Data sources

This study is based solely on DHS. DHS surveys use a stratified two-stage cluster sampling design. Samples are typically stratified by regions and urban/rural areas within each region. In each stratum, enumeration areas (EAs) are selected with a probability proportional to their population size, using a sampling frame usually obtained from the most recent census. Households are selected in each of the clusters. Within each household, all women aged 15-49 are selected for in-depth individual interviews. In recent surveys, geospatial information is collected to identify the central point of each cluster. To ensure that respondent confidentiality is maintained, coordinates are displaced by up to 2 km in urban clusters, 5 km in 99% of rural clusters, and 10 km in a random sample of 1% of rural clusters [21].

For this study, we retained only surveys conducted in Sub-Saharan countries that had at least 3 DHS surveys representative at a sub-national level, with a consistent set of sub-national boundaries across DHS surveys. This requirement was necessary to pool information from different surveys across time. When sub-national boundaries changed across DHS surveys, we used the GPS location of clusters to re-locate them into coherent sub-national units over time. This led us to select 96 surveys (see S1 Table in S1 Appendix) conducted in 20 Sub-Saharan African countries between 1990 and 2018. Our study covers the following countries: Benin, Burkina Faso, Cameroon, Ethiopia, Ghana, Guinea, Kenya, Lesotho, Madagascar, Malawi, Mali, Namibia, Niger, Nigeria, Rwanda, Senegal, Tanzania, Uganda, Zambia and Zimbabwe. The population-weighted average of the risk of dying in the age group 5-14 was 20 per 1000 in 2019 in these 20 countries, which is slightly higher than the regional estimate for Sub-Saharan Africa (16 per 1000) [9]. In 2019, 65% of all deaths that occurred in Sub-Saharan Africa in the age group 5-14 where in these 20 countries. The selected surveys contained data on about 11 million child-years of exposure and 376,015 deaths of children under the age of 5 years, against 9.2 million child-years of exposure and 20,605 deaths in the age group 5-14.

Estimation of sub-national mortality rates

We used the complete birth histories of women included in the DHS to obtain child-month data where the event of interest was the death of the child. We followed the methodology developed in Mercer et al. (2017) [17] and Li et al. (2019) [3] to estimate sub-national-period specific mortality rates, both for children aged 5 to 14 and for those aged less than five. The methodology consists of two steps. First, we obtained place-time probabilities of dying for both age groups. Then, we used these as inputs in a space-time Bayesian model to smooth estimates across space and time.

In the first step, we used discrete time survival analysis (DTSA) [22] to estimate age-specific monthly probabilities of dying using weighted logistic regression [23]. For under-five mortality, we used 6 age bands: [0, 1) to identify neonatal deaths, [1, 12), [12, 24), [24, 36), [36, 48) and [48, 60]. For children aged 5-14, we used 10 one-year age groups: [60, 72), [72, 84), … until [168, 180) months. The DHS sample weights associated with each mother account for the sampling probability and a non-response adjustment. Hence, the survey design and the related uncertainty is propagated through to the estimates and their associated standard error. This resulted in survey-place-time-age-specific estimated monthly probabilities of dying with variance estimates that fully account for the design of the survey. In our context, place consisted of Admin 1 level, which corresponds to administrative boundaries at the first sub-national level. DHS surveys are powered such that most indicators can be considered representative at this aggregation level. The time dimension of our mortality estimates was defined as a series of 4-year periods, for which multiple surveys could provide information. Because we need to combine surveys whose fieldwork dates have varied over time, we decided to let the most recent survey defines the time breaks (see S1 Appendix). For each survey, we went backwards in time by periods of 4 years. Information contained in full birth histories and referring to a period located more than 24 years before data collection was discarded when estimating under-five mortality. Similarly, this period was restricted to a maximum of 12 years for mortality in children aged 5-14 to reduce biases associated with the excess mortality of first-born children [12]. In other words, a survey contributed to a maximum of 6 periods for ₅q₀ and 3 periods for ₁₀q₅. Each country has its own series of 4-year periods. Note that if a survey provided only 1 year of exposure in a given 4- year period, it did not inform estimates in this period. We constructed the estimate of interest $$ {}_{10}{\widehat{q}}_5^{its}$$, in Admin 1 i, time period t and survey s using standard life table techniques:

Each of our selected countries had a least 3 DHS, thus for a given period, and a given Admin 1, $$ {}_{10}{\widehat{q}}_5^{it}$$ and $$ {}_5{\widehat{q}}_0^{it}$$ were potentially derived from multiple surveys. We combined estimates from different surveys into a single estimate for each age group using weighted average, where weights were given by the inverse of their variances. Resulting estimates can be seen as standard meta-analysis estimates. Similarly to Li et al. (2019) [3], we adjusted the estimates of $$ {}_5{\widehat{q}}_0^{it}$$ to account for downward biases introduced by mother-to-child transmission of HIV in Kenya, Lesotho, Tanzania, Zambia, Zimbabwe, Malawi, Namibia and Rwanda [24]. We used the correction factors made publicly available by Li and colleagues [3]. We did not correct for HIV-related biases in $$ {}_{10}{\widehat{q}}_5^{it}$$ as these biases has not been evaluated in older children and there is currently no method to adjust for these.

