Clinics register based HIV prevalence in Jimma zone, Ethiopia: applications of likelihood and Bayesian approaches

The data for this study was obtained from Jimma zone districts’ Public Health center offices. The Jimma zone is in the Oromia regional state and located in the South-Western part of Ethiopia between Latitude 6\(^{\circ }\) and 9\(^{\circ }\) North and Longitude 34\(^{\circ }\) and 38\(^{\circ }\) East, and between altitude ranges of 880 to 3340 meters above sea level. The total coverage area of Jimma Zone is 15,568.58 km\(^2\) [15].

When patients visit government clinics due to illness, not necessarily HIV related, they get orientation about HIV including means of transmission and advantage of knowing their HIV status by health professionals. Then they are asked if they are willing to do HIV test and those agreed to do the test fill a consent form. The clinics give pre- and post test counselling. All patients aged 15 years and older, who visited the clinics and tested for their HIV status in the 22 districts of Jimma zone during the periods September 2018 to August 2019 were included in this study. The data that were extracted from the patient register include patient HIV status (negative or positive), patient gender, age (15–19, 20–24, 25–49 or \(\ge\) 50), marital status (single, married, divorced or widowed), educational level (no education, primary, secondary or superior), condom use (no or yes), religion (protestant, muslim, orthodox and other), occupation (no job, daily worker, farmer, merchant or government employee) and place of residence (rural or urban).

Permission of the study was obtained from Postgraduate research office of Jimma University, College of Natural Sciences. In addition, permission to use the data for this study was obtained from Jimma Zone Ministry of Health Office. The data provided to the authors do not contain any personal identifiers, therefore the anonymity of the patients were assured.

The Bayesian approach to GLMMs differs from likelihood methods that it treats all unknown parameters in the model as random variables and have probability distributions called posterior distributions, in contrast to how likelihood methods treat parameters as fixed constants. The likelihood of the model describes the data generating process given the parameters, and the prior usually reflects any previous knowledge about the model parameters. When the prior knowledge is scarce, vague or non-informative priors are assumed so that the posterior distribution is driven by the observed data [22]. The Bayesian aim is to estimate the joint posterior distribution and inference is conducted through the posterior distribution, which combines information from the probability of the data given the parameters, essentially via the likelihood and the prior distributions of the parameters.

Using the Bayes’ theorem, the marginal likelihood can be expressed as [23]where \(f(y_{ij} | {\varvec{\beta }}, b_i)\) denotes a probability mass function for HIV status of a patient. As discussed in the previous section, for GLMM this integral usually has no closed form. In such case, it is necessary to resort to other methods to estimate the posterior distribution or, alternatively, to draw samples from it. The classical approach for Bayesian inference is to use the Markov Chain Monte Carlo (MCMC) simulation techniques [24]. However, the MCMC is computationally expensive. Therefore, we have modeled the HIV prevalence using the integrated nested Laplace approximation (INLA) numerical method [25]. For more detailed discussion on the INLA computational approach see, for example, [23, 25, 26].

A Bayesian approach requires the specification of prior distributions for all the random elements of the model. For the GLMM in Eq. (1), it involves choosing priors for the regression coefficients and the hyperparameter of district random effects standard deviation (SD), \(\sigma _b\) . In this paper, since empirical information on parameters, \({\varvec{\beta }}\) and \(\sigma _b\) or relevant to the study data is not available, non-informative priors were used [27]. Specifically, we have used non-informative priors, i.e. N(0, 1000) for the regression coefficients since this is common practice and it allows to compare Bayesian method with maximum likelihood method. Since the choice of priors can have an important impact on posterior distributions of the model parameters and model performance can be sensitive to the choice of the district-specific random effects variance priors [27], we have considered three priors or hyperpriors for \(\sigma _b\) precision parameter \(\tau _b\), which are based on the inverse gamma distribution [28] and selected the best for the current data using sensitivity analysis. These priors are The fitted models were compared using the Deviance Information Criterion (DIC) [31] and the widely applicable information criteria (WAIC) [32].

For the statistical software program implementation, R codes were written for the likelihood and Bayesian approaches. The GLMMs were fitted using the glmer() function from the lme4 R package [33], whereas the Bayesian analyses were done using the inla() function from the R-INLA package [34]. In addition, ggplot2 and lattice packages of R were used for the graphical outputs. The detailed results are presented in Results section.

