Small area estimation of under-5 mortality in Bangladesh, Cameroon, Chad, Mozambique, Uganda, and Zambia using spatially misaligned data

We estimated U5MR at the second administrative level in Cameroon (departments), Chad (departments), Mozambique (districts), Uganda (districts), and Zambia (districts); and at the third administrative level in Bangladesh (sub-districts). Table 1 describes the administrative hierarchy for each country.

Annual under-5 population estimates were compiled for each unit of analysis in each country from 1980 to 2015 from a combination of census microdata, census reports, and existing population series (Table 2) with the primary goal of describing the relative population size of different areas within a country rather than the precise absolute population size in any one area. For Bangladesh, Mozambique, Uganda, and Zambia, we sought age-specific census microdata or tabulations at the finest geographic level available. Where the appropriate level of geographic detail was not available – either because census counts were tabulated at a higher level (e.g., province or region) or because administrative boundary definitions had changed over time – the counts were split according to the proportions observed in the next census year. When only total (all ages combined) tabulations were available, these were age-split according to the age patterns observed in the closest census year with age-specific data available in the same country. For Mozambique, we supplemented the available census data with an existing district-level population series available via the Spatial Data Repository [12]. Total and age-specific populations for each area were interpolated and projected from 1980 to 2015 assuming geometric growth and then age-specific populations were scaled to match the total population in each year in each area. For Cameroon and Chad, where less age- and area-specific census data were readily available, we instead utilized the gridded population estimates from WorldPop [13, 14]. We overlaid the population grid on a current shapefile and then aggregated the population within each area to generate department-level population estimates for each country. WorldPop estimates were available for 2000, 2005, 2010, and 2015, and we utilized these four sets of estimates for 1980–2000, 2001–2005, 2006–2010, and 2011–2015, respectively, holding the population constant within each time period.

For this analysis, we identified surveys and censuses in all six countries which contained summary birth history (SBH) or complete birth history (CBH) data (Table 3). For surveys that contained the latitude and longitude of each survey cluster, we overlaid these coordinates with a current district, department, or sub-district (depending on the country) shape file and identified the area each survey cluster belonged to. In all other surveys, as well as in censuses, we extracted the finest geographic level identified in the available data. Birth history data for each geographic area included in a given survey or census were analyzed independently using the complete or summary birth history methods described below. When both complete and summary birth history data were available from a single survey, only complete birth history data were analyzed and included in the analysis.

We extracted child-level data on date of birth, survival status, age at death (if applicable), and sampling weights from surveys with complete birth histories. The data for each child were expanded to encompass each month that the child started alive prior to reaching age 5 or the date of the survey, whichever occurred first. All child-months were subdivided by calendar year and into six age groups (month 0, months 1–11, year 1, year 2, year 3, and year 4) and the monthly probability of death in each age-year group was calculated as the weighted proportion of child months which ended in death, where the weights were the survey sample weights. We then derived U5MR for each calendar year as:

where q_{j, t, s, a} is the monthly probability of death in area j, year t, age group a, source s; and n_a is the number of months in age group a.

We estimated \( {N}_{j,t,s}^{(CBH)} \), the number of births associated with a given \( U5{MR}_{j,t,s}^{(CBH)} \) estimate based on the number of child months contributing to that estimate. Specifically, the number of child months in area j, year t, and source s (summed across the six age groups) was divided by the mean number of months children in source s lived prior to death, reaching age 5, or the time of survey, whichever occurred first. We then estimated \( {Y}_{j,t,s}^{(CBH)} \), the number of deaths associated with each U5MR estimate, by multiplying the estimated number of births by the estimated U5MR:

We extracted woman-level data on the number of children ever born and the number of children died from surveys and censuses with summary birth histories. We then applied the combined summary birth history method described by Rajaratnam et al. [15] in order to generate preliminary estimates of U5MR by area and year from each survey. This method requires several inputs, indexed by mother’s age or reported time since first birth, including: number of women, total children ever born, total children died, and mean children born (per woman). Women-level sample weights were used to generate weighted estimates of all input parameters. The output of these summary birth history methods are annual estimates of U5MR for approximately 25 years preceding the date of the survey or census.

As an intermediate step in applying the summary birth history methods described by Rajaratnam et al. [15], all reported births are distributed on an annual basis to the years preceding the survey or census using empirical distributions of time since birth indexed by mother’s age and the reported number of children at the time of survey. The resulting annual estimates of births (\( {N}_{j,t,s}^{\ast (SBH)} \)) were taken as the starting point for estimating an effective sample size for each summary birth history estimate.

