Bayesian space-time smoothing of direct estimates
Prepare person-month data
We first demonstrate the method that smooths the direct estimates of subnational-level U5MR. For this analysis, we consider the \(8\) Admin-1 region groups. In order to calculate the direct estimates of U5MR, we need the full birth history data in the format so that every row corresponds to a birth and columns that contain:
- Indicators corresponding to survey design, e.g., strata
(
v023), cluster (v001), and household (v002) - Survey weight (
v025) - Date of interview in century month codes (CMC) format, i.e., the
number of the month since the beginning of 1900 (
v008) - Date of child’s birth in CMC format (
b3) - Indicator for death of child (
b5) - Age of death of child in months (
b7)
The birth history data from DHS is already in this form and the
getBirths function default to use the current recode manual
column names (as indicated above). The name of these fields can be
defined explicitly in the function arguments too. We reorganize the data
into the ‘person-month’ format with getBirths function and
reorder the columns for better readability. Notice that DHS reports age
in years for children over 2 years old. Thus the b7
variable contains ages mostly in multiples of 12 after 24 months. This
truncation is adjusted by the age.truncate = 24 argument,
which adds 5 months to the age of the children for any records at least
24 months old and the age being multiples of 12. We also specify the
surveyyear argument to remove any observations that fall
into time periods after 2014, which could exist in the adjusted
dataset.
dat <- getBirths(data = births, strata = c("v023"), surveyyear = 2014, year.cut = seq(1985,
2020, by = 5))
dat <- dat[, c("v001", "v002", "v024", "time", "age", "v005", "strata", "died")]
colnames(dat) <- c("clustid", "id", "region", "time", "age", "weights", "strata",
"died")
head(dat)## clustid id region time age weights strata died
## 1 1 6 nairobi 05-09 0 5476381 1 0
## 2 1 6 nairobi 05-09 1-11 5476381 1 0
## 3 1 6 nairobi 05-09 1-11 5476381 1 0
## 4 1 6 nairobi 05-09 1-11 5476381 1 0
## 5 1 6 nairobi 05-09 1-11 5476381 1 0
## 6 1 6 nairobi 05-09 1-11 5476381 1 0Notice that we also need to specify the time intervals of interest. In this example, we with to calculate and predict U5MR in 5-year intervals from 1985-1990 to 2015-2019. For U5MR, we will use the discrete survival model to calculate direct estimates for each region and time. This step involves breaking down the age of each death into discrete intervals. The default option assumes a discrete survival model with six discrete hazards (probabilities of dying in a particular interval, given survival to the start of the interval) for each of the age bands: \([0,1), [1,12), [12,24), [24,36), [36,48)\), and \([48,60]\).
We may also calculate other types of mortality rates of interest
using getBirths. For example, for U1MR,
And the smoothing steps can be similarly carried out.
Horvitz-Thompson estimators of U5MR
Using the person-month format data, we can calculate Horvitz-Thompson
estimators using getDirect for a single survey or
getDirectList for multiple surveys. The discrete hazards in
each time interval are estimated using a logistic regression model, with
weighting to account for the survey design. The direct estimates are
then calculated using the discrete hazards. In order to correctly
account for survey design, We need to specify the stratification and
cluster variables. In the Kenya DHS example, a two-stage stratified
cluster sampling design was used, where strata are specified in the
strata column, and clusters are specified by the cluster ID
(clusterid) and household ID (id).
years <- levels(dat$time)
direct0 <- getDirect(births = dat, years = years, regionVar = "region", timeVar = "time",
clusterVar = "~clustid + id", ageVar = "age", weightsVar = "weights")
head(direct0)## region years mean lower upper logit.est var.est survey logit.prec
## 1 All 85-89 0.093 0.079 0.109 -2.3 0.0082 NA 122
## 2 All 90-94 0.098 0.089 0.109 -2.2 0.0033 NA 306
## 3 All 95-99 0.088 0.081 0.097 -2.3 0.0024 NA 409
## 4 All 00-04 0.078 0.072 0.084 -2.5 0.0016 NA 630
## 5 All 05-09 0.059 0.054 0.064 -2.8 0.0020 NA 506
## 6 All 10-14 0.051 0.047 0.056 -2.9 0.0021 NA 472Adjustments using external information
Sometimes additional information are available to adjust the direct estimates from the surveys. For example, in countries with high prevalence of HIV, estimates of U5MR can be biased, particularly before ART treatment became widely available. Pre-treatment HIV positive women had a high risk of dying, and such women who had given birth were therefore less likely to appear in surveys. The children of HIV positive women are also more likely to have a higher probability of dying compared to those born to HIV negative women. Thus we expect that the U5MR is underestimated if we do not adjust for the missing women.
Suppose we can obtain the ratio of the reported U5MR to the true U5MR, \(r_{it}\), at region \(i\) and time period \(t\), we can apply the adjustment factor to the direct estimates and the associated variances. The HIV adjustment factors were calculated for the 2014 Kenya DHS survey and included in the package.