A total of 8440 patients were tested for HIV infection in government clinic at the 22 Districts between September 2018 to August 2019 in Jimma zone, of them 4228 (50.1%) were men and 4212 (49.9%) were women (see Additional file 1: Table S1). A large per cent (35.1%) of them in the age category 25–49. Over half of them (54.8%) lived in urban areas, only 6% were government employees, only 7.6% of them had above secondary schooling, i.e. superior education level, about 45% of them were muslims, more than a third (38%) were married, and almost half of them (50.4%) did not use condom during sex (Additional file 1: Table S1).

The overall HIV prevalence among tested patients at government clinics in the Jimma zone in the period between September 2018 to August 2019 was 22.1% in women and 24.3% in men (Table 1). In the age group 15–19 years, prevalence was 11.1% in women and 11.7% in men, whereas in the age groups 20–24, 25–49 and 50 years and older the HIV prevalence, respectively were 11.9% in women and 12.2% in men, 14.7% in women and 17.4% in men, and 6.6% in women and 7.2% in men. The HIV prevalence in Jimma zone increased with age, except for age group 50 years and older and was consistently higher among men across all age groups than among women (Table 1). The HIV prevalence was high among married compared to other marital status groups, in those individuals completed primary schooling, daily laborers, among muslims, among urban residents and those who did not use condom during sex (Table 1).

To identify factors associated with HIV infection, a multivariable logistic regression model with district random effects was applied on the data. The covariates were checked for multicollinearity using the variance inflation factor (VIF) before adding them to the model. None of these VIFs (the values are between 1.007 and 1.455) were greater than 5 suggesting the collinearity is not strong to affect the statistical inference in the analysis. To assess effect of a covariate on gender specific prevalence, the analyses were done for each sex separately in addition to analysis done using the combined data.

The estimated variances of the district random effects were \({\hat{\sigma }}^2_b = 0.094, 0.087\) and 0.080 for women, men and full data sets ( Table 2), and for the data sets, the asymptotic chi-square mixture distribution [18] test statistic for testing \(H_0: \sigma ^2_b = 0\) against \(H_1: \sigma ^2_b > 0\) take the value 19.86, 10.01 and 35.25 with p-value \(< 0.0001\), 0.0008 and \(< 0.0001\) ( Table 2), respectively. The very small p-values strongly suggest a rejection of the null hypothesis \(H_0: \sigma ^2_b = 0\) that no district-specific random effects should be included in the model. Therefore, these results imply the need for the district (cluster)-random effects in each model fitted to the data, which suggest that individuals with the same characteristics in different districts may have different HIV status in Jimma zone. The tests for fixed effects ( Table 3) show that except occupation across the three data sets and religion in men data set, the other fixed effects or covariates were significantly associated with odds of patients HIV infection because the 95% confidence intervals do not include 1. The results in Table 3 also show that the test on intercept, \(\beta _0 = 0\), is significant suggesting that a patient of 50 years of age or older, who was single, no formal education, muslim, had no job, used a condom during sex and lived in a rural area whose district had a random effect equal to zero had a log-odds of HIV infection different from zero.

Note that the odds ratios for the individual variables reported in Table 3 are conditional or cluster-specific measures of association. That is, they are interpreted as having an effect conditional on a district random effect being held constant [35]. Therefore, the interpretation of odds ratios are done here for within district comparisons, that is as district adjusted associations between patient characteristics and HIV prevalence.

Controlling for marital status, education, occupation, religion, place of residence and use of condom during sex, the odds of HIV infection was significantly higher in age group 15–19 years (aOR 1.734, 95% CI 1.190–2.527 for women; aOR 1.819, 95% CI 1.271–2.602 for men; and aOR 1.781, 95% CI 1.374–2.308) in the full data) compared to patients with age 50 years or older but who share the same district average risk, i.e. the same value of the random effect ( Table 3). Whereas the odds of HIV infection for women in age group 20-24 years was significantly lower than women patients with age 50 years or older and lived in the same district (aOR 0.707, 95% CI 0.514–0.971). When comparing patients who lived in the same district and who share identical values on the covariates age, education, occupation, religion, place of residence and use of condom during sex, the odds of HIV infection was 67.4 and 51.4% higher in married women and men compared with single women and men, respectively. Similar trend also observed among divorced and widowed, where the odds of infection among divorced women and men was 51.3 and 42.5% higher than single women and men, respectively; and among widowed it was 65.9 and 37.1% higher than single women and men, respectively. Further, except for widowed men group, the statistical tests on adjusted odds ratio for each of marital status category in the three datasets analyses were statistically significant at 5% level ( Table 3) because the corresponding 95% confidence intervals of the married, divorced and widowed groups for women and full datasets and married and divorced groups for men dataset do not include 1, whereas the confidence interval for widowed men group in mean dataset includes 1.

