Global Estimation of Neonatal Mortality using a Bayesian Hierarchical Splines Regression Model
GGlobal Estimation of NeonatalMortality using a BayesianHierarchical Splines Regression Model ∗ Monica Alexander † University of California, Berkeley
Leontine Alkema ‡ University of Massachusetts, Amherst
Abstract
In recent years, much of the focus in monitoring child mortality has beenon assessing changes in the under-five mortality rate (U5MR). However, as theU5MR decreases, the share of neonatal deaths (within the first month) tendsto increase, warranting increased efforts in monitoring this indicator in additionto the U5MR. A Bayesian splines regression model is presented for estimatingneonatal mortality rates (NMR) for all countries. In the model, the relation-ship between NMR and U5MR is assessed and used to inform estimates, andspline regression models are used to capture country-specific trends. As such,the resulting NMR estimates incorporate trends in overall child mortality whilealso capturing data-driven trends. The model is fitted to 195 countries usingthe database from the United Nations Interagency Group for Child MortalityEstimation, producing estimates from 1990, or earlier if data are available, until2015. The results suggest that, above a U5MR of 34 deaths per 1000 live births,at the global level, a 1 per cent increase in the U5MR leads to a 0.6 per centdecrease in the ratio of NMR to U5MR. Below a U5MR of 34 deaths per 1000live births, the proportion of deaths under-five that are neonatal is constantat around 54 per cent. However, the relationship between U5MR and NMRvaries across countries. The model has now been adopted by the United NationsInter-agency Group for Child Mortality Estimation. ∗ The authors would like to thank Danzhen You, Lucia Hug, Simon Ejdemyr, Jon Pedersen, JingLiu and Jin Rou New for their work on constructing the database. We are also grateful all membersof the (Technical Advisory Group of the) United Nations Inter-agency Group for Child MortalityEstimation for helpful feedback and discussions on the model. † [email protected] ‡ [email protected] a r X i v : . [ s t a t . A P ] D ec Introduction
In order to evaluate a country’s progress in reducing child mortality, it is important toobtain accurate estimates; be able to project mortality levels; and have some indicationof the uncertainty in the estimates and projections. In practice, obtaining reliablemortality estimates is often most difficult in developing countries where mortality isrelatively high, well-functioning vital registration systems are lacking and the data thatare available are often subject to large sampling errors and/or of poor quality. Thissituation calls for the use of statistical models to help estimate underlying mortalitytrends.In recent years, much of the focus on monitoring child mortality has been on assessingchanges in the under-five mortality rate (U5MR), which refers to the number of deathsbefore the age of five per 1000 live births. The focus was driven by Millennium Devel-opment Goal (MDG) 4, which called for a two-thirds reduction in under-five mortalitybetween 1990 and 2015. A report on MDG progress released in 2015 by the UnitedNations showed that, although this target was not met in most regions of the world,notable progress has been made (UN 2015). The global U5MR is less than half ofits level in 1990, and despite population growth in developing regions, the number ofdeaths of children under five has declined.As the U5MR decreases, the share of neonatal deaths, i.e. deaths occurring in thefirst month, tend to increase. Globally, the estimated share of under-five deaths thatwere neonatal in 2015 was 45 per cent, a 13 per cent increase from 1990 (IGME 2015).Indeed, in most regions of the world, the majority of under-five deaths are neonatal; forexample, the share is 56 per cent in Developed regions; 51 per cent in Latin Americaand the Caribbean; and 54 per cent in Western Asia. The share is still less than 50 percent, however, where the U5MR is relatively high: in Sub-Saharan Africa, the share isonly 34 per cent.The neonatal equivalent to the U5MR is the neonatal mortality rate (NMR), whichis defined as the number of neonatal deaths per 1,000 live births. The increasingimportance of neonatal deaths in over child mortality has warranted increased effortsin monitoring NMR in addition to the U5MR (e.g. Lawn et al. 2004; Mekonnen et al.2013; Lozano et al. 2011; Bhutta et al. 2010). The United Nations Inter-agency Groupfor Child Mortality Estimation (IGME) publishes estimates of NMR for all 195 UNmember countries (IGME 2015). IGME uses a statistical model to obtain estimatesfor countries without high quality vital registration data, with U5MR as a predictor(Oestergaard et al. 2011). While the method has worked well to capture the maintrends in the NMR, it has some disadvantages. Most notably, trends in NMR withina country are driven by the U5MR trends, rather than being driven specifically by theNMR data.In this paper, we present a new model for estimating NMR for countries worldwide,overcoming some of the concerns with the current IGME model using a Bayesian hier-2rchical model framework. From the point of view of modeling mortality levels acrosscountries, a Bayesian approach offers an intuitive way to share information across dif-ferent countries and time points, and a data model can incorporate different sourcesof error into the estimates. Increases in computation speed as well as the developmentof numerical methods have enabled a more widespread use of the Bayesian approachin many fields, including population estimation and forecasting (e.g. Girosi and King2007; Raftery et al. 2012; Alkema and New 2014; Schmertmann et al. 2014). In thisapplication, the proposed Bayesian model is flexible enough to be used to estimate theNMR in any country, regardless of the amount and sources of data available. Resultswere produced for 195 countries for at least the years 1990–2015, which covers theMDG period of interest, using a dataset with almost 5,000 observations from variousdata sources.The remainder of the paper is structured as follows. Firstly, the dataset and model aresummarized in the next two sections. Some key results are then highlighted, includingmodel validation results, followed by a discussion of the work and possible futureavenues. Additional details about the model are provided in the Appendix.
