In the literature, diarrhea prevalence is attributed to factors of two kinds: household and geographical 39. Among household variables, low mother's education is very widely identified as major factor to explain higher diarrhea occurrences in households 244041. Studies such as 42 established high correlation of diarrhea with household sanitation and hygiene conditions. Access of the household to good water has a positive effect in reducing diarrhea 910. Distance to hospitals is also important, as access to medication is effective in controlling repeated occurrence of diarrhea 43. Diarrhea occurrence is higher in lower income households where educational standards tend to be lower as well, access to good water source more limited, and hygiene practices poorer 444546. Many studies also emphasized the role of geographic variables such as climate 27, and especially rainfall 47, though the findings were not unambiguous. Studies such as 4849 found higher infection rates during dry seasons, while others reported the same for the humid period 5051. For West Africa, the rotavirus infections, to which children below 5 years are most sensitive (49 and references within), show a seasonal cycle that peaks mostly in drier months (52 and references within), when water scarcity affects hygienic conditions and favors transmission via fecal-oral route 53.

The connection between accessibility to good water source and diarrhea prevalence is well documented 9. For example, under ecological stress conditions, households tend to procure low quality water and cut down on water use for sanitary purposes 45. Similarly, higher population densities coincide with intensified diarrhea prevalence, due to transmission of diseases via social networks that are obviously more connected when people are living closer together 14. Households also often tend to substitute unreliable or inaccessible sources of good quality water by more reliable sources with poorer quality 5455, the effect of which may even get more accentuated under higher population densities. Such tradeoffs appear when households dependent on groundwater find their aquifer unreliable or suffer from lower rainfalls on average.

Summing up, major factors explaining diarrhea prevalence would be household conditions such as income, mother's education, distance to hospital and sanitation conditions; and geographic conditions such as household's access to good water, population density, rainfall, ground water quality and suitability, and population density.

We report on how such variables were obtained for joint analysis in the present study.

Figure 1 shows that population density distribution in the ORB is representative for the Middle Belt in West Africa: relatively low in the Northern part of the Basin and in higher in the South nearing the coastal areas.

Figure 2 has been created using Demographic and Health Survey data of USAID 25. Data on diarrhea prevalence are obtained as the frequency in the survey 25 of households with a child that suffered diarrhea in the past 24 hours, computed within a cluster. The figure shows mean household prevalence of diarrhea averaged at the municipality level (Figure 2a) as well as cluster level (Figure 2b). From hereon, maps at cluster level refer to "municipality constrained" Thiessen polygon maps, which are created using nearest neighbor mapping from cluster locations to grids belonging to the same municipality. Here the prevalence rates reach even higher values, as much as 480 per 1000 children at municipality level. Overall, we observe almost the same spatial pattern in diarrhea prevalence as in Figure 1. The most affected municipalities lie in the mid-northwest and to a lesser extent in the south. Some municipalities in the south and mid-south, however, show lower diarrhea prevalence than in Figure 1. Yet, both figures concur on the parts of the country that face high prevalence of diarrhea. Figure 2c) makes this explicit by showing the differences of cluster level diarrhea rates from national average via t-statistics. The t-value for a cluster indicates whether this cluster value differs significantly from the national average, with a positive sign when cluster level value is higher than the national average. Hence, all three figures suggest some geographic pattern of diarrhea prevalence, with the middle and the northeast part of the country facing the least favorable conditions. However, since the survey data are collected at household level, there also is a social, in the sense of non-spatial component to be distinguished, that represents the variation across households at every location. Therefore, before considering possible explanations of these variations in terms of biophysical and "behavioral" factors, we decompose the diarrhea occurrence into a spatial and a non-spatial component.

The data set compiled from DHS is geo-referenced at cluster level. We estimate the part of total variance that is spatial by calculating the variance, across various clusters, of within clusters prevalence means, and similarly for clusters aggregated to municipality level.

Table 1 presents various descriptive statistics on prevalence of diarrhea in Benin, including this variance decomposition into I^b the "variance between" and I^w, the "variance within" municipalities. Both variances are significantly different from zero. Only 14 percent of the variance is associated with variance across clusters, whilst for municipality classes this percentage falls down to 8 percent, which could be expected as this enlarges the spatial class but it remains that even at cluster-level, the presumably non-geographic and non-biophysical "within-variation" is far superior to the not even exclusively geographic "between-variation". The next sections further develop this point.

In this section, we show that despite the importance of within-variation, the between-variation that accounts for spatial variation cannot be discarded as this reflects quality differences among public institutions and cultural practices as well as of geographic conditions.

To represent the spatial variation of good access to water we averaged this binary variable at cluster level and depicted the results in Figure 3(a).