The odds of HIV infection of women patients who had primary, secondary and superior education levels were 1.633 (95% CI 1.286–2.074), 1.327 (95% CI 1.007–1.749) and 1.319 (95% CI 0.871–1.998) times higher, respectively than the odds of HIV infection of patients with no formal education but who share identical values on the covariates age, marital status, occupation, religion, place of residence and use of condom during sexual intercourse and who also share the same district average risk. Whereas for men patients, the odds 1.471 (95% CI 1.162–1.864), 1.125 (95% CI 0.854–1.482) and 1.158 (95% CI 0.767–1.748) times higher in those who were in primary, secondary and superior education levels compared with those who had no formal education. However, the result was statistically significant only for those in primary education level ( Table 3). For men patients who were farmers, the odds of HIV infection 1.446 (95% CI 1.053–1.987) times significantly higher than those patients who had no job. Compared to muslim patients who share identical values on the covariates age, marital status, education, occupation, place of residence and use of condom during sex and who also shared the same district average risk, the odds of HIV infection was 10.4 (95% CI 0.682–1.172) and 30.8 (95% CI 0.556–0.861) per cent lower in orthodox and protestant women patients, respectively, whereas in men patients it was 13 (95% CI 0.665– 1.40) and 27.5 (95% CI 0.585–0.897) per cents lower in orthodox and protestant patients, respectively. However, the results were statistically significant for protestant patients only (Table 3). Women and men patients lived in urban areas more likely to had HIV infection than those lived in rural areas but who share identical values on the covariates age, marital status, education, occupation, religion and use of condom during sex, and who also share the same district average risk (aOR 1.132, 95% CI 0.931–1.377 for women and aOR ratio 1.290, 95% CI 1.061–1.567 for men) but the result was only significant at 5% level for men patients because its the 95% CI for odds ratio does not include 1. When comparing two patients within the same district and who shared identical values on the remaining six covariates, women and men patients who had not used condom during sex were 74.009 (95% CI 59.651–91.823) and 67.026 (95% CI 54.395–82.590) times more likely to had HIV infection and these results were statistically significant.

Additional file 1: Tables S2 to S4 report on the results of priors influence on model parameters (posterior means and standard deviations). The three priors of hyperparameter for variance of district-specific random effects yielded similar results for the regression coefficients in models fitted to women, men and full data sets. However, there were slight differences in the point estimates of district random effects variances. Moreover, there were variations in the posterior densities of precision \(\tau _{b}\), where \(\tau _b = \sigma ^{-1}_b\) across three priors (see Fig. 1). The figure reveals that the Lunn et al. [29] prior was slightly more informative compared to Fong, Rue and Wakefield [30] prior but the inla default prior was the least informative.

Table4 displays the DIC and WAIC for the models fitted to the women, men and full data sets using different priors. The DIC and WAIC values of Lunn et al. prior were slightly smaller for models fitted to women and full data sets, however the DIC and WAIC values of Fong, Rue and Wakefield prior were slightly smaller for model fitted to men data than those of the inla default prior and the Lunn et al. prior ( Table 4). These results suggesting that the Lunn et al. prior to be a preferred prior for women and full data sets, whereas the Fong, Rue and Wakefield prior to be a preferred prior for the men data set. Therefore, in what follows, we only report results based on these priors for respective data.

The posterior means of the coefficients (Additional file 1: Tables S2 to S4) are very similar to the Adaptive Gaussian Quadrature or the maximum likelihood estimates ( Table 3), which is in line with the theory that non-informative priors should not have effect on the posterior. Table 5 displays adjusted odds ratios (aOR) and the corresponding 95% credible interval (CI) for covariates. Unlike the posterior means, the credible intervals for Bayesian analysis were different from the likelihood confidence intervals. The credible intervals provide a measure of uncertainty and since for the models fitted in this paper the posterior distributions of regression coefficients are symmetric these intervals are more robust than their likelihood counterpart [23]. Note in Table 5 that the Lunn et al. prior was used for women and full data sets and the Fong, Rue and Wakefield prior for men data set.

