Bayesian Estimation Modelling Heart Failure Lorenz Curve.…

Bayesian Estimation for Modelling Congestive Heart Failure Deaths and Using Lorenz Curve Abdulbari Bener *1,2, Adil Yousif 3, Omer Faruk Bener 4, Shahariar Huda 5 1

Dept. of Medical Statistics & Epidemiology, Hamad General Hospital & Hamad Medical Corporation, Doha, Qatar 2 Dept. of Evidence for Population Health Unit, School of Epidemiology and Health Sciences, The University of Manchester, Manchester, UK 3 Dept. of Mathematics and Statistics, Faculty of Science and Art, University of Qatar, Doha, Qatar, 4 Dept. of Electronics and Computer Engineering, Faculty of Engineering, University of Nottingham, Nottingham, United Kingdom, 5 Dept. of Statistics, Faculty of Science, University of Kuwait, State of Kuwait, Running Title : Empirical Bayesian estimation of Modeling Congestive Heart Failure and using Lorenz Curve

Key Words: Bayesian MCMC, Mortality, Congestive Heart Failure, Lorenz Curve, Qatar

*Correspondence to: Prof. Abdulbari Bener Advisor to WHO Consulatnt and Head of Dept. of Medical Statistics & Epidemiology Hamad General Hospital, Hamad Medical Corporation and Weill Cornell Medical College Qatar PO Box 3050, Doha STATE OF QATAR Office Tel: 974- 439 3765 Office Tel: 974- 439 3766 Fax: 974-439 3769 e-mail: [email protected] e-mail: [email protected]

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Summary Background: Coronary Heart Disease (CHD) and specifically Congestive Heart Failure (CHF) are among the most common causes of death for both men and women throughout the world. Several studies have indicated that women sustaining Acute Myocardial Infarction

(AMI) have similar

mortality than men. Although some studies state that CHF affects women in greater numbers and the short-term outcomes for women are worse. Aim: The aim of this study is to use/evaluate Lorenz Curve and associated Gini Index to assess whether gender is an independent factor related to in-hospital mortality due to CHF in a geographically defined population. Statistical Method: Smoothed age-specific rates are obtained by employing Bayesian methods and Markov Chain and Monte Carlo (MCMC) techniques. Bayesian MCMC method was used to calculate the estimates. A Lorenz curve for the age specific rates is constructed by plotting empirical cumulative distribution of CHF deaths against cumulative distribution of incidence. Application: A total of 20,856 patients were treated with various cardiovascular disorders during a tenyear period from 1991-2002 at the Hamad General Hospital in Doha, Qatar. Of the patients, 3712 patients (2476 males and 1236 females) were diagnosed with CHF, and hence were studied to evaluate whether female gender was an independent predictor for poor prognosis. Results : The Lorenz curve obtained for the age specific rates showed variation in the incidence of CHF by gender. The present study shows that women are likely to have CHF similar to their male counterpart. Also the incidence rate for CHF seems to increase with age. The Lorenz curve showed that the age inequality was comparable among male and female patients. This fact was also proved by the Gini index being 0.15 for males and 0.17 for females. Conclusion: It was found that the Bayesian approach is a promising technique for the estimation of mortality rates for small geographic areas such as the State of Qatar.

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Introduction Congestive heart failure (CHF) is a debilitating condition that affects the lives of patients and their families. Despite advances in medical treatment, CHF is still associated with high morbidity, high mortality, high hospitalization rates and impaired quality of life (QoL) (Cowie et al., 2002; Juenger et al., 2002; Mostred and Hoes, 2002). Recent studies have shown that epidemic of cardiovascular diseases in women has attained increasing recognition. (Adams et al., 1996; Adams et al., 1999; Simon et al., 2001). Little is known about sex differences in baseline characteristics and outcomes in patients with acute congestive heart failure (Ritter et al. 2006). Women make up 50% of the population of heart failure patients. However, they are significantly under-represented in he art failure clinical trials (Lindenfeld et al., 1997; Heiat et al, 2002; Petrie et al., 1999). An African American Heart Failure Trial also reported that at baseline, women had lower hemoglobin and creatinine levels; less renal insufficiency; and higher body mass indexes, diabetes prevalence and systolic blood pressures; but worst QoL scores. (Taylor et al., 2006). But when mortality is concerned newer studies have reported that female sex is not an independent predictor of long-term mortality in CHF (Ritter et al., 2006) while others claim better survival for females than men (Levy et al., 2002; van Jaarsveld et al., 2006). Previous results along with these show important difference between men and women with chronic CHF where women seems to have better sur vival than men. (Ghali et al., 2002; Levy et al., 2002; Rathore et al., 2002; DeMaria et al., 1993; Hunt et al., 2001.)

