Proximate Determinants of Fertility in Eastern Africa: The case of Kenya, Rw...
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Longitudinal analysis of the under-5 mortality rate in Africa
1. Longitudinal analysis of under-five
mortality rates in Northern and sub-
Saharan Africa
STAT825: Research Project
Jane Shrapnel
Student Number: 41597133
Supervisor: Ken Beath
2. Abstract
As part of a review of the progress made by the United Nations Millennium Development Goals, this
research examined the fourth goal: the under-five mortality rate across sub-Saharan and Northern
Africa. Since sub-Saharan Africa looks to not achieve the goal of reducing child mortality rates by two-
thirds between 1990 and 2015, this paper focused on investigating if there was a significant changes in
the under-five mortality over the last 20 years, as well as across regions. To further explore any possible
circumstances or initiatives that contributed to the reduction or increase in child mortality, a model was
developed that explored the impact of possible explanatory variables on under-five mortality rates.
The methodology chosen for this research took into account the correlated nature of the data, choosing
a research technique that was appropriate for longitudinal data analysis. The data was modelled using a
generalised linear mixed model, with a random intercept and random slope of time, to take into the
account the variation in mortality rates across countries as well as over time. The covariates that were
added into the model were included as fixed effects.
The initial results showed a decrease in under-five mortality rates over time and across regions. Once
additional explanatory variables were added into the model, the region effect was no longer significant.
The final model showed four variables to have a significant impact on the under-five mortality rate: the
prevalence of tuberculosis; the proportion of the population with access to improved drinking water in
rural areas; the percentage of one year olds immunised against measles; and the proportion of the
population undernourished.
To conclude, this analysis revealed that investing in decreasing prevalence of preventable illness,
decreasing undernourishment within the population and increasing access to clean drinking water in
rural areas would have a significant impact on the under-five mortality rates across sub-Saharan and
Northern Africa.
3. Contents
Abstract.........................................................................................................................................................2
Introduction ..................................................................................................................................................4
Data...............................................................................................................................................................5
Variable definitions and formulas.................................................................................................................6
Data pre-processing......................................................................................................................................9
Preliminary data analysis ............................................................................................................................11
Statistical analysis .......................................................................................................................................15
Stage 1.....................................................................................................................................................15
Stage 2.....................................................................................................................................................18
Results.........................................................................................................................................................20
Discussion....................................................................................................................................................21
References ..................................................................................................................................................23
Appendix .....................................................................................................................................................25
Q-Q plots for under-five mortality rate over five year intervals.............................................................25
Stage 1 β Model codes and outputs .......................................................................................................26
Stage 2 β Model selection.......................................................................................................................38
Stage 2 β Model code and output...........................................................................................................39
4. Introduction
In 2000, the United Nations Millennium Declaration was signed by 189 heads of state, committing them
to achieve a set of goals to eradicate poverty in developing nations (United Nations General Assembly
2000). The Millennium Development Goals (MDGs) specified eight key goals that were thought to be
instrumental in reducing the poverty gap between developed and developing nations. These goals
included topics such as education, gender equality, maternal and child mortality, health and
international partnerships. Targets, to be reached by 2015, were set for each goal and a number of
different indicators were used to measure the progress of those goals (Waage et al 2010).
The focus of this paper will be on the fourth MDG which is to reduce child mortality rates. A target was
set to reduce rates of child mortality by two-thirds, from 1990 to 2015, for children under the age of
five. While there has been a great deal of progress made in achieving the target in many areas of the
world, parts of Asia and sub-Saharan Africa have fallen behind (Waage et al 2010; United Nations 2013).
While worldwide the under-five mortality rate has dropped by 47 per cent in 2012, 18,000 children a
day still died from preventable illness, with sub-Saharan Africa and Southern Asia accounting for 81 per
cent of these deaths. In fact, a sub-Saharan African child has a one in 10 chance of dying before the age
of five, 15 times higher than the average for developed regions (United Nations 2013). Therefore this
research will focus specifically on the African region, comparing the child mortality rates of Northern
and sub-Saharan African countries. The aim of this research is to assess whether there has been a
significant reduction in child mortality rates over time, and between regions, whilst also investigating
potential predictors of child mortality.
Within the fourth MDG, there are three indicators used to measure progress: under-five mortality rate;
infant mortality rate; and child immunisation rates of measles (United Nations n.d.). This research paper
will look specifically at the under-five mortality rate over time, as well as examine the effect other
indicators within the MDGs have had on reducing the under-five mortality rate.
There are a number of factors that are thought to contribute to the reduction in childhood mortality
rates. Preventable illness, such as pneumonia and malaria, under nutrition and length of the child being
breastfed are key health factors that contribute to higher mortality rates (United Nations 2013). The
characteristics and health of the mother are also key contributors to child mortality. Two studies, one
researching child mortality rate in Ethiopia and the other in Tanzania, identified the age of the mother as
a significant factor, with teenage mothers more likely to experience mortality risks than older mothers
(Susuman 2012; Susuman et al 2012). HIV positive mothers are more likely to experience child mortality,
than mothers not infected with HIV (Rajaratnam et al 2010). Another factor is the time between children
being born to the same mother, with children being born within 24 months of each other more likely not
to survive (Susuman 2012; Susuman et al 2012).
Previous studies in this area have either analysed causes of high child mortality rates at a point in time,
(Susuman 2012; Susuman et al 2012), or have developed a model applicable across over a one hundred
countries, to assess whether each countryβs child mortality rates has decreased (Rajaratnam et al 2010;
Lozano et al 2011). The two studies from Susuman (2012) and Susuman et al (2012) have analysed the
effect different covariates have on child mortality rates through linear and logistic regression
respectively, for two different countries. This type of research is important to investigate the effects of
specific policies and practices within countries, as this knowledge can be applied to other countries
facing similar problems. However, in some instances, it may not translate well to other countries due
5. differences such as climate, cultural norms or distribution of the population in urban and rural areas,
amongst other reason. This research paper has therefore focused on the covariates across countries, as
this will show which explanatory variables have had an effect across the African region, while controlling
for these other effects by measuring a cross section of countries.
The other research papers looked at changes to child mortality rates across the world. Rajaratnam et al
(2010) developed a generalised linear mixed model that was applied across all countries with available
data to assess the changes in child mortality rates in each country. Regional and national effects were
modelled as a random intercept and slope and year was included as a covariate. The same model was
used in Lozano et al (2011) analysis on child mortality rates. This paper has developed upon the
longitudinal model used in the previous analysis, by including a random time effect, and has introduced
covariates, based on previous research, into the model that helps explain some of the reduction in the
child mortality rate across Africa.
The proceeding sections will firstly introduce the data set and provide definitions of all indicators used.
Following this, the data pre-processing and preliminary data analysis are described. Next, the statistical
analysis will be presented in two parts. The first stage will look at developing a model on the longitudinal
effect of under-five mortality rate disaggregated by sub-Saharan and Northern Africa. Stage 2 will utilise
the model developed in stage 1 and introduce different indicators to find a model that best predicts the
reduction in under-five mortality rates across Africa. The results will show there has been a significant
reduction in under-five mortality rates in Africa and detail the covariates that had a statistically
significant effect on this. To conclude, this paper will discuss limitations within the data set and the focus
of under-five mortality rates in the next instalment of development goals.