In a second step, we modeled the entire collection of place-time specific logit($$ {}_5{\widehat{q}}_0^{it}$$) and logit($$ {}_{10}{\widehat{q}}_5^{it}$$) for each country, using a Bayesian space-time smoothing model with appropiate design-based estimator of the variance, $$ {\widehat{V}}_{it}$$, obtained in the previous step. The smoothing allows sharing information between near neighbors in both space and time, leading to estimates that are characterized by a smaller variance and higher signal to noise ratio [3]. The model can be expressed as follows:

A constant overall level μ with a flat prior.

An unstructured spatial term with prior $$ {\theta }_i{\sim }_{iid}\mathcal{N}(0,{\sigma }_{\theta }^2),i=1,\dots ,n$$.

A smooth spatial term [ϕ₁, …, ϕ_n] where its prior is expressed as an intrinsic conditional autoregression (ICAR) model [25]. This model is a generalization of the random walk of order 1 (RW1) to the space dimension.

An unstructured temporal term where the prior is expressed as $$ {\alpha }_t{\sim }_{iid}\mathcal{N}(0,{\sigma }_{\alpha }^2),t=1,\dots ,T$$.

A smooth temporal term [γ₁, …, γ_T] where its prior is a random walk of order 2. A random walk of order 2 (RW2) model generally leads to more smoothing than a random walk of order 1 (RW1) model. Indeed, in the later, each value depends on its nearest neighbors while in the former, it also depends on neighbors’ nearest neighbors.

δ_it, an interaction term (type IV) where we assumed that temporal trends differ between areas and that they are more likely to be similar for adjacent areas [18]. Its associated prior is expressed as the product of the random walk and the intrinsic conditional autoregression.

Hyperpriors on the dipersion parameters have Gamma(a,b) distribution. For a detailed explanation of how a and b are specified, we refer the reader to [17, 18]. As emphasized by Li et al. (2019) [3], for both space and time dimensions, there are two terms capturing different sources of variation. The first term refers to unobserved risk factors that vary smoothly (ϕ_i and γ_t), the other reflects random shocks that are specific to that place or time only (θ_i and α_t). The Bayesian estimation process takes place separately for each country. The amount of smoothing is determined by the variances of ϕ_i and γ_t that are directly estimated from the data. The space-time interaction term (δ_it) allows the trend in time to vary across spatial areas. It was modeled assuming that the temporal (RW2) and spatial (ICAR) structured effects interact [18]. In summary, each $$ logit\left({}_{10}{\widehat{q}}_5^{it}\right)$$ (or $$ logit\left({}_5{\widehat{q}}_0^{it}\right)$$) depends on its nearest neighbors in space and its nearest two neighbors in both the past and future in the time dimension.

In the S1 Appendix, we compare the estimates of ₁₀q₅ and their associated precision before and after space-time smoothing (that is, after the first and second step). This comparison gives a sense of the reduction in uncertainty induced by the smoothing model. We also compare our final estimates at the national level with those provided by the UN Inter-agency Group for Child Mortality Estimation (UN IGME), using other data series and methods [9].

Precision of sub-national estimates of $$ {}_{10}{\widehat{q}}_5$$

Our first objective is to assess if Admin 1 estimates are sufficiently precise for children aged 5-14 when derived from multiple DHS. In the frequentist framework, the coefficient of variation (i.e. the standard deviation divided by the mean) is regularly employed to decide on the usability of estimates [16]. Hansen et al. (1953) and Kish (1995) consider that a coefficient of variation larger than 0.2 is a sign of unstable estimate [26, 27]. Working in a Bayesian framework, we used the posterior distributions of sub-national mortality estimates and computed a coefficient of variation by dividing their standard deviations by their means. We multiplied these ratios by 100 to obtain percentages. Dong and Wakefield (2004) showed that there is a direct relationship between the posterior CV obtained in this way and the ratio of higher end to lower end of the posterior credible interval [28]. If the coefficient of variation was higher or equal than 20%, we considered that the associated sub-national mortality estimate for children aged 5-14 was not sufficiently precise.

