1 Introduction
An estimate of the underfive mortality rate (U5MR) over time is crucial for determining effectiveness of public health interventions and for better allocating resources. In countries without vital registration systems that track births and deaths, the data often come from surveys that are typically of one of two types: full birth history (FBH) and summary birth history (SBH). In surveys that collect FBH data, women are asked to recall the birth and, if applicable, death dates for each of their children. In contrast, in surveys that collect SBH data, women are only asked for the total number of children they have had and the number of those children that died. Given the temporal information contained in FBH data, it is relatively straightforward to obtain estimates of child mortality and U5MR using survey or modelbased approaches (Mercer et al., 2015; Pezzulo et al., 2016; Rutstein and Rojas, 2006; Wakefield et al., 2018). Unlike FBH data, SBH data do not directly provide temporal information on when births and deaths occurred, and thus require more specialized methods. Overall, SBH data are easier and faster to collect, and thus widely available. In a recent study of U5MR in Africa, approximately 44% of the surveys contained only SBH data and 90% of the births were from SBH (Golding et al., 2017).
1.1 Existing SBH Methods
A common approach to obtaining U5MR estimates based on SBH data is to use variations of the Brass method (Brass, 1964; Feeney, 1980; Trussell, 1975), which involve model life tables. This approach provides an estimate of U5MR for fiveyear age groups of women and additionally assigns each estimate a reference time in the past using a series of complex formulas. These formulas make use of quantities that are observed in SBH data: the number of children that died, the number of children born, and number of women surveyed along with women’s age. Since this approach does not use a statistical model, it is not immediately clear how to obtain estimates of uncertainty. One proposed approach is to use the jackknife (Pedersen and Liu, 2012). A major shortcoming of the Brass method is that the mortality rates in the most recent time periods are based on the youngest women. Children born to younger mothers tend to have worse outcomes, resulting in bias. Therefore, these estimates are typically excluded from analyses. Alternatively, Brasstype methods using time since first birth or marriage are used instead of age of mother (Hill and Figueroa, 1999; United Nations, 1983). Currently, the United Nations produce national estimates of U5MR that incorporate SBH data via indirect estimates obtained from the time since first birth variant of the Brass method (Alkema et al., 2014; Hill et al., 2012). Hill et al. (2012) summarizes the historical approach of combining the estimates using a loess smoother in time. The bootstrap can be used to obtain uncertainty intervals (Alkema and New, 2012). A more recent approach, proposed by Alkema et al. (2014) and Alkema and New (2014), combines the different sources of data and uses a Bayesian penalized regression spline to model trends over time. Uncertainty in the SBH data is incorporated by using the jackknife, or survey weights if microlevel data is available, and otherwise fix the uncertainty to a predetermined number.
Other methods (Hill et al., 2015; Rajaratnam et al., 2010) that do not rely on the demographic models used in the Brass method have been proposed, but critically do not assume a full probability model. Rajaratnam et al. (2010)
describe a number of methods, including one that uses FBH data to derive empirical distributions of births and deaths prior to the survey and then matches SBH women to the relevant empirical distribution. This provides a yearly estimate of the ratio of children that died to children ever born. The ratio is then related via a logistic regression model to the probability that a child dies within five years, calculated using FBH data. In practice, FBH data from surveys in different countries and time periods are pooled together to build the regression model and empirical distributions.
Hill et al. (2015)propose a birth history imputation approach in which SBH women are matched to FBH women, who are typically available from an earlier survey in that country. Women are matched by age, number of births, and number of deaths. The FBH data are then used to impute births and deaths to the SBH women. This approach gave disappointing results when validating mortality estimates obtained from this method and comparing to estimates computed from a later FBH survey
(Brady and Hill, 2017; Hill et al., 2015). The authors attribute this to incompatibility of the SBH and FBH data, stemming from data quality issues with SBH data.1.2 Malawi Data Available
As our motivating example, we consider retrospective birth history data taken between 2004 and 2015 in Malawi, from five surveys and a census. Specifically, we use FBH data from the 2004 Malawi Demographic and Health Survey (DHS), the 2010 Malawi DHS, the 2015 Malawi DHS, the 2006 Multiple Indicator Cluster Survey (MICS), and the 2013 MICS. We include surveyed women who are aged 15–49 at time of the survey. We use SBH data from the 2008 Malawi Census. From the census, we include women who are aged 25–49 at time of the survey. Excluding women under 25 who provide SBH data is consistent with other analyses that incorporate SBH data (Hill et al., 2012). The DHS and MICS use 2stage stratified cluster sampling designs. The 2004 DHS was stratified by urbanrural classification with certain districts being oversampled. The 2006 MICS was stratified by district. The 2010 DHS, 2013 MICS, and 2015 DHS were stratified by urbanrural classification crossed with district. We use available microdata, which is a 10% sample of the 2008 Census.
