother is that proportions dead for women aged 1519 and 2024 are higher than for subsequent age groups, despite the fact that they reflect shorter average exposure times of children. The problem arises because young women who have children are generally of below average socioeconomic status, and their children are disproportionately first births, both of which are known risk factors for high child mortality. The experience of the children of such young mothers is therefore not representative of all children born in the population. Partly to address this bias, methods have been developed classifying women by duration of marriage (Sullivan 1972) and time since first birth (Hill and Figueroa 2001). It is also argued that these methods are less affected by fertility change.
The method was originally developed by Brass without explicit consideration of the effects on the estimates of mortality change, though he notes that under conditions of change “the estimates of q(2) and q(3) would be representative of the average mortality for a short period (less than a decade) before the census or survey” (Brass and Coale, 1968: 116). It is clear today that child mortality has been declining globally, and very rapidly in some populations. Following pioneering work by Feeney (1976, 1980), methods were developed to estimate a 'time reference' for the estimate derived from each age/duration group (Coale and Trussell (1977); Palloni and Heligman (1985); Hill and Figueroa (2001)). The proportion dead of CEB for a group of women represents an average of mortality risks across all the birth cohorts of their children. The older the women, or the longer their exposure, the further back in time the cohorts stretch, and the further back the reference time for a child mortality estimate based on the proportion dead. Since child mortality risks are highly concentrated at young ages, the reference date is not in practice very different from the average number of years ago that the births occurred if mortality trends have been fairly stable over time. The exact calendar year reference dates for the child mortality estimates thus have to be treated as central points in time with a distribution of deaths around them. As a result, the child mortality estimates derived from SBHs and variants of the Brass indirect methods cannot be used to identify mortality changes or crises located at a particular point in time. The method provides a good description of general trends in child mortality but smoothed in comparison with the true yeartoyear fluctuations seen in almost every population. Other methods of measurement (such as FBH) are thus needed to estimate the temporal impact of, for example, health interventions on child survival.
The proportion dead of children ever borne by women of exact age x, PD(x), is a birthweighted average of cohort probabilities of dying:
where f(y) is the fertility rate at age y, _{xy}q^{c}_{0 }is the probability of dying by age (xy) for the cohort born (xy) years earlier, and α is the earliest age of childbearing. The expression is exactly the same for duration of marriage and time since first birth, except that α becomes 0. Proportions dead for fiveyear age or duration groups can then be estimated by averaging point PD estimates across the group, weighting for assumed population distributions N_{x} at each x. It is typically assumed that the underlying population can be regarded as stable, with a growth rate appropriate for the demographic parameters underlying the PD(x), and that the underlying life table at adult ages is that used in the calculation of the PD(x). Such calculations use discrete age groups, for example, using single years of age:
Model schedules of f, q and N are used to model the _{5}PD_{x} values, which are then related to appropriate q values by regression analysis using parity ratios as the independent variables (see estimating equations above).
As described above, though not illustrated in detail, variants of the original method have been developed, classifying women by duration of marriage (Sullivan 1972) or time since first birth (Hill and Figueroa 2001). The methods were developed to address two potential sources of error in the agebased method: effects of changing fertility (primarily through distorting the parity ratios) and the aboveaverage mortality risks of children born to young mothers.