Considering the credible interval results in the multivariable model with Lunn et al. prior for variance precision, overall the odds of HIV infection among women patients visited district government clinics for various health related problems between September 2018 and August 2019 in Jimma zone was associated with all the covariates. Controlling for other covraiates, the odds of HIV infection among women in age group 15-19 years was 74% (aOR 1.74, 95% CI 1.195–2.536) more than those women with age 50 years or older ( Table 5). However, for those in age groups between 20 and 24, and between 25 and 49, the odds were 29.4 (aOR 0.706, 95% CI 0.513–0.969) and 9.2 (aOR 0.908, 95% CI 0.667 - 1.236) per cents lower than those of adults with age 50 years or older, respectively. Under the Fong, Rue and Wakefield prior, the odds of HIV infection for men in age group 15–19 years was 82% (aOR 1.82, 95% CI 1.273–2.604) higher than those men with age 50 years or older. Similarly, for men patients in age group 25-49 years the odds was 11.9% higher than those men with age 50 years or older. However, in age group 20–24 the odds was 7.7% (aOR 0.923; 95% CI 0.677–1.257) lower compared with the odds of HIV infection in age group 50 years or older. The results revealed that odds of HIV infection among women was 14% (aOR 0.860, 95% CI 0.750–0.986) lower than males (Table 5). Controlling for age, education, occupation, religion, place of residence and use of condom during sexual intercourse of patients, married, divorced and widowed women patients, respectively were 68, 51.8 and 66.5% more likely to had HIV infection compared to unmarried or single women patients (aOR 1.680, 95% CI 1.258–2.245 for married; aOR 1.518, 95% CI 1.665–2.087 for divorced; and aOR 1.665, 95% CI 1.162–2.387 for widowed). Similarly, the odds of HIV infection among married, divorced and widowed men patients were 51.4 (aOR 1.514; 95% CI 1.143–2.005), 42.4 (aOR 1.424, 95% CI 1.045–1.940) and 37.1 (aOR 1.371, 95% CI 0.959–1.962) per cent higher than single men, respectively.

In addition, women with primary, secondary and superior education levels were 63.7, 32.9 and 32.0% more likely to had HIV infection, respectively compared to those women with no formal education (aOR 1.637, 95% CI 1.290–2.079 for primary; aOR 1.329, 95% CI 1.009–1.753; and aOR 1.320, 95% CI 0.872–2.001). However, the increased in odds of HIV infection was statistically nonsignificant for superior education level. Among men patients, there were higher odds of HIV infection in primary (aOR 1.474, 95% CI 1.164–1.866), secondary (aOR 1.125, 95% CI 0.854–1.482) and superior (aOR 1.157, 95% CI 0.767–1.747) education levels relative to men with no formal education but the result was significant only for primary education level. Also, women daily laborers, farmers, government employees and merchants were 16.3, 8.1, 0.6 and 23.2% more at risk of HIV infection, respectively compared to those who were without job, whereas men patients in these categories were 24.6, 45.1 and 16.2 cents more at risk of HIV infection, respectively compared to those who were without job but the increase in odds of HIV infection was statistically significant only in men farmers (aOR 1.451, 95% CI 1.057–1.993). Further, for women the odds of HIV infection among orthodox and protestant were 10.7 and 30.1% lower than muslims (aOR 0.906, 95% CI 0.754, 1.090 for orthodox; and aOR 0.698, 95% CI 0.602, 0.809 for protestant), whereas for men the odds were 13 and 27.6% lower in orthodox and protestant relative to muslims (aOR 0.870, 95% CI 0.664–1.140 for orthodox and aOR 0.724, 95% CI 0.584–0.896 for protestant). For women who were urban residents compared to those who were in rural areas, the odds of HIV infection was 13.3% more (aOR 1.133, 95% CI 0.932–1.378), whereas for men urban residents the odds of HIV infection was 29.1% higher compared to rural residents (aOR 1.291, 95% CI 1.063–1.569). The results also showed that women who had sex without condom were 76.47 times more at risk of having HIV infection compared to those who used condom during sex (aOR 70.611, 95% CI 61.068, 81.872) and the risk was also very high for men who had sex without condom (aOR 68.848, 95% CI 56.093–84.997).

By transforming the quantiles of \(\sigma _b\), i.e. by exponentiating median and a 95% credible interval, one can easily interpret the standard deviation on an odds scale. For the results in Table 5, in odds scale median and a 95% credible interval for women are 1.372 and (1.199, 1.673), for men are 1.329 and (1.145, 1.642), respectively. This suggests that one standard deviation in district variation would multiply the odds of HIV infection for women on average by about 1.372, or equally possible divide them by 1.372, and the multiplication also would vary between 1.199 and 1.673 with 0.95 probability. Therefore, the variation in odds of HIV infection among districts was heterogeneous for both women and men in Jimma zone.