A geographically weighted regression approach to examine the dynamics of fertility differentials across Africa

2.1Data source and limitation

The challenge of data has been a major concern highlighted in almost all studies about Africa due to the cost in time and money to conduct censuses and surveys. Many international donors sponsor censuses and surveys in Africa which are carried out in many countries and done in different time periods in different countries. So it is quite difficult to get dataset for all countries in the same period of time from the same survey (such a survey would be very huge and too expensive). The dataset used in this study on TFR and the other factors such as infant mortality rate (IMR), adolescent fertility rate (AFR), life expectancy at birth, GDP per capita and female labour participation were from the World Bank Open Data [9] for the period of 2017. Data on Contraceptive prevalence rate, demand for family planning and unmet need for family planning were obtained from United Nations, Department of Economic and Social Affairs, Population Division [10].

Disputed territories of Western Sahara and Somaliland are not included in the analysis as data on these places were not available. The observations of each variable reported in the data represent specific geographical locations where the information on each variable had been taken. The dataset does not contain observations for the geographical divisions in each country. These observations are taken to representative the centroids of the countries. In this case, the underlying assumption is that the distribution of the variable within each country is sufficiently homogeneous to approximate it to a single observation.

The several sources of data used, means different samples and experimental conditions as opposed to data set from a single source which would have been collected under the same samples and experimental conditions, may account for some margin of error. Albeit, the situation may be a blessing in disguise, when the assumption of randomization for statistical analyses is considered, the data set are of the same period but different samples and experimental conditions, so it can be said that sampling fluctuation will be reduced.

The metadata for each of the variables are given thus:

2.2Spatial autocorrelation

Autocorrelation is the association of a variable with itself when observations on the variables are obtained in different time periods or in space. Spatial autocorrelation has been defined as the positive or negative correlation of a variable with itself due to the spatial location of the observations [11]. Analysis of spatial autocorrelation enables quantified analysis of the spatial structure of the studied variable whose value is related to its other values from neighbouring regions. The presence of spatial autocorrelation either means that similar values of the variables are geographically clustered together or values far away are more similar to values in nearby locations. Spatial autocorrelation indices such as the Moran index and the Geary index make it possible to assess spatial structure of variables.

This study employs the Moran index which takes into account the variances and covariances with the difference between each observation and the average of all observations. In the literature, Moran’s index is often preferred to that of Geary due to greater general stability [11, 12].

Given observations, i=1,…,n, on a variable, the Moran index is given as [11]

where wi⁢j are normalized weights. Iw>0 implies positive autocorrelation, Iw<0 implies negative autocorrelation and Iw=0 means no autocorrelation.

Spatial autocorrelation value is obtained using the Moran’s test which tests the null hypothesis of absence of spatial autocorrelation for a variable, in which case the values of the variable are randomly assigned to the spatial units in order to calculate the test statistic. The Moran’s index from the test is a single measure which represents the entire data.

2.3Geographically weighted regression

In order to seek the factors that relate with TFR and examine the spatial consistency in the relationship between the factors and TFR across Africa, GWR is employed on the dataset. GWR is a regression analysis technique that employs the traditional regression framework to geographical data by allowing the estimation of parameters at each location instead of parameters at a single summary for the entire space. In other words, GWR runs a regression for each location, instead of a sole regression for the entire study area.

The coefficients of GWR are not fixed but depend on coordinates of observations, that is, the coefficients of the explanatory parameters form continuous surfaces that are assessed at certain points in space [8]

where the parameters β0 and βk are dependent on the coordinates of the location. In this model, the coefficients vary and are considered identical across the study area. However, the hypothesis of spatial uniformity of the effect of explanatory variables on the dependent variable is often unrealistic [7]. If the parameters vary significantly in space, a global estimator will hide the geographical richness of the phenomenon. Spatial heterogeneity corresponds to this spatial variability in the model’s parameters or its functional form [8]. Local linear regressions can be used examine spatial heterogeneity and this is where the need for GWR comes into play.