Correlation between $$ {}_{10}{\widehat{q}}_5$$ and $$ {}_5{\widehat{q}}_0$$

Our second objective is to examine the relationship between $$ {}_{10}{\widehat{q}}_5$$ and $$ {}_5{\widehat{q}}_0$$ at the sub-national level. To do so, we retained only the periods for which at least three quarters of the sub-national mortality estimates of $$ {}_{10}{\widehat{q}}_5$$ were considered as sufficiently precise according to the CV. This threshold was set according to the countries with fewer Admin 1 entities. In the case of Malawi (consisting of three Admin 1 entities), applying a 75% rule, a period could not be retained if one sub-national estimate within this period was not robust. For Rwanda (characterized by five Admin 1 entities), applying the same rule, if more than one sub-national estimate in a given period was not robust, this period was not retained. Having our set of selected periods for each country, we first computed Pearson correlation coefficients between $$ {}_{10}{\widehat{q}}_5$$ and $$ {}_5{\widehat{q}}_0$$ by country. We further stratified by period. Working in a Bayesian inference setup, $$ {}_{10}{\widehat{q}}_5^{it}$$ and $$ {}_5{\widehat{q}}_0^{it}$$ were associated to their respective posterior distributions. This allowed us to obtain 95% credible intervals around Pearson correlation coefficients, accounting for the uncertainty in our sub-national estimates.

Analyses were run with the R software version 3.6.1. We used the SUMMER package [29] that uses the svyglm function from the survey package [30] and the R-INLA package [31] to fit the models. All the R codes used to perform the analysis is available on Github: https://github.com/benjisamschlu/Subnational-5-14-MR.

Sub-national mortality estimates for children aged 0-4 and 5-14 years old

The estimates presented in this section are obtained after space-time smoothing of meta-analysis estimates. Figs 1 and 2 present Admin 1 mortality rates and their associated 95% credible intervals for Nigeria and Kenya. The corresponding graphs for other countries are the S1 Appendix. For most countries and Admin 1 units, mortality rates declined in young children aged 0-4 over the periods analysed. There were fewer cases with a significant decline over the period of observation in older children and young adolescents (40.2% of admin-1 units over the 20 countries against 81.7% in under-five mortality, when comparing the first and last periods). However, the low percentage in 10q5 is also a consequence of the wide confidence intervals for the first and last periods. The sub-national patterns observed in under-five mortality cannot be readily generalised to older children. There is substantial heterogeneity in the ₁₀q₅-to-₅q₀ relationship.

Fig 1

Sub-national estimates of the probabilities ₅q₀ and ₁₀q₅ in Nigeria, based on DHS conducted in 1990, 2003, 2008, 2013 and 2018 (log scale).

Fig 2

Sub-national estimates of the probabilities ₅q₀ and ₁₀q₅ in Kenya, based on DHS conducted in 1993, 1998, 2003, 2008 and 2014 (log scale).

The uncertainty around sub-national mortality rates for children aged 5-14 is much higher than for children aged less than 5. First, deaths counts are subject to more stochastic variation as the mortality rates for older children are smaller. Second, in the older age group, mortality estimates associated to a period are generally informed by less DHS surveys. This comes from the fact that 3 and 6 periods in the past are retained from each DHS in the case of older and younger children, respectively. Uncertainty is particularly large for the first and last periods. This is because these estimates are built on fewer DHS surveys and oldest surveys are characterized by smaller sample sizes. When the Admin 1 refers to a capital city, estimates tend to be associated with large credible intervals due to the smaller sample size and smaller count of deaths in this area. This can be observed for Nairobi in Fig 2.

Precision of sub-national mortality estimates of children aged 5-14

Fig 3 displays the coefficient of variation for the probability $$ {}_{10}{\widehat{q}}_5$$ estimated for each Admin 1-period combination. These CVs are compared to the threshold value of 20%. All countries exhibit a U-shape, as sub-national mortality estimates for the first and last periods are less precise than the ones in the middle. Out of the 1,132 space-time estimates of $$ {}_{10}{\widehat{q}}_5$$, 62.3% fall below the 20% threshold. By contrast, this is the case of 99.3% space-time estimates of under-five mortality (see S1 Appendix). There is important heterogeneity in the amount of sub-national estimates that fulfill our criterion. In Zimbabwe, Kenya, Ghana, Lesotho and Namibia, sub-national estimates of the probability $$ {}_{10}{\widehat{q}}_5$$ are not sufficiently precise as most values of the CV fall above the 20% threshold. These countries had generally small sample sizes and a large number of Admin 1 entities which increases the amount of noise in the data.