We focus on the 9 districts in the Central region: Dedza, Dowa, Kasungu, Lilongwe, Mchinji, Nkhota Kota, Ntcheu, Ntchisi, and Salima. In this region, birth history information is available from 109,713 women (43% of women had SBH information only). The number of clusters, women surveyed, births, and deaths for each of the surveys and census is in Table A.1. Across all surveys and the census, the age of the mother at time of survey, district, and strata are available. The 5 surveys that contain FBH information also provide reported birth and death dates for each child. The census contains only the reported number of births and deaths for each woman. An exploratory analysis, comparing the age of surveyed women, total number of reported births, and ratio of number of deaths to number of births, is shown in Figure A.1.
Weighted estimates (Horvitz and Thompson, 1952)
and 95% confidence intervals (CIs) going back to 1980 were computed for each of the FBH surveys and are shown in Figure
1 (we include 3 districts for illustration, with the remaining 6 districts being given in Figure 1). Overall, underfive mortality is decreasing, but there is significant heterogeneity over districts and time periods.The organization of this paper is as follows. In Section 2 we describe our method, which is based on data augmentation (DA). In Section 3 we describe an alternative approach based on combining estimates from using weighted, also known as direct, estimators on FBH data, and indirect estimates from using the Brass method on SBH data. In Section 4 we show the usefulness of the approaches in a simulation. We apply our approach to the survey and census data from Malawi in Section 5. Finally, we conclude with a discussion in Section 6.
2 Data Augmentation Method
We propose using a DA approach within a Bayesian framework. This is implemented via a Markov chain Monte Carlo (MCMC) algorithm, where each iteration is divided into two major steps. In the first step, the missing birth and death dates (available for FBH data) are introduced as auxiliary variables for the SBH data. In the second step, mortality and fertility parameters are then updated conditional on this imputed FBH data and combined with the existing FBH data.
In modeling SBH data, we will model fertility rates and mortality rates, which we specify as probabilities. The forms for these models are heavily driven by context. We define the fertility rate as the probability a woman gives birth at age and time and denote it as , where contains the covariates at time associated with fertility. The mortality rate, or probability a child dies between age and , is denoted by , where again contains covariates at time associated with mortality.
2.1 Simple Example
To motivate our approach, we first consider a simple scenario where we suppose that fertility and mortality do not change with time. Suppose we survey a woman who is age 18 at the time of the survey and has had two children, one of whom has died. Further suppose that the youngest age she could have given birth was 15 and that it is not possible to have multiple births. Additionally, we will discretize time and work on a yearly scale. Therefore, assuming the woman could have given birth when she was 15, 16, or 17, there are 3 options for when the children were born relative to the woman’s age (i.e., at ages 15 and 16, 15 and 17, or 16 and 17). We also know that one of the children has died. This means there are 6 options for when the children were born and which of the children died (e.g., the child born when the mother was 15 died and the child born at 16 survived). The probabilities of each of these 6 options will depend on the fertility rates and the mortality rates. A visual depiction of these scenarios is given in Figure 2.
Now suppose we knew when the children were born and which of those children died (i.e., we selected one of the 6 scenarios). For the child who died, we can enumerate the options for age of death. In this case, suppose the woman was 15 when the child who died was born. Again considering a yearly time scale, there are 3 scenarios (i.e., the child died between ages 0 and 1, ages 1 and 2, or ages 2 and 3). The probability associated with each of these options is based on the mortality rates (see Figure 2).
We have given an overview of the DA that links SBH information to FBH information for a simple example. This forms the backbone of the DA step of our method, and we now give details of the probability model that is used in this step.
2.2 Step 1: Data Augmentation
Let denote calendar time in years. Consider a woman with births, deaths, and covariates . Let be the age of the woman at time of survey and be the year the survey was completed. Define
to be a vector of length
which contains the (unknown) years of birth of the children. Define to be a vector of length which contains the “death indicators” for the children (i.e., if child dies). Therefore, . It will also be helpful to define to be the age of the woman when she gave birth to child . Note that .The probability of children being born in particular years and surviving or dying by the time of the survey is proportional to the probability of giving birth to the children in those years, observing them to survive through the survey or die at some point prior to the survey, and not having children born in the other years:
(1) 
This probability is conditional on births, deaths, covariates , survey taken in year , a woman of age at time of survey, and functions and , but for notational ease we do not explicitly state this. As expressed, this probability does not allow for multiple births and assumes that births in separate years are independent. However, these assumptions could be relaxed.
