This article has Open Peer Review reports available.
Applications of Bayesian approach in modelling risk of malaria-related hospital mortality
- Lawrence N Kazembe^{2, 2}Email author,
- Tobias F Chirwa^{1},
- Jupiter S Simbeye^{1} and
- Jimmy J Namangale^{1}
https://doi.org/10.1186/1471-2288-8-6
© Kazembe et al; licensee BioMed Central Ltd. 2008
Received: 17 October 2007
Accepted: 19 February 2008
Published: 19 February 2008
Abstract
Background
Malaria is a major public health problem in Malawi, however, quantifying its burden in a population is a challenge. Routine hospital data provide a proxy for measuring the incidence of severe malaria and for crudely estimating morbidity rates. Using such data, this paper proposes a method to describe trends, patterns and factors associated with in-hospital mortality attributed to the disease.
Methods
We develop semiparametric regression models which allow joint analysis of nonlinear effects of calendar time and continuous covariates, spatially structured variation, unstructured heterogeneity, and other fixed covariates. Modelling and inference use the fully Bayesian approach via Markov Chain Monte Carlo (MCMC) simulation techniques. The methodology is applied to analyse data arising from paediatric wards in Zomba district, Malawi, between 2002 and 2003.
Results and Conclusion
We observe that the risk of dying in hospital is lower in the dry season, and for children who travel a distance of less than 5 kms to the hospital, but increases for those who are referred to the hospital. The results also indicate significant differences in both structured and unstructured spatial effects, and the health facility effects reveal considerable differences by type of facility or practice. More importantly, our approach shows non-linearities in the effect of metrical covariates on the probability of dying in hospital. The study emphasizes that the methodological framework used provides a useful tool for analysing the data at hand and of similar structure.
Background
Plasmodium falciparum malaria is a major public health problem in most tropical countries in the world. Between 300 and 500 million cases of clinical episodes occur each year, and 1–3 million people die of the disease [1, 2]. The sub-Saharan African region has the greatest burden with over 90% cases and 80% malaria-attributable deaths [3]. Measuring malaria burden in a population is a challenge in most developing countries [1, 2], because most disease incidences and deaths occur outside of the formal health care, particularly at home [4, 5]. Instead, routine hospital data provide a proxy for measuring the incidence of severe malaria and for crudely estimating morbidity rates or equivalent clinical indicators [6].
Analysis of these data may allow to assess, compare and ultimately improve the care provided at all levels of health care. It may assist in monitoring and planning resource needs in a health system and designing appropriate interventions, tailored towards communities at high risk or lead to further investigations to identify important risk factors [7]. Variability in these indicators is a well known issue, and is a function of various covariates, at both patient or group level, some observed and others unobserved, and maybe spatially correlated or time-varying [7–10]. Geographical differences are driven by socio-economic determinants, availability and access to health care or health seeking behaviour [7, 10]. Temporal variation may again be a factor of access to care and malaria transmission [9], for example, there can be increased access in dry season and yet fewer cases in the same season. Adequate statistical modelling and analysis is, therefore, of epidemiological interest.
This paper is motivated by the analysis of malaria-related hospital mortality data collected at patient's level, covering a period of two years among children admitted to a referral district hospital in Malawi. The response variable is binary (whether died of malaria in hospital or not) and is linked to several covariates which are categorical or continuous, spatial and temporal. Unobserved heterogeneity due to, for example, differences in practice style or type of hospital, inequalities in utilisation or access, may exist and should be explored. Hierarchical regression modelling provides a general framework to investigate the effect of these cofactors.
We apply a geoadditive logistic model as proposed by Fahrmeir and Lang [11]. Applications of such models are many and literature is growing. These models can be estimated through a fully or empirical Bayesian approach, and are implemented in BayesX [12]. For example, Augustin et al. [13] employed the model to study the relationship between needle losses of pine-trees and various covariates. Inference was performed with a full Bayes (FB) approach making use of Markov Chain Monte Carlo (MCMC) simulation techniques. Tutz [14] developed a class of generalised semiparametric mixed models and proposed penalized marginal likelihood approach for the estimation of parameters. Fahrmeir et al. [15] considered a penalised geoadditive model for space-time data with inference performed using an empirical Bayesian (EB) approach.