In estimating the coefficients, using fixed-coefficient model and including only observations close to point i, “the more points included in the sample, the lower the variance but higher the bias. The solution is therefore to reduce the importance of the most remote observations by giving each observation a decreasing weight with the distance to the point of interest.” [8]. Given the model

at any given point, if weights are given to observations with weighted least squares, then β^ will minimise the sum

and

where Y^=S⁢Y and S is hat-matrix defined as

xi′=(1,xi⁢1,xi⁢2,⋯,xi⁢p) is column i of explanatory variable of the matrix X of variables and W contains weights of each observation corresponding to its distance to the point i and it is assumed that observations close to point i have more influence over the parameter estimates at i than remote observations.

The weight of observations decreases with the distance of the observation to the point i and this decrease in the weight is determined by a kernel function with parameters shape, kernel and bandwidth given as Gaussian Kernel, adaptive kernel and adjusted Akaike Criterion respectively, where

Gaussian Kernel, w⁢(di⁢j)=𝑒𝑥𝑝⁢(-12⁢(di⁢jh)2) and adjusted Akaike Criterion,

where di⁢j are the distances between the observations and h is the bandwidth. The AIC criterion favours a compromise between the predictive power of the model and its complexity [8]. This process has been fully described in [13] and is implemented with the Comprehensive R Archive of Network (CRAN) package ‘spgwr’ by [14] and is used in this study because of its ease and flexibility.

2.3.1Defining neighbourhoods

The definition of neighbourhood consists in selecting the kclosest points as neighbours, which leaves no point without a neighbour and this best offers a reflection of reality of including islanded countries of Madagascar, Mauritius, Comoros, Cape Verde and Sao Tome and Principe (non-contiguous areas), which are part of the landmass called Africa. The choice can also be made to keep only the points located at a certain distance. The choice of k here is 1, one nearest neighbour.

The nbdists function of the ‘spgwr’ package in CRAN is used to calculate the vector of distances between neighbours. It makes it possible to determine the maximum distance dmax below which all points have at least one neighbour. The dnearneigh function allows us to then keep as neighbours only the points between distances 0 and dmax.

Figure 1.

Nearest neighbours spatial relationship of observations of total fertility rate across Africa.

Figure 2.

Spatial distribution of variables across Africa including non-contiguous areas (a) shows the 2017 data on total fertility rate (b) shows the 2017 data on adolescent fertility rate (c) shows the 2017 data on infant mortality rate (d) shows the 2017 data on percentage contraceptive prevalence rate by women of reproductive age.

Figure 3.

Spatial distribution of variables across africa including non-contiguous areas (e) shows the 2017 data on gross domestic product per capita (US Dollars) (f) shows the 2017 data on life expectancy in years (g) shows the 2017 data on percentage of women of reproductive age with unmet need for family planning (h) shows the 2017 data on percentage of female labour force participation.

The observations are centroids which are specific locations representative of each country of Africa. In this case, the underlying assumption is that the distribution of the variable within each country is at least homogeneous enough to be estimated by a single observation. Figure 1 shows the neighbourhood distribution across space showing the links of neighbours of the observations taken for Africa as defined by the nearest neighbours distance spatial relationship.

2.4Spatial distribution of variables

Figures 2 and 3 show the spatial distribution of the 2017 observations of the eight variables including TFR across Africa. These observations are a single value for each of the 53 countries for the analysis on the eight different variables, and represent the average values for all the divisions of a country. It can be seen that as at 2017, the West African sub-region had the highest TFR in Africa, with Middle Africa following. Southern and Northern Africa have the lowest fertility rates in the continent. This pattern can also be seen in the distribution of AFR as the highest rates are within Western, Middle and Eastern Africa, while Northern and Southern Africa have the lowest adolescent births.

IMR, CPR, and unmet need for family planning show the same spatial distribution pattern across Africa. Northern and Southern Africa have the lowest infant mortality rates and unmet need for family planning compared to other sub-regions.

The gross domestic product per capita for countries in Western, Middle and Eastern Africa is on the average less than US$2,000, with exception of the Island of Mauritius that recorded over US$10,000 in gross domestic product per capita. Southern and Northern Africa recorded gross domestic product per capita in 2017 between US$4,000 and US$8,000, except for Libya, which has been in unrest and political crises since 2011. Life expectancy at birth is much higher in Northern Africa than other regions, Fig. 2f. Figure 2h shows that the percentage of female labour force participation in 2017 was almost evening out across Western Middle, Eastern and Southern Africa, with exceptions of countries like Guinea, Burundi, Rwanda, Mozambique and Zimbabwe where the percentage of women in labour force is between 50% and 60%.