Given birth in year , , covariates , and the mortality function , the probability that child dies in a particular year is proportional to the probability of surviving to year and subsequently dying in the next year,
(2) 
These are the necessary probabilities for updating the birth and death years for all SBH women’s children (an example of which is shown in Figure 2). In terms of computation, it is possible to enumerate all options thus computing the denominators for (1) and (2), and sample from the full posterior conditional distribution. This can be efficiently achieved by grouping women together that have the same age, number of births, number of deaths, and covariates so that the desired probabilities need only be computed once. However, computational gains can be made using a MetropolisHastings algorithm. This is especially useful for older women, who tend to have more children. One simple approach is to iterate through all of a woman’s children and sample the birth years onebyone, where on the left hand side represents the proposal probability,
In the data analysis, this independence chain proposal resulted in acceptance ratios .
2.3 Step 2: Parameter Updates
After performing the DA step, the imputed FBH from the SBH is combined with available FBH and the fertility and mortality probabilities are updated. Let be an indicator for birth at woman’s age and year , then .
3 Weighted Estimation with the Brass Method
First, we briefly provide a review of the Brass method as it pertains to obtaining estimates of the U5MR and a description of how to obtain estimates of uncertainty. Lastly, we describe how to combine Brass and weighted estimates and incorporate an adjustment for HIV.
3.1 The Brass Method
Define , and to be, respectively, the total number of children dead, the total number of children born and the proportion that died, to women aged at the time of survey. Then
(3) 
where is the proportion of births to women who are at the time of the survey years prior to the survey and is the probability that a child born years before the survey dies before the survey. The Brass method treats (3) as deterministic, replacing the expectation on the left side with the observed proportion, . By the mean value theorem, there exists an such that,
The key idea of the Brass method is to identify and thus use the observed proportion of children dead to find (Chapter 11 of Preston et al., 2000).
Brass (1964) achieved this by using model life tables and a polynomial fertility model to numerically integrate (3) using 5year age groups of women, . The proportion dead in age group , denoted , is then compared to a curve obtained from model life tables to find the such that . These times, , were then adjusted to whole number of years. For example, correspond approximately to (see Table 4 of Brass, 1964). These correspondences are not exact; therefore, adjustment terms are needed. Ideally, (the proportion of births to women in age group ) would be used. However, with those proportions unknown, fertility information from cohorts of women are used instead. The Brass method makes use of observed parity measures, taken to be the mean number of children born to women in each age group. Comparing parity measures across age groups can provide a sense of the earliness of fertility. Define , , and to be the mean number of children ever born to all women in age groups , , and , respectively. Trussell (1975) developed the model that is commonly used to adjust the observed proportion of children dead to ,
where the set of coefficients, and were estimated via simulation. In fact, 4 sets of coefficients were derived, one for each of the Coale and Demeny (1983) regional model life tables: “North”, “West”, “South”, and “East”.
To acknowledge changing mortality, Coale and Trussell (1977) assume mortality declines linearly. Using (3), the reference time would equal the mean length of time since birth of children. Again using simulation, Coale and Trussell (1977) develop another formula involving parity measures and coefficients, and , to identify a time for which the Brass estimate is most relevant,
Finally, to obtain estimates of , the Brass estimates are converted using numbers based on life tables. To determine which model life table to follow (and thus which set of coefficients to use), we follow the suggestion of Hill (2013) and first obtain direct estimates of and from the 2010 DHS and 2006 MICS (FBH surveys taken near to the time of the census). We compare the direct estimates at the regional level with the 4 Coale–Demeny regional model life tables and select the model life table that the direct estimates most closely follow (see Figure A.4); we use the “North” regional model life table. The coefficients for both formulas and adjustment procedure can be found in the United Nations’ Manual X (United Nations, 1983).
3.2 Estimates of Uncertainty for the Brass Method
Define where are the Brass (indirect) estimates. We assume,
where is the true (unadjusted for HIV) logit U5MR in time period containing the reference time and
is the variance. Note that there are seven estimates, as there are seven fiveyear age groups of women. We will assume they are independently distributed.
To obtain variance estimates when using the Brass Method we use a technique similar to Pedersen and Liu (2012) based on the jackknife. We adapt their approach, which is based on deleting clusters, since cluster level information is not available for the census SBH data, and instead define jackknife samples based on women. For the th sample (where the th woman is removed from the dataset), we compute where is computed using the Brass method on this reduced sample. For total women, we calculate,
where the reference time is computed using the reference time Brass equation and is held fixed during this procedure. Theoretically, there is uncertainty in the reference time, but as with all previous implementations of Brass we do not consider that here.
