The Poisson-Gamma model for relative risk estimation uses a gamma prior distribution for the relative risk combines with the Poisson likelihood function for the disease counts which gives a gamma posterior distribution for the relative risk. The Poisson-Gamma model is widely used in disease mapping to account for extra variability in the data through the prior distribution [18] .

Suppose that the unknown risk of TB in region i is given as. Let and denote the number of TB cases and the population at risk respectively in region i. The expected number of TB cases in

region i can then be written as, where is the overall disease risk in the study population.

Under the frequentist paradigm, we assume that. Based on the sample, the likelihood function and the corresponding log-likelihood function are expressed as

and

The maximum likelihood estimator of is obtained via and is given by

This estimator, is referred to as the standardized mortality ratio in region i. Under the Bayesian paradigm, assume that, where is is the Poisson mean and has a gamma distribution with shape and scale parameters a and b respectively. The likelihood function for is denoted by

and prior distribution for is denoted by

We used Bayesian hierarchical methods for parameters estimation in the Poisson-Gamma model. That is, if in addition, a and b are given prior distributions such that and, where and are the hyperprior distribution with hyper-parameters and for a and b re- spectively, then we can obtain parameters using Bayesian hierarchical methods. This is a second stage hierarchical modeling using the Poisson-Gamma model.

In this paper, we defined (as exponential distribution) and as a gamma distribution. Therefore, the posterior distribution is given by

Parameters estimation of the Poisson-Gamma model was carried out using MCMC via Gibbs Sampling. Convergence of the Chain occurs at 40,000 iterations after a burnin period of 1000 sample and thinning of every 30^th element of the sample. Figure 1 presents the MCMC convergence diagnostics plots.

Figure 1 presents Gelman and Rubin convergence diagnostics of the Poisson-Gamma model: Column-wise from the top left, Figures 1(a)-(j) are trace plots for a, b, the mean and variance respectively. Figures 1(b)-(k) are posterior marginal density plots for b, a, the mean and the variance respectively. Figures 1(c)-(l) are auto- correlation plots for b, a, the mean and the variance respectively. The Gelman and Rubin trace plots show the convergence of the two parallel chains (Chains with different initial values). “Vanishing” autocorrelation function (ACF) plots show that there is low correlation among parameters that constitute the chain. More satis- factory kernel density plots for parameters of interest would more bell-shaped or symmetric. The density plots for the parameters show that convergence of the chain has reached.

The posterior statistics of the Poisson-Gamma model are shown in Table 2. Table 2 shows that the mean of

the posterior relative risk is 0.93 (95% credible interval = 0.82 - 1.08). The posterior mean is approximately the same as the mean of the standardized mortality ratio 0.93 (95% credible interval = 0.62 - 1.05) (Equation (8)). The standard deviation of the relative risk, 0.42 (95% credible interval = 0.35 - 0.56), is lower than the standardized mortality ratio's standard deviation 0.49 (95% credible interval = 0.62 - 1.05). Thus their standard deviation has been reduced by 82% by the Poisson-Gamma model. The significance of the variance indicates variation in risk among counties. In a situation of rare cases, standard deviation of the Poisson-Gamma model is expected to be much lower than that of the standardized mortality ratio [18] .

Figure 2 shows standardized mortality ratio for TB prevalence in the counties of Kenya for 2002-2009. The

SMRs vary around their mean, 0.93 with standard deviation, 0.49 (as discussed in Table 2). There is some suggestion of high TB prevalence in the North, West, North-West and Central counties of Kenya and low TB prevalence in the South-East counties except Mombasa (SMR > 2.0).

From Figure 3, we observed high risk of TB prevalence in the North, West, North-West and Central counties of Kenya and low risk in the South-East counties except Mombasa. Nairobi and Mombasa have the highest relative risk (RR > 2.0) and Laikipia, Nandi, Narok, Nyamira, and Vihiga have the lowest risk (RR < 0.5). The mean of the posterior relative risk and the SMR are the same. The range of the posterior relative risk of the Poisson-Gamma remains the same as the SMR, each having lowest relative risk estimated at 0.40 and the highest risk at 2.38. Variability in risk remains the same due to abundant of information or data.

The map of the exceedence probability in Figure 4 revealed 13 counties that exhibit high risk of TB above the national risk (RR > 1). These counties are: Nairobi, Mombasa, Kisumu, Turkan, Migori, Homa bay, Uasin

Gishu, Isiolo, Marsabit, Siaya, Tharaka-Nithi, Mandera, and Embu. This map confirms with Figure 3 concern- ing high and low TB prevalence areas.