Fig 3

Coefficient of variation associated with sub-national ₁₀q₅ estimates.

Fig 4 highlights the uncertainty in the sub-national estimates of $$ {}_{10}{\widehat{q}}_5$$ for two countries. In Ethiopia (top panel), a majority of coefficient of variations are above 20% for the period 2008-2011. This is due to wide posterior distributions for the sub-national mortality estimates, making it impossible to detect variations in mortality in older children within the country. On the contrary, sub-national estimates for Nigeria in the period 2007-2010 are all considered robust, which is due to much narrower posterior distributions, allowing detection of significant variations in $$ {}_{10}{\widehat{q}}_5$$. This is less so for the period 2015-2018 as it consists of the last period, and hence, estimates are built on fewer data points.

Fig 4

Ridge plots of sub-national estimates of $$ {}_{10}{\widehat{q}}_5$$ for Ethiopia (2008-11) and Nigeria (2007-10, 2015-2018).

In order to study the relationship between $$ {}_5{\widehat{q}}_0$$ and $$ {}_{10}{\widehat{q}}_5$$, we retained periods where at least 75% of the $$ {}_{10}{\widehat{q}}_5$$ had a coefficient of variation below 20% for a given period. We also checked how many periods were considered sufficiently precise when using a threshold of 15% (see S61 Fig in S1 Appendix).

The ₁₀q₅-to-₅q₀ relationship at the sub-national level

Fig 5 displays the $$ {}_5{\widehat{q}}_0$$ estimates against the $$ {}_{10}{\widehat{q}}_5$$ values, each with their associated 95% credible intervals (note that the scale is defined specifically for each country and that different point shapes for a country corresponds to different periods). This Figure is based only on the periods for which we have sufficient precision in estimates of $$ {}_{10}{\widehat{q}}_5$$ according to our criterion. Except Lesotho and Zimbabwe, each other retained country has at least 4 periods considered as sufficiently precise. The figure allows to compare uncertainty in sub-national $$ {}_{10}{\widehat{q}}_5$$ estimates across countries. It shows the long term improvement in mortality for both age groups over time.

Fig 5

The ₁₀q₅-to₅q₀ relationship at the sub-national level, based on space-time estimates considered sufficiently precise (scales varying by country).

Pooling all sub-national estimates across retained periods, there is a clear positive and statistically significant Pearson correlation for all countries except Burkina Faso and Niger (see Fig 6). The correlation coefficient ranges from 0.14 in Niger (95% CI: -0.02—0.31) to 0.92 in Malawi (95% CI: 0.86—0.96). However, One should not conclude from these high coefficients that the spatial variation in mortality in older children is well reflected by that in under-five mortality. Indeed, the relationship captured here is driven by the long-term improvement in mortality in both age groups. Stratifying by period, this relationship is much less obvious. Out of 82 retained periods, 46 have a Pearson correlation where its 95% credible interval crosses zero. In other words, in the majority of periods, the Admin 1 characterized by the highest under-five mortality rate are not the ones facing the highest mortality rate in older children. We can differentiate three groups of country. The first group consists of Cameroon, Ethiopia, Nigeria, Senegal and Zambia where correlations are most of the time positive and significantly different from zero. The second group contains Madagascar, Mali, Niger, Rwanda, Tanzania and Uganda when occasionally, a period is characterized by a positive and significant correlation. Finally, in the last group encompassing Benin, Burkina Faso, Ghana and Guinea, we did not find any significant correlation. We did not compute the Pearson correlation estimates stratifying by period for Malawi as the country consists of only three Admin 1 entities. Zimbabwe and Lesotho are not plotted as they only have one period considered sufficiently precise. Correlations associated with these periods are not significantly different from zero.

Fig 6

Pearson correlations between Admin-1 estimates of ₅q₀ and ₁₀q₅.

Comparison of space and time heterogeneity among age groups

The total variance in $$ logit\left({}_{10}{\widehat{q}}_5\right)$$ (or $$ logit\left({}_5{\widehat{q}}_0\right)$$) can be split into the variance of the five components of the model (see Eq (4)). Pooling countries together, the variance associated to the RW2 time trend ($$ {\sigma }_{\gamma }^2$$) makes up for an average of 58.86% and 46.44% of the total variance for $$ {}_5{\widehat{q}}_0$$ and $$ {}_{10}{\widehat{q}}_5$$, respectively (see Table 1). Countries with specifically high values for $$ {\sigma }_{\gamma }^2$$ such as Benin, Rwanda, Ethiopia and Malawi, are the ones where all Admin 1 experienced a similar temporal trend, explaining the majority of the variation in sub-national estimates. Lesotho, Namibia and Niger are countries where $$ {\sigma }_{\gamma }^2$$ for $$ {}_{10}{\widehat{q}}_5$$ is below 10% which, in the case of Lesotho and Niger, is in contrast with what is found for $$ {}_5{\widehat{q}}_0$$.