3.3 Combining Brass and Weighted Estimates
Using FBH data, we can construct weighted (direct) estimates and associated variances for each FBH survey on the 5year period scale. For survey in period , denote the estimate of the logit U5MR as and estimated variance . Again we assume,
We follow Wakefield et al. (2018) and use HIV multiplicative correction factors on the U5MR, obtained using the approach of Walker et al. (2012) to adjust for HIV bias. This correction is needed to adjust for women who died during the HIV epidemic in Malawi and are therefore not included in the survey. These women would tend to have children with worse outcomes, thus mortality would be biased downwards and the aim of the adjustment terms is to correct for this. They are survey specific and the HIV correction factors are depicted in Figure A.3. For an unadjusted estimate and associated variance, we sample 100,000 realizations and transform them using the HIV correction factors, denoted ,
The means and variances of the HIV adjusted samples are computed and denoted and , respectively. The same procedure is used for the Brass estimates and variances.
These corrected estimates and variances are then combined over all surveys and times that fall in period via inverse variance weighting. Therefore, the overall HIVcorrected estimate and HIVcorrected variance are respectively,
4 Simulation Study
We illustrate the gains that can be made by incorporating SBH data using our approach in a simulation study. We hold fertility fixed over time, but set mortality to vary in 5year increments. Fertility probabilities and discrete hazards for each period are presented as horizontal lines in Figures 3 and B.2, respectively. As evident from the figures, fertility was set to be constant over fiveyear age groups of women and closely resembles fertility patterns observed in the 2010 Malawi DHS (National Statistical Office  NSO/Malawi and ICF Macro, 2011). Additionally, we use three distinct discrete hazards for each fiveyear time period with and for roughly following the North regional model life table. Birth histories for a total of 5,000 women aged 15 to 49 were simulated. The distribution of ages roughly followed the 2010 Malawi DHS. We will consider two surveys taken in 2010, with one survey containing FBH information for 1,000 women and the other survey containing SBH information for the remaining 4,000 women. More details on simulating the data are provided in Appendix B.
We take as our model for fertility, , where the probability of birth depends on the mother’s age, via 5 factors,
(4) 
and assign independent priors . For mortality, we define 3 age groups
(5) 
and set where is a temporal random effect that follows a random walk of order 2 (RW2) model that depends on a precision
. We take the penalized complexity prior for precision as a hyperprior for
(Simpson et al., 2017).Several different analyses are considered. First, a complete case analysis is performed (i.e., using the 1,000 women with FBH data). Second, we use our proposed DA approach and include the SBH data on 4,000 women and combine with the FBH data. In both analyses, we use the aforementioned models and use a Hamiltonian Monte Carlo (HMC) algorithm (Neal, 2011) to sample from the posterior during the parameter update step. This approach is explained further for the Malawi example in Appendix D. Finally, we follow Section 3 and combine the Brass method fit on SBH data with estimates from fitting a logistic regression model to the FBH data; that is,
where a separate model is fit for each 5year time period .
Results are depicted in Figures 3 and B.2. Uncertainty from the method that uses Brass is substantially greater in earlier periods. This is because the SBH data only contributes to the 3 most recent time periods. However, this approach tends to produce reasonable estimates in the more recent time periods (diamond points). This is interesting since the Brass method is based on assuming a demographic life model, which may not hold in this simulation. However, since many of the numbers used in the simulation are based on observed fertility and mortality data, this likely explains the good performance of Brass. We note that, as expected, estimates that incorporate the women with SBH data to the FBH data tend to be better (closer to the truth) and have narrower credible intervals than those obtained from the complete case analysis (see Tables B.1 and B.2).
5 Application to Central Malawi
5.1 Model
Based on exploratory analysis and well known differences in fertility across different covariate groups (National Statistical Office  NSO/Malawi and ICF Macro, 2011), we propose the following model for fertility,
(6) 
where is a mother’s age group specific RW2 in (roughly) 5year time periods . We take to be the same as in the simulation (see Equation (4)). However, since we observe women in the available FBH data to have given birth between ages , we adjust accordingly: another group for women aged is added and the oldest age group is truncated at age . Since older women are not observed in earlier time periods, women over age 35 are grouped together for the RW2 model. There are 11 time periods : . We include a fixed effect for each age (ages were grouped together because of many 0 counts). Thus, women in age groups defined by have the same trend in fertilities, but different overall levels of fertility by age. We also include a fixed effect for strata (urban and rural).