In this paper, we use the fully Bayesian approach via MCMC simulation techniques. The advantages of FB inference is that the functionals of the posterior can be computed without relying on large sample Gaussian justifications, and the approach is computationally feasible for large datasets. Moreover, the uncertainty in the parameters is easily quantified [15]. Furthermore, Bayesian methods are more flexible in that empirical information, when available, can be incorporated with the data through an informative prior distribution. When this information is not available, a non-informative prior can be chosen. The methodology is of substantive interest since the effects of other covariates are jointly estimated with the random effects, e.g., spatially structured and unstructured heterogeneity effects [16]. This is extended to incorporate nonparametric terms for nonlinear continuous covariates and time-varying coefficients, for example, time trend and seasonal variation of calendar time. In addition, space-time interactions are assessed within the varying-coefficient models framework [17].
The rest of this paper is organised as follows. We first describe the data. Next, we specify the model and outline the Bayesian approach used for model estimation. This is followed by the application of the model to the data, and then results are presented. Discussion on the results and limitations of the study conclude the article.
Methods
Data
Data were obtained from discharge records of all paediatric hospital admissions at Zomba district hospital, Malawi, between 1 January 2002 to 31 December 2003. Each case was confirmed as malaria on admission through microscopic identification of parasites in blood samples. Zomba district hospital, with over 500 beds is the largest health facility in the district and serves both as the first consultation point for patients within its catchment, and as a referral centre for other 23 primary health centres. These facilities are managed by the Ministry of Health and the Christian health association of Malawi, and variations in health care management is expected.
Descriptive summary of variables used in the study.
Description | n (per cent)^{§} | |
---|---|---|
Binary variables | ||
Sex | 1 = female | 1683 (7.7) |
0 = otherwise | 2286 (7.6) | |
Day | 1 = if admitted over weekend | 2418 (7.5) |
0 = otherwise | 1492 (7.5) | |
Season | 1 = if admitted during dry season | 1128 (5.4) |
0 = otherwise | 1128 (5.4) | |
Distance | 1 = if distance travelled is ≤ 5 km | 1938 (7.4) |
0 = otherwise | 1999 (8.8) | |
Referral | 1 = if referred to hospital from networking PHC | 1895 (8.8) |
0 = otherwise | 1494 (6.1) | |
Metrical variable | Mean (SD^{‡}) | |
age | Age of child | 30.5 (30.7) |
los | Length of hospital stay | 78.9 (264.1) |
ct | Calendar time | 44.8 (30.1) |
Spatial/heterogeneity variables | ||
v | 21 structured residential wards effects | |
u | 21 unstructured residential wards effects | |
h | 23 unstructured primary health care (PHC) facility effects | |
N | Total number of observations | 3969 |
A total of 302 deaths were registered among 3,969 children hospitalised for malaria, between January 2002 to December 2003, resulting in an overall case fatality ratio (CFR) of 7.6%. Table 1 shows the proportion who died in different covariate levels. The proportion varies with age, referral status, season, distance from the hospital and length of hospital stay (LOS). The CFR drops from 8.5% in the age of <1 year to 6.2% at age of between 1–4 years and increases in the 5–14 years groups to 10.5. This suggests a curvature in the association of age and the probability of in-hospital mortality. The number of cases are relatively more in the wet season (October-March) compared to the dry season (April-September), with a similar pattern of CFR. Boys are more frequently hospitalised than girls (58%), but the CFR is not different. The hospital receives relatively more patients from a distance of more than 5 kms (52%), with distant patient likely to die in hospital. As for LOS, CFR is very high on day 1, drops and then increases as the stay is prolonged. Again there is an indication of curvature in the relationship between LOS and the risk of inpatient mortality. Children referred to the hospital are most likely to die in hospital (CFR = 8.8%). Further detailed descriptive and exploratory analyses presented elsewhere clearly show spatial and temporal variations [10].