Table 1

Share (%) of total variance from model components.

Country	RW2 ($$)		ICAR ($$)		RW2xICAR ($$)		Time ($$)		Space ($$)
	$$	$$	$$	$$	$$	$$	$$	$$	$$	$$
Benin	81.7	84.4	9.8	1.7	3.2	10.3	3.1	1.9	2.2	1.7
Burkina Faso	47.5	18.7	28.0	9.0	17.0	63.1	5.3	5.3	2.2	3.8
Cameroon	37.6	33.6	56.0	47.0	3.2	16.5	2.6	2.2	0.7	0.8
Ethiopia	72.6	73.7	21.7	4.9	3.7	18.7	1.6	1.1	0.4	1.6
Ghana	56.0	36.1	34.4	3.0	6.4	53.0	2.7	5.1	0.5	2.7
Guinea	52.7	75.4	38.7	2.4	4.8	16.7	3.2	2.7	0.6	2.8
Kenya	30.3	20.7	50.5	29.1	15.9	38.0	2.2	2.4	1.1	9.8
Lesotho	43.0	9.6	6.8	1.9	33.8	84.6	8.5	1.9	8.0	2.0
Madagascar	76.4	67.6	2.4	3.8	5.6	20.6	14.6	5.4	1.0	2.5
Malawi	92.5	83.6	2.0	4.5	0.6	4.7	4.3	5.3	0.5	1.9
Mali	68.4	48.3	27.7	21.2	2.1	22.4	1.1	5.5	0.7	2.7
Namibia	16.9	4.1	38.1	24.6	35.3	68.3	4.5	2.3	5.2	0.8
Niger	54.6	3.8	36.3	6.8	7.0	81.2	1.3	5.3	0.8	2.8
Nigeria	33.9	17.3	60.1	46.5	3.5	24.7	1.6	1.7	0.9	9.7
Rwanda	89.9	92.2	1.2	1.2	0.8	2.7	7.5	3.2	0.6	0.7
Senegal	69.9	66.8	24.8	2.8	3.7	20.8	0.9	5.1	0.7	4.4
Tanzania	66.1	62.9	27.8	11.2	3.4	17.6	2.1	5.2	0.7	3.1
Uganda	88.6	63.0	2.5	2.7	2.6	23.3	5.7	2.3	0.6	8.7
Zambia	82.7	53.3	12.4	16.6	2.9	24.8	1.2	2.2	0.8	3.1
Zimbabwe	16.1	13.5	57.0	10.2	14.1	56.0	5.4	3.7	7.4	16.6

The average share of total variance accounted for by the ICAR structured space term ($$ {\sigma }_{\varphi }^2$$) is 26.9% and 12.54% for $$ {}_5{\widehat{q}}_0$$ and $$ {}_{10}{\widehat{q}}_5$$, respectively. In all countries but Madagascar, Malawi, Uganda and Zambia, our estimates of $$ {\sigma }_{\varphi }^2$$ for children under the age of five are higher or equal than for older children, sometimes by a large margin (e.g. Ghana, Guinea, Kenya, Niger and Zimbabwe). This is a sign that sub-national patterns of mortality declines that are consistent over time account for less variance in sub-national mortality estimates for children aged 5 to 14 in comparison to children aged < 5. It seems that most of the “lost” variation from this component in the older age group is now accounted for by the time×space interaction (RW2xICAR). The average percentage of variability captured by $$ {\sigma }_{\delta }^2$$ equals 8.48% in children under the age of 5 and 33.40% in older children and young adolescents. This suggests that particular Admin 1 within a given country had mortality estimates that deviated from levels expected based on the secular trend (applying to all Admin 1) and the sub-national pattern (applying to all periods). However, we have to be cautious as most countries with the highest values in $$ {\sigma }_{\delta }^2$$ are also the ones that had few precise sub-national estimates (Burkina Faso, Ghana, Lesotho, Namibia and Zimbabwe). Hence, this might also comes from stronger noise in sub-national estimates for theses countries. Finally, the variability explained by the unstructured terms ($$ {\sigma }_{\alpha }^2,{\sigma }_{\theta }^2$$) is within 1-5%.