For the mortality model, we will assume for simplicity that the probability of death within one year is the same for children ages 5 and older. We propose the following model for mortality,
(7) 
where is a term for HIV bias that varies by year and depends on the year the survey was conducted. The term allows for bias in SBH (census data) by urban/rural. In an exploratory analysis, we found that women living in rural areas in the census reported a higher proportion of their children dead than similarly aged women in the surveys taken either side of that time period (see Figure A.1). When making predictions, we do not include the HIV or SBH bias terms. We include a RW2 in years, , by age group (see Equation (5)). We include fixed effects for strata and district and also for age group with
Therefore, the trend in logit hazards is the same for ages 1–4, but each age has its own level of mortality.
5.2 Computation
In the simulation, we had supposed that women could not give birth in the year of the survey, thus a woman’s fertility history and child’s life trajectory are fully observed up to the time of the survey (see Appendix B). Clearly, this is not a reasonable simplification for applying our method to the Malawi data. We describe the special considerations regarding the year of the survey in Appendix C. Briefly, because we are working on a discrete time scale, it is possible for women to give birth during the year of the survey. We accommodate this and acknowledge the shortened observation time by creating adjustment terms for the fertilities and hazards.
We again implement an HMC algorithm for the parameter update step. Forms for the negative log posteriors and gradients along with hyperpriors are given in Appendix D.
5.3 Results
We fit four different models to the data. First, we computed weighted estimates based only on FBH data. Second, we used the approach described in Section 3 to combine the direct estimates with indirect estimates obtained using the Brass method at the districtlevel on SBH data. Both of these were done on the period (5year time scale) since there were insufficient data to support separate hazards on the yearly time scale. Third, we fitted (6) and (7) to all available FBH data (DHS and MICS). Finally, we used our DA approach to incorporate SBH data and fitted models (6) and (7) to all data. Trace plots for the mortality model are in Figure F.1.
Results for fertility are found in Figures F.2–F.4 and show similar results between FBH only and FBH and SBH analyses. Figure 4 (left and middle panels) compares posterior medians and 95% credible intervals for the U5MR in three districts for illustration (results for the other districts are in Figures F.5–F.6). The posteriors tend to be similar when we add in the census data and use our proposed model. We aggregate to five year time periods in order to compare with the other methods. To do this, we sample births using the fertility samples for our FBH only and FBH and SBH models. We then use them to average the samples of over strata and years within 5year time periods. More details are in Appendix E.
In the right panel of Figure 4 and Figures F.5–F.6, we see that the indirect estimates from Brass contribute to 3 time periods: 1990–1994, 1995–1999, and 2000–2004 and often result in higher . In general, the estimates are similar across the other three methods. Uncertainty is much higher when using the weighted estimates on FBH data as compared to using our proposed model.
To assess the accuracy of the four models, we divide the FBH data into training and holdout data. We split the 2006 MICS and 2010 DHS into two roughly equalsized groups: a training and validation set. The validation set consisted of 331 clusters and the remaining 314 were used in the training set (see Table A.1). We refit the models to the training data (2004 DHS, 2008 Census, 2013 MICS, 2015 DHS, and training sets from the 2006 MICS and 2010 DHS) and assess the accuracy of estimation using a weighted mean squared error (MSE). Denote the estimate of logit of U5MR in district time period for model , . For the weighted and weighted and Brass estimates, we use the asymptotic variance of the estimate and for the Bayesian analyses, the posterior variances. These four estimates and variances are compared with the weighted estimates of the logit of U5MR from the holdout clusters, taken to be the truth. We assess the accuracy by time period over all districts by using a weighted MSE:
where and , with denoting the variance of the weighted estimates in district and time period . This allows us to upweight the MSE in districts and periods where the “truth” is more certain.
Models involving direct estimates tend to perform worse (Table 1 and Figure F.7). Adding SBH via the Brass method or the DA approach in general improves the predictions. In our smoothed model, the variance component of the MSE is lower when SBH data are included; however, the bias tends to be higher. In Appendix F we carry out another comparison based on the percentage absolute relative error (PARE) of the probabilities . The substantive results are changed slightly with the DA procedure producing a PARE of 8.8%, compared to 9.0% for the FBH analysis only, the direct estimate PARE was 10.5% and the direct and indirect combined was 11.1%. Figure F.8 and Table F.1 contain the PARE values by period.