The Model
Here γ is a p-dimensional vector of unknown regression coefficients.
where v _{ i }, v ∈ {1, ⋯, V} are spatially structured effects for child i; u _{ i }, u ∈ {1, ⋯, U} and h _{ i }, h ∈ {1, ⋯, H} model unstructured heterogeneity at area and health facility levels respectively, f _{ i }are unknown functions for nonlinear effects of continuous covariate x _{ i }(e.g., age of the child), or calendar time effect t _{ i }. Note that the spatially structured effects and unobserved heterogeneity tries to capture all sources of unmeasured influential factors, some that occur locally or at large scale, or those that may vary with time.
Several extensions to the additive predictor (3) are possible. For example, the calendar effect f(t _{ i }) can be decomposed into time trend f(r _{ i }) and time-varying seasonal component f(s _{ i }), i.e.,
η _{ i }= ⋯ + f(r _{ i }) + f(s _{ i }) + ⋯,
In addition, the model can be extended to include interaction surfaces within the varying coefficient framework proposed by Hastie and Tibshirani [17]. Here the effect of some covariate z is assumed to vary smoothly over the range of a second covariate x, giving the predictor
η _{ i }= ⋯ + f(x _{ i })z _{ i }+ ⋯,
The function f _{4} quantifies the deviations from the effect at some specified reference or baseline time period. This will be discussed in detail in a separate analysis.
Estimation: fully Bayesian approach
Prior distributions for covariate effects
where f= (f _{1}, ⋯, f _{ p }) and ${\tau}_{f}^{2}$ is the variance, with diffuse priors f _{1} ∝ const, f _{2} ∝ const for initial values.
again assuming diffuse priors for initial values, s _{1}, ⋯, s _{11}, and ${\tau}_{s}^{2}$ is a variance that controls the degree of smoothness. The unstructured spatial heterogeneity term, u _{ i }is assumed to follow an exchangeable Gaussian prior with zero mean and variance, ${\tau}_{u}^{2},\text{i}.\text{e}.,{u}_{i}~N(0,{\tau}_{u}^{2})$. A similar prior is assigned to the heterogeneity term for the health facility, i.e., ${h}_{i}~N(0,{\tau}_{h}^{2})$.
where i ~ j denotes that area i is adjacent to j, and assumes that parameter values v _{ i }and v _{ j }in adjacent areas are similar. The degree of similarity is determined by the unknown precision parameter ${\tau}_{v}^{2}$.
with an appropriate penalty matrix K _{ j }. Its structure depends on the covariate and smoothness of the function. In most cases, K _{ j }is rank deficient and hence the prior for β _{ j }is improper. For the variances ${\tau}_{j}^{2}$ we assume inverse Gamma priors IG(a _{ j }, b _{ j }), with hyperparameters a _{ j }, b _{ j }chosen such that this prior is weakly informative.
Posterior distribution
For updating the full conditionals of parameters, we use a hybrid MCMC sampling scheme of the iteratively weighted least squares (IWLS) proposals, developed for generalised linear mixed models by Gamerman [19], and Metropolis-Hastings algorithm. Full details are presented elsewhere [11, 14, 15, 18].
Applications
We analyse the following logistic models,
M0: η _{ i }= ${{w}^{\prime}}_{i}\gamma $
M1: η _{ i }= ${{w}^{\prime}}_{i}\gamma $ + f _{1}(age) + f _{2}(los) + f _{3}(ct)
M2: η _{ i }= ${{w}^{\prime}}_{i}\gamma $ + f _{1}(age) + f _{2}(los) + f _{3}(ct) + v _{ i }+ u _{ i }+ h _{ i }
M3: η _{ i }= ${{w}^{\prime}}_{i}\gamma $ + f _{1}(age) + f _{2}(los) + f _{3}(trend) + f _{4}(season) + v _{ i }+ u _{ i }+ h _{ i }
Model M0 is a basic regression model of fixed covariates only (Table 1). Model M1 assumes nonlinear functions for the continuous factors, i.e., age f _{1}(age), and LOS f _{2}(los) and calendar time f _{3}(ct) measured in weeks, and tries to assess the gains of fitting a semiparametric model. The choice of estimating age and LOS using nonlinear smoothing priors is motivated by preliminary results, see Ref. [10], which clearly suggest a nonlinear relationship in LOS and possibly one in age. Model M2 considers all possible risk factors, i.e., we simultaneously analyse nonlinear effects of age, time trend of calendar time, structured spatial effects, v, for the 21 residential wards, unstructured spatial effects, u, heterogeneity effects, h, for the 23 health facility, and fixed effects, w'γ, for the categorical variables. In model M3, we extend model M2 to consider further temporal effects, whereby the effect of calendar time is decomposed into a time trend, f _{3}(trend) and seasonal component, f _{4}(season).