Despite the fact that assigning a gamma prior distribution for is mathematically convenient, it is likely to be restrictive since covariate adjustment is difficult and there is no possibility for allowing spatial correlation between risk in nearby areas [18] . We therefore present models that nullify theses limitations in the next sections.

Consider a response random variable, where is the variance-covariance matrix. The linear

regression function of on is defined by. Therefore, the Gaussian process of

regression density or likelihood function for is given by

where is a design matrix of the covariates. Just to simplify analytic calculation, we can alternatively write the Gaussian linear model Equation (18) as

where, , and are the diagonal covariance matrix with the

standard model:. Consider a general likelihood function, and let us take a second order Taylor expansion of the log-likelihood around its maximum, then we have

where is the maximum likelihood estimator of. Letting and taking of Eq-

uation (19), we have

Finding the derivative of Equation (21), it follows that

Solving Equation (22), we have

Now finding the second derivative of Equation (25), we have:

Hence, we can rewrite Equation (22) as

From the Taylor’s expansion in Equation (20), and by the Maximum Likelihood Principle (MLP) that

it follows that

Taking exponent on both side of Equation (26), we have

or

where

Assuming that the prior distribution of is

Then writing only the terms from the likelihood and prior which depend on the weights, and “completing the squares” for Multiple parameters model, the posterior function is defined as

Simplifying Equation (31), it follows that

where and are the covariance of the likelihood function and the prior function respectively. The Equation (32) above is indeed Gaussian with the constant term. In “completing the squares”, we are given a quadratic form defining the exponent terms in a Gaussian distribution, and we need to determine the corresponding mean and covariance. To avoid computational complexity with “completing of squares”, we sort to using “kernel’s trick” TB324. The exponent of a general Gaussian distribution defined as where is the precision matrix can be expressed as

Comparing Equation (32) with Equation (33), we have

That is. Note that must be invertible, that is. The maximiser of the likelihood is the mean which is again the mode of the likelihood. Therefore, the Maximum Posterior (MAP) is given by

In fact the MAP is similar to the maximum likelihood value. Therefore, the posterior mean and Covariance are respectively defined by

It follows that the posterior distribution for is also Gaussian defined by

Therefore, the prior distribution of is assumed to be normally distributed as

and the prior distribution for the area-specific random effect defined by

Therefore, the posterior distribution is defined as

Hence,

The prior distribution for the linear regression coefficients is given by. The corresponding conjugate prior distribution for is the inverse-gamma [16] [19] defined as

The Equation (42) is the hyperprior distribution for with hyper-parameters. We defined and modeled the random effect and hyper-prior distribution for the precision parameter.

We first consider the LN without covariates. Parameter estimation was carried out using Bayesian Markov Chain Monte Carlo via Gibbs Sampling. MCMC convergence was reached at 100,000 iterations after a burnin period of 10,000 sample and thinning of every 30^th element in the sample. Figure 5 presents convergence diagnostics plots of this model. Table 3 presents the Posterior statistics of the LN model.

Table 3, revealed that the overall mean of the posterior relative risk is −0.17 (95% credible interval = (−0.31, −0.046). This indicates that the overall TB risk effect in Kenya estimated by the Poisson Log-Normal model decreases keeping all other determinants of TB constants. The standard deviation of the relative risk is 0.46 (95% credible interval = 0.37 - 0.56) with precision variation (95% credible interval = 3.18 - 7.24) indicating significance of variability of TB risk among counties. In a situation of rare TB cases, standard deviation of the Log-Normal model is expected to be much lower than that of the standardized mortality ratio 0.49 [18] .

Figure 6, revealed that TB risk is expected to be high in the North,West, North-West and central counties of

Kenya and low risk in the South-West counties. According to this model, Nairobi and Mombasa are expected to have the highest TB risk (RR > 2.0) and Nandi, Narok, Nyamira, Vihiga, and laikipia are expected to have the lowest TB risk (RR < 0.5). Counties with relative risk above the national risk (RR = 1) are apparent from Figure 7. Again, due to abundant of data or information, the range of the posterior relative risk of the Poisson Log- Normal remains the same to that of the SMR. The lowest estimated risk is 0.40 and highest estimated risk is 2.38. There is no reduction of relative risk range compare to SMR's risk as would be expected in a case of rare information or data.

