Modelling malaria treatment practices in Bangladesh using spatial statistics

This study was carried out in the remote, forest-covered Chittagong Hill Tracts (CHT) of Rajasthali, Bangladesh (Figure 1), with a population of 24,097 [4]. All households and health facilities of Rajasthali were mapped using Global Positioning System (GPS) devices, distances from households to health facilities were computed, and a survey was administered asking about malaria-related treatment-seeking behaviour. In addition, 1,400 of the 5,322 households were screened for malaria using a rapid diagnostic test (Falci-vax). A full description of the data gathered, sampling techniques and logistics is provided in Haque et al, [4, 22].

For this research, the sampled household data were aggregated into village totals and analysed within a Geographic Information System (GIS) environment [23]. Twenty-one villages with less than five household samples were excluded from the analysis. The remaining 88 villages were represented geographically by the mean centre [24] of their sampled households. Distance variables, such as distance to regional hospitals, were computed by summing, then averaging individual household distances.

The local Getis-Ord Gi statistic [25] provides a picture of malaria-related treatment-seeking variations in the study area. This statistic produces a hot-spot map (Figures 2 and 3), and was applied to both the vendor usage and the village control programme usage rates. The local Getis-Ord Gi statistic works by comparing the local mean rate (the rates for a village and its nearest neighbouring villages) to the global mean rate (the rates for all villages). It produces a z-score and p-value for each village, reflecting whether the differences between the local and global means are statistically significant or not. A statistically significant positive z-score indicates a hot spot for high rates where it is very unlikely that the spatial clustering of high values is the result of random spatial processes. Similarly, a statistically significant negative z-score for a village indicates a local spatial clustering of low rates (a cold spot) [25‐27].

With a picture of the spatial patterns of control programme and vendor use for villages within the study area, the next step was to model those treatment preferences using regression analysis. Results from ordinary least squares regression (OLS) can only be trusted, however, if they are derived from a properly specified model. A properly specified model is one that meets all of the requirements of the OLS method [28]: coefficients for model explanatory variables should be statistically significant and have the expected sign (+/-); explanatory variables must be free from multicollinearity; the model should not be biased (heteroscedasticity or non-stationarity); residuals must be normally distributed with a mean of zero; the model cannot be missing key explanatory variables; and residuals must be free from spatial autocorrelation [29].

A data-mining tool called Exploratory Regression was used to find a model that met all of the requirements of the OLS method. Exploratory Regression is similar to stepwise regression; instead of looking only for models with a high-adjusted R², however, it identifies models that meet all of the requirements outlined above. The candidate variables used with the Exploratory Regression tool are described in Table 1. This method yielded two strong, properly-specified OLS models (Tables 2 and 3) [30]: one explaining the number of people using vendor services in each village (Adj R² = 0.90) and one explaining the number of people in each village using control programme services (Adj R² = 0.81). Both of these models were validated using bootstrapping and cross validation. For cross validation, the model was fit to a random sample of half the villages and validated against the remaining villages. For bootstrapping, 100 random samples (with replacement) were extracted and both models were tried. All of the explanatory variable coefficients were statistically significant in more than 95% of these random samples.

Geography matters: a variable that might be a strong predictor in one village may not necessarily be a strong predictor in another. To better understand this type of regional variation, Geographically Weighted Regression (GWR) was applied [31]. GWR is one of several spatial regression techniques used increasingly in geography and other disciplines. Rather than fitting a single linear regression equation to all of the data in the study area, GWR creates an equation for every feature (each village, in this case) and calibrates it using nearby features; closer features have a larger impact on calibration than features that are further away. Because each feature has its own equation, coefficients are allowed to vary over space [31]. Variations in tribal affiliations, education, economics, and access to treatment resources, for example, are likely to foster a variety of treatment preferences and community practices. Particularly useful outputs from GWR are maps of the coefficients associated with each explanatory variable. Where an explanatory variable has remediation implications, understanding where it is a strong predictor can provide guidelines for targeted interventions.