We implement the models in BayesX ver 1.4 – a public domain software for computing complex Bayesian techniques [12]. For the four models, 40,000 iterations are carried out after a burn-in sample of 10,000. We thin every 20th iteration, yielding 2,000 samples for parameter estimation. Convergence is monitored by plotting trace and autocorrelation plots of the samples. Quantiles, median, mean and standard deviation for all parameters, estimated from the posterior distributions, are used to assess model fit. In particular, credible intervals are used to assess the significance of parameters.
We also monitored the posterior deviance, and compared the set of plausible models using the Deviance Information Criterion (DIC) [20]. Specifically, we compare the structured additive models (i.e., M1, M2 and M3) with the simpler parametric alternative (M0). The DIC is given by DIC = $\overline{D}$ + p _{ D }, where $\overline{D}$ is the posterior mean of the deviance, which is a measure of goodness of fit, and p _{ D }is the effective number of parameters, which is a measure of model complexity and penalises overfitting. Since small values of $\overline{D}$ indicate good fit while small values of p _{ D }indicate a parsimonious model, small values of DIC indicate a better model. Models with differences in DIC of <3 compared with the best model can not be distinguished, while those between 3–7 can be weakly differentiated [[24], p.613].
Results
Model assessment
Comparison of the four fitted models using the Deviance information criteria. See text for details.
Models | ||||
---|---|---|---|---|
M0 | M1 | M2 | M3 | |
Model fit | ||||
$\overline{D}$ | 1360.41 | 1347.58 | 684.75 | 651.54 |
p _{ D } | 11.98 | 21.48 | 61.01 | 59.84 |
DIC | 1372.39 | 1369.06 | 729.07 | 711.38 |
ΔDIC ^{§} | 661.01 | 657.68 | 17.69 | 0 |
Fixed effects
Estimates of fixed parameters based on Model M3.
Model coefficients | Odds Ratio | ||||
---|---|---|---|---|---|
Covariate | Mean^{‡} | 95% CI^{§} | Mean | 95% CI | |
Sex | Female child | -0.09 | -0.35, 0.14 | 0.92 | 0.79, 1.19 |
Male | 0 | 1.00 | |||
Day | Weekend | 0.19 | -0.06, 0.38 | 1.19 | 0.94, 1.46 |
Weekday | 0 | 1.00 | |||
Season | Wet | 0 | 1.00 | ||
Dry | -0.48 | -0.77, -0.24 | 0.63 | 0.49, 0.86 | |
Distance | ≤ 5 kms | -5.15 | -7.73, -2.60 | 0.005 | 0.0006, 0.28 |
> 5 kms | 0 | 1.00 | |||
Referral | Yes | 4.59 | 3.45, 5.85 | 98.49 | 21.33, 383.75 |
No | 0 | 1.00 |
Nonlinear effects
Temporal effects
Spatial effects
Sensitivity analysis of model M3. Relative changes of fixed effects, deviance information criterion, and variance components for different choices of hyperparameters for ${\tau}_{v}^{2}$, ${\tau}_{u}^{2}$ and ${\tau}_{h}^{2}$.