Data on NMR are derived from either vital registration (VR) systems; sample vitalregistration (SVR) systems; or survey data. Data for a particular country may comefrom one or several of these sources, and the source may vary over time. Table 1summarizes the availability of data by source type.Data from VR systems are derived directly from the registered births and deaths in acountry. The observed NMR for a particular country and year is the number of regis-tered deaths within the first month divided by the number of live births. SVR systemsrefer to vital registration statistics that are collected on a representative sample of thebroader population.As well as using civil registration systems, NMR observations can also be derived fromdata collected in surveys. During a survey, a mother is asked to list a full historyof all births (and possible deaths) of her children. A retrospective series of NMRobservations can then be derived using the birth histories (Pedersen and Liu 2012).For the majority of survey data series (72 per cent), microdata are available and it ispossible to estimate the sampling error associated with each of the observations. Forthe remaining 28 per cent, there is not enough information to calculate the samplingerrors from the data and values are imputed (see Methods section). All rates, ratios ofrates, and corresponding standard errors were calculated from the survey microdatausing the software ‘CMRJack’ (Pedersen and Liu 2012). Optimized time series arecalculated such that all estimates have a coefficient of variation of less than 10 percent. 3he majority of survey data come from Demographic and Health Surveys (DHS) (Ta-ble 1). The category ‘Other DHS’ refers to non-standard DHS, that is, Special Interimand National DHS, Malaria Indicator Surveys, AIDS Indicator Surveys and WorldFertility Surveys (WFS). National DHS are surveys in DHS format that are run by anational agency, rather than the external DHS agency. The Multiple Indicator ClusterSurvey (MICS), developed by UNICEF in 1990, was originally designed to addresstrends in goals from the World Summit for Children, and has since focused on as-sessing progress towards the relevant MDG indicators. The ‘Other’ category includessurveys such as the Pan-Arab Project for Family Health and the Reproductive HealthSurveys.Data availability varies by country and by year. For most developed countries, a fulltime series of VR data exists. For other countries with VR data, the time series is oftenincomplete and is supported by other sources of data. Of the 105 countries where VRdata are available, 44 countries have incomplete VR times series. For some smallercountries with VR data, observations were recombined to avoid issues with erratictrends due to large stochastic variance. See Appendix B.3 for more details. SVR dataare only available for Bangladesh, China and South Africa. Most developing countrieshave no vital registration systems and so observations of the NMR are derived entirelyfrom surveys. A total of twelve countries had no available data.In terms of data inclusion, we follow the same inclusion exclusion rules as the UNIGME-estimated U5MR (IGME 2015). These exclusion rules are based on externalinformation which suggests some NMR observations are unreliable, due to, for example,poor survey quality or under coverage of VR systems. A total of 16 per cent of the4,678 observations were excluded.Table 1: Summary of the NMR data availability by source type and whether or notsampling errors were reported. The totals include observations that were excludedfrom the estimation.Source No. ofSeries No. ofCountries No. ofObs No. ofCountry-yearsVR 105 105 2607 2607SVR 3 3 79 78DHS (with sampling errors) 239 81 1212 934DHS (without sampling errors) 16 15 50 48Other DHS (with sampling errors) 52 42 251 251Other DHS (without sampling errors) 26 21 78 75MICS (with sampling errors) 16 14 81 73MICS (without sampling errors) 12 12 49 46Others (with sampling errors) 24 16 119 111Others (without sampling errors) 72 36 152 151Figure 1 illustrates examples of the data available for four countries. The shaded area4round the observations has a width of two times the sampling or stochastic error.The NMR for Australia (Figure 1a), as estimated from a full VR data time series, hasa trend over time which is relatively regular and the uncertainty is low. Data for SriLanka indicate NMR are roughly five times as high as Australia. Data are availablefrom 1950, but the VR data series is incomplete. The rest of the data comes fromWFS, DHS and National DHS. There are multiple estimates for some years, and theuncertainty around the estimates varies by source and year. The uncertainty aroundthe VR data is much less than for the survey data. The National DHS series do nothave estimates of sampling error. Iraq (Figure 1c) has no VR data, and the estimatesare constructed from MICS and two other surveys: the Infant and Child Mortalityand Nutrition Survey, and the Child and Maternal Mortality Survey. Again, there aremultiple estimates for some time points, and uncertainty level and availability varies.Finally, Vanuatu (Figure 1d) has only three observation points from one National DHS.In the NMR model, country-year specific U5MRs were used as explanatory variablesand to obtain final estimates. Point estimates, given by posterior medians, as well asposterior samples were obtained from the UN IGME (IGME 2015).
The aim is to produce estimates of the NMR for all countries in the world, and reportthe associated uncertainty around these estimates. The model needs to be flexibleenough to estimate NMR in a variety of situations, as illustrated in Figure 1. Theestimates should follow the data closely for countries with reliable data and low un-certainty. On the other hand, the model estimates need to be adequately smooth incountries with relatively large uncertainty and erratic trajectories. The model alsoneeds to be able to estimate NMR over the period 1990-2015 for all countries, includ-ing those countries where there are limited or no data available. To do so, the modelutilizes the relationship between the U5MR and NMR: as the level of U5MR decreases,the proportion of deaths under 5 that are neonatal increases. The model also allowsfor country-specific effects and time trends to capture data-driven trends in data-richcountries.
Write N c,t and U c,t as the NMR and U5MR for country c at time t , respectively, with U c,t given by the IGME U5MR estimate for that country-year. We explain the modelset-up in terms of the ratio R c,t = N c,t U c,t − N c,t ,
970 1980 1990 2000 2010
Year N M R VR (a) Australia Year N M R VR (Excluded)VRDHS 1987NDHS 1993*NDHS 2000*NDHS 2006*WFS 1975 (b) Sri Lanka
Year N M R MICS 2006MICS 2011Others 1991*Others 1999*Others 2004* (c) Iraq
Year N M R NDHS 2013* (d) Vanuatu
Figure 1: Neonatal Mortality Data (deaths per 1,000 births) for selected countries.Data series where sampling errors were not reported are marked with a *.6hich refers to the (true) ratio of neonatal deaths compared to deaths in months 2 to59. We constrain R c,t > ≤ N c,t U c,t ≤ R c,t is modeled as follows: R c,t = f ( U c,t ) · P c,t , (1)where f ( U c,t ) is the overall expected ratio given the current level of U5MR and P c,t isa country-specific multiplier to capture deviations from the overall relationship.The observed ratio r c,i , which refers to the i -th observation of the ratio in country c ,is expressed as a combination of the true ratio and some error, i.e. r c,i = R c,t [ c,i ] · (cid:15) c,i (2)= ⇒ log( r c,i ) = log( R c,t [ c,i ] ) + δ c,i for c = 1 , , . . . , C and i = 1 , . . . , n c , where C = 195 (the total number of countries)and n c is the number of observations for country c . The index t [ c, i ] refers to theobservation year for the i -th observation in country c , (cid:15) c,i is the error of observation i and δ c,i = log( ε c,i ). The first step in modeling the ratio of neonatal to non-neonatal deaths is to findan appropriate function f ( · ) in Equation 1, which captures the expected value of theratio given the current level of U5MR. Figure 2 shows a scatter plot of log-transformedobserved ratios log( r c,i ) versus log( U c,t [ c,i ] ). The relationship between the two variablesappears to be relatively constant up to around log( U c,t ) = 3, after which point thelog ratio decreases linearly with decreasing log( U c,t ). Given this observed relationship, f ( · ) is modeled as follows: f ( U c,t ) = β + β · (log( U c,t ) − log( θ )) [ U c,t >θ ] , with indicator function [ U c,t >θ ] = 1 for U c,t > θ and zero otherwise such that f ( U c,t ) = β for U c,t ≤ θ and f ( U c,t ) changes linearly with log( U c,t ) with slope β for log U c,t > log θ . Parameters β , β , and θ are unknown and are estimated.The fitted relationship between the ratio and the level of U5MR is illustrated in Fig-ure 2. The cutpoint θ is estimated to be around 34 deaths per 1,000 births (95% CI:[33, 57]). At U5MR levels that are higher than θ , the β coefficient suggests that a 1%increase in the U5MR leads to a 0.65% decrease (95% CI: [0.61, 0.71]) in the ratio R c,t .The fitted line is quite similar in shape to the loess curve fitted to the data, shown bythe red line in Figure 2. 