The exceedence probability map Figure 7 confirmed with the Poisson-Gamma model that 13 counties have their TB risk above the national risk. These counties are: Nairobi, Mombasa, Kisumu, Turkan, Migori, Homa bay, Uasin Gishu, Isiolo, Marsabit, Siaya, Tharaka-Nithi, Mandera, and Embu. Figure 7 confirms with Figure 6 that high risk of TB prevalence is observed in the North, West, North-West and central counties and low risk in the South-East counties except Mombasa.

We now present the results of the Poisson log-normal model with with UH and covariates effects. Figure 8 presents the Rubin and Gelman convergence diagnostics plots of this mode. Table 4 revealed that the overall level of relative risk effect estimated is (95% credible interval = (−0.30, −0.06)). The overall risk effect is significantly different from zero and negative. This indicates that overall TB risk effect would be decreasing keeping all other determinants of TB constants. Among the cova- riates considered as TB determinants, only HIV parameters is significantly different from zero 1.20 (95% credible interval = 0.50 - 2.60) but positive. This indicates that TB risk increases with increasing HIV prevalence. [18] noted that, the higher the, the higher the variability of TB risk among counties and and the lower it is the lower the variability. This means that, a very small will indicate possibility of risk similarity between neighbouring counties. The precision for the UH (95% credible interval = 3.35 - 7.76) is significant, indicating that there exist variation in risk among counties. The UH revealed high variability of relative risk compare to the Poisson Log-Normal without random and covariate effects (95% cre- dible interval = 3.18 - 7.24).

Figure 9 also confirmed that out of the 47 counties in Kenya, 13 exhibit TB relative risk higher than the national risk (RR = 1). Table 5 presents counties in groups according to their relative risk level. High TB risk can again be observed that in the North, West, North-West and central counties of Kenya and low TB risk in the South-East counties except Mombasa.

Table 5 presents the results of the UH model with counties categorised according to their range of relative risk. The results showed that 14 counties have their relative risk above 1 and the lowest risk counties are 5. The exceedence probability map in Figure 10 visually presents counties with risk above 1. Figure 10 shows 14 counties having elevated risk of TB. These maps again confirmed high TB risk in the North, West, North-West and central counties of Kenya and low risk in South-East counties of Kenya except Mombasa.

Figure 11 is the maps of the area-specific random effect, which shows variation of risk among counties in Kenya. This map captures and displays true TB excess risk surface after covariates and confounding factors are considered TB21. Excess risk of TB is observed in Marsbit, Embu, Migori and Kisumu.

Though the Poisson-Gamma model provides good information about the TB prevalence in Kenya, one of its shortcomings is that it is unable to handle problem of spatial correlation and incorporation of covariates [18] . The Poisson Log-Normal model provides specifications that allow for incorporation of covariates. The Poisson Log-Normal model also enable us to capture the area random effect and to explain the extend of risk variability among counties through the unstructured random effect term.

Though thinning reduces the speed of the MC but it significantly reduces the number of iterations and solves the issue of autocorrelation among parameters that form the chain. Thinning reduces storage demand while preserving the integrity of the MC process [19] . [19] noted that the value of every element to be sampled is determined by the researcher and out most care must be taken since extremely large k value may results in lost of potentially an important information.

The slower the chain to converge, the more careful one should be about the burn-in period. However, it should be noted that there is no standard, systematic or guaranteed way of determining the length of the burn-in period [19] . Nevertheless, considerable work on convergence diagnostics has been done to make specific recommendations and identify tests [19] .

HIV is identified as significant among the covariates considered. Reason being that HIV patients have their immune system weakened or destroyed by the HIV rendering the body natural defence incapable of carrying out its function of protecting the body against other diseases. Since HIV has this capacity to weaken the immune system, it also implies that it has the effect of re-activating latent TB to active TB in individuals who are latently infected. Models that allow for handling spatial correlation are discussed in Chapter 6.

Though conditional autoregressive models where introduced decades ago by [11] , they were not widely used until the 1990s. Since they are defined conditionally, they are particularly suited for use with the Gibbs sampler [21] . The Conditional Autoregressive (CAR) models have been used extensively to identify and detect clustering in diseases risk. In these models, risks of disease at any given area is affected by the risk in the neighbouring areas. These models have been referred to as the structured model or the Correlated Heterogeneity (CH) models. That is, estimation of risk in any given area depends on risk at neighbouring areas [18] . The distances or boundaries between the regions are often used in determine neighbourhood properties within CAR models [22] .

Generally, the CAR model is a continuous Markov random field with a conditional probability density function characterization and designed to model spatial phenomena that are highly related to a specific local context [23] [24] . Application of CAR models can found in [23] [15] [25] . Let represents the

area to be studied. Let denote the set of all regions that are neighbouring region i. Let

be a random variable. We define the corresponding random field as the vector.