National estimates of U5MR
The direct estimates calculated using getDirect contains
both national estimates and subnational estimates for the \(8\) regions, over the \(6\) time periods and the projection period
2015-2019. We first fit a model with temporal random effects only to
smooth the national estimates over time. In this part, we use the subset
of data region variable being “All”. We can fit a Random Walk 2 only
model defined on the 5-year period.
fit1 <- smoothDirect(data = direct, Amat = NULL, year_label = years, year_range = c(1985,
2019), time.model = "rw2", m = 1)We can also estimate the Random Walk 2 random effects on the yearly scale.
fit2 <- smoothDirect(data = direct, Amat = NULL, year_label = years, year_range = c(1985,
2019), time.model = "rw2", m = 5)The marginal posteriors are already stored in the fitted object. We use the following function to extract and re-arrange them.
We can compare the results visually. Notice to correctly display the period estimates, the reference year in each period needs to be specified. Here we simply take the median year in each period.
years.ref <- c(1987, 1992, 1997, 2002, 2007, 2012, 2017)
g1 <- plot(out1, year_med = years.ref) + ggtitle("National period model") + ylim(c(0,
0.17))
g2 <- plot(out2, year_med = years.ref) + ggtitle("National yearly model") + ylim(c(0,
0.17))
g1 + g2
The national model also allows us to benchmark the estimates using other published national results. For example, we take the 2019 UN-IGME estimates and find the median under-5 mortality rates in each periods. We can then calculate the ratio of the estimates from national models to the published UN estimates.
data(KenData)
UN <- KenData$IGME2019
UN.period <- data.frame(period = c("85-89", "90-94", "95-99", "00-04", "05-09", "10-14"),
median = NA)
for (i in 1:6) {
UN.period$median[i] <- median(UN$mean[which(UN$years %in% (c(1985:1989) + (i -
1) * 5))])
}
ratio <- subset(out1, region == "All")$median[1:6]/UN.period$median
print(ratio)## [1] 0.98 0.95 0.91 0.95 0.92 1.00We will use this adjustment ratio to correct the bias from our direct estimates. We organize the adjustment ratios into a matrix of two columns, since the adjustment factor only varies over time. We can then perform the benchmarking to the UN estimates similar to the HIV adjustment before.
benchmark <- data.frame(years = c("85-89", "90-94", "95-99", "00-04", "05-09", "10-14"),
ratio = ratio)
direct <- getAdjusted(data = direct, ratio = benchmark, time = "years", region = "region",
est = "mean", logit.lower = "logit.lower", logit.upper = "logit.upper")After benchmarking, we can fit the smoothing model again on the adjusted direct estimates, and see if they align with the UN estimates.
fit2.benchmark <- smoothDirect(data = direct, Amat = NULL, year_label = years, year_range = c(1985,
2019), time.model = "rw2", m = 5)
out2.benchmark <- getSmoothed(fit2.benchmark)
g1 <- plot(out2, year_label = years, year_med = years.ref, data.add = UN, option.add = list(point = "mean"),
label.add = "UN", color.add = "orange") + ggtitle("National yearly model: original") +
ylim(c(0, 0.17))
g2 <- plot(out2.benchmark, year_label = years, year_med = years.ref, data.add = UN,
option.add = list(point = "mean"), label.add = "UN", color.add = "orange") +
ggtitle("National yearly model: benchmarked") + ylim(c(0, 0.17))
g1 + g2
Subnational estimates of U5MR
The syntax to fit subnational smoothing model is similar. Similar to
the national model, we can choose to estimate temporal random effects on
either yearly or period level. We can also choose the four types of
space-time interaction terms using the st.type argument.
The default hyper priors on the precision of random effects are now PC
priors.
fit3 <- smoothDirect(data = direct, Amat = Amat, year_label = years, year_range = c(1985,
2019), time.model = "rw2", type.st = 4, m = 1)
out3 <- getSmoothed(fit3)Similarly we can also estimate the Random Walk 2 random effects on the yearly scale.
fit4 <- smoothDirect(data = direct, Amat = Amat, year_label = years, year_range = c(1985,
2019), time.model = "rw2", type.st = 4, m = 5)
out4 <- getSmoothed(fit4)
g1 <- plot(out3) + ggtitle("Subnational period model") + ylim(c(0, 0.3))
g2 <- plot(out4) + ggtitle("Subnational yearly model") + ylim(c(0, 0.3))
g1 + g2
We can also add back the direct estimates for comparison.
plot(out4, data.add = direct, option.add = list(point = "mean", by = "survey")) +
facet_wrap(~region, scales = "free")
We can show the estimates over time on maps. We revert the color scale so that higher mortality are represented by darker colors.
mapPlot(data = subset(out4, is.yearly == FALSE), geo = geo, variables = "years",
values = "median", by.data = "region", by.geo = "REGNAME", is.long = TRUE, ncol = 4,
direction = -1, legend.label = "U5MR", per1000 = TRUE)
In order to also illustrate uncertainties of the estimates when presented on maps, we can use hatching to indicate the width of the 94% posterior credible intervals.
hatchPlot(data = subset(out4, is.yearly == FALSE), geo = geo, variables = "years",
values = "median", by.data = "region", by.geo = "REGNAME", lower = "lower", upper = "upper",
is.long = TRUE, direction = -1, legend.label = "U5MR", per1000 = TRUE, ncol = 4)