7 l lllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllll llllllllllll lllllllll lllllllllllllllllll lllllllllllllll lll llllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllll lllllllll lllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllll lllll lllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllll llllll llllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllll lllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllll llllllllllllllllllllll llllllllllllllll llllll llllllllllllllllllllllllllllllllll llllllllllllllllll lllllllllllllllllllllllllllllllll llll llllll llllllllllllllllllllllllllllll llllllllllllllllllllll llllllll lllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllll lllllllllllllllllllllll lllll llllllllllllllllllllllllllllllllllllllll ll llllllllllllllll lllllllllllllllllllllllllllllll llllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllll llllll lllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllll llllllllllll lllllllllllllllll − − Estimated log(U5MR) O b s e r v ed l og (r a t i o ) lllll lllll llll lllllllllllllll lllll llll ll llllllllll llllllllllllllllll lllll llllllllllllllllllll lllllllllllllllllllllllllllllllll lll lllllllllllllllllllllllllllll lllllllllllll lllll lllllllllllllllllllllllll llllllllll ll lllllllll llllllllllllllllllllllllllll lllll lllllllllllllll lllll llllllllllllllllllll llllllllllllllllllll lllll llllllllllllllllllllllllllllll llllllll lllll llllll llllllllllllllllllllllllllllllllll lllll llllllll lllllllllll lllll llllllllllllllllll lllll lllllllllllllllllllllllllllllllllllll llllllllllll lllll lllllllllllllllll lllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllll lllll lllll llllllllllllllllll llllllll lll llllllllll lllll llllll lllll llllllll lllll lllllllllll lllll llllllllllllllllllll l llllllllllllllllllllllllllllllllllllllll lllll llllllllllllllll llllllllllllllllllllllllllll llllllllll lllllllllllll lllll lllllllllllllllllllllllll lllll lllllllllllllll llllllllllllllllllllllllllll llllllllllllll llllllllllllllllll lll lllll lllll llllllllllll lllllllllllllll lllllllllll lllllllllllll lllll lllllllllllll lllll llllllllllllllllllll llllll llllllllll lllll llllllllllllllllllllllll lllllllll llllllllllllllllllllllll lll llllllllll lllll lllll lllll lllll lllll llllllllllllllllllll llllllllll llllllllllllllllllll llllllllllllllllllll lllll llllllllllllllllllll lllllllllllllll lllll llllllllllllllllllll llllll llllllll llllllll llllllllllllll llllllllll lllllllll lllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllll lllll lllllllllllllllllll llllllllll lll llll llllllllllllll lllllllll ll lll lllllllllll llll lllll lllllllllllllllllllllllllllllllllllllllllllll lllll llllllll llllllll llllll lllllllllll llllllllllllll lllllllllllllll lllll llllllllll lllllllllllllll lllll lllll lllll lll lllll llllll lllll lllll llllll llllllllll llllllllllll llllllllllllll lll llllllll lllll lllllllllllllllllllllllll lllllllllllllllllllllllll llllllllll llll lllll lllllllllll llllllll lll lllllllllllllllllllll lllll lllll llllllllllllll lllllllllllllllllllllllll llllllllllllllllllllllllllllll Estimated log(f(U))Loess
Figure 2: Observed and estimated relation between the ratio of neonatal and non-neonatal deaths and under-five mortality. Observations r c,i are displayed with greydots and plotted against U c,t [ c,i ] . The Loess fit to the observations is shown in red, andthe estimated relation (function f ( U c,t )) is added in blue (dashed line).8 .2.2 Country-specific multiplier Although there is a relationship between the neonatal ratio and U5MR at the aggregatelevel, the relationship between R c,t and U c,t is likely to differ from country to country.For instance, some countries may have higher levels of NMR than what we expectgiven the level of U5MR. The model needs to account for this higher ratio of NMRto U5MR for these countries. However, within a particular country, the relationshipbetween NMR and U5MR may not be constant over time, so the model should beflexible enough to also allow for temporal changes. This is the purpose of the country-specific term P c,t in Equation 1: to capture data-driven differences across countriesand also within countries over time. The effect of the addition of P c,t for Nigeria isillustrated in Figure 3a. The blue dashed line represents the fit with just the the globalrelation f ( . ), as described above. The inclusion of the P c,t changes the fit from theblue line to the red line, which allows the trajectory to follow the data more closely.The country-year multiplier P c,t was modeled on the log-scale with a B-splines regres-sion model: log( P c,t ) = K c (cid:88) k =1 B c,k ( t ) α c,k , where B c,k ( t ) refers to the k th B-spline function for country c evaluated at time t and α c,k is the k -th splines coefficient for country c . The B-splines B c,k ( t ) were constructedusing cubic splines. Country c has a total of K c splines and K c knot points defined by t < t < · · · < t K c ; K c is the number of B-splines needed to cover the period up to2015 and back to 1990 or the start of the observation period, whichever is earlier. Interms of knot spacing, the same interval length of 2.5 years was used in each countryregardless of the number or spacing of observations. The consistent interval lengthwas chosen to be able to exchange information across countries about the variabilityin changes between spline coefficients.The country-specific multiplier P c,t for Nigeria is illustrated in Figure 3b. Each splineis represented in a different color at the bottom of the figure, and the gray dottedvertical lines indicate knot positions. The splines regression is fit to residual patternin the data on the log scale once the global relation is taken into account.A penalty was imposed on the first-order differences between spline coefficients α c,k to ensure a relatively smooth fit. To better see how the smoothing penalty is imple-mented, we rewrite the k -th splines coefficient for country c , α c,k , in the form of anintercept and fluctuations around the intercept (Eilers and Marx 1996; Currie andDurban, 2002): α c,k = λ c + [ D (cid:48) K c ( D K c D (cid:48) K c ) − ε c ] k , (3)where D K c is a first-order difference matrix: D K c i,i = − D K c i,i +1 = 1 and D K c i,j = 0otherwise; and ε c is a vector of length Q c : ε c = ( ε c, ...ε c,Q c ) (cid:48) , Q c = K c −
1. Each element of the vector is the first order difference of thecoefficients α c,q i.e. ε c,q = α c,q +1 − α c,q = ∆ α c,q for q = 1 ...Q c .The λ c represent country-specific deviations in level from the overall global relationshipbetween R c,t and U c,t . As such the λ c ’s are modeled centered at zero so that λ c ∼ N (0 , σ λ ) . The λ c is similar to a country-specific intercept. If λ c >
1, then country c ’s levelof NMR is generally higher than expected given its U5MR level, and vice versa for λ c <