Summary birth history estimates are subject to both sampling error and model error, and we wanted to reflect this in our estimates of the effective number of births and deaths associated with each U5MR estimate derived from a summary birth history. To approximate the sampling variance of \( U5M{R}_{j,t,s}^{(SBH)} \), we assumed the number of children that die is approximately binomially distributed:

To approximate the model variance, we utilized the results of the validation exercise reported by Rajaratnam et al. [15] Five-fold cross-validation was used to assess the performance of the summary birth history methods in reproducing estimates derived from complete birth histories. We calculated the variance of the residuals from this comparison for each year prior to survey – i.e., the difference between the summary birth history estimate and corresponding complete birth history estimate, on a probability scale – as an approximation of the error introduced by using the summary birth history data and methods compared to the complete birth history data and methods. For each estimate of \( U5M{R}_{j,t,s}^{(SBH)} \), we used this variance, matched for appropriate number of years prior to survey, as \( {\sigma}_{j,t,s\ \left[ model\right]}^2 \). We assumed that the model error and sample error were independent, and calculated the total variance for each estimate of \( U5M{R}_{j,t,s}^{(SBH)} \) as

and calculated the corresponding effective sample size \( {N}_{j,t,s}^{(SBH)} \) again assuming that the number of children who die is approximately binomially distributed:

This procedure was carried out both at the national level as well as each subnational area at the finest level available in given survey or census, and the resulting values of \( {N}_{j,t,s}^{(SBH)} \) for all subnational areas were scaled to sum to the estimated value of \( {N}_{j,t,s}^{(SBH)} \)for the country as a whole. Finally, we calculated the effective number of deaths by multiplying the estimated U5MR by the effective number of births:

We started with the following hierarchical generalized linear model defined for data stratified by area (department, district, or sub-district, depending on the country), year, and data source (a single survey or census in a particular area):

where

N_{j, t, s} and Y_{j, t, s} are the number of births and deaths respectively in area j, year t, and source s (equivalent to \( {N}_{j,t,s}^{(CBH)} \) and \( {Y}_{j,t,s}^{(CBH)} \) or \( {N}_{j,t,s}^{(SBH)} \) and \( {Y}_{j,t,s}^{(SBH)} \), depending on the birth history method used to analyze source s);

p_{j, t, s} is the underlying U5MR in area j, year t, and source s;

β₀ and u_{0, j} are the country-level fixed intercept and the area-level random intercept, respectively;

S_i(t) is basis i of a natural cubic spline [16] with four equally-spaced interior knots evaluated at time t;

β_i and u_{i, j} are the country-level fixed slopes and area-level random slopes on S_i(t), respectively;

and γ_s is a source-level random intercept.

We then added a second component which allows us to incorporate data defined for other geographic levels (i.e., higher level or historical administrative units):

Where p_{k, t, s}, N_{k, t, s}, Y_{k, t, s} are defined analogously to p_{j, t, s}, N_{j, t, s}, Y_{j, t, s} but for some area k made up of multiple area j’s (e.g., a province, containing multiple districts). p_{j, t}, the underlying U5MR in area j and year t is defined analogously to p_{j, t, s} above, but with γ_s set to 0, and p_{k, t}, the true prevalence in area k and year t, is given by the population (P) weighted average of p_{j, t} across all areas j contained within area k.

The random effect terms u_{0, j} and u_{i, j} (i = 1–5) were assigned conditional autoregressive priors as described by Leroux et al. [17] These priors allow for spatial smoothing based on the neighborhood structure of the areas being modeled, specifically by assuming that the prior mean for a given area is a function of the values in neighboring areas. The full conditional distribution implied by this prior is:

where m~j indicates that area m is a neighbor of (i.e., shares a boarder with) area j and n_j is the number of neighbors of area j. The two hyperparameters for each random effect were estimated as part of the model fitting process: σ² determines the overall amount of variation and ρ, which varies between 0 and 1, determines the degree of spatial autocorrelation. Lower values of ρ indicate less spatial relatedness, while higher values of ρ indicate a high degree of spatial relatedness; at the extremes, this prior reduces to a Normal(0, 1) prior when ρ is 0 and to an intrinsic conditional autoregressive prior [18] where the prior mean is equal to the mean of all neighbors when ρ is 1. Normal(0, 10) hyper-priors were specified for logit-transformed ρ and half-Normal(0, 1) hyper-priors were specified for σ for all random effects. Common σ² and ρ parameters were estimated for all random effects on the spline bases, u_{i, j} (i = 1–5). Posterior estimates of these hyperparameters are listed in Additional file 1.