Data
The data was sourced from the official United Nations site for the Millennium Development Goals
(MDG) Indictors (United Nations n.d.), which is managed by the United Nations Statistics Division. This
dataset has been compiled from multiple international sources that collate data from available national
statistical services for specific indicators. Available data sources include household or population-based
surveys, national population censuses and vital registration systems (United Nations n.d.). The data was
collected on a yearly basis within the period 1990 β 2012.
A subset of the full data set was extracted to only include Northern and sub-Saharan African countries
where the under-five mortality rate was collected. In total, 53 countries across Northern and sub-
Saharan Africa out of a potential 57 countries in these regions had statistics available on this indicator.
The potential predictors of the mortality rate were selected. These included adolescent birth rate, HIV
indicators, health care received during and after pregnancy and access to clean water and sanitation
facilities. However, due to either no data being collected or less than 30% of the data points were
collected for the time series, a high number of these indicators could not be included in the analysis. The
final explanatory variables were the percentage of the population who were undernourished, the
prevalence of tuberculosis, the child immunisation rate for measles, access to clean drinking and access
to improved sanitation facilities, both in urban and rural areas.
6. While under-five mortality rate had been collected up to 2012 in the 53 countries selected, all other
indicators included in the analysis contain data to 2011 only. Therefore the time series for the analysis
spans from 1990 to 2011.
The final list of variables that were used in the analysis is present in Table 1 below. The outcome variable
for the analysis was the under-five mortality rate of each country. The coding used for this, along with
the potential covariates used in the model, are detailed in the table, in addition to summary statistics for
each variable. Where needed, detailed definitions of some variables are given in the next section.
Table 1. Full set of indicators and corresponding coding used in analysis
Variable Coding Mean Std Dev.
Under five mortality rate U5MR (parts per 1,000) 119.58 58.88
Country Country
African region Region ID (1=sub-Saharan,
2=Northern Africa)
Year Year (1990 β 2011)
Percentage of population
undernourished
UNourish 26.61 17.73
Proportion of 1 year olds immunised
against measles
ChMeaImm 69.30 20.17
Prevalence rates associated with
tuberculosis
TuberPrev (parts per
100,000)
358.59 259.21
Proportion of urban population using
improved drinking water sources
WaterUrban 84.01 14.18
Proportion of rural population using
improved drinking water sources
WaterRural 55.18 21.64
Proportion of urban population using
improved sanitation facilities
SanUrban 49.71 24.93
Proportion of rural population using
improved sanitation facilities
SanRural 29.09 26.45
Variable definitions and formulas
Under-five mortality rate
Under-five mortality rate is the probability for a child born in a specified year to die before reaching the
age of five, if subjected to current age-specified mortality rates.
It is calculated as:
π5ππ (π) =
π·(0 β 4, π)
π΅(π)
Γ 1000
Where:
π = calendar year
π·(0 β 4, π) = number of deaths of children aged 0 β 4 in calendar year n
π΅(π) = number of live births in calendar year n
(United Nations n.d.)
7. Region
Africa is divided into two regions, the Northern region, which covers six countries across the Northern
part Africa, and the sub-Saharan region which includes all other countries in the continent. Sub-Saharan
Africa can be split into the north, south, east and west regions but for the purpose of this research, it
will compare the two broad regions. In this research paper, the Northern Africa region includes five of
the possible six countries and sub-Saharan Africa contains 48 countries out of 51 possible countries.
Population undernourished, percentage
This is defined as the proportion of the population below the minimum level of dietary energy
consumption. It is calculated under a probability distribution:
π(π) = π(π₯ < ππΏ) = β« π(π₯) ππ₯ = πΉπ₯(ππΏ)
.
π₯<π πΏ
Where:
π(π) = the proportion of the total population that is undernourished
(π₯) = dietary energy consumption intake
ππΏ = the cut-off point reflecting the minimum acceptable dietary energy consumption
π(π₯) = the density function of dietary energy intake
πΉπ₯ = cumulative distribution function
It is derived from three measures:
ο· The average amount of food available for human consumption per person
ο· The level of inequality in access to that food
ο· The minimum number of calories required for an average person.
Undernourishment is the proportion that falls below the cut-off point (ππΏ) on the cumulative distribution
curve (United Nations n.d.).
Immunisation rates of measles
The immunisation rate of measles is calculated as the proportion of children one year of age that have
received at least one dose of a measles vaccine.
The calculation for this indicator is:
1 π¦πππ β ππππ ππππ’πππ ππ‘πππ πππ‘π =
π
π
Γ 100
Where:
π = Number of vaccinations administered
T = Number of children in the target group (children 1 year of age)
The target group number is taken from population estimates. In some instances, if the rate of
vaccinations administered is high and/or the population numbers are underestimated the proportion
8. can exceed 100 per cent. In these instances, the data points have been recoded to 99 per cent (United
Nations n.d.).
Tuberculosis Prevalence
There are three indicators for tuberculosis in the 2015 millennium goals: incidence, prevalence and
death. There is strong collinearity between the three variables, particularly between prevalence of and
death from tuberculosis so only one indicator for tuberculosis was used in this study. Prevalence of
tuberculosis was chosen as this measures the current number of people suffering from the infection.
Over time, prevalence measures monitor the burden of the disease as duration time changes, rather
than incidence which records new incidence of tuberculosis. Death rates are also a good measure
however, in developing countries, death rates of the disease are not as reliable (United Nations n.d.).
Tuberculosis prevalence is calculated as the number of cases of bacterial infections of the disease at a
point in time. It is presented as per 100,000 of the population and is usually based upon estimates of the
incidence of tuberculosis.
ππ’πππππ’πππ ππ ππππ£ππππππ (π) = πΌππππππππ (π) Γ ππ’πππ‘πππ ππ ππππππ‘πππ (π)
Where:
π = calendar year
πΌππππππππ (π) = Incidence estimates derived from various methods based on available data in calendar
year n
π·π’πππ‘πππ ππ ππππππ‘πππ(π) = duration of tuberculosis in calendar year n. This is assumed to vary based
mainly on individual treatment received.
Proportion of the population using an improved drinking water source (urban and rural)
This indicator is expressed as a percentage and is derived from the proportion of the population that is
using improved drinking water, including piped or protected water supplies, compared to the population
that is using any type of drinking water.
This variable can be measured as a combined total of urban and rural percentages or it can be
disaggregated by these broad regional categories. For this analysis, it has been disaggregated to see if
there is a significant difference between urban and rural usage of drinking water. It must be noted that
there is no set definition for what constitutes an urban or rural area, so there may be variance between
countries due to each country specifying these areas themselves (World Health Organisation/UNICEF
n.d.). One limitation which specifically affects some African countries is that this indicator does not
include time travelled to access water or means of transporting water, which would vary greatly
between urban and rural regions.
This indicator was not collected on a yearly basis, but rather the available data has been plotted from
1980 to present. Using the least-squares method, a linear trend line has been drawn where two or more
points were present on the time scale, and were spaced five or more years apart. In some instances
where data points were not available across the whole time period, the regression line has been
inferred up to two years before or after the first or last data point, respectively. In instances where the
data was not available outside of these times, a flat line has been plotted for up to four years to
continue the time series (World Health Organisation/UNICEF n.d.).