In the Gaussian CAR model, we often assume that each observation of the outcome variable has a conditional distribution defined by

These are full conditionals where is the weight of each observation on the mean of and also denotes the spatial dependence parameter. The is non-zero only if. Conventionally, we set since we do not want to regress any observation on itself. Hence no region is a neighbour of itself. The denotes a vector of all observation except. Note that depends only on a set neighbours only if location j is a neighbourhood set of. The is a potential unique variance for. For instance, if state i has M

neighbours and for every state that is a neighbour, and otherwise, then the conditional

expectation of a state's observation is the mean of all neighbours observations [25] . The Gaussian processes are specified by their mean and covariance function [26] . Assuming that each conditional distribution is Gaussian, we will need the mean and the variance-covariance to define the CAR model. The mean and the variance- covariance are respectively defined as

Therefore, conditional probability density function of a CAR random variable is has the form [25]

where, ,. The conditional joint probability density function of all the observations is

where, , invertible matrix defined as

diagonal matrix; such that is symmetric. It follows that the joint multivariate Gaussian distribution for with has covariance matrix which is

symmetric such that. Thus, a conditional autoregressive model in (45) has a pro-

bability density function defined as

and the joint probability density function in (46) becomes

The necessary and sufficient condition for (49) to be a valid joint probability density function is that its covariance matrix should not only be symmetric, but also positive definite (that is, its eigenvalues) TB223. For to be a Gaussian random variable, we need to show that is symmetric.

To show that is symmetric, we defined a symmetric weighted adjacency matrix, where

where is a measure that quantifies the proximity between region i and region j; if then i and j share a common boundary (neighbours). The could be the distance between the centroids of region i and j. Also, if, then j is one of the h nearest neighbours of i. Let be the diagonal of the adjacency matrix. The adjacency matrix of normalization or standardization is defined as

Suppose

then we define a matrix of interaction, to be a normalized adjacency matrix defined as

Suppose again that the matrix corresponding to a constant diagonal matrix normalized as (53), then we have

It follows that the conditional joint probability density function can be rewritten as

where and. Hence, the CAR model structure for is defined as

and the Equation (49) can be alternatively defined as

The controls the overall variability of, while represents the overall effect of spatial dependence. The value of is should be chosen appropriately [25] .

The row stochasticity of indicates that the distribution is improper. This impropriety can be fixed by the parameter. Redefining and choose such that is non singular, preferably with, where are the ordered eigenvalues of. Simplifying

the bounds, we replace by. It follows that

If, then is non singular. Non-singularity is guaranteed if where is

the minimum eigenvalue of. The bound mostly preferred is. This is a proper Intrinsic

Autoregressive model which add parametric flexibility and is an indication of independence. is the additional parameter which makes independent when it is equal to zero. An improper choice of may enable wider scope for posterior spatial pattern and may be preferable [27] .

In this study, we estimate parameters in the CAR model using Bayesian hierarchical methods. In disease mapping, we assumed that disease counts

The relative risk is defined as

Fitting the generalized linear model with a log-link function, we have. Under the Bayesian method, given the likelihood function of defined as

the prior distribution for is

and the prior distribution for the CAR random effect is defined by

The posterior distribution is defined as

Therefore,

The hyperperprior distribution for the precision parameters and are and respectively. The linear regression coefficient distribution is defined by. Parameter estimation was carried out using Bayesin Markov Chain Monte Carlo via Gibbs Sampling. Con- vergence of the MCMC was reached at 11,000 iteration after a burnin period of 10,000 sample and thinning of every 30^th element of the chain. Convergence diagnosis plots are presented in Figure 12 and posterior statistics of parameters presented in Table 6. We compare these results with the [11] model results in Section 6.3.

Among the models proposed for performing risk smoothing which have appeared in literature, the [11] model has found most extensive application. The BMY model is divided into two components; the CAR model component, and the UH component, (discussed in Section 6.1 and Section 5.1.1 respectively). The BYM model was introduced by [10] and latter developed by [11] . The BYM or convolution model is defines as

As noted earlier, we assume

The linear link function. The log relative risk. Therefore, the relative risk for

the i area is defined by

The log log-link function is defined as

where and are vectors of the covariate, the associated parameters, the expected number of cases, and the relative risks of TB prevalence respectively. The is the county level random effect capturing the residual log RR of disease in county i. The (UH) is sometime thought of as a latent variable which captures the effect of unknown or unmeasured area level covariates and has a CAR model structure.