Hyperparameters for ${\tau}_{v}^{2}$, ${\tau}_{u}^{2}$ and ${\tau}_{h}^{2}$ | ||||
---|---|---|---|---|
a = 0.5, b = 0.0005 | a = 1, b = 0.005 | a = 0.001, b = 0.001 | a = 0.01, b = 0.01 | |
Model fit | ||||
$\overline{D}$ | 651.54 | 650.12 | 652.66 | 650.06 |
p _{ D } | 59.84 | 61.01 | 61.26 | 61.53 |
DIC | 711.38 | 711.13 | 713.92 | 711.59 |
Fixed effects | ||||
Intercept^{§} | -2.78 (-4.04, -1.39) | -3.11 (-4.58, -1.69) | -3.15 (-4.88, -1.43) | -3.16 (-4.72, -1.87) |
Sex (female) | -0.09 (-0.35, 0.14) | -0.07 (-0.32, 0.18) | -0.08 (-0.34, 0.17) | -0.08 (-0.32, 0.10) |
Season (dry) | -0.48 (-0.77, -0.24) | -0.46 (-0.77, -0.14) | -0.45 (-0.74, -0.18) | -0.48 (-0.77, -0.22) |
Distance (≤ 5 km) | -5.15 (-7.73, -2.60) | -5.21 (-8.34, -2.49) | -5.19 (-8.36, -2.22) | -5.17 (-10.02, -1.62) |
Referral (yes) | 4.59 (3.45, 5.88) | 4.61 (3.30, 6.37) | 4.51 (3.54, 6.48) | 4.65 (3.17, 6.65) |
Day (weekend) | 0.19 (-0.06, 0.38) | 0.20 (-0.01, 0.41) | 0.20 (-0.03, 0.43) | 0.19 (-0.04, 0.40) |
Random effects ^{†} | ||||
Areal: structured (${\tau}_{v}^{2}$) | 26.27 (10.86, 58.77) | 26.58 (12.18, 69.59) | 26.44 (8.86, 57.28) | 27.34 (11.33, 72.11) |
Areal: unstructured (${\tau}_{u}^{2}$) | 2.26 (0.003, 9.23) | 2.19 (0.008, 2.11) | 2.34 (0.008, 6.93) | 2.22 (0.008, 2.27) |
HF: heterogeneity (${\tau}_{h}^{2}$) | 1.21 (0.48, 2.69) | 1.24 (0.43, 2.82) | 1.23 (0.29, 1.84) | 1.24 (0.46, 3.80) |
Sensitivity analysis
Table 4 reports on the results investigating the influence of hyperpriors since the performance of the model can be sensitive to the choice of the variance components priors [21]. We therefore consider alternative specifications, and carry out sensitivity of our model assuming an IG with scale and shape parameters a and b respectively. We assume four alternatives a = 0.5, b = 0.0005, a = 1, b = 0.005, a = 0.001, b = 0.001 and a = 0.01, b = 0.01. The first specification was suggested by Kelsall and Wakefield [22], for modelling the precision of the spatial effects in an MRF model. The second alternative was proposed in Besag and Kooperberg [23]. The remaining two priors with equal scale and shape parameters, especially a = b = 0.001, have often been used as standard choice on the variances of random effects [24]. Re-running MCMC simulations based on these specifications, using model M3 for simplicity, yield relatively similar inference on risks of dying in hospital, variance components and model fit. Therefore our choice of IG(a = 0.5, b = 0.0005) is appropriate for all the analyses.
Discussion
This study apply Bayesian techniques to analyse patterns and risk factors of malaria attributable case fatality data. We develop and use logistic regression models to have an in-depth understanding of factors associated with the probability of dying of malaria in hospital, building on the existing methodological contributions by Fahrmeir and Lang [11], Fahrmeir et al. [15].
A number of variables are used to explain the variation in the response and include spatial, continuous, categorical, and heterogeneity terms. The spatially structured variation and unstructured heterogeneity are modelled using MRF prior and zero mean Gaussian heterogeneity priors as proposed by Besag et al. [16]. The continuous variables are estimated non-parametrically by applying second order Gaussian random walk prior, which permits enough flexibility while avoiding over-fitting the data [18]. The proposed methodology allows all these factors to be estimated in a single framework. Because the models are highly parameterised and analytically intractable, the maximum likelihood approach is not be feasible. Thus, the Bayesian inference, making use of MCMC simulation techniques, offers a viable alternative.