1. For Nigeria (Figure 3a), the estimated country specific intercept is negative,and so the addition of the country-specific intercept lowers the fit from the blue line(global relation only) to the green line.The ε c term represents fluctuations around this country-specific intercept in the formof first-order differences in adjacent spline coefficients. These fluctuation terms allowfor the P c,t term to be influenced by the changes in the level of the underlying data.The first-order differences in adjacent spline coefficients are penalized to guarantee thesmoothness of the resulting trajectory as follows: ε c,q ∼ N (0 , σ ε c ) , where variance σ ε c essentially acts as a country-specific smoothing parameter. Thesmoothness of a particular country’s trajectory depends on the regularity of the trendin the data and also the measurement errors associated with the data points. As σ ε c decreases, the fluctuations go to zero, and the α c,k ’s become a country-specific interceptwith no change over time. The σ ε c is modeled hierarchically:log( σ ε c ) ∼ N ( χ, ψ σ ) , (4)where e χ can be interpreted as a ‘global smoothing parameter’ and ψ σ reflects theacross-country variability in smoothing parameters. The hierarchical structure of themodel allows information on the amount of smoothing to be shared across countries.The countries with fewer data points and thus less information about the level ofsmoothness borrow strength from countries with more observations.10
970 1980 1990 2000 2010
Year N M R Global relationGlobal relation + interceptFinal fit (Global relation + intercept + fluctuations) (a) The three components of fit: the global relation f ( . ), and P c,t , which consists of a country-specifc in-tercept and fluctuations. l l l l l − . − . − . . . Year l og ( da t a ) − l og ( e x pe c t ed ) l l l l ll l l l ll l l l ll l l l l log ( P ct ) (b) Estimate of log P c,t for Nigeria using splines re-gression. The B-splines have been scaled verticallyfor display purposes. The gray dotted vertical linesindicate knot positions (every 2.5 years). Figure 3: Illustration of the global relation, splines regression and combined compo-nents for Nigera 11 .3 Data model
Equation 5 indicates the observed ratio r c,i is modeled on the log-scale as the trueratio R c,t plus some error term δ c,i . This error term δ c,i is modeled differently basedon the source of the data. VR data
For VR data series, the error term δ c,i is modeled as δ c,i ∼ N (0 , τ c,i ) , where τ c,i is the stochastic standard error. These are obtained based on standardassumptions about the distribution of deaths in the first month of life. Details aregiven in the Appendix. Non-VR data
For the non-VR data, the error term δ i is modeled as δ c,i ∼ N (0 , ν c,i + ω s [ c,i ] ) , where ν c,i is the sampling error and ω s [ c,i ] is non-sampling error of the series type s of observation i in country c . Non-sampling error variances are estimated separatelyfor each of the series types listed in Table 1: DHS, Other DHS, MICS, and Others.The distinction by series type was made to allow for the possibility that a particularsurvey group may run a survey in a similar fashion across all countries, and as suchmay display similar characteristics in terms of non-sampling error.Sampling error variances were reported for the majority of the non-VR observations(see Table 1). For those observations ( c, i ) where sampling error was not reported,the sampling error was imputed based on the median value of all observed samplingerrors of series type s [ c, i ] within the group-size category of country c , which is small orother. A country was categorized as ‘small’ if the annual number of births was in thelowest quartile of all countries (corresponding to a maximum of around 25,000 birthsper year). The distinction between small and other countries was made due to thelarge differences in observed standard errors. The imputed values for missing standarderrors for each size category and series type are shown in Table 2. The model described so far produces estimates of R c,t . The corresponding estimate of N c,t is obtained by transforming the ratio and combining it with U c,t : N c,t = logit − ( R c,t ) · U c,t . U c,t estimates, the N c,t estimatesare generated by randomly combining posterior draws of R c,t and of U c,t . The result isa series of trajectories of N c,t over time. The best estimate is taken to be the medianof these trajectories and the 2.5th and 97.5th percentiles are used to construct 95%credible intervals. Other aspects of the method, including the projection method, estimation for countrieswith no data and crisis and HIV/AIDS adjustments are detailed in Appendix B.
The hierarchical model detailed in the previous sections is summarized in AppendixA. The model was fitted in a Bayesian framework using the statistical software R.Samples were taken from the posterior distributions of the parameters via a MarkovChain Monte Carlo (MCMC) algorithm. This was performed through the use of JAGSsoftware (Plummer 2003).In terms of computation, three chains with different starting points were run with atotal of 20,000 iterations in each chain. Of these, the first 10,000 iterations in eachchain were discarded as burn-in and every 10th iteration after was retained. Thus 1,000samples were retained from each chain, meaning there were 3,000 samples retained foreach estimated parameter.Trace plots were checked to ensure adequate mixing and that the chains were past theburn-in phase. Gelman’s ˆ R (Gelman and Rubin 1992) and the effective sample sizewere checked to ensure a large enough and representative sample from the posteriordistribution. The value of ˆ R for all parameters estimated was less than 1.1.13 Results
Estimates of NMR were produced for the 195 UN member countries for at least theperiod 1990 – 2015, with periods starting earlier if data were available. In this sectionsome key results are highlighted. Results are also compared to those produced by themethod previously used by the IGME.The estimated global relation (Table 3) suggests that the relationship between theratio and U5MR is constant up to a U5MR of 34 deaths per 1,000 births, the ratioof neonatal to other child mortality is constant at around 1.20 (95% CI: [1.03, 1.25]).This is equivalent to saying the proportion of deaths under-five that are neonatal isconstant at around 54 per cent (95% CI: [50, 55]). Above a U5MR 34/1000, theestimated coefficient suggests that, at the global level, a 1 per cent increase in theU5MR leads to a 0.6 per cent decrease in the ratio.Table 3: Estimates for parameters in global relationMedian 95% CI β β -0.62 (-0.61, -0.71) U cut The fits for the four countries illustrated in Section 2 are shown in in Figures 4. ForAustralia (Figure 4a), the estimated red line follows the data closely, given the smalluncertainty levels around the data. There has been a steady decrease in NMR since1970. In earlier time periods, the level of NMR was higher than the expected level(that is, the red line is higher than the blue). This switched in the 1980s and 1990s,and more recently, the estimated and expected levels are close.For Sri Lanka (Figure 4b), the estimates of NMR are informed by the combinationof VR and survey data. The VR has a greater influence on the trajectory because ofthe smaller associated standard errors. In the earlier years, the uncertainty intervalsaround the estimate are larger due to the higher uncertainty of the data. Thereis a small spike in the estimate in the year 2004, which is a tsunami-related crisisadjustment.No VR data were available for Iraq (Figure 4c), and the larger sampling errors aroundthe survey data have led to relatively wide uncertainty intervals over the entire period.This is in contrast to Sri Lanka, where uncertainty intervals became more narrow onceVR data was available. The larger sampling errors in Iraq have also led to a relativelysmooth fit (high smoothing parameter), and the shape of the trajectory essentiallyfollows the shape of the expected line. 14or Vanuatu (Figure 4d), the trajectory is driven by the expected trajectory givenVanuatu’s trend in U5MR. The available data determine the country-specific interceptfor Vanuatu, which is lower than the expected level. However, the relative absence ofdata for this country means that the uncertainty around the estimates is high.