3.1Spatial autocorrelation of total fertility rate

Table 4 shows the output Moran autocorrelation index from Moran’s test as indicated by R.

The Moran I statistic value is 0.345, showing that TFR is only weakly and positively autocorrelated across Africa and the p-value is significant (0.00005978) for the rejection of the null hypothesis of absence of autocorrelation among the observations and hence the conclusion that similar values of TFR do spatially cluster.

With the presence of autocorrelation in TFR across space in Africa, the spatial clustering across space can be examined for patterns that are not obvious in the global analysis given by OLS models. First a Moran plot is created which looks at each of the values plotted against their spatially lagged values. It explores the relationship between the data and their neighbours as a scatter plot.

Figure 4.

Moran’s diagram of observed total fertility rate versus spatially lagged values of total fertility rate.

From Fig. 4 it can be seen that the observations have a particular spatial structure and the linear regression slope is non-null indicating a correlation between TFR and its spatially lagged values. The regression line shows a positive relationship (its slope is defined by the value of the value of Moran’s I given in Table 4 as 0.345) and most of the observations are in the upper right corner of the plot, indicating values of the TFR that are higher than the mean value (high-high). The lower left corner which has observations values lower than the mean (low-low) has the second highest numbers of observations after the upper right corner, followed by the top left (low-high) and lastly by the bottom right (high-low) corners in that order. The implication of this structure is that most countries with higher TFRs have positive spatial autocorrelation and high index value. Other countries are surrounded a mix of countries with high and low TFRs. There are six observations that do not follow the spatial structure as can be seen in the Moran’s diagram.

Figure 5.

Map of local moran index (iw) for spatial autocorrelation of total fertility rate at each location.

A local Moran output in Fig. 5 explores the local spatial autocorrelation and shows the intensity and significance of local autocorrelation between TFR value in a location and its value in the surrounding locations. Significant groupings of similar TFR values (positive local Moran statistic (Ii) values) can be seen around Western, Northern, parts of Middle and Southern Africa. While groupings of dissimilar values (negative local Moran statistic (Ii) values) can be observed mostly in Eastern Africa and Northern Africa and some countries around the coastlines.

From the map it is possible to observe the variations in autocorrelation across space. It can be interpreted that there seems to be a geographic pattern to the autocorrelation. In large sample observations, it may not be possible to understand if groupings are of high or low values. To obtain the local spatial autocorrelation structure for a more clearer picture, a map of the p-values may be used to observe variances in significance across space as is given in Fig. 6, which labels the features based on the type of relationships they share with their neighbours, that is high-high, low-low, high-low or low-high.

Figure 6.

Map of p-values of local moran index.

From Fig. 6, it is apparent that there is a statistically significant geographic pattern to the grouping of TFR across Africa. High-high locations have positive spatial autocorrelation and high index value, that is, a country with high TFR value is surrounded by countries with high values. Low-low locations have positive space autocorrelation and low index value. High-low locations have negative spatial autocorrelation and high index value. Low-high locations have negative spatial autocorrelation and low index value.

Since there is global spatial autocorrelation, the local Moran statistic of spatial autocorrelation shows that the areas with high-high local autocorrelation patterns are the ones with strong influence on the overall total fertility rate in Africa. It can also be seen that the distribution of local Moran’s I is centred on the global Moran’s I (Fig. 7). These zones have a significantly similar spatial association structure with the global structure and from Fig. 6, it can be seen that it is the region made up of countries with high-high local autocorrelations. From this it can be seen that the hypothesis of independence of observations does not hold as a result of spatial autocorrelation, hence analysis of the data with the usual statistical techniques may not be sufficient, thus the need of GWR technique that puts into consideration the influence of space in the dynamics of the total fertility rate is highlighted.

Figure 7.

Density plot of local moran index for total fertility rate (2017 data) in Africa.

The study continues with the analysis of the dataset on total fertility rate with respect to its determinants using the GWR, but first examines the results of ordinary least square (OLS) regression before looking at the GWR results and compares it with that of OLS regression, and lastly examines the GWR model fit.