In this paper we find evidence that the risk of dying in hospital due to malaria is lower in the dry season, and for children travelling less than 5 kms to the hospital. However, for those referred to the hospital the risk increases. These results seem to suggest that when health care is accessible or available lives can be saved. Malaria is a preventable disease, but delayed treatment or lack of effective treatment can lead to fatal malaria within days [1]. Children are particularly vulnerable because of lack of immunity against the disease [2]. The risk decreases with age, again infants being the most vulnerable, but overall children under five years are the most at risk. The increase in risk for those aged 6–14 years, although these are supposed to be protected through acquired immunity, may reflect some aspects of health seeking behaviour, and emphasize the need for prompt and effective management of malaria for all children including those aged over five years even if such cases may not frequently occur in the general population [7, 9, 10].
The lower risk in the dry season should be interpreted with care. While the risk of infection is reduced during this period, this effect is directly linked to few cases being hospitalised, hence fewer deaths, and if anything such death should reflect disease management other than severity of the disease. Another possible explanation is that during the dry season access to the hospital is easier than during rainy season, leading to early treatment, and therefore fewer avoidable deaths. Referral children are at increased risk because probably these are already worse-off when they arrive at the hospital. Disease management differences or inaccessibility of care may contribute towards this finding.
The spatial effects are often a surrogate of underlying unobserved information, and may give leads for further epidemiological research or assist in designing malaria interventions. For example, the increased risk in rural areas may be an influence of different factors, such as unavailability or inaccessibility of health facilities resulting in increased risk for such children. These effects may also reflect health seeking behaviour, which plays a critical role in accessing prompt and effective care. Since most antimalarial remedies at first taken at home, effective care may be delayed, leading to increased risk for rural children. Scaling-up of interventions such as insecticide-treated nets or health promotions on appropriate and effective treatment in home or community based care should be emphasized in rural areas [25].
The significance of health facility effects further suggests that management of health care differs in the 23 referring facilities in Zomba district. Indeed, as these are public or private operated, resources such as drugs or ambulatory support may be lacking mainly in government-run health centers. Moreover, some facilities, for example, dispensaries and clinics have limited capacity to treat severe malaria, and may not refer severely sick in time because of lack of communication. There is need to ascertain actual factors contributing to such discrepancy, e.g. using health facility surveys on malaria case management. If indeed, these are the underlying factors, resources need to be committed to improve primary health care. The seasonal variations indicate that malaria transmission processes may explain the variation in the probability of dying in hospital. This is because malaria transmission is highly seasonal and may change within the same area as the year progresses. Essentially, interventions or health promotion campaigns should be tailored in recognition of these varying risk patterns.
The data-driven approach we have taken in this analysis has a greater advantage in that the nonlinear effects of continuous variables are estimated, and avoids ad hoc categorizations although the effect of age can as well be estimated as linear (Figure 1). Indeed, the methodological framework we have applied provide useful tools for handling this type of data, and in similar conditions. Our application demonstrates that spatial and temporal analysis may reveal some salient features of the data, which may be overstepped by the classical regression models (Model M0) or the purely spatial models (Model M2). Flexible modelling, via nonparametric or semiparametric model enable us establish a better epidemiological relationship existing between the response and continuous explanatory variables.
Model selection in this paper is based on the DIC, which is a Bayesian analogue to the Akaike Information Criterion. Although the DIC is now widely used for model choice in complex hierarchical Bayesian models, its usage is at least debatable [[24], pp.612-633]. The DIC measures only the relative goodness of fit among a collection of models. It does not provide information on the adequacy of the model. A model diagnostic tool based on the posterior predictive distribution can be used to assess model adequacy by comparing the observed data with the samples drawn from the posterior predictive distribution. Different approaches for validation, when DIC is not appropriate for decision making on which model to choose have already been employed, for example see Gosoniu et al [26]. Nevertheless, some simulation results [27] suggest that the DIC gives reasonable results even in complex nonparametric regression models.