Figure 3 shows that many clusters in the northwest, mid-north and middle, and in the south of the country that have poor access to a good water source correspond to those with high diarrhea prevalence. However, not all the clusters with high diarrhea prevalence correspond to poor accessibility, as can be seen from boxes A, B, and C in the figures 2b and 3a. The same boxes, however, show in Figure 3(b) and 3(c) that high stress conditions prevail. Thus, households equipped with good source of water but unfavorable geographic conditions may face a similar situation as households with poor water accessibility. They tend to substitute their quantitatively unreliable sources with a less unreliable source, often of poorer quality. This increases their propensity to suffer from various water-borne diseases, such as diarrhea.

Recall from Table 2 that only 14 percent of variance in diarrhea prevalence at cluster level is spatial while the remainder is "within-cluster" variance. We now look into factors such as mother's education (ED), hygiene condition (HG), income level (HC) and distance to hospitals (DH) that may help explaining this "within-cluster" variance in diarrhea prevalence.

Figure 4(a) shows [1 – Ind1] with higher values implying poorer household conditions, while figure 4(b) shows a box plot of [1 – Ind1] with increasing class number identifying poorer household conditions. This plot rejects any suggestion of diarrhea prevalence not being connected to household conditions. It also indicates that the variance in diarrhea occurrence across the classes remains the same but that the median of classes is non-decreasing (almost always) with poorer household conditions, and doubles from one extreme of household conditions to the other.

This sub-section discusses application of the mixed effect logit model based on data for the entire country at household level, used to predict diarrhea prevalence in the Oueme River Basin (ORB).

The discussion so far has decomposed the prevalence of diarrhea into factors occurring at two levels, between and within clusters, with geographic or biophysical factors associated to the between-part, and socio-economic factors mainly but not exclusively confined to the within-part explanation. We now seek to go one step further, by estimating a logit model of diarrhea prevalence that includes socio-economic variables (such as DA, GW, ED, HG, DH) at household levels and geographic variables (geographic stress conditions Geo₁ and Geo₂ and rainfall, Rf) at cluster level, treating the household variables as exogenous as opposed to instrumentalizing them, for the reasons given earlier.

Regarding the linearity of the functional form of the mixed logit model (linear in three variables: geographic stress indicator scaled by accessibility conditions, rainfall and socio-economic indicator) we find that this specification is reasonably stable when nonlinear (square) terms are being allowed for. Likelihood estimates were found slightly lower with lower significance when covariates entered in square instead of linear forms. However, when covariates entered in both linear and square forms, corresponding parameters were found to be insignificantly different from 0.

Another linear functional specification ("Form2") with scaled geographic stress indicator as the only independent variable was also tested. While the resulting parameters were significant at 95% confidence level, the likelihood value was considerably lower than the linear form in variables functional specification ("original") with the latter additionally containing rainfall and socio-economic indicator as independent variables. This clearly suggests that additional variables explain additional variance in diarrhea prevalence. Furthermore, regarding our choice of explanatory variables, the likelihood improves as the variable "access to good water source" is supplemented by other covariates. This suggests that household variables influence diarrhea prevalence through other channels than accessibility and should therefore not be treated as instrument variables but rather as explicit explanatory variables. Next, we looked into the trade-off between joint estimation that improves the overall explanatory power as expressed by the likelihood value and possible bias on the treatment effect, as expressed by the Wu-Hausman test that might call for instrumentalization. For this, we conduct a two stage estimation with the first stage as a linear regression between the two covariates (rainfall and socio-economic indicator as independent variables) and geographic stress indicator scaled by "access to good water source" (as dependent variable); and use this regression function in the second stage mixed logit. First stage parameters are significant but the r-square (~0.072) is very poor, which, as reported by 67, tends to invalidate the use in IV-regression that is outperformed by OLS. The poor fit actually suggests that 'access to water' has ample room for variation at given level of the instruments, and can be used as an exogenous variable next to these instruments in the second stage regression, albeit that the Wu-Hausman test on first-stage residuals significantly deviates from zero, pointing to possible endogeneity.

In sum, given the low fit of the first stage estimation, the lower likelihood in the second stage estimation, the fact that no other variables of sufficient quality were obtainable from the DHS survey that could substitute as instruments and that the instruments have to appear in the structural equation anyhow, we conclude that joint estimation is the preferable approach to explain the diarrhea prevalence.

Table 3 shows the estimation results. Coefficients are all at least significant at 0.05 significance level and the hit ratio of correctly predicted classes is 76 per cent. All the fixed effects have expected signs. Improved household conditions reduce prevalence of diarrhea and its effect is significant. Geographic stress conditions accentuate prevalence of diarrhea, as households with similar socio-economic conditions and even similar access to good water source but less favorable geographic stress conditions have higher chance of facing diarrhea. Similarly, households with comparable socio-economic and geographic stress conditions have higher rates of diarrhea under circumstances of poorer access to good water source. Effect of rainfall on diarrhea prevalence is in line with other studies in West Africa as lower rainfall increases diarrhea prevalence.

We conclude that the data show evidence of higher diarrhea transmission under scarcer water conditions in Benin, due to higher chance of human consumption of contaminated water.