Individuals have multiple options for obtaining malaria treatment in Rajasthali and they commonly utilize more than one treatment strategy. For the 1,400 individuals surveyed, 42% indicated they would use vendor-provided services as well as other types of services, including control programme treatments; 73% indicated they would use control programme services as well as other types of services, including treatment from local vendors; 26% indicated they would seek treatment only from non-control programme options. These variations in treatment-seeking preferences were especially interesting when analysed spatially. Applying hot-spot analysis to vendor usage rates for each village (n = 88) revealed statistically significant (p < 0.01; z > 2.58) hot spots for the 10 western-most villages in the study area (Figure 2). In contrast, statistically significant hot spots for high control programme usage were located in the centre of the study area around the region's largest hospital and supplier of control programme services (Figure 3).

Table 2 shows the results from an OLS model of vendor usage. The dependent variable was the number of people in each village indicating they would use vendor services for their malaria treatment. This model explained more than 90% of the variation in vendor usage (Adj R² = 0.9011), and met all of the requirements of the OLS method [30]. The robust probabilities (based on White Standard Errors [32]) for the explanatory variable coefficients were statistically significant (p < 0.01), variance inflation factor (VIF) values were low (VIF < 7.5) indicating no problems with multicollinearity, the Joint Wald Statistic indicated overall model significance (p < 0.01), and the non-significant (p > 0.10) Jarque-Bera diagnostic indicated model residuals were normally distributed. The factors influencing vendor use include education (Figure 4), homes constructed from straw roof (poor economic status) (Figure 5), and an interaction term relating village size with distance to the regional urban centre where the largest regional hospital is located (Figure 6). Larger villages further from the regional hospital (the interaction term), with very little schooling (education), and better economic status (fewer homes constructed using straw roofing materials), were more likely to use vendor services. Factors influencing preference for control programme services, on the other hand, include household densities (Figure 7), tribal affiliation (Figure 8), housing construction materials (Figure 9), and, again, the interaction term relating village size to regional hospital proximity (Figure 6). This model (Table 3) explained 81% of the variation for people preferring control programme services (Adj R² = 0.8102), and indicated that the smaller, more compact, villages closest to the regional hospital, with wood flooring, and larger numbers of Bengali residents were more apt to take advantage of these services.

While OLS effectively identified the factors contributing to both control programme and vendor usage, OLS is a global model that assumes the relationship between each explanatory variable and the dependent variable is consistent (stationary) across the study area. In the case where relationships are non-stationary, model fit will improve by using GWR [31]. The adjusted R² value for vendor usage, for example, increased from 0.90 using OLS (Table 2) to 0.95 using GWR (Table 4). The corrected Akaike's Information Criterion (AICc) value from the OLS model (AICc = 465.79) was larger than the AICc value from the GWR model (AICc = 423.89). AICc is an effective way to compare models [33] and a drop of even three points indicates an important improvement in model fit [31]. Similarly, the adjusted R² value for control programme usage (Table 3) increased from 0.81 using OLS to 0.85 using GWR (Table 5), with a corresponding drop in AICc value from 532.97 with OLS to 515.66 with GWR.

In addition to improving model fit, GWR also provides useful information about explanatory variable stationarity [31]. Mapped coefficients for each village (Figures 10, 11, 12, 13 and 14) indicated where the explanatory variables were effective predictors of treatment preferences and where they were not. Education, for example, had a positive relationship to vendor use: as the number of people with no schooling increased, preference for vendor treatment also increased. In Figure 10, the villages rendered using the darkest colours indicate where the coefficient for the education variable is largest. The larger the coefficient is, the stronger the relationship is. The education variable is a strong predictor in the western-most villages only. Other variables exhibit this non-stationarity as well. The interaction term relating distance to the regional urban centre and village size also had a positive relationship with vendor use preference. Figure 11 shows that this explanatory variable is a very weak predictor in the eastern-most villages where the coefficient is near zero and even slightly negative; it is a strong predictor throughout the rest of the study area, however. Figure 12 looks at the coefficients for household densities found to be a key explanatory variable for predicting preference for control programme services. This variable predicts most strongly in the eastern- and also western-most villages; it is a less effective predictor in the central villages. Villages with larger numbers of Bengali people were also found to be associated with increased control programme preference. Figure 13 presents a map of the coefficients for that variable, indicating that the Bengali population is an especially good gauge of control programme preference in the northern central villages.