The set-up of the model allows for the intuitive interpretation of results in comparingthe estimated level of NMR to the expected level given the U5MR. We define a countryto be outlying if the the estimated NMR in 2015 was significantly higher or lower thanthe expected level by at least 10 per cent. That is, the ratio of estimated-to-expectedwas at least 1.1 or less than 0.9 in 2015, and statistically significant at the 5 per centlevel. Figure 5 illustrates these countries, and the values of estimated-to-expected in1990 and 2015.Countries that have a lower-than-expected NMR include Japan, Singapore and SouthKorea, and some African countries such as South Africa and Swaziland. Countriesthat have a higher-than-expected NMR include several Southern Asia countries, suchas Bangladesh, Nepal, India and Pakistan. The former Yugoslavian countries Croatia,Bosnia and Herzegovina, and Montenegro also have higher-than-expected NMR.Figure 6 shows estimates through time for two contrasting countries, Japan, which haslower-than-expected NMR and India, with higher-than-expected NMR. In each of thefigures, the red line represents the estimated fitted line (with 90% CIs). The blue linerepresents the estimation with the f ( U c,t ) only (without the country-specific effect, P c,t ). The blue line can be interpreted as the expected level of NMR in a particularyear given the level of U5MR. The gap between the expected and estimated is beingsustained through time for Japan, and has widened since the 1970s. The change inNMR levels for India has been dramatic. Not only is the current NMR around 30 percent of what it was in 1970, the discrepancy between the expected and estimated levelsis decreasing through time, and is no longer significant. The smoothness of the fluctuations σ ε c is modeled hierarchically, assuming a log-normaldistribution with a mean parameter χ (see Equation 4). Smoothing parameters canalso be expressed in terms of precision, 1 /σ ε c ; Figure 7 shows the distribution ofestimated precisions for all countries. The larger the value of the smoothing parameter(precision), the smoother the fit. The estimate of the mean smoothing parameter wasaround 59 (95% CI: [40, 83]).Larger values of smoothing parameters were estimated for countries that had no avail-15
970 1980 1990 2000 2010
Year N M R EstimatedExpectedVR (a) Australia
Year N M R EstimatedExpectedVR (Excluded)VRDHS 1987NDHS 1993NDHS 2000NDHS 2006WFS 1975 (b) Sri Lanka
Year N M R EstimatedExpectedMICS 2006MICS 2011Others 1991Others 1999Others 2004 (c) Iraq
Year N M R EstimatedExpectedNDHS 2013 (d) Vanuatu
Figure 4: Observed and estimated neonatal mortality (deaths per 1,000 births) forselected countries 16 .5 1.0 1.5
Ratio of Estimated − to − Expected NMR
RatioPakistanRepublic of MoldovaDominicaDominican RepublicBosnia and HerzegovinaDenmarkIndiaBangladeshNepalMaltaKuwaitNigerBrunei DarussalamCubaBelarusSingaporeSwazilandJapanSouth AfricaBahrain ●●●●●●●●●●●●●●●●●●●● ●●●● ●● ●●●●●● ●● ●●●● ● ● ●●
Figure 5: Ratio of estimated to expected NMR for outlying countries
Year N M R EstimatedExpectedVR (a) Japan
Year N M R EstimatedExpectedDHS 1992−1993DHS 1998−1999DHS 2005−2006SVR (b) India
Figure 6: Higher and lower-than-expected countries17ble VR data but many observations from survey data. Senegal, which had the highestsmoothing parameter at a value of 582 (95% CI: [85, 5828]), had a total of 55 obser-vations over a 45-year period (Figure 8a). The effect of having many observationswith relatively large standard errors is a relatively smooth fit. In contrast, one of thesmallest smoothing parameters occurred for Cuba, at around 4 (95% CI: [2, 8]). Cubais a country with good quality VR-data, which has relatively small standard errors.This means the fitted line follows the data more closely (Figure 8b). s e c2 D en s i t y . . . . MeanCubaSenegal
Figure 7
It is useful to compare the results of this new model to the NMR results from the modelpreviously used by the IGME. The previous model is described in Oestergaard et al.(2011). In this method, NMR estimates for countries with complete VR series are takendirectly from the data. For countries without a complete VR series, a multilevel modelis fit using U5MR as a predictor. A quadratic relationship with U5MR is specified. Inaddition, the model allows for country-level and region-level random effects:log(
N M R c,t ) = α + β log( U c,t ) + β (log U c,t ) (cid:124) (cid:123)(cid:122) (cid:125) log( f ( U c,t )) + α country [ i ] + α region [ i ] (cid:124) (cid:123)(cid:122) (cid:125) log( P c,t )
960 1970 1980 1990 2000 2010
Year N M R EstimatedExpectedDHS 1986DHS 1992−1993DHS 1997DHS 1999−2000DHS 2005DHS 2010−2011DHS 2012−2013DHS 2014MIS 2008−2009WFS 1978 (a) Senegal
Year N M R EstimatedExpectedVR (b) Cuba
Figure 8: Countries with small and large amount of smoothingFor comparison, the new model is:log( R c,t ) = β + β · (log( U c,t ) − log( θ )) [log( U c,t ) > log( θ )] (cid:124) (cid:123)(cid:122) (cid:125) log( f ( U c,t )) + K c (cid:88) k B c,k ( t ) α c,k (cid:124) (cid:123)(cid:122) (cid:125) log( P c,t ) The existing model is similar in that it estimates NMR as a function of U5MR, plussome additional country-specific effect, i.e. there is an f ( U c,t ) and a P c,t . However, oneof the main differences between the two models is that for countries with non-VR data,estimates from the new model can be driven by the data, while the previous model isrestricted to follow the trajectory of the U5MR in a particular country, plus or minussome country-specific intercept. The other differences between the two models arehighlighted in Table 4. Table 4: Comparison of two modelsIGME 2014 New modelModel used for non-VR countries Model used for all countries f ( U c,t ) is quadratic f ( U c,t ) is linear with changing slope P c,t is a country and region-specific intercept P c,t is a country-specific int + fluctuationsCountry-specific effect constant over time Country-specific effect can change over timeNo data model Data modelFigure 9 compares the results of four countries to the estimates from the current IGMEmodel. The charts are focused to just show the period 1990–2015, which is the period19or which the IGME publishes estimates.The estimates from the existing IGME modelgenerally follow the same trajectory as the expected line, as determined by U5MRpatterns, and is shifted up or down depending on the estimate of the country-specificeffect. In contrast, the estimates from the new model follow the data more closely.The fluctuation part of the country-specific multiplier P c,t allows the estimated line tomove above or below the expected line, as is the case with the Dominican Republic(Figure 9d). In addition there is generally less uncertainty around the estimates in thenew model, especially