Period  Weighted Estimates  Weighted Estimates  Smoothed Model:  Smoothed Model: 

+ Brass  FBH  FBH + SBH  
1985–1989  3.84  3.83  2.69  4.13 
1990–1994  2.65  2.08  1.86  2.04 
1995–1999  2.92  1.64  1.56  1.78 
2000–2004  2.13  2.62  1.21  0.33 
2005–2009  5.83  5.82  5.10  4.19 
Average  3.07  2.79  2.06  1.99 
6 Discussion
We have presented a novel framework for analyzing SBH data in the context of U5MR estimation. Using our approach, data from a variety of sources that contain birth history information at varying degrees of detail can be analyzed together to produce an estimate of U5MR over time with a measure of uncertainty. Our method falls under the umbrella of data augmentation, where we impute FBH information for the SBH data.
We make some simplifying assumptions since we opt to work on a discrete time scale; however, modifications to our approach can be made. For example, multiple births and having a lag time between births could be accommodated by adjusting equation (1) and the model for fertility. Further, the approach could be extended to a monthly time scale or continuous time. We also define the fertility patterns by the age of the women rather than time since fist birth or marriage; however, both of these variations are possible with our framework.
In our application, we included fixed effects for strata and districts and did not have these vary in time. One future consideration is using information on migration and allowing the district of the children and women to change over time. Other terms, such as a spatial random effect could be included.Additionally, terms for mother’s age, number of births, and if available, sex of the child, which may be informative of U5MR, could also be included. However, producing an overall estimate of U5MR at an administrative level would be more complicated since the estimate would need to be marginalized over the age, birth number, and sex population distributions. For understanding individuallevel associations, including these terms into the hazard model would be useful and straightforward to implement in our approach.
In our simulation, we found that by incorporating SBH data uncertainty is reduced. When we applied our method to actual data, adjustments to our model were made to accommodate differences between the SBH census data and FBH survey data. In general, we assumed that FBH data were the “gold standard” and adapted our model to reflect this. A fixed effect term for SBH data by strata was added into the mortality model to adjust for the proportion of deaths as reported by SBH women being higher in rural areas than expected (compared to FBH women). We also used a portion of the FBH data as holdout data to validate the models. It is not immediately clear what the truth is in this scenario, however. There are well known biases in both types of data, making model assessment difficult (Silva, 2012). For SBHs, women may omit live births since they are not asked to systematically recall every birth. In the DHS and MICS, women are asked detailed questions about all of their children (including those that no longer live with them) to limit this type of bias. In FBHs, women may misreport when births and deaths occurred. Furthermore, additional questions on pregnancy and postnatal care are asked if a woman has had recent births, thus displacement of births or omission of recent births may occur. Special consideration for the data sources and outcome of interest is needed when developing the fertility and hazard models, and this is highly context specific.
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Appendix A Descriptives for Malawi Application
Table A.1 contains the number of clusters, women, births, deaths by survey type.
Survey  No. Clusters  No. Women  No. Births  No. Deaths  

Census 2008  71,618  332,055  62,884  
DHS 2004  186  4,199  13,394  2,532  
DHS 2010  Training  140  3,879  12,626  2,125 
Holdout  145  3,983  12,873  2,066  
DHS 2015  284  8,417  23,240  2,621  
MICS 2006  Training  174  4,518  14,505  2,448 
Holdout  186  4,832  15,069  2,397  
MICS 2013  381  8,267  24,798  3,028 
Results from an exploratory analysis, similar to the diagnostic measures reported by Hill et al. (2015), where FBH information was transformed to SBH information are depicted in Figure A.1. The proportion of women interviewed tends to be similar across surveys. There appears to be some age heaping present, where women report their age as ending in “5” or “0.” There also do not appear to be systematic differences between the number of total births women report having had across surveys. However, there appear to be differences in the ratio of children dead (CD) to children ever born (CEB). Importantly, the differences tend to vary in rural and urban areas. We would expect the proportion CD among women in the census to be similar to the proportion CD among women in the 2006 MICS and 2010 DHS since these surveys were taken closest in time. These surveys are much noisier, but we can see that in rural areas the proportion CD is higher than both the 2006 MICS and 2010 DHS across almost all women. This is not the case in urban areas. This trend motivates including a term in our model that can capture this pattern.
Figure A.2 shows the HIVadjusted direct estimates by district. Figure A.3 shows the HIV bias as a function of survey. HIV bias factors were computed following Walker et al. (2012).
To determine which model life table to use in analyses involving the Brass method, we computed the direct estimates of and for the 2006 MICS and 2010 DHS and compared them to the 4 CoaleDemeny regional models (Figure A.4) as suggested by Hill (2013) . Although none of the models fit the observed values exactly, the North model appeared to fit best.
Appendix B Simulation Details
Birth histories for 5,000 women were simulated on a discrete, yearly time scale. For simplicity, we allowed the year prior to the survey to be completely observed and did not allow for births during the survey year, which follows the simple example provided in Section 2.1. Thus, children could be born at any point prior to and including (when the woman was aged ), and could die in where is the year of the survey and is the age of the woman at time of survey.