While these variants are illustrated below, the two refinements suffer from limitations of their own. First, many population censuses do not routinely collect the information needed to tabulate women and their children by duration of marriage or time since first birth. Second, with regard to the duration of marriage approach, in many developing countries marriage is not a prerequisite for commencement of childbearing. Further, the mortality of children of unmarried mothers may well be higher than that of mothers who are married. Thus distortions may also arise in the results of these variants. Where such data are collected, as in many Arab countries with low proportions of births outside marriage, they can provide important insights into the relative performance of the different approaches.
Some surveys, for instance the MICS (Multiple Indicator Cluster Surveys, conducted under the auspices of UNICEF) carried out before about 2010, have collected summary birth histories and the necessary information on date of marriage and first birth. The question thus arises as to which variant of the methodology should be preferred.
To explore this question, we have used a full birth history survey, the 19992000 Bangladesh Demographic and Health Survey. The data have been tabulated in all three formats, and analysed using the Princeton ‘South’ family of model life tables. Figure 3 shows the estimates of _{5}q_{0} derived from each approach. (It may be noted that this particular data set does not show a strong overestimation of mortality based on the most recent (1519) age group.)
If one ignores in each series the most recent point (1519 for age, 04 for both duration of marriage and time since first birth) the general trends of the estimates are very similar. The marital duration and time since first birth methods show similar levels, but from age group 2529 onwards, the age estimates are always highest, averaging 20 per 1,000 higher (155 per 1,000) than either of the other two methods. Straight averages of all the duration of marriage and time since first birth estimates for all points except the most recent give almost the same result, 133 for marriage duration and 136 from time since first birth, although the estimates span slightly different periods.
There are several points of interest in this application. First, the marriage duration and time since first birth methods were developed in part to try to circumvent the problem of selection bias in the agebased points for women aged 1519 and 2024. In this application, however, these two methods show jumps for the most recent point as large as or larger than that for the agebased estimates. This could be the case if the dominant selection bias is for first births, all (time since first birth) or almost all (marriage duration) of which will be concentrated in the first category, whereas they will be more spread out across age groups.
A second issue is the use of parity ratios as observed at the time of the survey to reflect the time distribution of births in the past. As noted above, if fertility is declining (even if the relative age pattern of fertility is not changing), the parities for the younger women will be lower and those for the older women higher, reducing the parity ratios below the values for any true cohort. This will give the appearance of more recent childbearing (and thus lower average duration of child exposure to the risk of dying) than is really the case, and thus lead to overestimates of child mortality.
Bangladesh experienced a substantial fertility decline from the mid1980s to the time of the 19992000 DHS, so it is interesting to explore the size of this possible bias. Since the data come from a full birth history, we can calculate cohort parities at points 5, 10, 15 etc. years in the past by subtracting recent births from children ever born, and then calculate parity ratios for true cohorts. We can only use the standard methodology (using ratios P(1)/P(2) and P(2)/P(3)) for cohorts that have reached the third age or duration group, thus age groups 2529 and up or time since first birth/marriage duration categories of 1014 and up.
The first panel of Table 12 compares parity ratios at the time of survey and cohort parity ratios. The expected pattern is quite clear for the age method, with the time of survey ratios being clearly below any of the cohort ratios, which are themselves rather stable across cohorts. The picture is less clear for the marital duration and time since first birth methods however: the P(2)/P(3) ratios are generally larger than the time of survey ratios, whereas the P(1)/P(2) ratios are uniformly lower.
Table 12: Parity ratios P(1)/P(2) and P(2)/P(3) calculated at the time of survey versus for true cohorts, Bangladesh 19992000 DHS
Age method  
Parity ratio 
Time of survey 
Cohort aged 2529 
Cohort aged 3034 
Cohort aged 3539 
Cohort aged 4044 
Cohort aged 4549 
P(1)/P(2) 
0.269 
0.342 
0.311 
0.327 
0.361 
0.336 
P(2)/P(3) 
0.544 
0.660 
0.639 
0.603 
0.632 
0.619 
Duration of marriage method  
Parity ratio 
Time of survey 
Cohort 1014 
Cohort 1519 
Cohort 2024 
Cohort 2529 