9. Proportion of the population using improved sanitation facilities (urban and rural)
Similarly to the previous indicator this variable is measured by the population that has access to
sanitation facilities that hygienically separate human waste from human contact, such as flushing toilets,
piped sewage systems and sanitised pit latrines, divided by the total population that is using any type of
sanitation facilities. It is expressed as a percentage (World Health Organisation/UNICEF n.d.).
This indicator can also be presented in two ways, as a total proportion of the population that has access
to improved sanitation facilities, or disaggregated by urban and rural regions. Similarly to the percentage
that have access to improved drinking water, this indicator has been disaggregated in this paper to
analyse any differences between the two regions.
The numbers of people using improved sanitation facilities is collected from household surveys and
censuses which are not collected on a yearly basis. The same methodology in estimating the improved
used of drinking water facilities for the yearly proportions was used in this variable (World Health
Organisation/UNICEF n.d.).
Data pre-processing
While compiling this dataset, the United Nations conducted routine data pre-processing steps, for
example, imputing missing variables, testing and recoding for illogical values and validating data
collection methods. There were still further steps that needed to be conducted before the dataset was
ready for analysis. Within this dataset there were some variables that have missing data. These are
shown in Table 2 below.
Table 2. Variables with missing observations within the dataset
Variable # Missing % Missing
Percentage population, undernourished 53 4.55
Proportion of 1 year oldβs immunised against measles 22 1.89
Proportion of urban population using improved drinking
water sources
40 3.43
Proportion of rural population using improved drinking
water sources
46 3.95
Proportion of urban population using improved sanitation
facilities
47 4.03
Proportion of rural population using improved sanitation
facilities
44 3.77
From inspecting the data, the percentage of the population that was undernourished was missing for
each country for 1990, the first year of the time series. Since it was the consistent across all countries
and did not cause a break in the time series, no treatment of the missing data on this variable was
required. The impact of including these missing variables means that, if the percentage of
undernourished was included in the final model, the time series would span 20 years rather than 21
years. Reducing the time series by one year at the start of the series would have a negligible impact on
the relationship between the proportion of the population that is undernourished and the under-five
mortality rate.
Figure 1 shows the pattern of the missing data for the percentage of the population accessing improved
drinking water in the urban region. The other three variables: access to improved drinking water in the
rural area; and access to improved sanitation facilities in the urban region and rural region, show an
10. almost identical pattern. The missing observations fall at the start or end of the time series. As explained
in the definition section of this report, these variables were estimated using a linear regression when
two or more data points were available, and were only continued two points past the first or last
collected data point in the time series. The dataset does not provide enough information to ascertain
why this information was not recorded for each country, i.e. poorer countries couldnβt spend the money
on this, funding went to some countries and not others, or the Government or Statistical Office of the
country deemed this information not important to collect. Therefore, it is not possible to assume the
data is missing at random and can legitimately be ignored without affecting outcomes (Gelman et al
2006). However, there would be little value in using an estimation method to input missing values, such
as multiple imputation, as these estimates could potentially be quite unstable. They would contain large
confidence intervals as a result of estimating the missing data points from estimates in the linear
regression. Given that the impact would be quite minimal on the dataset, it would be reasonable to
continue the analysis with these variables missing.
Figure 1: Pattern of missing observation across the time series for the proportion of population
with access to improved drinking water, urban region.
The other noted data issue was the instance of the proportion of measles immunisations being over 100
per cent. There were 17 cases in the data set where the proportion of the 1 year-old population that is
immunised against measles was 99 per cent, which covers 1.46% of all records in this variable. Some of
these instances may be legitimate. However, there is no way of knowing which values were or were not
legitimate so, for this reason, these values have been left as they are.
11. Preliminary data analysis
On inspecting the longitudinal trend of the under-five mortality rate by country and region (Figure 2),
there was a strong pattern, showing that overall the mortality rate had decreased for most countries.
There was also a regional difference, with Northern Africa countries showing a lower mortality rate to
begin with (all below 100 parts per 1000), less variance between countries and all rates trending
downwards. In comparison, the sub-Saharan base mortality rates range from approximately 10 to 320
parts per 1000, with most countries mortality rates decreasing. In some instance this rate has increased
for some parts of the time series, showing a potential quadratic relationship over time.
Figure 2: Under-five morality rates of countries in the sub-Saharan and Northern African regions
from 1990 - 2011
Figure 2 showed a couple of countries with peaks and troughs and on further inspection of the data, the
most extreme line belonged to Rwanda (purple line in Figure 2) which suffered a brutal civil war
resulting in genocide during the time period. Since this would have had a destabilising effect on the
model, Rwanda was taken out. The Democratic Republic of Congo and Sudan also suffered civil wars
resulting in high mortality rates over the time period so these two countries were also removed from
the model (Rajaratnam et al 2010). The replotted line graph displayed fewer countries with peaks and
troughs of mortality rates over time (Figure 3). Removing these three countries from the dataset also
reduces the number of countries in the sub-Saharan region from 48 to 45.
12. Figure 3: Under-five morality rates of countries in the sub-Saharan and Northern African regions
from 1990 β 2011; excluding war torn countries
To model an outcome variable in a generalised linear mixed model requires an assumption to be made
about the probability distribution, so that the data is modelled with the best fitting distribution and link
function (Dobson & Barnett 2008). The response variable, under-five mortality rates, were measured on
a yearly basis and were assumed to be normally distributed. To assess this assumption graphically, a
histogram was plotted and showed a slightly right skewed curve (Figure 4). To improve the fit, a log
transformation of the under-five mortality rate was taken (Figure 4). This dramatically decreased the fit
to a normal curve and showed a left-skewed distribution. Since the histogram of under-five mortality
rate did not deviate significantly from a normal curve, Q-Q plot were graphed for each year (Appendix).
These supported the normality assumption, only showing very slight curvature towards at the bottom
end of each plot. Therefore a model with a Gaussian distribution and identity link function seems to be
suitable.
In the second stage of analysis, when the predictor variables are added into the model, the distribution
of the under-five mortality rate may not necessarily be normal, as it may be affected by the distribution
of the explanatory variables. Of course, the assumption of this model is that the residuals follow a
normal curve and with the correct fitting distribution on the data, this means the residuals will be
approximately normal with constant variance (Dobson & Barnett 2008).
13. Figure 4: Histogram plots of the under-five morality rate (left) and its log transformation (right)
From the dataset there were seven potential covariates identified, that could potentially explain some
of the reduction in the under-five mortality rate over the past two decades. These were the percentage
of the population who were undernourished, the child immunisation rate for measles, the prevalence of
tuberculosis, and access to clean drinking and access to improved sanitation facilities, both in urban and
rural areas. Table 3 shows the mean values of each of the indicators, disaggregated by region. As
expected, sub-Saharan Africa did not fare well compared to Northern Africa under all variables, with the
average under-five mortality rate difference of 69.71%.
Table 3. Variable means disaggregated by region
Variable Mean
Sub-Saharan Africa Northern Africa
Under-five mortality rate 127.15 38.51
Percentage of population undernourished 28.85 2.08
Proportion of 1 year olds immunised against measles 67.19 92.00
Prevalence rates associated with tuberculosis 386.78 85.48
Proportion of urban population using improved drinking
water sources
83.42 91.00
Proportion of rural population using improved drinking
water sources
53.52 75.30
Proportion of urban population using improved sanitation
facilities
44.78 93.65
Proportion of rural population using improved sanitation
facilities
24.28 71.68
Figure 3 shows the relationship between the outcome and explanatory variables plotted individually.