Figure B.1 illustrates how FBH data were simulated for a woman who was at the time of the survey. In the top left panel, when the woman is , the probability she gives birth is (fertility does not change over time). In this example, she does not give birth. In the middle top panel, when the woman is , the probability she gives birth is . Here, she does give birth. In the following year (top right panel), the probability the woman gives birth is , and the probability the child dies is . As the woman and her children age, we observe her to have 3 children at ages 16, 18, and 23. One child dies between age 1 and 2 and another dies between age 2 and 3. Her other child survives through the time of the survey. For women with FBH data, this information is completely observed. For women with SBH data, we only observe the total number of children the woman had and the number of those children that died (in this example, 3 births and 2 deaths).
Discrete hazards used in the simulation are in Figure B.2 (solid lines). Results from fitting the model to FBH data only and all data are also shown. The posterior medians are similar and close to the truth and uncertainty is reduced when we incorporate SBH data, as expected.
To assess accuracy we use where is the truth for a generic parameter (i.e., or ) and is an estimate of (e.g. posterior median) obtained from approach/model . To compare measures of uncertainty we take the ratio of the 95% credible/confidence intervals (CIs). Results are in Tables B.1 and B.2. In general, absolute bias and the width of the uncertainty intervals is lower when we include SBH data.
Absolute Bias  Width of CIs  Ratio of CIs  

FBH  SBH + FBH  FBH  SBH + FBH  
15–19  0.44  0.16  1.9  1.5  0.77 
20–24  0.10  0.015  2.9  2.3  0.81 
25–29  0.87  0.76  2.9  2.8  0.97 
30–34  0.38  0.50  3.0  3.4  1.12 
35–49  0.85  0.76  1.4  2.3  1.59 
Absolute Bias  Width of CIs (Ratio of CIs)  

FBH  SBH + FBH  GLM + Brass  FBH  SBH + FBH  GLM + Brass  
1975–79  7.3  4.3  8.5  22  23 (1.0)  42 (1.9) 
1980–84  2.6  0.29  8.1  13  12 (0.94)  18 (1.4) 
1985–89  1.5  0.72  0.37  7.7  6.5 (0.84)  12 (1.6) 
1990–94  2.4  0.45  2.3  5.8  4.6 (0.81)  8.7 (1.5) 
1995–99  2.3  0.93  0.92  4.4  3.9 (0.88)  2.3 (0.52) 
2000–04  0.29  0.088  1.1  3.6  2.8 (0.78)  2.1 (0.58) 
2005–09  0.54  0.70  0.50  4.2  2.6 (0.62)  2.1 (0.49) 
Appendix C Date Setup for Malawi Application
Calendar time is first defined on yearly scales, based on the month and year of the survey. In order to more accurately align with the woman’s age and to reduce the number of corrections (as described later), we divide time into the most recent 6 months prior to the survey and 12 month intervals preceding that. For example, if a survey was taken in February 2009, the time intervals would look like the following:

September  December 2008, January and February 2009

September  December 2007, January  August 2008

…
Based on the month and year reported for the birth of any children, we determine in what time interval the child was born.
The next step is to align the age of the woman during each interval. This is based on assuming that, on average, the woman will be the age in years reported plus six months. For example, if a woman reports that she is 35 at the time of survey, on average she will be 35.5. Thus, the most recent interval will correspond to when she is 35, second most recent when she was 34, etc.
Finally, we discretize the intervals by assigning them a year according to the year of survey. The most recent time period would correspond to the year of the survey and each subsequent interval would be one year prior. Since we smooth over time, the misalignment between the time intervals and assigned year is unlikely to be a major issue. We note that if the survey was taken in June there would be no misalignment.
We will need to make corrections to the model based on the yearly discretization of time. Suppose we are in the most recent time interval (which corresponds to the year of the survey). Since this interval only corresponds to 6 months, the fertility, or probability of birth should be, approximately half of what it normally is. This is assuming constant fertility throughout the year. Mathematically, if and then . All other years will not require this correction.
Finally, we need to consider a hazard correction. Suppose a child is born in the most recent time period. At most, they have 6 months of exposure time. On average, we might expect them to be born at the midpoint of that time interval, meaning we are interested in the probability they die within 3 months. Clearly, this is less than the probability of dying within 12 months, but the factor is likely to be since children are at highest risk in the first month of life. We base the correction on what is observed in the FBH data and use .