3.2Analysis with GWR

Firstly, an ordinary least square (OLS) regression model was fitted for TFR given the predictors as displayed in Table 5. This shows that the variables in the model explain 72.6 percent of the variations in TFR (adjusted R-squared value 0.726). Of the factors considered in the regression only GDP per capita, CPR and AFR were the significant in the model. But it is known that unmet need for family planning and modern family planning have strong correlations with CPR, so one can be safe to say that this correlation was reflected in the model by leaving these two factors out.

It is worthnoting that female labour force participation, life expectancy at birth and IMR were not significant. To be sure of the result, several other models were fitted with different combinations of all the predictors to see how each fair comparing them with their adjusted R-squared values and AIC values. AIC takes in account the number of independent variables in each model and the maximum likelihood of the models, but penalises models with more parameters. According to AIC, the best-fit model (models with lower AIC scores) explains the greatest amount of variation using the fewest possible independent variables. Models with higher adjusted R-squared values and lower AIC values are preferred and selected among groups of models. The model given in Table 6 was found to be the best having adjusted R-squared value of 0.739 and AIC of 99.90. This means that this model has the fewest number of independent variables which account for 73.9% of the variations in TFR.

3.3GWR model coefficients

GWR was applied for the identified variables from the OLS model in Table 6 to take into account the influence of location on TFR as it fits different model at each point of the space. The output of interest here included a summary of the regression coefficient estimates across the Africa and a number of attributes which corresponded with each unique output area such as local R-squared and residuals. With AIC of 90.4 and Quasi-global R-squared value of 0.777 based on the GWR, these variables explained 77.7% variation in TFR.

Figure 8.

Map of variable coefficients and coefficient standard errors (a) the plot of coefficients of adolescent fertility rate across Africa (a(i)) the plot coefficients standard errors for adolescent fertility rate across Africa (b) the plot of coefficients of percentage of contraceptive prevalence rate across Africa (b(i)) the plot coefficients standard errors for percentage of contraceptive prevalence rate across Africa.

The output from the GWR model in Table 7 reveals how the coefficients vary across the 53 countries. The global coefficients are exactly the same as the coefficients in the earlier OLS model. In this particular model, multiplying the coefficients by 10000 it can be seen that GDP per capita range from a minimum value of -1.16 (1 unit change in GDP per capita results in a drop in average TFR by -1.16) to -0.882 (1 unit change in GDP per capita results in a drop in TFR by -0.882). For half of the countries in the dataset, as GDP per capita rises by 1 point, TFR will decrease between -1.07 and -0.919 points (the inter-quartile range between the 1st Quartile and the 3rd Quartile). It is also observed that negative relationship in coefficient of percentage contraceptive usage is also here with total fertility rate. The coefficient for CPR ranges from a minimum of -236.69 to a maximum of -200.30 across Africa.

The coefficients for IMR and ADR are positive with TFR. As IMR rises by 1 point for half of the countries in the dataset, TFR will increase between 40.681 and 74.871. Similarly, a unit change in AFR results in an increase in TFR between a minimum of 103.40 and a maximum of 123.49.

Table 8 shows a comparison of the measures of model fit for GWR and OLS models and one can assess the benefits of moving from a global model (OLS) to a local regression model (GWR) as is evident in the measures’ values.

The distribution across space of the coefficients and their standard errors can be visualized as shown in the plots of GWR coefficients in Figs 8 and 9 for all the variables and are suggestive of spatial patterning.

Figure 9.

Map of coefficients and coefficient standard errors (c) the plot of coefficients of infant mortality rate across Africa (c(i)) the plot coefficients standard errors for infant mortality rate across Africa (d) the plot of coefficients of gross domestic product per capita across Africa (d(i)) the plot coefficients standard errors for gross domestic product per capita across Africa.

Looking at Fig. 8a plot, the coefficients for adolescent fertility rate can be seen to be highest in Northern Africa and some part of Western Africa (Mali, Niger and Nigeria), including Chad (Middle Africa) and Eritrea (Eastern Africa). The coefficients for adolescent fertility rate are lower in the other parts of Africa. Yet all across Africa, adolescent fertility rate has a positive relationship with total fertility rate. The CPR in Fig. 9c, the lowest coefficients are seen across Western Africa (Cote d’Ivoire, Liberia, Guinea, Sierra Loene, Guniea-Bissua, The Gambia, Senegal, Mali, Cabo Verde and Mauritania), including Morocco (Northern Africa). The highest CPR coefficients are seen in Middle and Western, while Southern Africa has lower coefficient compared to other regions.