There were three variables that did not have a strong linear relationship with the outcome variable:
tuberculosis prevalence; access to improved sanitation facilities in rural areas; and access to improved
drinking water in urban areas. Tuberculosis prevalence also displayed a fan like effect indicating a non-
constant variance, which would also improve through transformation. Tuberculosis prevalence and
access to improved sanitation in the rural region were both right skewed so a log transformation was
applied to both variables. This was successful for the first variable, increasing the linear relationship
between the outcome and explanatory variable. This created a new variable called ltubprev. The log
14. transformation did not improve the linear relationship of access to sanitation facilities so a different
transformation method was needed. A square root transformation was applied and this improved the
linear relationship. This new variable was called ssanrural. Finally, access to improved drinking water in
the urban region was left skewed so squared and cubed transformations were applied with very little
improvement in linear relationship so this variable was not transformed. The other variables appeared
moderately normal with a generally moderate linear relationship between the explanatory and outcome
variables.
Figure 5: Scatterplot matrix of the outcome and explanatory variables
15. Statistical analysis
The majority of the analysis was undertaken in SAS 9.3 software, with analysis on missing variables
conducted in SPSS. This analysis has been split into two parts. Firstly, a model was fitted to analyse the
regional effect on the under-five mortality rate over time. In the next stage, explanatory variables were
added to this model to see which other indicators helped explain some of the reduction in the under-
five mortality rate so this could be a focus for future aid programs to help reduce childhood mortality
rates.
Stage 1
The response variable was estimated for each country on a yearly basis. The repeated measures nature
of this data set meant there was a strong within-subject correlation between under-five mortality rates
for each country (Twisk 2003). An appropriate model must be chosen to allow for this correlation
without affecting the usual assumption of independence between observations expected for general
linear models. There are two main types of generalised linear models that account for this within-
subject correlation. These are fixed or random effects models.
A fixed effect model assumes consistency across subjects and models the within-subject correlation
using a serial correlation structure. The covariate between-subject effects on the outcome variable
remain constant over time (Fitzmaurice et al 2011). It is often modelled with a marginal model using the
generalised estimating equation approach. This works by fitting a serial correlation structure to the
correlated effects which produce weights to aid in estimation, as well as fitting a generalised linear
model to produce marginal means and regression coefficient estimates. A basic model is:
πππ‘ = π½0 + π½1 π‘ + πππ‘
Where πππ‘ are the observations for subject π at time π‘, π½0 is the intercept, π‘ is time, π½1is regression
coefficient for time and πππ‘ is the error for subject π at time π‘ . The intercept and slope for this model are
constant across subjects.
One thing to note about marginal models is that as only one variance parameter is estimated, it does
not account for changing variance over time (Twisk 2003). The correlation structure chosen can account
for this though based on the within-subject correlation. For example, longitudinal data often shows
decreasing within-subject correlation as time points move further away from each other so a correlation
function that accounts for the decaying correlation would assume to be the best fit.
In contrast, a random effects model allows the regression coefficient to vary between subjects. This is
generally referred to as a generalised linear mixed model as it contains a combination of fixed and
random effects. The simplest form is a random intercept model where the intercept differs for each
subject (Fitzmaurice et al, 2011). This occurs when the intercept for each subject starts at varying points
along the Y-intercept. A simple form of this model is:
πππ‘ = π½0π + π½1 π‘ + πππ‘
Where πππ‘ are the observations for subject π at time π‘, π½0π is the random intercept, π‘ is time, π½1is
regression coefficient for time and πππ‘ is the error for subject π at time π‘ (Twisk 2003). This model
includes a fixed effect for time.
16. The π½0π represents the intercept and this can vary between subjects. There are also situations where the
time slope will vary across subjects as well. This is shown when the rate of change in the outcome
variables are different across subjects. In this instance a random intercept and random slope model
would be fitted. This can be shown when the Y-intercept and slope for each country varies from each
other. This is represented in a model as:
πππ‘ = π½0π + π½1π π‘ + πππ‘
Where coefficients are almost identical to the random intercept model, however, π½1π π‘ is a random
regression coefficient for time (Twisk 2003), rather than a fixed coefficient. The main assumption with
the random effects is that the variance in the intercept or slope are normally distributed with π~(0, π).
It must be noted that adding in a random slope can account for some change in variance over time,
however, in some instances it will not account for all changing variance and a variance function must be
added to the model (Jones 1993).
To estimate the coefficients in a generalised linear mixed model there are two common estimation
methods, maximum likelihood estimation (MLE) and restricted maximum likelihood estimation (REML).
REML is a less biased estimator as it accounts for the loss of degrees of freedom when estimating π½
compared to the MLE method (Muller & Stewart 2006). In situations where there are a high number of
number of high number of fixed effects compared to observations, the REML is needed to ensure an
unbiased estimation (Jones 1993). However, as this is not the case in the data, the difference in the
coefficient estimate would be negligible. Therefore, this estimation method used in the models was
MLE.
As Figure 1 shows, the intercept of the under-five mortality rate varied greatly across countries. A
random intercept model may fit the data better than a fixed effect model, on the assumption that
under-five mortality rate varies randomly from one country to the next. By fitting a random intercept
this is accounting for the assumed natural heterogeneity across countries. It also assumes the intercept
follows a normal distribution, ππ ~ π(0, π£2
) (Fitzmaurice et al 2011).
The model is:
π¦ππ = (π½0 + π½0π) + π½1 π‘1π + π½2 π‘2π + π½3 π‘3π + π½4 π‘4π + π½5 π‘5π + π½6 π‘6π + π½7 π‘7π + π½8 π‘8π + π½9 π‘9π + π½10 π‘10π
+ π½11 π‘11π + π½12 π‘12π + π½13 π‘13π + π½14 π‘14π + π½15 π‘15π + π½16 π‘16π + π½17 π‘17π + π½18 π‘18π
+ π½19 π‘19π + π½20 π‘20π + π½21 π₯π1 + πππ
Where:
π¦ππ = under-five mortality rate of the πth country at πth year π = 1,β¦, 50; π = 1,β¦, 21
π‘ π π was a dummy variable for year π , with 1990 being the base category, and
π‘1π = {
1 ππ π = π
0 ππ‘βπππ€ππ π
for s = 1991, β¦, 2011
π₯π1 = {
1 ππ πππππππΌπ· = 1
0 ππ‘βπππ€ππ π
π½0π ~ π(π½0π, π£2) and πππ ~ π(0, π2
); and are mutually independent
17. The model was created in SAS using glimmix, comparing the longitudinal and regional effect on the
under-five mortality rate. Both year and regionID were significant (p < 0.0001). The random intercept
variance was 2039.53, and the test of independence was rejected which confirms a significant random
effects variance (p < 0.0001). The within-subject correlation matrix showed a high correlation in the
correlation matrix (0.8509) and the residual estimate was high at 357.31.
While the previous model confirmed the need for a random intercept, the year effect was assumed to
be fixed across all countries (Rajaratnam et al 2010). Figure 2 suggests this is not the case, as not all
countries have the same slope, for example, some having a steeper slope than others. Assuming year is
a continuous variable with a random slope i.e. differing rate of change of under-five mortality rates
across countries, a new model can be fitted.