Models were estimated separately for each country. All models were fit using the TMB package [19] in R version 3.2.4 [20]. We extracted point estimates and the variance-covariance matrix for logit(p_{j, t}) and used these to generate 1000 draws of p_{j, t} by drawing from a multivariate normal distribution and then inverse-logit transforming each draw. These draws were scaled to match existing national-level estimates of U5MR from the Global Burden of Disease (GBD) Study: [3] for each year t and each draw, we calculated the ratio of the national estimate from the GBD to the national estimate derived from population-weighting p_{j, t} and multiplied p_{j, t} by this ratio. Finally, we calculated point estimates of p_{j, t} from the mean of these draws and the lower and upper bounds of the 95% uncertainty interval from the 2.5th and 97.5th percentiles, respectively. Relative change in p_{j, t} over time was also calculated for each draw, and 95% uncertainty intervals were derived from the 2.5th and 97.5th percentiles.

Figure 1 shows birth history and small area estimates for four example areas. Figures 2, 3, 4, 5, 6 and 7 show the predicted U5MR for each country in 2015 and the relative change from 1980 to 2015. Results for all years with associated uncertainty intervals are provided in Additional file 2. We found significant variation in U5MR at the second (or third, in Bangladesh) administrative level in all six countries.

In 2015, U5MR in Bangladesh was 36 (95% uncertainty interval: 33, 41) deaths per 1000 live births; however, U5MR varied among sub-districts from 23 (10, 44) in Barisal sub-district to 74 (32, 141) in Derai sub-district. In general, the highest U5MRs were found in northeastern sub-districts, while the lowest were found in eastern and central-western sub-districts. Between 1980 and 2015, U5MR improved in all sub-districts, with relative declines ranging from 70% (43, 87) in Jhikargachha sub-district to 87% (75, 94) in Barisal sub-district.

U5MR in Cameroon in 2015 was 91 (74, 112) deaths per 1000 live births. At the department level, there was a more than two-fold difference between U5MR in Hauts Plateauxand (57 [34, 87]) and Mayo Louti (150 [99, 218]). Overall, the highest mortality rates were found in northern departments while departments in the central and eastern parts of the country experienced more intermediate U5MRs, and departments in the northwest experienced the lowest U5MRs. U5MR declined in all departments in Cameroon between 1980 and 2015, but the magnitude of these declines varied widely, from 39% (12, 60) in Diamare to 52% (31, 69) in Mayo Danay.

Chad experienced an U5MR of 117 (103, 136) deaths per 1000 live births in 2015. At the department level, U5MR ranged from 54 (29, 91) in Kobe to 180 (121, 260) in Loug Chari. Departments in the southwest generally had the highest mortality rates while departments with relatively low mortality rates were concentrated in the northeast and northwest. Declines in U5MR between 1980 and 2015 ranged from 33% (8, 53) in Loug Chari to 63% (45, 76) in Guera.

U5MR in Mozambique was 81 (70, 93) deaths per 1000 live births in 2015. At the district level, U5MR varied from 45 (20, 94) in Cidade de Lichinga in Niassa province to 144 (79, 232) in Angonia in Tete province. In general, the lowest U5MRs were found in northern districts while the highest U5MRs were found in central-western districts. U5MR improved in all districts between 1980 and 2015, with declines ranging from 42% (− 27, 78) in Distrito Urbano 7 in Maputo to 77% (65, 86) in Mueda in Cabo Delgado province. There were distinct regional patterns in the decline in U5MR between 1980 and 2015, with the largest improvements found in northern districts and the smallest improvements found in central-western and southern districts.

In 2015, U5MR in Uganda was 66 (58, 75) deaths per 1000 live births; however, U5MR varied among districts from 42 (28, 62) in Kampala to 92 (60, 135) in Moroto. Large-scale spatial patterns in U5MR in 2015 were less prominent in Uganda compared to other countries. However, there were clusters of districts with higher than average U5MR in the northeast and southwest, and clusters of districts with lower than average U5MR in the central part of the country. Between 1980 and 2015, U5MR declined in all districts in Uganda. The rate of decline was more uniform in Uganda than the other countries considered, varying between 64% (48, 76) in Kaabong and 70% (57, 80) in Kampala.

Zambia experienced a U5MR of 62 (49, 76) deaths per 1000 in 2015. U5MR varied among districts by more than a factor of three, with the lowest rate found in Chavuma (39 [21, 66]) and the highest in Chiengi (134 [87, 190]). In general, lower mortality rates were found in the Copperbelt region and in more central districts, while higher mortality rates were found in districts in the east, north, and southwest. U5MR declined in all districts in Zambia between 1980 and 2015, with declines ranging from 53% (34, 68) in Chiengi to 66% (56, 76) in Kasama.

Figure 8 depicts how the distribution of U5MR at the second (or third) administrative level changed over time in each country. In Bangladesh, Chad, Mozambique, and Uganda, the distribution shifted downwards over the course of each decade in our analysis. In contrast, this distribution shifted upwards slightly in Cameroon between 1990 and 2000 and in Zambia between 1980 and 1990, though over the analysis period as a whole the same downwards shift was observed.