Next, we apply the estimated model for prediction within the ORB. To compare the actual cluster-wise mean rate of diarrhea prevalence with the predicted one, we define the cluster-wise arithmetic mean of modeled prevalence: π¯j=∑ik,k=jπ¯jm MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafqiWdaNbaebadaWgaaWcbaGaemOAaOgabeaakiabg2da9KqbaoaalaaabaWaaabuaeaacuaHapaCgaqeamaaBaaabaGaemOAaOgabeaaaeaacqWGPbqAcqWGRbWAcqGGSaalcqWGRbWAcqGH9aqpcqWGQbGAaeqacqGHris5aaqaaiabd2gaTbaaaaa@3EFE@, where π¯j MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafqiWdaNbaebadaWgaaWcbaGaemOAaOgabeaaaaa@2F33@ is the prevalence of diarrhea (1 or 0) and m is the number of households in jth cluster.

Figure 5 compares the actual cluster-wise mean rate of diarrhea prevalence to the mean of the modeled one. Figure 5(a) shows the comparison of cluster-wise average of modeled prevalence within Benin and Figure 5(b) predicted cluster averages of prevalence rates in the ORB. We see in Figure 5a) that the model fits the observed average prevalence rates reasonably well with a correlation coefficient of 0.89. However, it appears to smoothen prevalence rates, under-predicting high levels while over-predicting the low ones.

The relationship between inter- and intra-cluster variables and diarrhea prevalence is induced from DHS survey data. However, as this data set contains information only for selected households that lie within a sample of cluster locations, we interpolate the relationship to all the settlements within the ORB by nearest-neighbor rule. Figure 6a) shows a population map by settlements and Figure 6b) a mean prevalence map. This settlement map will serve as reference for the remainder of the paper. The logit model clearly points to significant adverse effect of poor water access on prevalence of diarrhea, which depends among others on the stress conditions of the aquifers. However, the strength of this effect appears to depend on households' socio-economic conditions. In other words, the model confirms that at the same location the households with poorer socio-economic conditions have higher prevalence rates.

For the other explanatory variables in the logit model we use the values at the nearest cluster location, and given these values for explanatory variables evaluate the corresponding diarrhea prevalence. Figures 6c) and 6d) identify disparities in prevalence of two types. One is that while the highest as well as the lowest socio-economic tertiles (sub-section 'Analysis') face conditions of bad access to good water source, they suffer to a different degree.

Figure 6c) indicates that there are some households belonging to the lowest tertile in almost all settlements with poor water access and figure 6d) that very few settlements have their highest class with poor access. Besides this disparity in access, diarrhea rates are also higher for lower household class under conditions of poor access, suggesting once more that sanitation practices and education level of the households play a major role. This suggests that increase in accessibility conditions, irrespective of household class, reduces the probability of diarrhea occurrence but the effect of increased accessibility on diarrhea reduction is different for different household classes.

We have seen already that a policy intervention that improves accessibility reduces the probability of diarrhea occurrence. However, as is evident from figures 6c) and 6d), households belonging to the lowest household class are the most affected. To simulate the effects of such an intervention, we reduce the fraction of population within the lowest socio-economic tertile without access to good water from zero intervention levels to half of that, while keeping access frequencies of the other two household classes constant.

Figure 7(a) presents the result measured as prevalence of diarrhea for the lowest household class in the entire basin, against the cost of drilling wells, supposing that the increased accessibility is realized through addition of wells. Each such well is taken to serve at most 250 people in each of the settlements in the basin.

One remarkable outcome is the reduction of 0.17 percent only achieved at maximal intervention, from the average diarrhea prevalence in the ORB of approximately 18 percent. This leads us to conclude that, while necessary, drilling of wells that in principle can provide access to good water is far from sufficient to tackle the problem.

Figure 7(b) gives the associated spatial distribution of costs needed to deliver the stated improvement in access, depending on the number of people covered by the intervention, and location-specific information on aquifer properties 57. The cost estimate for the construction of a well is based on drilling costs per meter depth as 5000 FCFA (1 EUR = 656 FCFA; 1 EUR ~1.6 USD) (community contributions; personal communication with experts from Direction de l'Hydraulique-Benin, 68 and 69), multiplied by the drilling depth. For drilling depth, we use nation-wide groundwater inventories of 57 and selected from the maximum of water table depth or observed drilling depths. Finally, we consider a multiplicative factor that represents the aquifer suitability (rock type) conditions for each grid. Thus, total costs of drilling a single well, C_s, at any grid location, s, can be formulated as,

C_s = 5000.η_s(q_s).max(d_s, w_s),

where for each grid s, d_s: drilling depth, w_s: depth to the water table, η_s(q_s): aquifer suitability multiplicative factor and q_s: aquifer suitability.

The total cost for each location is based on the cost estimate of drilling a single well and the assumption that each well serves at most 250 persons. The figure shows that costs are highest in the south and some places in the mid-north as well as north-west. For the majority of the remaining basin, the costs of the intervention remain low.