Now, consider a child born in the second most recent interval. On average, the child will be born halfway through the interval. Since the last interval is only 6 months, their exposure time is, on average, one year. We will not use any adjustments for the earlier periods. However, we note that for a small portion of women, they have children that, in terms of the newly defined discretized time, die after the survey. For example, consider a child born 1.5 years before the survey, which using our discretized intervals is 1 year before the survey. In our setup, they are only at risk for one year, but could theoretically die after 1 year and we would observe this death in the data. To balance the fact that some children in the interval may only be observed for 6 months, we will count this as a death within the first year. While the approach is not perfect and will result in slight biases, the impact should be minimal and relatively restricted to the most recent time periods.
Appendix D Hamiltonian Monte Carlo Algorithm Details
We derive the negative log posterior and corresponding gradient for our fertility and mortality models, which are used in a Hamiltonian Monte Carlo (HMC) algorithm (Neal, 2011) to update the various model parameters.
For the fertility model, define where is a matrix ( is the number of observations) and is a matrix, and where is a column vector of length and is a column vector of length . In the simulation, we do not have or . For the hazard model, define where is a matrix, and is a matrix, and where is a column vector of length and is a column vector of length . Since we will be using a random walk model in time, which does not specify an overall level, we opt to drop the unidentifiable terms from Models (6) and (7) instead of imposing a constraint.
For general and , with data model where is known (and includes the HIV bias in the mortality model and an adjustment for the final year; see Appendix C) and , we have
We assign independent priors,
where is taken to be a identity matrix, , and is the random walk of order 2 (RW2) precision matrix, scaled such that the generalized variance of is 1, following Sørbye and Rue (2014).
We use independent penalized complexity priors for and , that is for general ,
where with (Simpson et al., 2017). We set and .
Therefore, the negative log posterior (up to a constant) for the hazard model is
where is a vector containing the number of children that died, is a vector containing the number of children at risk, is a vector containing the multiplication factors , is the rank of matrix , and denotes elementwise multiplication. Define so that,
The gradient is then,
where (a column vector of length ) and (also a column vector of length ).
The negative log posterior and gradient is similar for the fertility model. In the simulation the vector is set to 1.
Appendix E Aggregation Algorithm
Define to be woman’s age, to be strata (rural, urban), to be district, to be the year, and to be the 5year time period. Define to be the total number of women of age at time in district and strata . Denote the number of births these women experience (Children Ever Born) as . Let index the samples. To obtain samples of on the 5year time period and district level, we employ the following steps:

Simulate

Transform:
Appendix F Additional Results for Malawi Application
Trace plots for the fixed effects and precision of the RW2 models suggest convergence (Figure F.1). Posteriors for the fertility parameters look similar. Posteriors for the hazards look similar for the age of the child,
. However, we do see some differences in posteriors for the districtlevel effects. In terms of the urban effect, the posterior medians are 0.397 (95% CI: 0.458, 0.338) for FBH data only and 0.377 (95% CI: 0.458, 0.338) for FBH + SBH. In terms of the bias terms, the posterior median for the indicator for SBH data is 0.171 (95% CI: 0.153, 0.191) and for the indicator for SBH and urban data is 0.228 (95% CI: 0.294, 0.162). The interpretation is that the SBH data result in an increase in the yearly hazard odds of 18.6% in rural areas and a decrease in the yearly hazard odds of 5.5%.
Fertility probabilities are similar when SBH data are added (Figures F.2–F.4). Fertility tends to be higher in rural areas, to be highest among women in their twenties, and to decrease over time for all age groups.
Results for U5MR comparing the 4 methods in the remaining 6 districts are presented in Figures F.5 and F.6.
To assess accuracy of the various models we fit the models to the training data and compared the results to the direct estimates obtained from the holdout data. Define to be the logit of U5MR in district time period . Figure F.7 provides a sense of the uncertainty in the numbers used as the “truth” and displays the estimate and 95% uncertainty interval of for each model on the probability scale, where we approximate the distribution of
with a normal distribution.
We also calculate the percentage relative error (PARE; Table F.1),
where and is the variance of the direct estimates. These (unweighted) values are shown for each district and period in Figure F.8.
Period  Weighted Estimates  Weighted Estimates  Smoothed Model:  Smoothed Model: 

+ Brass  FBH  FBH + SBH  
1985–1989  10.5  10.4  10.4  12.4 
1990–1994  9.82  10.2  9.21  9.91 
1995–1999  9.69  8.18  7.42  9.03 
2000–2004  9.41  12.7  7.62  3.24 
2005–2009  16.2  16.2  13.8  16.7 
Average  10.5  11.1  8.96  8.76 