in periods where there are data. Model performance was assessed through an out-of-sample model validation exercise.In creating a training dataset, rather than removing observations at random, theprocess of removing data was chosen to emulate the way in which new data may bereceived. Mortality databases are updated as least once a year as more data becomeavailable. These updates may include not only data for the most recent time periodbut may also include, for example, retrospective estimates from a survey. Ideally themodel should not be sensitive to updates of historical data so estimates do not changefrom year to year.The training set was constructed by leaving out the most recent survey data series,and for countries with only one series (including VR countries), the most recent 20per cent of data observations were removed. The resulting training data set was madeof around 80 per cent of the total data available.For the left-out observations, the absolute relative error is defined by e i = | n i − ˜ n i | ˜ n i , where where ˜ n i denotes the posterior median of the predictive distribution for a left-outobservation n i based on the training set. Coverage is defined by1 N (cid:88) n i ≥ l c [ i ] ( t [ i ])]1[ n i < r c [ i ] ( t [ i ])]where N is the the total number of left-out observations considered and l c [ i ] ( t [ i ]) and r c [ i ] ( t [ i ]) the lower and upper bounds of the predictions intervals for the i th observation.Coverage at the 80, 90 and 95 per cent levels were considered.The validation measures were calculated for 100 sets of left-out observations, whereeach set consisted of a random sample of one left-out observation per country. Table5 shows the median and standard deviation of each validation measure. The medianabsolute relative error between the observations and estimated value was less than 10per cent, and the prediction intervals roughly correspond to a similar coverage of leftout observations. 20
990 1995 2000 2005 2010 2015
Year N M R EstimatedExpectedIGMEDHS 1993−1994DHS 1996−1997DHS 1999−2000DHS 2004DHS 2007DHS 2011DHS 2014Special DHS 2001SVRWFS 1975−1976 (a) Bangladesh
Year N M R EstimatedExpectedIGMEDHS 1993DHS 1998−1999DHS 2003DHS 2010 (b) Burkina Faso
Year N M R EstimatedExpectedIGMEDHS 1997 (Excluded)DHS 2003−2004DHS 2011MICS 2008 (c) Mozambique
Year N M R EstimatedExpectedIGMEVR (Excluded)DHS 1986DHS 1991DHS 1996DHS 1999 (Excluded)DHS 2002DHS 2007DHS 2013DHS 2014MICS 2006WFS 1975 (d) Dominican Republic
Figure 9: Estimated Neonatal Mortality (deaths per 1,000 births); new versus IGME2014 model 21able 5: Validation measures, left-out dataMedian Std. DevAbsolute relative error 0.086 0.02480% coverage 0.788 0.26790% coverage 0.878 0.21695% coverage 0.926 0.178A similar set of validation measures was calculated comparing the model estimatesbased on the training data set with the model estimates based on the full data set.Results in Table 6 are reported for estimates up to (and including) 2005, and post2005. Model performance is better prior to 2005. This is due to the most recent databeing removed, so the data prior to 2005 would be very similar between training andtest sets. However the post-2005 measures show estimates are reasonably consistentfrom using the reduced and full datasets.Table 6: Validation measures, model comparison ≤ > A new model was introduced for estimating neonatal mortality rates. The model canbe expressed as the product of an overall relationship with U5MR and a country-specific effect. The overall relationship with U5MR is a simple linear function, whilethe country-specific effect is modeled through B-spline regression as a country-specificintercept plus fluctuations around that intercept. Estimates of the NMR were producedfor 195 countries, spanning at least the period 1990–2015.The model appears to perform reasonably well in a wide variety of situations whereextent and type of data available varies. In many developed countries, where VR dataseries are complete and uncertainty around the data is low, NMR estimates follow thedata closely. On the other hand, where there is limited data available or if uncertaintyaround the data is high, estimates are more influenced by the trends in U5MR.Model estimates were compared to estimates from the existing IGME model. Thenotable advantage of this model is that trends in NMR for countries without VR dataare driven by the data itself, rather than just reflecting trends in U5MR, as is the casewith the existing model. Another advantage of this model is that it is along the same22ethodological lines as the current model used by IGME to estimate U5MR (Alkemaand New, 2014).There are several avenues worth investigating further research. The choice of a linearfunction with changing slope for f ( U c,t ) was a data-driven decision, based on theobserved relationship in Figure 2. It would be interesting to compare the performanceof models which have a functional form that draws upon existing demographic models.For example, an extended version of the Brass relational logistic model (Brass 1971)and Siler models (Siler 1983) can be used to predict survival in the first months of life,as a function of the survivorship at older ages.The potential for bias in estimates from survey data is always a concern. Bias mayoccur from interviewing a sample that is not representative of the overall population,from selective omission of answers, and can even be influenced by the length of thesurvey administered (Curtis 1995; Bradley 2015). The data model included an estima-tion of an overall level of non-sampling error for each survey type, which may accountfor some biases. However, there is scope to further extend the data model to try tobetter estimate potential bias in survey data estimates.The focus of this paper was on the methodology. Future work will also focus oninterpretation of results. More investigation is needed on what is potentially causingNMR to be higher- or lower-than expected in outlying countries and whether theseare real effects or artifacts of data issues. This distinction is an important one andwill become even more so as the focus on child mortality continues to shift towardsthe early months of life. 23 References