The model is:
π¦ππ = (π½0 + π½0π) + (π½1 + π½1π)π‘π + π½2 π₯π1 + πππ
Where:
π¦ππ = under-five mortality rate of the πth country at πth year π = 1,β¦, 50
π‘π = year
π₯π1 = {
1 ππ πππππππΌπ· = 1
0 ππ‘βπππ€ππ π
π½0π ~ π(π½0π, π£0
2), π½1π ~ π(π½1π, π£1
2) and πππ ~ π(0, π2
); and are mutually independent
Output omitted
Covariance Parameter Estimates
Cov Parm Subject Estimate Standard Error
Intercept Country 132465
Year Country 0.01792 0.01129
Residual 312.96 15.8673
Type III Tests of Fixed Effects
Effect Num DF Den DF F Value Pr > F
RegionID 1 1000 0.92 0.3373
Year 1 49 1212.57 <.0001
Output omitted
The estimated random slope for this model is approximately zero which means the longitudinal effect of
each country was constant i.e. there was no change in the under-five mortality rate across all countries.
The first model, along with Figure 3, showed this was not the case so this model does not fit the data
well. This model assumes that the random intercept and random slope are independent. This may not
be the case and they may in fact be correlated. If they are not independent, this assumes that higher
18. rates of under-five mortality at the intercept, the steeper the slope and therefore showing a greater
reduction in under-five mortality rates over time. From looking at Figure 3 this does not seem to be an
unreasonable assumption. In this circumstance, a covariance structure needs to be fitted to the random
intercept and slope. The covariance structure generally used for longitudinal data is an unstructured
covariance matrix as it does not assume the matrix to be a particular form (Verbeke et al 2009). A
variation on this is the Cholesky root which parameterises the unstructured covariance matrix through
its Choleksy root. This has good statistical properties compared to the unstructured covariance matrix,
as the matrix is at least positive semidefinite, so therefore, this covariance structure will be used (SAS
2014).
The new model is:
π¦ππ = (π½0 + π½0π) + (π½1 + π½1π)π‘π + π½2 π₯π1 + πππ
Where:
π¦ππ = under-five mortality rate of the πth country at πth year π = 1,β¦, 50
π‘π = year
π₯π1 = {
1 ππ πππππππΌπ· = 1
0 ππ‘βπππ€ππ π
β
π½0π
π½1π
β ~ π (
π½0π
π½1π
,
π£0
2
π£01
π£01 π£1
2 ), and πππ ~ π(0, π2
); and are mutually independent.
Where π£01 is the covariance between the π½0π and the π½1π random effects.
The model output showed a regional difference between sub-Saharan and Northern Africa, which was
consistent with the random intercept model. The estimated difference in under-five mortality between
the two regions reduced from 89.52 to 76.71 between the two models. The residual estimate has also
substantially lowered from the other models from 357.31 to 38.34, providing a much better fit to the
data than the random intercept model only. The AIC also favours the correlated random intercept model
with an AIC of 7737.11 compared to 10469.99.
The residuals were plotted to test the assumption of normality and a constant variance of the residuals.
While the histogram showed a fairly symmetrical shape, the Q-Q plot shows some curvature towards
the bottom suggesting a slight variation from the normal curve towards the left hand side of the
distribution. The residuals plotted against the linear predictors do not show any particular pattern so the
assumption of constant variance does not seem to be violated.
The final model that was used to test the covariates in the next stage was the random intercept and
slope model with a Cholesky covariance matrix on the random effects. SAS code and output for relevant
models can be found in the Appendix.
Stage 2
The next stage involved adding the covariates into the model to see if any of them could help explain
the reduction in child mortality rates across Africa. To begin with, a model selection method was chosen
to find the best fitting model. There are two popular criteria that can be used for this purpose. The
Akaike Information Criterion (AIC) penalises the number of parameters in the model so avoids over
19. fitting the data (Jones 1993). The Bayesian Information Criterion (BIC) works in a similar way (Stroup
2013). The model with the smallest AIC and BIC is selected as the best fitting model.
Another method of selecting the best fitting model is to use the likelihood ratio test using the likelihood
value, however this method is more complicated. It requires the models to be nested and in some
instances two models may not be nested or it may not be obvious which model is a nested version of
the other. Since, in this instance the nested model was not obvious between the covariates in this
research, the AIC and BIC were used.
One issue that arose as a result of using the AIC and the BIC was the problem of using a different
number of observations in different models. Both calculations require each model to use the same
number of observations. Since most covariates have some missing variables, modelling them as they are
would create an incorrect comparison of the AIC or BIC between models and make model selection
difficult. To resolve this issue a new variable was created called notmissing that coded each row with a
one if no observations were missing. This meant that only rows that had all observations were used
when deciding which model to use from the model selection criteria. Once the optimal model was
selected, all available observations where used in the final output.
Initially, each explanatory variable was modelled with the outcome variable individually. This was to
establish which covariates had a significant relationship when modelled with child mortality rates on
their own. The model was then developed in an iterative process, whereby the model in each stage that
had the lowest AIC and BIC was used in the next stage as the base model, and each covariate in the
previous stage was added to the new model, only if it had a significant p-value below 0.20. The appendix
shows the model selection process. The optimal model with the lowest AIC and BIC included the log of
tuberculosis prevalence, the proportion of the rural population using improved drinking water sources,
the proportion of 1 year olds immunised against measles and the percentage of the population that was
undernourished.
Once all the appropriate explanatory variables were added into the model, the regional effect was not
significant anymore (p-value=0.25). In fact, the AIC and BIC favoured the model without controlling for a
regional effect. This means that the other factors had a stronger predictive effect on child mortality
rates than the region the country was in. However, the small sample in the Northern region may have
had an impact on the lack of significance as well.
As a result, the final model did not include the region variable. With the final group of explanatory
variables added into the model the residual variance reduced from 38.34 to 26.91, indicating the
covariates aided in explaining some of the variance in the reduction of the under-five mortality rate. The
AIC also favoured the model with the covariates added from the random intercept and slope model with
a correlation function, reducing from 7737.11 to 6460.47. Most variables were highly significant with a
p-value less than 0.001, with percentage of population undernourished having the highest p-value at
0.018.
The residuals were plotted to test the assumptions of normality and constant variance. The residual
plots showed a similar pattern to the model without the explanatory variables added to the model.
There was some curvature at the bottom of the Q-Q plot suggesting a slight variation from normality
curve. This may have been due to some outlying variables, as there are some points with large residuals.
20. Overall, the normality assumption was reasonable. The residuals plotted against the predicted values
did not show a distinct pattern, indicating that the assumption of constant variance was met.
Results
The initial analysis conducted showed that there was a significant difference between the two regions,
as well as over time, when modelled using a random intercept and slope. Once the covariates were
added to the model, the regional effect was no longer significant. There were four explanatory variables
that were significant. These were the log of tuberculosis prevalence, access to improved drinking water
in rural regions, one year old immunisation rates of measles and the percentage of the population
undernourished. Table 4 displays the parameter estimates, standard errors and p-values of all predictors
in the final model. The full output can be found in the Appendix.