P(1)/P(2) 
0.353 
0.352 
0.288 
0.274 
0.287 

P(2)/P(3) 
0.635 
0.682 
0.650 
0.598 
0.603 

Time since first birth method  
Parity ratio 
Time of survey 
Cohort 1014 
Cohort 1519 
Cohort 2024 


P(1)/P(2) 
0.553 
0.544 
0.495 
0.505 


P(2)/P(3) 
0.709 
0.775 
0.753 
0.723 


To get a sense of the magnitude of this effect on the estimates, we can use cohort parity ratios for each proportion dead of children ever born (for age groups 2529 and over, and duration groups 1014 and over). The three panels of Figure 4 show the original and cohortbased estimates by reference date, for age groups, marital duration groups and time since first birth groups respectively. The use of cohort ratios reduces the estimates based on age groups 2529 and 3034, but has little effect on the estimates from the other two methods. One possible explanation would be that the fertility decline in Bangladesh over this period was primarily an increase in age at childbearing, with small effects by duration of marriage or time since first birth. The net effect of using cohort ratios is to improve consistency across methods, but at the rather high cost of losing the most recent estimates.
The bottom line is that the agebased method will tend to overestimate child mortality if fertility is changing rapidly, whereas the effects on the other two approaches appear to be small. If the data are available, it is advisable to use one of the other two variants rather than the agebased method, but it may not be worth including an extra question in a census just to get this information.
The indirect estimation of child mortality is discussed in all the classic manuals on indirect estimation (Sloggett, Brass, Eldridge et al. 1994; UN Population Division 1983).
Brass W and AJ Coale. 1968. "Methods of analysis and estimation," in Brass, W, AJ Coale, P Demeny, DF Heisel, et al. (eds). The Demography of Tropical Africa. Princeton NJ: Princeton University Press, pp. 88139.
Brass W, AJ Coale, P Demeny, DF Heisel et al. (eds). 1968. The Demography of Tropical Africa. Princeton NJ: Princeton University Press.
Coale AJ and J Trussell. 1977. "Estimating the time to which Brass estimates apply; Annex I to Preston SH and Palloni A "Finetuning Brasstype mortality estimates with data on ages of surviving children"", Population Bulletin of the United Nations 10:8789.
Collumbien M and A Sloggett. 2001. "Adjustment methods for bias in the indirect childhood mortality estimates," in Zaba, B and J Blacker (eds). Brass Tacks: Essays in Medical Demography. London: Athlone, pp. 2042.
Feeney G. 1976. "Estimating infant mortality rates from child survivorship data by age of mother", Asian and Pacific Census Newsletter 3(2):1216. http://hdl.handle.net/10125/3556 [16].
Feeney G. 1980. "Estimating infant mortality trends from child survivorship data", Population Studies 34(1):109128. doi: http://dx.doi.org/10.1080/00324728.1980.10412839 [17]
Hill K and ME Figueroa. 2001. "Child mortality estimation by time since first birth," in Zaba, B and J Blacker (eds). Brass Tacks: Essays in Medical Demography. London: Athlone, pp. 919.
Palloni A and L Heligman. 1985. "Reestimation of structural parameters to obtain estimates of mortality in developing countries", Population Bulletin of the United Nations 18:1033.
Preston SH and MR Haines. 1991. Fatal Years: Child Mortality in Late Nineteenthcentury America. Princeton, NJ: Princeton University Press.
Sloggett A, W Brass, SM Eldridge, IM Timæus, P Ward and B Zaba. 1994. Estimation of Demographic Parameters from Census Data. Tokyo, Japan: United Nations Statistical Institute for Asia and the Pacific.
Sullivan JM. 1972. "Models for the estimation of the probability of dying between birth and exact ages of early childhood", Population Studies 26(1):7997. doi: http://dx.doi.org/10.1080/00324728.1972.10405204 [18]
UN Population Division. 1983. Manual X: Indirect Techniques for Demographic Estimation. New York: United Nations, Department of Economic and Social Affairs, ST/ESA/SER.A/81. http://www.un.org/esa/population/techcoop/DemEst/manual10/manual10.html [19]
UN Population Division. 1991. Child Mortality in Developing Countries. New York: United Nations, Department of Economic and Social Affairs, ST/ESA/SER.A/123.
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[19] http://www.un.org/esa/population/techcoop/DemEst/manual10/manual10.html
[20] http://demographicestimation.iussp.org/sites/newdemographicestimation.iussp.org/files/sites/demographicestimation.iussp.org/files/CM_Indirect_6.xlsx
[21] http://demographicestimation.iussp.org/sites/newdemographicestimation.iussp.org/files/sites/demographicestimation.iussp.org/files/CM_Indirect_FR.xlsx