Alkema, L, and New, JR (2014). ‘Global estimation of child mortality using aBayesian B-spline bias-reduction method.’ The Annals of Applied Statistics8(4): 2122–2149.Bhutta ZA, Chopra M, Axelson H, Berman P, Boerma T, Bryce J, Bustreo F, Cav-agnero E, Cometto G, Daelmans B, de Francisco A, Fogstad H, Gupta N, LaskiL, Lawn J, Maliqi B, Mason E, Pitt C, Requejo J, Starrs A, Victora CG, Ward-law T (2010). ‘Countdown to 2015 decade report (2000–10): taking stock ofmaternal, newborn, and child survival.’ The Lancet, 375, 2032–2044.Bradley, SEK (2015). ‘More Questions, More Bias? An Assessment of the Qualityof Data Used for Direct Estimation of Infant and Child Mortality in the De-mographic and Health Surveys.’ Paper presented at PAA 2015. Available at: http://paa2015.princeton.edu/uploads/152375 .Brass, W (1971). ‘On the Scale of Mortality.’ In: Brass, W (ed.). Biological Aspectsof Demography. London: Taylor & Francis.Centre for Research on the Epidemiology of Disasters (CRED) (2012). EM-DAT:The CRED International Disaster Database. Available at: .Currie, ID and Durban, M. (2002). ‘Flexible smoothing with P-splines: A unifiedapproach.’ Statistical Modelling, 4: 333–349.Demographic and Health Surveys (DHS) (2012). Guide to DHS Statistics: Demo-graphic and Health Surveys Methodology. Available at: .Eilers, PHC and Marx, BD (1996). ‘Flexible smoothing with B-splines and penalties.’Statistical Science, 11 89–121.Gelman, A and Rubin, D (1992). ‘Inference from iterative simulation using multiplesequences.’ Statistical Science, 7: 457–511.Girosi, F and King, (2007). ‘Demographic Forecasting.’ New Jersey: PrincetonUniversity Press.Lawn J, Cousens S, Zupan J and the Lancet Neonatal Survival Steering Team (2004).‘4 million neonatal deaths: When? Where? Why?’ The Lancet, 365, 891–900.Liu L, Johnson HL, Cousens S, Perin J, Scott S, Lawn J, Rudan I, Campbell H,Cibulskis R, Li M, Mathers C, Black RE (2012). ‘Global, regional, and nationalcauses of child mortality: an updated systematic analysis for 2010 with timetrends since 2000.’ The Lancet, 379, 2151–2161.24ozano R, Wang H, Foreman KJ, Rajaratnam JK, Naghavi M, Marcus JR, Dwyer-Lindgren L, Lofgren KT, Phillips D, Atkinson C, Lopez AD, Murray CJL (2011).‘Progress towards Millennium Development Goals 4 and 5 on maternal and childmortality: an updated systematic analysis.’ The Lancet, 378, 1139–1169.Mahy, M (2003). ‘Measuring child mortality in AIDS-affected countries.’ Paperpresented at the workshop on HIV/AIDS and adult mortality in Developingcountries. Available at: .Oestergaard, M.Z., Inoue, M, Yoshida, S., Mahanani, W.R., Gore, F.M., Cousens, S.,Lawn, J.E, and Mathers, C.D., (2011). ‘Neonatal Mortality Levels for 193 Coun-tries in 2009 with Trends since 1990: A Systematic Analysis of Progress, Projec-tions, and Priorities’. PLoS Medicine. DOI: 10.1371/journal.pmed.1001080Pedersen, J and Liu, J (2012). ‘Child mortality estimation: appropriate time pe-riods for child mortality estimates from full birth histories.’ PLoS Medicine.DOI:10.1371/journal.pmed.1001289Plummer, M. (2003). ‘JAGS: A Program for Analysis of Bayesian Graphical ModelsUsing Gibbs Sampling.’ In Proceedings of the 3rd International Workshop onDistributed Statistical Computing (DSC 2003), March 20-22, Vienna, Austria.ISSN 1609-395X. Available at: http://mcmc-jags.sourceforge.net/ .Price M, Klingner J, Ball P (2013) ‘Preliminary Statistical Analysis of Documentationof Killings in Syria. United Nations Office of the High Commissioner for HumanRights (OHCHR) technical report. Available at:
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The full model is summarized below. r c,i ∼ N ( R c,t [ c,i ] , δ i ) δ i = (cid:40) τ c,i for VR data ,ν c,i + ω s [ c,i ] for non-VR data R c,t = f ( U c,t ) · P c,t log( f ( U c,t )) = β + β · (log( U c,t ) − log( θ )) [ U c,t >θ ] log( P c,t ) = K c (cid:88) k =1 B k ( t ) α c,k α c,k = λ c + [ D (cid:48) K c ( D K c D (cid:48) K c ) − ε c ] k λ c ∼ N (0 , σ λ ) ε c,q ∼ N (0 , σ ε c )log( σ ε c ) ∼ N ( χ, ψ )where • R c,t is the true ratio in country c at time t , R c,t = N c,t U c,t − N c,t , where N ct and U c,t are the NMR and U5MR for country c at time t , respectively. • r c,i is observation i of the ratio in country c . • τ c,i is the stochastic standard error, ν c,i is the sampling error and ω s [ c,i ] is non-sampling error for series type s . • β is global intercept; β is global slope with respect to U5MR; θ is the level ofU5MR at which β begins to act. • P c,t is country-specific multiplier for country c at time t . • B k ( t ) is the k th basis spline evaluated at time t and α c,k is splines coefficient k . • λ c is the splines intercept for country c . • ε c,q are fluctuations around the country-specific intercept. • σ ε c is country-specific smoothing parameter, modeled hierarchically on the log-scale with mean χ and variance ψ . 27he model was fit in a Bayesian framework. Priors are given by ω ∼ U (0 , β ∼ N (0 , β ∼ N (0 , θ ∼ U (0 , σ λc ∼ U (0 , χ ∼ N (0 , ψ ∼ U (0 , B Other aspects of the method
B.1 Stochastic errors for VR model
Recall that the observed ratio r c,i , which refers to the i -th observation of the ratio incountry c , is expressed as a combination of the true ratio and some error, i.e. r c,i = R c,t [ c,i ] · (cid:15) c,i = ⇒ log( r c,i ) = log( R c,t [ c,i ] ) + δ c,i for c = 1 , , . . . , C and i = 1 , . . . , n c , where C = 195 (the total number of countries)and n c is the number of observations for country c . The index t [ c, i ] refers to theobservation year for the i -th observation in country c , (cid:15) c,i is the error of observation i and δ c,i = log( ε c,i ).For VR data series, the error term δ c,i is modeled as δ c,i ∼ N (0 , τ c,i ) , where τ c,i is the stochastic standard error. These can be obtained once some standardassumptions are made about the distribution of deaths in the first month of life. Weassume that deaths in below age five d are distributed d ∼ P ois ( B × q )where B = live births and q = the probability of death between ages 0 and 5.Additionally, we assume deaths in the first month of life d n are distributed d n ∼ Bin ( d , p )where p = n q / q and n q is the probability of death in the first month of life. Notethat the values of n q and q come from the raw data.The stochastic error was obtained via simulation. For each year corresponding toobservation i in country c , 28 A total of 3,000 simulations of under-five deaths d were drawn from a Poissondistribution d ( s )5 ∼ P ois ( B × q ); • A total of 3,000 simulations of neonatal deaths d n were drawn from a Binomialdistribution d ( s ) n ∼ Bin ( d ( s )5 , p ); • The ratio y ( s ) = logit (cid:18) d ( s ) n d ( s )5 (cid:19) was calculated for each of the simulated samplesand the standard error τ c,i was calculated as σ ( Y ) where Y = ( y (1) , y (2) , ...y ( s ) ), s = 3 , B.1.1 SVR data
For SVR data, the value for the sampling error was imputed based on the samplingerror for U c,t [ c,i ] SVR data, and the observed ratio between the stochastic error of r c,i and the stochastic error of U c,t [ c,i ] . On average, the stochastic error of r c,i was twice aslarge as the stochastic error of U c,t [ c,i ] . In addition, the sampling error for U c,t [ c,i ] SVRdata was assumed to be 10 per cent. As such a value of 20 per cent was imputed forthe sampling error for r c,i SVR data.