Table 4. Parameter estimates and 95% confidence intervals
Parameter Estimate Standard error p-value
Year -2.0459 0.3477 <.0001
Ltuperprev 24.5963 1.5257 <.0001
WaterRural -0.9210 0.1825 <.0001
ChMeaImm -0.09588 0.02598 0.0002
UNourish 0.1388 0.05838 0.0177
From this, the fitted model is:
π¦ππ = (4128.26 + π½0π) + (β2.046 + π½1π)π‘π + 24.596ππππ₯π1 β 0.921π₯π2 β 0.096π₯π3 + 0.139π₯π4 + πππ
Where:
π¦ππ = under-five mortality rate of the πth country at πth year π = 1,β¦, 50
π‘π = year
π₯π1 = log of tuberculosis prevalence
π₯π2 = proportion of population with access to improved drinking sources in rural regions
π₯π3 = proportion of one year olds immunised against measles
π₯π4 = percentage of the population undernourished
β
π½0π
π½1π
β ~ π (
π½0π
π½1π
,
π£0
2
π£01
π£01 π£1
2 ), and πππ ~ π(0, π2
); and are mutually independent.
Where π£01 is the covariance between the π½0π and the π½1π random effects.
Parameter interpretation:
ο· For every 10% increase in tuberculosis prevalence, the under-five mortality rate increased by
1.02 (24.596*log(1.1) =1.02), meaning an increase of 1.02 deaths per 1,000.
ο· For each proportion increase of the population that had access to improved drinking sources in
rural areas the expected mortality rate would decrease by 0.921. This means that every 10%
21. increase in the proportion of the population with access to improved drinking sources, child
mortality rates would decrease by 9.21 deaths per 1,000.
ο· For every per cent increase in the proportion of one year olds immunised against measles the
expected mortality rate would drop by 0.096, so for every 10% increase in immunisation rates,
mortality rates would drop by 0.96 deaths per 1,000.
ο· For each unit increase in the percentage of the population that was undernourished the
mortality rate would increase by 0.139. This means that for every 10% increase in the
percentage, would increase mortality rates by 1.39 deaths per 1,000.
Discussion
The fourth millennium development goal aimed to reduce under-five mortality rates by two-thirds,
between 1990 and 2015. Parts of the world, particularly sub-Saharan Africa will not achieve this target
within the next couple of years. As a result, this paper has investigated the under-five mortality rate
across both the sub-Saharan and Northern African region. By using a generalised linear mixed model,
modelling varied mortality rates and changes across countries, this research has shown that there was a
significant decrease in under-five mortality rates over the last 20 years across both regions, as well as a
significant difference in under-five mortality rates across the regions. So, despite sub-Saharan Africa not
achieving the MDG goal by 2015, significant progress has been made in reducing the under-five
mortality rates in this region.
Additional covariates where added into the model to ascertain if they contributed to the reduction in
under-five mortality rates. There were a number of indictors that impacted the mortality rates over the
years, which were mostly related to health factors. One reasons for this may be due to the increase in
funding for health related assistance, increasing from US$6.8 billion to US$16.7 billion, with most of it
focused on infectious diseases including measles and tuberculosis (Waage et al 2010). Increases in
tuberculosis prevalence of the population of the country as well as increases to the percentage of
population that were undernourished increased the under-five mortality rate by 1.02 and 1.39 per
1,000, respectively, for each 10% increase. Immunisation rates of measles for child one year of age was
another important factor, showing that as immunisation rates increased the rates of child mortality
decreased very marginally, at 0.96 deaths per 1,000, for every 10% increase.
The last significant indicator was access to improved water facilities in the rural region. This had the
biggest impact on under-five mortality rates, decreasing the number of deaths by 9.21 per 1,000 deaths
for every 10% increase. One noted limitation to this measure was that it did not account for the time
travelled to access improved water facilities nor did it include methods of transporting water. In rural
areas, this would have had a considerable impact, and if it had been included, may have resulted in a
greater decrease in the mortality rate.
Another noted limitation was that there was no set definition of a rural or urban region. As this relied on
countries to specify these regions themselves there may have been high variance within these indictors.
Since this variable was significant, this may not have had a huge impact. However, it may have effect on
the three variables that were not included in the final model, due to not providing any additional
information to the final model. These variables were the proportion of the population with access to
improved drinking water in urban regions and the percentage of the population that had access to
improved sanitation facilities in both rural and urban areas.
22. The final model also did not include the regional effect. However, this may be due to the small sample
size of Northern Africa, as it only contained five countries. Table 3 showed a considerable difference in
under-five mortality rate between the two regions, along with stronger figures on all other covariates,
which suggests that had the sample been larger, there may have been a significant difference between
the two regions with all other covariates added into the model. The other possible explanation is that
once other variables had been added to the model, region was not a strong predictor of mortality rates
compared to the other explanatory variables.
While this research was able to show a number of variables that had an impact on rates of child
mortality, the biggest limitation in this research paper was the availability of data for all indictors. There
were a number of indictors related to the motherβs health and demographic information that were not
sufficiently collected across all countries and across the time series, so therefore could not be used in
this analysis. Within the indicators used in this study, there was also some missing values, and some
imputation methods used prior to accessing the data, which may have affected the outcome of the final
model as well.
The other main limitation was the complexity of some of the indictors. Access to clean water and
sanitation is hard to measure so a proxy measure was developed. This measured the proportion of the
population that shared an improved type of water and sanitation facility instead, which meant it relied
heavily on population and household ratios. As already discussed, it also did not account for time taken
to access facilities, meaning rural areas may have a higher impact than is captured in these measures.
The proportion of the population that is undernourished was also another complex measure with
methodological issues (Waage et al 2010).
With these limitations in mind, further analysis could look at investigating data sets outside of the
MDGs, to investigate other potential indicators that may have an impact on child mortality rates across
Africa.
In conclusion, the focus for future aid programs to help reduce childhood mortality rates should include
health related factors, such as preventable illness and nutrition, as well as increasing access to improved
drinking sources in rural regions. Additionally, increasing funding and resources for data collection and
management would assist in allowing for comprehensive analysis of all the predictors of child mortality
rates.
23. References
Dobson, AJ & Barnett, AG 2008, An Introduction to Generalized Linear Models, Third Edition, CRC Press;
Taylor & Francis Group, United States of America.
Fitzmaurice, GM, Laird, NM & Ware, JH 2011, Applied Longitudinal Analysis, Second Edition, John Wiley
& Sons, Inc., New Jersey, United States of America.
Gelman, A & Hill, J 2006, Data Analysis Using Regression and Multilevel/Hierarchical Models, Cambridge
University Press, United Kingdom.
Jones, RH 1993, Longitudinal Data with Serial Correlation: A State-space Approach, Chapman & Hall,
Suffolk, Great Britain.
Lozano, R, Wang, H, Foreman, KJ, et al 2011, Progress towards Millennium Development Goals 4 and 5
on maternal and child mortality: an updated systematic analysis, Lancet 2011; 378: 1139-1165.
Muller, KE & Stewart, PW 2006, Linear Model Theory, Univariate, Multivariate, and Mixed Models, John
Wiley & Sons, Inc., United States of America.
Rajaratnam, JK, Marcus, JR, Flaxman, AD, et al 2010, Neonatal, postnatal, childhood and under-5
mortality for 187 countries, 1970-2010: a systematic analysis of progress towards Millennium
Development Goal 4, Lancet 2010; 375: 1988-2008.
SAS 2012, SAS/STAT (R) 9.2 Userβs Guide, Second Edition,
http://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_glimmi
x_sect009.htm, viewed, 10 June 2014.