Among these six countries, only Bangladesh and Uganda have no overlap between the distribution of U5MR in 1980 and in 2015. In each of the remaining countries – Cameroon, Chad, Mozambique, and Zambia – there are areas where U5MR was higher in 2015 than in other areas of the same country three and half decades earlier. The comparison between 1980 and 2015 is particularly extreme in Cameroon, where the highest U5MR observed in any department in 2015 was similar to the median observed among departments in 1980.

Figure 8 shows the MDG target for each country as a solid black line. Among the six countries considered, only Bangladesh achieved the goal of a two-thirds reduction overall in U5MR between 1990 and 2015; most (87.2%), but not all, sub-districts in Bangladesh had U5MR below the country-wide MDG target in 2015 (based on the point estimates of U5MR in each area). In Chad, Mozambique, Uganda, and Zambia, a minority of areas reached the country-wide MDG target (6.5, 37.8, 16.1, and 48.6%, respectively). No department in Cameroon achieved the country-level MDG target in 2015.

Figure 8 also shows the Sustainable Development Goal (SDG) target for 2030 (25 deaths per 1000 for all countries). As of 2015, there was no district or department in Cameroon, Chad, Mozambique, Uganda, or Zambia where U5MR was below this threshold, while in Bangladesh four sub-districts had U5MR just below this threshold. Moreover, in all six countries there were areas where U5MR in 2015 was many times higher than the SDG target: e.g., in every country other than Bangladesh, a large majority (76.4–100%) of districts and departments had U5MR at least twice the SDG target.

A comparison of U5MR estimates from this analysis to those in Golding et al. [11] is shown in Additional file 3. The point estimates from these two analyses are reasonably consistent for most countries in most years. However, there are important differences, particularly in Chad and Mozambique in 2015. The estimates from both analyses are associated with considerable uncertainty as indicated by wide uncertainty intervals; in nearly all cases (95.5% of area-years) these uncertainty intervals overlap between the two analyses.

This analysis is subject to a number of limitations. First, the birth history data we used as the primary source of information on U5MR are subject to misreporting and survival biases [26, 27]. Previous research has suggested that the impact of survival bias can be significant in populations with high HIV prevalence. Several methods have been proposed for correcting for this bias, and additional research is warranted to investigate integrating these methods alongside small area estimation methodologies [28‐30]. Second, surveys that include GPS coordinates generally have these coordinates displaced to some degree to protect respondent’s confidentiality (for example, Demographic and Health Surveys randomly displace GPS coordinates up to 1 km in urban areas, up to 5 km in rural areas, and up to 10 km in a random 1% subset of survey clusters [31]). This displacement is not accounted for when mapping GPS data to administrative boundaries, potentially leading to some misclassification in areas near administrative borders. Third, the population data which we used to inform how different levels of geographic aggregation relate to each other in the small area estimation models are also subject to error, both due to errors in the underlying census data (e.g., omissions or age misreporting) as well as potential violation of the assumptions used to interpolate these data (e.g., due to migration). Fourth, the methods used to account for both sampling and model error and estimate the effective number of births associated with U5MR estimates from summary birth histories require a number of assumptions and approximations that may not be appropriate in all cases. Fifth, the small area estimation models smooth over space and time by making assumptions about the temporal and spatial structure of U5MR that may not always hold. Sixth, the small area models assume that the estimated deaths from the birth history data are binomially distributed for all areas, and this assumption may not be valid in all cases. Future research, potentially including simulation studies, should explore the implications of these assumptions as well as possible alternatives. Seventh, while the methods developed here allow for including spatially misaligned data, they do require that this spatial misalignment is of a particular type: specifically, that all larger areas are composed of simple combinations of the smaller areas of interest. Eighth, the estimates we derive are uncertain, as reflected by the uncertainty intervals that are often quite wide. Given this uncertainty, the results must be interpreted with caution. Ninth, we make a number of cross-country comparisons of the distribution of U5MR within each country, but this distribution is likely sensitive to how a given country is partitioned (i.e., the modifiable areal unit problem) [32]. Finally, it is difficult to assess the accuracy of these estimates, which may be compromised due either to violations of the modeling assumptions or quality issues in the underlying data sources. The estimates from this study generally have high face-validity given existing knowledge of the social and health context of each country. However, there are exceptions. In particular, estimates for recent years in Mozambique are difficult to explain, as some districts with relatively high and others with relatively low estimated U5MR are similar with regards to human and financial resource inputs (e.g., Macanga and Nangande). Future research should focus on formally validating these methods and comparing them to available alternatives.