B.2 Projection
NMR estimates need to be produced up until 2015, but no data are available up tothis year, and many countries also had other recent years missing. As such, countrytrajectories needed to be projected forward to the year 2015.The parameters β , β and θ , which make up the expected relation with U c,t , are fixedover time, as is the country-specific intercept, λ c . The component that needs to beprojected is the random fluctuations part, the ε c,q in equation 3. Above, these ε c,q were assumed to be normally distributed around zero, with some variance σ εc . Thisassumption is used to project the ε c, (and thus the splines) forward.Start at the first α c,k that is past the last year of observed data. For each time periodto be projected: • Draw ε c,k ∼ N (0 , σ εc ) to obtain α c,k = ε c,k + α c,k − • Repeat to generate α k for k up to K c , where K c is knot number needed to coverthe period up to 2015.The simulated ε c,k are generally close to zero, so the method essentially propagatesthe level of the most recent α c,k that overlaps with the data period with the slope of29he expected trajectory, as determined by f ( U c,t ). The projection exercise is necessaryin order to maintain a consistent level of uncertainty in the estimates. B.3 Recalculation of VR data for small countries
There are several island nations and other small countries that have vital registrationdata available to calculate NMR. However, observations from these small countries areprone to large stochastic error, which can create erratic trends in NMR over time.To help avoid this issue, observations from adjacent time periods in a particular countryare recombined if the coefficient of variation of the observation is greater than 10 percent. The result is a smaller set of observations with smaller standard errors whichdisplay a smoother trend. Figure 10 shows the example of Saint Vincent and theGrenadines on which this process was applied.NMR is recalculated using the original NMR observations and annual number of livebirths. For two adjacent years that are to be recalculated: • The number neonatal deaths in each year is first calculated as NMR × live births. • The combined NMR for the two years is then total neonatal deaths divided bytotal births over the two years. • The standard error of the new NMR estimate is then recalculated based on theprocess described in 3.3.After recalculation, the coefficient of variation is calculated for the new estimate. If itis still >
10 per cent, the NMR is recalculated again, recombining with the previousadjacent year.
B.4 Crisis deaths
For some countries, there are known natural or political crises that have caused anexcess of deaths; for example the Rwandan genocide or, more recently, the Haiti earth-quake and conflict in Syria. For the crisis years, the survey data is unlikely to berepresentative of the actual number of deaths.Adjustments were made to the relevant crisis country-years, using estimates compliedby the World Health Organization (WHO). The WHO uses external data sourceson the number of deaths, including the Centre for Research on the Epidemiology ofDisasters International Disaster Database (CRED 2012) and estimates from UN Office30 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
Saint Vincent and the Grenadines Original data year N M R (a) Original data ● ● ● ● ● ● ● ● ● ● ● ● Saint Vincent and the GrenadinesRecalculated data
Year N M R (b) Recalculated data Figure 10: Recalculation of VR data: Saint Vincent and the Grenadinesof the High Commissioner for Human Rights for the Syrian conflict (Price 2013). TheWHO estimates the proportion of deaths that occur under the age of five (WHO 2013).From there, the best guess of the number of crisis deaths that occur within the firstmonth is simply 1/60th of the total deaths under five years.Estimation of crisis countries was firstly done without any crisis adjustments. Inaddition, the global relation with U5MR, f ( U c,t ), is fit to crisis-free U c,t estimates.The relevant adjustments to country-years were then made post estimation. This wasto ensure that the crisis deaths, which are specific to particular years, do no have aneffect on the splines estimation. B.5 HIV/AIDS countries
Although there have been vast improvements in recent years, many countries in Sub-Saharan Africa still suffer from relatively high levels of HIV/AIDS-related deaths. Thishas a substantial effect on the child mortality – if children living with HIV are noton antiretroviral treatment, a third will not reach their 1st birthday, and half will notreach their 2nd birthday (UNAIDS 2014). However, it is unlikely that children withHIV will die within the neonatal period, and so HIV/AIDS itself does not have andexplicit effect on the NMR (although there may be indirect effects on mortality, forexample through losing their mother to HIV) (Mahy 2003).Due to this disproportionate effect of HIV/AIDS on U5MR compared to NMR, thereare several adjustments made to the data used in the model, which leads to NMRbeing modeled as a function of ‘HIV-free’ U5MR. Firstly, the U5MR data used inthe ratio observations are adjusted to incorporate reporting bias. This adjustment31ccounts for the higher maternal mortality, which leads to under-estimation of childmortality from surveys (Walker et al. 2012). Once adjusted, the HIV/AIDS deathsare removed from U5MR, using estimates of deaths provided by UNAIDS (UNAIDS2014). The result is a ratio of neonatal to other child mortality which is free of HIVdeaths. In addition, the global relation with U5MR, f ( U c,t ), is fit to HIV-free data.Unlike the crisis adjustments, no HIV deaths were added in post-estimation, becauseit is assumed no neonatal deaths are due to HIV/AIDS. B.6 Countries with no data
There were twelve UN-member countries for which the IGME produces NMR estimatesfor, but where there are no available data. For these countries, the estimates of NMRare based on the global relation with U5MR, f ( U c,t ). Additionally, some steps areneeded to obtain the appropriate uncertainty around these estimates. For country c : • Draw λ c ∼ N (0 , σ λ ); • Set α = λ c ; • Draw ε ∼ N (0 , σ ε ); where σ ε = e χ is the global smoothing parameter, based onequation 4; • Set α = α + ε ; • Repeat to generate α k for k = 3 , . . . , K cc
There were twelve UN-member countries for which the IGME produces NMR estimatesfor, but where there are no available data. For these countries, the estimates of NMRare based on the global relation with U5MR, f ( U c,t ). Additionally, some steps areneeded to obtain the appropriate uncertainty around these estimates. For country c : • Draw λ c ∼ N (0 , σ λ ); • Set α = λ c ; • Draw ε ∼ N (0 , σ ε ); where σ ε = e χ is the global smoothing parameter, based onequation 4; • Set α = α + ε ; • Repeat to generate α k for k = 3 , . . . , K cc . K cc