Modeling Age Patterns of Under-5 Mortality

Papers:

Guillot M., J. Romero Prieto, A. Verhulst, P. Gerland. Modeling Age Patterns of Under-5 Mortality: Results From a Log-Quadratic Model Applied to High-Quality Vital Registration Data. Demography.

J. Romero Prieto J. , A. Verhulst, and M. Guillot. Estimating 1a0 and 4a1 in a Life Table: A Model Approach Based on Newly Collected Data. Demography.

Overview

This repository contains the R package logquad5q0 associated with the above-mentioned article published in the journal Demography.

The package uses the method of Lagrange to implement a log-quadratic model able to estimate the age pattern of under-5 mortality by detailed age. A variety of mortality inputs between 0 and 5 years can be used in order to predict a series of 22 cumulative probabilities of dying (q(x)) and mortality rates (nMx) for the first 5 years of life.

Based on these outputs, the package also computes the average number of years lived in the age intervals 0-1, 1-4 and 0-5 (i.e., 1a0, 4a1, and 5a0).

Data and R code needed to replicate the coefficients of the log-quadratic model are available on the website of the Under-5 Mortality Database (U5MD): https://web.sas.upenn.edu/global-age-patterns-under-five-mortality/

BETA VERSION: The package includes new coefficients derived from Demographic and Health Surveys (DHS) collected between 1985 and 2022 in sub-Saharan Africa and south Asia. These coefficients will be updated with the most recent DHS available before publication of the paper.

To report issues or suggest improvements, please contact andrea.verhulst@ined.fr

Authors of the package

• Andrea Verhulst - Development of the R code and package. - web page

• Julio Romero - Analytical development of the method of Langrage for solving the log-quadratic model (under MATLAB). - web page

Installation

You can install the released version of logquad5q0 with:

install.packages("devtools")
library(devtools)

install_github(repo = "verhulsta/logquad5q0")
library(logquad5q0)

Main function: lagrange5q0

The function lagrange5q0 predicts a series of 22 cumulative probabilities of dying (q(x)) and mortality rates (nMx) for the first 5 years of life, based on one or more mortality inputs between 0 and 5 years.

This function allows four types of mortality inputs:

1. A single mortality input (either nqx or nMx) for any age interval. The value of k will be assumed equal to 0 (average outcome of the model) and the value of the mortality input will be matched exactly.

2. A single mortality input (either nqx or nMx) for any age interval and a value of k. Both the mortality input and the value of k and will be matched exactly.

3. Two mortality inputs (either nqx or nMx) for any age interval. For the purpose of estimating nax–the average number of years lived in the age interval x,x+n–with high precision, the proportion of infant deaths before 28 days or 3 months (z(28d) or z(3m)) can be used as second input along with another mortality input (either nqx or nMx) starting at age 0. The two mortality inputs will be matched exactly and the value of k will be estimated.

4. More than two mortality inputs (either nqx or nMx) for any age interval. One of them must be selected for matching. The selected input will be matched exactly, the root mean square error will be minimized over the remaining mortality inputs, and the value of k will be estimated.

The computation of the root mean square error (RMSE) can be weighted. When minimizing a series of q(x), i.e. cumulative probabilities starting at age 0, we recommend using weights proportional to the length of the last age interval (see example 5 below). This approach was used to produce the results of the paper.

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Use the function format_data to prepare and verify the mortality inputs.

Age intervals must be defined with two exact ages in days (1 year = 365.25 days, 1 month = 30.4375 days). The log-quadradic model deals with the following age breakdowns:

Age	in Days
0	0
7 days	7
14 days	14
21 days	21
28 days	28
2 months	60.8750
3 months	91.3125
4 months	121.7500
5 months	152.1875
6 months	182.6250
7 months	213.0625
8 months	243.5000
9 months	273.9375
10 months	304.3750
11 months	334.8125
12 months	365.2500
15 months	456.5625
18 months	547.8750
21 months	639.1875
2 years	730.5000
3 years	1095.7500
4 years	1461.0000
5 years	1826.2500

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The function lagrange5q0 returns a list including:

• The predicted value of q(5y).
• The predicted value of k.
• A data frame with predicted values of q(x) and nMx.
• The predicted value of 1a0, 4a1, and 5a0.

Different age intervals are provived for the values of q(x), i.e. the cumulative probabilities of dying between 0 and age x, and for the values nMx, i.e. the central death rates between x and x+n.

Two types of 95% confidence intervals (CIs) are automatically provided when:

• CIs are based on the patterns of prediction errors observed in the underlying data when using a single mortality input (k = 0).
• CIs are based on the uncertainty of k when minimizing the RMSE over a full series of 22 q(x).

Caveat

The log-quadratic model is based on a set of vital records from Western countries. Predictions based on values of q(5y) above 0.150 and on values of k below -1.1 or above +1.5 are extrapolations. A warning message will be produced when such cases occur. Beyond these limits, it also might be impossible to converge to a solution. An error message will be produced in such case.

BETA VERSION

To use the new coefficients for sub-Saharan Africa and south Asia, specify the ‘B’ model in the function lagrange5q0 (see example 6 below).

In the case scenario of having a single mortality input and no information on the value of k, the user can specify a region (‘Eastern Africa’, ‘Middle Africa’, ‘Western Africa’, or ‘South Asia’) in order to benefit from a regional prior of k instead of assuming k = 0 (see example 6 below).

Examples

#1. One input (k = 0): q(28d,5y) (VR Jordan 2015)
input <- format_data(
   rate      = 0.00804138,
   lower_age = 28,
   upper_age = 365.25*5,
   type      = "qx",
   sex       = "total")

lagrange5q0(data = input)


#2. One input and k = 0.5: q(28d,5y) (VR Jordan 2015)
input <- format_data(
   rate      = 0.00804138,
   lower_age = 28,
   upper_age = 365.25*5,
   type      = "qx",
   sex       = "total")

lagrange5q0(data = input, k = 0.5)


#3. Two inputs: q(0,1y) and q(1y,4y) (VR Belgium 1984)
input <- format_data(
  lower_age = c(0,365.25),
  upper_age = c(365.25,365.25*5),
  rate      = c(0.00996356, 0.00207788),
  type      = c("qx", "qx"),
  sex       = c("male", "male"))

lagrange5q0(data = input)


#4. Two inputs: M(0,1y) and z(28d) (VR Australia 1935)
input <-  format_data(
  lower_age = c(0,0),
  upper_age = c(365.25, 28),
  rate      = c(0.04196866, 0.68614291),
  type      = c("mx", "zx"),
  sex       = c("total", "total"))

lagrange5q0(data = input)


#5. 22 inputs + 1 match: q(x) (VR Finland 1933)
data(fin1933)
fin1933$weight <- c(fin1933$n[1:22]/(365.25*5), NA) 
input <- format_data(
  lower_age = fin1933$lower_age,
  upper_age = fin1933$upper_age,
  rate      = fin1933$rate,
  type      = fin1933$type,
  sex       = fin1933$sex,
  fit       = fin1933$fit,
  weight    = fin1933$weight)

lagrange5q0(data = input)


#6. One input (Model B & regional k) : q(5y) (DHS DR Congo 2013-14)
input <- format_data(
   rate      = 0.10924,
   lower_age = 0,
   upper_age = 365.25*5,
   type      = "qx",
   sex       = "total")

lagrange5q0(data = input, model = 'B', region = "Middle Africa")