A major limitation of our analysis is that data used comes from hospital registers. In most African countries, most malaria cases occur at home, and the pattern may be biased towards urban areas that are well covered by health facilities. Moreover, one may argue that much of this data represent severe forms of malaria, because studies on health seeking behaviour for malaria report that biomedical care is sought when the disease is nearly fatal [5]. Health facility data can best be described as providing proxies for prevalence or morbidity and hence health need. A more representative data is through cross-sectional household surveys, e.g. the demographic and health surveys (DHS), however, these are often carried out every four years, thus the periodicity is not frequent enough for surveillance and to inform immediate decision making [4].
Conclusion
In many resource-poor African countries, collection of population-based health data is a challenge and hospital data provide a critical source of information for decision making. In this paper, we set out to analyse risk factors of malaria mortality, using hospital register data. Our model, using the Bayesian approach, shows that malaria mortality is associated with both individual and group level factors, as well as observed and unobserved risk factors, some of which exhibit spatial and seasonal variation. From a public health perspective, with a goal of prevention and control, our results highlight that reducing malaria burden may require integrated strategies encompassing improved availability and access to effective care at primary facilities; reinforcing home and community case management where prompt care is inaccessible, and encouraging early referral, as well as inducting health promotion interventions aimed at interrupting malaria transmission. Methodologically, this model can easily be adapted to analyse other health indicator of similar structure and in like settings.
Declarations
Acknowledgements
LNK would like to acknowledge the research training grant received from WHO/TDR and support from Medical Research Council, Durban, South Africa. We thank the HMIS specialist and hospital management at Zomba district for making the records accessible. This study was approved by the Ministry of Health Ethics Committee.
Authors’ Affiliations
References
- Breman JG: The ears of the hippopotamus: manifestations, determinants, and estimates of the malaria burden. Am J Trop Med Hyg. 2001, 64 (S): 1-11.PubMedGoogle Scholar
- Breman JG, Alilio MS, Mills A: Conquering the intolerable burden of malaria: what's new, what's needed: a summary. Am J Trop Med Hyg. 2004, 71 (S2): 1-15.PubMedGoogle Scholar
- WHO/UNICEF: World malaria report 2005. 2005, World Health Organization/UNICEF. Report series: WHO/HTM/MAL/2005.1102Google Scholar
- De Savigny D, Binka F: Monitoring future impact of malaria burden in sub-Saharan Africa. Am J Trop Med Hyg. 2004, 71 (S): 224-231.PubMedGoogle Scholar
- De Savigny D, Mayombana C, Mwageni E, Masanja H: Care-seeking patterns for fatal malaria in Tanzania. Malaria J. 2004, 3: 27-10.1186/1475-2875-3-27.View ArticleGoogle Scholar
- United Nations Millenium Project: Coming to grips with malaria in the new millenium. 2005, Task force on HIV/AIDS, Malaria, TB, and Access to Essentail Medicines, Working Group on MalariaGoogle Scholar
- Gething PW, Noor AM, Gikandi PW, Ogara EAA, Altkinson PW, Hay SI, Nixon MS, Snow RW: Improving imperfect data from health managenemet information systems in Africa using space-time geostatistics. PLoS Med. 2006, 3: e271-10.1371/journal.pmed.0030271.View ArticlePubMedPubMed CentralGoogle Scholar
- Schellenberg JA, Newell JN, Snow RW, Mung'ala V, Marsh K, Smith PG, Hayes RJ: An analysis of the geographical distribution of severe malaria in children in Kilifi District, Kenya. Int J Epidemiol. 1998, 27: 323-329. 10.1093/ije/27.2.323.View ArticlePubMedGoogle Scholar