We defined the likelihood function as

The prior distribution for is

prior distribution for the area-specific random effect is defined by

and prior distribution for the CAR structure is

Therefore the posterior distribution for the parameters of interest is defined as

Therefore,

The hyperprior disribution for the precision parameters, and are, and respectively. The linear regression coefficient are as- sumed to have normal distribution defined by. The reflects the amount of extra-poisson variation in the data [18] . The precision variances and control the variability of and respec- tively. Parameter estimation was carried out using Bayesin Markov Chain Monte Carlo via Gibbs Sampling. Convergence of the MCMC was reached at 11,000 iteration after a burn-in period of 10,000 sample and thinning of every 90^th element of the chain.

Posterior statistics of the CAR and the BYM model are presented in Table 6. Table 6 revealed that the estimated overall relative risk effect of the CAR model is (95% credible interval = (−0.180, −0.174)) and BYM model (95% credible interval = (−0.267, −0.0908)). Each model’s overall risk effect is significantly different from zero and negative. These models confirmed with the UH model that overall TB risk would be decreasing keeping all determinants of TB constant. Again, only the HIV variable is signi- ficant and positive for the CAR model and the BYM model with parameter estimates 1.812 (95% credible interval = 0.7735 - 2.758) and 1.41 (95% = 0.488 - 2.34) respectively. We therefore infer that the relative risks of TB increases as HIV prevalence increases.

As noted previously, the smaller the precision variance, the risk in any given area is similar to that in the neighbouring areas. The CAR model’s precision variance, (95% credible interval = 0.94 - 2.19) indicates high similarities of TB relative risk between neighbouring counties than the BYM model’s precision variance, (95% credible interval = 2.1 8- 40.80). High variation of TB risk exhibited by in the BYM model could be due to the presence of the term with precision variation (95% credible interval = 5.79 - 40.00).

Although the CAR model and BYM model each provides important information about TB relative risk behaviour, we recommend the CAR model as the best fitting spatial model to Kenya TB data since it yields lower DIC (627.21) and lower pD (49.19) than the BYM model with DIC (630.76) and pD (50.97). The BYM model, though robust, its robustness as a spatial model is lost a in situation where there is over-fitting [28] . That is, adding spatially structured extra-variability to the data when such variability doe not actually exit, con- ditionally on the covariates included in the model, leads to over-fitting, and may bias the estimations of the medical association between covariates and relative risk towards the hypothesis that it has no significant effect. In other words, not accounting for an actual spatial variability may lead to major biases but if spatially, variability of health indicators is completely explained by that of the socio-economic and environmental factors taken into consideration, regression residuals could results to a biased estimate of the medical association. We therefore presents detailed results of the CAR model (43). Figure 12 shows the convergence diagnostics plots of the CAR model. The convergence diagnostics of the BYM model can be found in Figure 13 and Figure 14.

Figure 15 also shows that out of the 47 counties in Kenya, 13 exhibit TB relative risk higher than the national risk (RR = 1). Table 7 grouped counties according to their respective relative risk ranges. The pattern of risk behaviour is similar to that reported in the previous models. High TB risk occurs in the North, West, North-West and central counties of Kenya and low risk in the South-East counties except Mombasa.

Table 7 shows 4 counties (Nairobi, Mombasa, Isiolo, and Marsabit) having highest relative risk (RR > 2.0). Out of the 47 counties, 13 counties show high relative risk above 1. Counties with high TB relative risk are visually shown by the exceedence probability map Figure 16.

Figure 16 confirms with the UH model’s results that there are 13 counties elevated risk (RR > 1). These counties exhibiting high relative risk are: Nairobi, Mombasa, Kisumu, Turkan, Migori, Homa bay, Uasin Gishu, Isiolo, Marsabit, Siaya, Tharaka-Nithi, Mandera, and Embu. Again, it can be observed that North, West, North- West and central counties of Kenya exhibit high TB prevalence and low prevalence in the South-West counties except Mombasa. Figure 17 captures areas with potential clusters of disease risk. Clusters of TB risk are suspected in Marsabit, Embu, Migori and Kisumu.

The CAR and the BYM models are used to capture clustering or clusters information about disease risk. Each model identified HIV as a major cause of high TB prevalence in Kenya. Each model revealed significance of risk similarities between neighbouring counties. Local clusters of TB risk occurs in neighbouring counties with high TB relative risk. Though each of the CAR model and the BYM provides interesting information about Kenya’s TB data, CAR model appears to provide best fit since it yields lower DIC (49.19) and lower pD (627.21) than the BYM model with DIC (50.97) and pD (630.76).