Stroup, WW 2013, Generalized Linear Mixed Models, Modern Concepts, Methods and Applications, CRC
Press; Taylor & Francis Group, United States of America.
Susuman, AS 2012, Child Mortality Rate in Ethiopia, Iranian J Public Health, Vol. 41, No. 3, pp. 9 β 19.
Susuman, AS & Hamisi, HF 2012, Under-5 Mortality in Tanzania: A Demographic Scenario, Iranian J
Public Health, Vol. 41, No. 12, pp. 8 β 18.
Twisk, JWR 2003, Applied Longitudinal Data Analysis for Epidemiology. A Practical Guide, Cambridge
University Press, United Kingdom.
United Nations General Assembly 2000, United Nations Millennium Declaration, A/RES/55/2 edn., New
York, NY: United Nations.
United Nations 2013, We Can End Poverty, Millennium Development Goals and Beyond 2015 Fact Sheet.
Goal 4: Reduce child mortality, http://www.un.org/en/mdg/summit2010/pdf/MDG_FS_4_EN.pdf,
viewed 30 March 2014.
United Nations n.d., Millennium Development Goals Indicators, Metadata,
http://mdgs.un.org/unsd/mdg/Metadata.aspx, viewed 17 May 2014.
United Nations, n.d. Millennium Development Goals Indicators, Country Level Data,
http://mdgs.un.org/unsd/mdg/Data.aspx, viewed 23 March 2014.
24. Verbeke, G & Molenberghs, G 2009, Linear Mixed Models for Longitudinal Data, Springer Verlag New
York, LLC, United States of America.
Waage, J, Banerji, R, Campbell, C, et al 2010, The Millennium Development Goals: a cross-sectoral
analysis and principles for goal setting after 2015, The Lancet 2010; 376: 991-1023.
World Health Organisation/UNICEF Joint Monitoring Programme for Water Supply and Sanitation,
Definitions and methods, http://www.wssinfo.org/definitions-methods/, viewed 18 May 2014.
25. Appendix
Q-Q plots for under-five mortality rate over five year intervals
Year=1990 Year=1995
Year=2000 Year=2005
Year=2010
26. Stage 1 β Model codes and outputs
Random intercept model
SAS Code:
proc glimmix data=cmr.u5mrfull method=mspl ;
class country regionID year ;
model u5mr = year regionID / dist=normal link=id s;
random intercept / subject=country v vcorr;
covtest indep;
run;
Output:
The GLIMMIX Procedure
Model Information
Data Set CMR.U5MRFULL
Response Variable U5MR
Response Distribution Gaussian
Link Function Identity
Variance Function Default
Variance Matrix Blocked By Country
Estimation Technique Maximum Likelihood
Degrees of Freedom Method Containment
27. Class Level Information
Class Levels Values
Country 53 Algeria Angola Benin Botswana Burkina Faso Burundi Cameroon Cape
Verde Central African Republic Chad Comoros Congo Cote d'Ivoire
Democratic Republic of the Congo Djibouti Egypt Equatorial Guinea
Eritrea Ethiopia Gabon Gambia Ghana Guinea Guinea-Bissau Kenya
Lesotho Liberia Libyan Arab Jamahiriya Madagascar Malawi Mali
Mauritania Mauritius Morocco Mozambique Namibia Niger Nigeria
Rwanda Sao Tome and Principe Senegal Seychelles Sierra Leone Somalia
South Africa Sudan Swaziland Togo Tunisia Uganda United Republic of
Tanzania Zambia Zimbabwe
RegionID 2 1 2
Year 22 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
2003 2004 2005 2006 2007 2008 2009 2010 2011
Number of Observations Read 1166
Number of Observations Used 1166
Dimensions
G-side Cov. Parameters 1
R-side Cov. Parameters 1
Columns in X 25
Columns in Z per Subject 1
Subjects (Blocks in V) 53
Max Obs per Subject 22
Optimization Information
Optimization Technique Dual Quasi-Newton
Parameters in Optimization 1
Lower Boundaries 1
Upper Boundaries 0
Fixed Effects Profiled
Residual Variance Profiled
Starting From Data
28. Iteration History
Iteration Restarts Evaluations Objective
Function
Change Max
Gradient
0 0 4 10419.98954 . 1.78E-13
Convergence criterion (ABSGCONV=0.00001) satisfied.
Fit Statistics
-2 Log Likelihood 10419.99
AIC (smaller is better) 10469.99
AICC (smaller is better) 10471.13
BIC (smaller is better) 10519.25
CAIC (smaller is better) 10544.25
HQIC (smaller is better) 10488.93
Generalized Chi-Square 416626.5
Gener. Chi-Square / DF 357.31
Estimated V Correlation Matrix for Country Algeria
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31. Solutions for Fixed Effects
Effect RegionID Year Estimate Standard Error DF t Value Pr > |t|
Year 1991 59.5585 3.6720 1092 16.22 <.0001
Year 1992 57.8925 3.6720 1092 15.77 <.0001
Year 1993 56.4962 3.6720 1092 15.39 <.0001
Year 1994 56.8887 3.6720 1092 15.49 <.0001
Year 1995 54.5585 3.6720 1092 14.86 <.0001
Year 1996 51.6019 3.6720 1092 14.05 <.0001
Year 1997 50.1547 3.6720 1092 13.66 <.0001
Year 1998 48.2887 3.6720 1092 13.15 <.0001
Year 1999 45.2321 3.6720 1092 12.32 <.0001
Year 2000 42.1283 3.6720 1092 11.47 <.0001
Year 2001 38.6849 3.6720 1092 10.54 <.0001
Year 2002 34.9491 3.6720 1092 9.52 <.0001
Year 2003 31.0453 3.6720 1092 8.45 <.0001
Year 2004 26.9887 3.6720 1092 7.35 <.0001
Year 2005 22.8887 3.6720 1092 6.23 <.0001
Year 2006 18.8472 3.6720 1092 5.13 <.0001
Year 2007 14.8792 3.6720 1092 4.05 <.0001
Year 2008 11.0774 3.6720 1092 3.02 0.0026
Year 2009 7.3170 3.6720 1092 1.99 0.0465
Year 2010 3.4962 3.6720 1092 0.95 0.3412
32. Solutions for Fixed Effects
Effect RegionID Year Estimate Standard Error DF t Value Pr > |t|
Year 2011 0 . . . .
RegionID 1 89.5208 21.3069 1092 4.20 <.0001
RegionID 2 0 . . . .
Type III Tests of Fixed Effects
Effect Num DF Den DF F Value Pr > F
Year 21 1092 59.45 <.0001
Tests of Covariance Parameters
Based on the Likelihood
Label DF -2 Log Like ChiSq Pr > ChiSq Note
Independence 1 12383 1962.68 <.0001 MI
MI: P-value based on a mixture of chi-squares.