- Snow RW, Schellenberg JR, Peshu N, Forster D, Newton CR, Winstanley PA, Mwangi I, Waruiru C, Warn PA, Newbold C: Periodicity and space-time clustering of severe childhood malaria on the coast of Kenya. Trans R Soc Trop Med Hyg. 1993, 87: 386-390. 10.1016/0035-9203(93)90007-D.View ArticlePubMedGoogle Scholar
- Kazembe LN, Kleinschmidt I, Sharp BL: Patterns of malaria-related hospital admissions and mortality among Malawian children: an example of spatial modelling of hospital register data. Malaria J. 2006, 5: 93-10.1186/1475-2875-5-93.View ArticleGoogle Scholar
- Fahrmeir L, Lang S: Bayesian inference for generalized additive mixed models based on Markov random field priors. J R Statist Soc C. 2001, 50: 201-220. 10.1111/1467-9876.00229.View ArticleGoogle Scholar
- Brezger A, Kneib T, Lang S: BayesX: Analyzing Bayesian structured additive regression models. J Statist Software. 2005, 14: 11-View ArticleGoogle Scholar
- Augustin NH, Lang S, Musio M, von Wilpert K: A spatial model for the needle losses of pine- trees in the forests of Baden-Wurttenberg: an application of Bayesian structured additive regression. J R Statist Soc C. 2007, 56: 29-50. 10.1111/j.1467-9876.2007.00563.x.View ArticleGoogle Scholar
- Tutz G: Generalized semiparametrically structured mixed models. Comput Statist Data Anal. 2004, 46: 777-800. 10.1016/j.csda.2003.10.001.View ArticleGoogle Scholar
- Fahrmeir L, Kneib T, Lang S: Penalized structured additive regression for space-time data: a Bayesian perspective. Statist Sinica. 2004, 14: 731-761.Google Scholar
- Besag J, York J, Mollie A: Bayesian image restoration with two applications in spatial statistics (with discussion). Ann Instit Statist Mathemat. 1991, 43: 1-59. 10.1007/BF00116466.View ArticleGoogle Scholar
- Hastie TJ, Tibshirani RJ: Varying-coefficient models (with discussion). J R Statist Soc B. 1993, 55: 757-756.Google Scholar
- Lang S, Brezger A: Bayesian P-splines. J Comput Graph Statist. 2004, 13: 183-212. 10.1198/1061860043010.View ArticleGoogle Scholar
- Gamerman D: Efficient sampling from the posterior distribution in generalised linear models. Statist Comput. 1997, 7: 57-68. 10.1023/A:1018509429360.View ArticleGoogle Scholar
- Spiegelhalter DJ, Best NG, Carlin BP, van der Linde A: Bayesian measures of model complexity and fit (with discussion). J R Statist Soc B. 2002, 64: 1-34. 10.1111/1467-9868.00353.View ArticleGoogle Scholar
- Gelman A: Prior distributions for variance parameters in hierarchical models. Bayesian Anal. 2006, 1: 515-534.View ArticleGoogle Scholar
- Kelsall and Wakefield: Discussion of "Bayesian models for spatially correlated disease and exposure data". Bayesian Statistics 6. Edited by: Best NG, Arnold RA, Thomas A, Conlon E, Waller LA, Bernado JM, Berger JO, Dawid AP, Smith AFM. 1999, Oxford University Press, Oxford, 151-Google Scholar
- Besag J, Kooperberg C: On conditional and intrinsic autoregressions. Biometrika. 1995, 82: 733-746.Google Scholar
- Spiegelhalter DJ, Thomas A, Best NG, Gilks WR, Lunn D: BUGS- Bayesian inference using Gibbs sampling. 2003, MRC Biostatistics Unit, Cambridge, England, [http://www.mrc-bsu.cam.ac.uk/bugs]Google Scholar
- World Health Organization: Scaling up home-based management of malaria: from research to implementation. 2004, Roll Back Malaria Department/TDR. Report series: WHO/HTM/MAL/2004.1096 TDR/IDE/HMM/04.1. orld Health Organization: GenevaGoogle Scholar
- Gosoniu L, Vounatsou P, Sogoba N, Smith T: Bayesian modelling of geostatistical malaria risk data. Geospatial Health. 2006, 1: 127-139.View ArticlePubMedGoogle Scholar
- Brezger A, Lang S: Simultaneous probability statements for Bayesian P-splines. Statistic Modelng. 2008Google Scholar
- The pre-publication history for this paper can be accessed here:http://www.biomedcentral.com/1471-2288/8/6/prepub
Pre-publication history
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.