From Fig. 9e plot, the coefficients for IMR is seen to be highest around Eastern Africa including Democratic Republic of the Congo (Middle Africa) and these values reduces as the circumference expands towards other part of Africa, with the least coefficients in parts of Western (Liberia, Guinea, Sierra Leone, Guinea-Bissau, The Gambia, Senegal, Mali, Cape Verde and Mauritania) and Northern Africa (Morocco and Algeria). Yet all across Africa, as IMR has a positive relationship with TFR.

Taking the Fig. 9g plot, which is for the GDP per capita coefficients, the spatial pattern is quite interesting. As can be seen from the plot, the lowest negative coefficient for GDP per capita is around Southern Africa and some parts of Eastern Africa and the values increases north-wards with the negative-maximum values seen in Northern countries of Libya, Tunisia, Algeria and Morocco, including Western countries of Benin, Niger, Burkina Faso, Mali, Senegal and Mauritania, plus Chad (Middle Africa).

Figure 10.

Map of GWR residuals.

3.4Assessing GWR model fit

The residuals are examined to assess model fit and residuals above or below zero indicate that the model under- or over-predicts total fertility rate. The Residuals are mapped as shown in Fig. 10. It is expected that the grouping of negative and positive residuals would be randomly distributed. From Fig. 10, it can be seen that the negative and positive residuals are distributed across Africa.

Figure 11.

Histogram of GWR residuals.

A histogram of the residuals in Fig. 11 shows the values are centered on zero and approximates a normal distribution across Africa.

Taking as threshold interval for residual (-0.49, 0.49), the largest over-predictions occur at Cape Verde, Guinea, Liberia, Sierra Leone (Western Africa), Mauritius, Mozambique, Djibouti (Eastern Africa), Equatorial Guinea, Central Africa Republic (Middle Africa) and Libya (Northern Africa) while the lowest under-prediction occur at Nigeria, Niger (Western Africa), Somalia, Burundi, Rwanda, Ethiopia (Eastern Africa), Democratic Republic of the Congo (Middle Africa) and Algeria (Northern Africa). These are summarized in Tables 9 and 10.

Figure 12.

Map showing locations of over-predictions and under-predictions of total fertility rate.

Figure 12 gives a pictorial overview of Tables 9 and 10 and it shows the map of Africa with the locations where the GWR over predictions and under predictions of TFR and also shows that they are random across space.

A Moran’s I Spatial Autocorrelation is run on the GWR residuals to confirm they are spatially random. The Moran’s I test shows that the residuals are significantly random with a p-value of 0.276 and a Moran I statistic value of 0.036 autocorrelation as shown in Table 11. As can be seen in Fig. 12, the clustering of high and/or low residuals is not statistically significant as reported by the spatial autocorrelation value and indicates that the GWR model is reasonable for the spatial distribution of TFR across Africa.

Figure 13.

Moran’s diagram for plot of GWR residuals against spatially lagged GWR residual values.

Figure 14.

Map of local R-squared values across Africa.

The Moran’s diagram in Fig. 13 shows a random distribution of the points on the plot of the GWR residuals against their spatially lagged values, which supports the Moran’s I test that there is no spatial autocorrelation of the residuals. The points are almost evenly distributed in each quadrant of the diagram and the regression line close to zero (horizontal to the x-axis) and the slope of the regression line is close to zero (0.036).

Figure 14 shows the map of the local R-squared values with the minimum and the maximum local R-squared values being 0.743 and 0.776 respectively at Democratic Republic of the Congo and Tunisia respectively. The minimum local R-squared value of 0.744 is above the R-squared value from the OLS regression; hence in this case, the geographical location in the evaluation in GWR improves the model’s ability to explain variability in TFR across Africa.

The map of the Local R-squared values also shows the locations where GWR predicts the best with highest values. GWR predicts well across Africa and shows that the predictor variables sufficiently explain the variability in TFR across Africa. They inherently capture many other difficult-to-track factors like religion and cultural practices.