Final model - Random intercept and slope model
proc glimmix data=cmr.u5mrfull method=mspl plot=residualpanel;
class country regionID;
model u5mr = regionID year / dist=normal link=id dfm=sat s;
random intercept year / subject=country type=chol vcorr;
covtest DiagG;
covtest GLM;
run;
The GLIMMIX Procedure
Model Information
Data Set CMR.U5MRFULL
Response Variable U5MR
Response Distribution Gaussian
Link Function Identity
Variance Function Default
Variance Matrix Blocked By Country
33. Model Information
Estimation Technique Maximum Likelihood
Degrees of Freedom Method Satterthwaite
Class Level Information
Class Levels Values
Country 50 Algeria Angola Benin Botswana Burkina Faso Burundi Cameroon Cape
Verde Central African Republic Chad Comoros Congo Cote d'Ivoire
Djibouti Egypt Equatorial Guinea Eritrea Ethiopia Gabon Gambia Ghana
Guinea Guinea-Bissau Kenya Lesotho Liberia Libyan Arab Jamahiriya
Madagascar Malawi Mali Mauritania Mauritius Morocco Mozambique
Namibia Niger Nigeria Sao Tome and Principe Senegal Seychelles Sierra
Leone Somalia South Africa Swaziland Togo Tunisia Uganda United
Republic of Tanzania Zambia Zimbabwe
RegionID 2 1 2
Number of Observations Read 1100
Number of Observations Used 1100
Dimensions
G-side Cov. Parameters 3
R-side Cov. Parameters 1
Columns in X 4
Columns in Z per Subject 2
Subjects (Blocks in V) 50
Max Obs per Subject 22
Optimization Information
Optimization Technique Dual Quasi-Newton
Parameters in Optimization 3
Lower Boundaries 2
Upper Boundaries 0
Fixed Effects Profiled
Residual Variance Profiled
34. Optimization Information
Starting From Data
Iteration History
Iteration Restarts Evaluations Objective
Function
Change Max
Gradient
0 0 4 7723.121338 . 404.7356
1 0 4 7723.114587 0.00675097 344.4614
2 0 7 7723.1137335 0.00085352 206.1668
3 0 4 7723.1116263 0.00210726 2.064337
4 0 2 7723.111626 0.00000025 0.012576
Convergence criterion (GCONV=1E-8) satisfied.
Fit Statistics
-2 Log Likelihood 7723.11
AIC (smaller is better) 7737.11
AICC (smaller is better) 7737.21
BIC (smaller is better) 7750.50
CAIC (smaller is better) 7757.50
HQIC (smaller is better) 7742.21
Generalized Chi-Square 42191.91
Gener. Chi-Square / DF 38.36
Estimated V Correlation Matrix for Country Algeria
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37. Covariance Parameter Estimates
Cov Parm Subject Estimate Standard Error
Residual 38.3563 1.7154
Solutions for Fixed Effects
Effect RegionID Estimate Standard Error DF t Value Pr > |t|
Intercept 6052.60 758.31 50.05 7.98 <.0001
RegionID 1 76.7113 17.7074 50 4.33 <.0001
RegionID 2 0 . . . .
Year -3.0009 0.3771 50.01 -7.96 <.0001
Type III Tests of Fixed Effects
Effect Num DF Den DF F Value Pr > F
RegionID 1 50 18.77 <.0001
Year 1 50.01 63.34 <.0001
Tests of Covariance Parameters
Based on the Likelihood
Label DF -2 Log Like ChiSq Pr > ChiSq Note
Diagonal G 1 8217.72 494.61 <.0001 DF
Independence 3 11703 3979.85 <.0001 --
38. Stage 2 β Model selection
Model p-value (last variable) AIC BIC
U5MR + Year + RegionID + Unourish 0.1532 6672.89 6687.86
U5MR + Year + RegionID + ChMeaImm <.0001 6637.61 6652.58
U5MR + Year + RegionID + ltuberprev <.0001 6445.97 6460.94
U5MR + Year + RegionID + WaterUrban 0.0133 6667.27 6682.24
U5MR + Year + RegionID + WaterRural <.0001 6653.09 6668.06
U5MR + Year + RegionID + SanUrban 0.0847 6670.54 6685.51
U5MR + Year + RegionID + ssanrural <.0001 6659.12 6674.09
Model p-value (last variable) AIC BIC
U5MR + Year + RegionID + ltuberprev + Unourish 0.0179 6440.9 6457.74
U5MR + Year + RegionID + ltuberprev + ChMeaImm 0.0001 6431.29 6448.14
U5MR + Year + RegionID + ltuberprev + WaterUrban 0.0043 6438.22 6455.06
U5MR + Year + RegionID + ltuberprev + WaterRural <.0001 6423.72 6440.56
U5MR + Year + RegionID + ltuberprev + SanUrban 0.2579 6444.91 6461.75
U5MR + Year + RegionID + ltuberprev + ssanrural 0.0401 6442.72 6459.56
Model p-value (last variable) AIC BIC
39. U5MR + Year + RegionID + ltuberprev + WaterRural +
Unourish
0.0221 6420.6 6439.31
U5MR + Year + RegionID + ltuberprev + WaterRural +
ChMeaImm
0.0002 6411.76 6430.47
U5MR + Year + RegionID + ltuberprev + WaterRural +
WaterUrban
0.2074 6423.77 6442.48
U5MR + Year + RegionID + ltuberprev + WaterRural +
ssanrural
0.3863 6424.5 6443.21
Model p-value (last variable) AIC BIC
U5MR + Year + RegionID + ltuberprev + WaterRural +
ChMeaImm + Unourish
0.0233 6408.61 6429.2
U5MR + Year + ltuberprev + WaterRural +
ChMeaImm + Unourish
0.0177 6407.92 6426.63
Stage 2 β Model code and output
Random intercept and slope model with covariates added
proc glimmix data=cmr.u5mrfull method=mspl plots=residualpanel;
class country ;
model u5mr = year ltuberprev WaterRural ChMeaImm Unourish / dist=normal
link=id dfm=sat s;
random intercept year / subject=country type=chol vcorr;
run;
The GLIMMIX Procedure
Model Information
Data Set CMR.U5MRFULL
Response Variable U5MR
Response Distribution Gaussian
Link Function Identity
Variance Function Default
Variance Matrix Blocked By Country
Estimation Technique Maximum Likelihood
Degrees of Freedom Method Satterthwaite
Class Level Information
Class Levels Values
Country 48 Algeria Angola Benin Botswana Burkina Faso Burundi Cameroon Cape
Verde Central African Republic Chad Comoros Congo Cote d'Ivoire
Djibouti Egypt Eritrea Ethiopia Gabon Gambia Ghana Guinea Guinea-
40. Class Level Information
Class Levels Values
Bissau Kenya Lesotho Liberia Libyan Arab Jamahiriya Madagascar Malawi
Mali Mauritania Mauritius Morocco Mozambique Namibia Niger Nigeria
Sao Tome and Principe Senegal Seychelles Sierra Leone South Africa
Swaziland Togo Tunisia Uganda United Republic of Tanzania Zambia
Zimbabwe
Number of Observations Read 1100
Number of Observations Used 960
Dimensions
G-side Cov. Parameters 3
R-side Cov. Parameters 1
Columns in X 6
Columns in Z per Subject 2
Subjects (Blocks in V) 48
Max Obs per Subject 21
Optimization Information
Optimization Technique Dual Quasi-Newton
Parameters in Optimization 3
Lower Boundaries 2
Upper Boundaries 0
Fixed Effects Profiled
Residual Variance Profiled
Starting From Data
Iteration History
Iteration Restarts Evaluations Objective
Function
Change Max
Gradient
0 0 4 6539.3314619 . 145332.2
1 0 18 6466.603406 72.72805587 6759.036