In an epidemiological study, characterizing exposure-disease association in human population is very important for revealing real cause of death or the disease. Reliable information on causes of death is essential to the development of national and international health policies for prevention and control of the disease as well as for patient management. The methods of comparing several conditions like gender, while taking into account possible differences in age structures has been proposed by Breslow and Day (1975). By assuming that a simple multiplicative relationship exists between the age-specific mortality and morbidity rate for several populations, one is led to the comparison of these populations using indirectly standardized mortality and morbidity ratios (SMR) where the age specific rates for all populations combined are used as standard. Adjustments to these ratios are needed in case of large age differences among the

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populations in both age -specific rates and structures. This method is appropriate when insufficient data are available for direct standardization. The identification of burden disease by exceptionally high rates of mortality is a fundamental step in the development of community-based intervention and prevention strategies. However, failure to identify correctly the most severely burdened countries can result in the misdirection of public health resources (Stevenson and Olson 1993). Nowadays with the high technology and Internet facilities, the vital statistics are commonly available. Unfortunately, the crude age-standardized event rates, if at all, very rarely provide an adequate basis for ranking countries with respect to mortality rates. Therefore, the problem arises from the variability of the vital health statistics rates, which are free of sampling error, but the counts and rates are not completely free of uncertainty. Assuming that number of deaths in an open population is distributed as a Poisson variable; this suggests several modeling approaches. Multiplicative Poisson regression models have been studied by Breslow and Day (1975). Similarly, empirical Bayesian models have been widely applied to Poisson distributed data (Madi and Bener 2003, Bener and Abdalla 2004). Lui et al (1990) also developed an empirical Bayesian model that yields a group of parameter estimates by using empirical distribution function. Furthermore, Manton et al (1989), Stevenson and Olson (1993) described different empirical Bayesian models to estimate county-level mortality rates. The purpose of their study was to review general classes of models applicable to the problem of ranking countries with respect to event rates. It should be noted that among the models that they discussed, Crude SMRs and age -specific rates are unstable and sensitive to changes in expected disease cases and population sizes. Recently, a Bayesian framework that assumes prior information of the variability of disease rates has been adopted to estimate specific rates. (Lawson et al, 2003, and Mollie, 1995, Bener and Abdalla 2004, Bener et al. 2006) . Finally, Lorenz curve and the Gini Index have been adopted by several epidemiologists to characterize and test seasonal variation of disease occurrence (Lee, 1996, 1997, 1999; Prakasam and Murthy 1992, Llorca and DelgadoRodriguez 2002, Bener and Abdalla 2004, Bener et al., 2006, Bener and Farooq, 2006). The aim of this study is to demonstrate how multiplicative and Bayesian models can be utilized to estimate Standardized mortality and morbidity ratios (SMRs) and age-specific rates for the CHF data collected and compare the results of the two models. Bayesian model may be readily implemented using standard software (WinBUGS), since it has the advantage over other

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approaches to estimating uncertainty of parameters in the model. In addition the Lorenz curve and Gini index will be implemented to measure gender as an independent predictor of mortality for the CHF patients.

Methodology Breslow and Day Multiplicative Model Breslow and Day (1975) suggest a multiplicative model to deal with a two-way classification. Let N ij and X ij be total cases of CHF among Qatari people and the number of deaths of CHF respectively, where in the (i , j ) th cell, i indicates sex factor being i = 1 for male and i = 2 for female, and j specifies age groups. In this study thirteen different age groups were considered as seen in table 1. (e.g.) j = 1 and j = 13 represent the age groups < 30 and =85 years, respectively. Let us assume that X ij is a binomial random variable with index N ij and parameters λ ij . Under the multiplicative model, λij = θ iφ j (i = 1,2; j = 1, 2,...13) . Where φ j is a parameter represents proportion of number of cases in age group j, and θ i is a parameter represent the proportion of death among the age group Our objective is to obtain maximum likelihood estimates (MLE’s) of the parameter θ i , φ j . Assuming the X ij is independent, the log likelihood of X ij is given by

= ∑ ∑ ln ( N ij C X ij ) + ∑ ∑ X ij ln θ i + ∑ ∑ X ij ln φ j + ∑ ∑ ( N ij − X ij ) ln( 1 − θ iφ j ) (1) i

j

i j

i j

i j

from which it readily follows that the MLE’s should satisfy ^

θi =

X i. ^

^

---------(2)

^

^

---------(3)

^

∑{( N ij − X ij ) φ j } /(1 − θ i φ j ) j

X.j

^

φj =

^

∑{( Nij − X ij ) θ i} /(1 −θ i φ j ) i

where X i . = ∑ X ij , X . j = ∑ X ij . j

i

^

^

Equations (2) and (3) can be solved iteratively. Initial values of the parameters θ i and φ j may be set to 5

^

θi =

X. j X i. ^ ,φj = where N i . = ∑ N ij , N . j = ∑ N ij . N. j N i. j i

In order to further simplify the calculation Poisson approximation to the binomial set-up can be used. This is justified since population size N ij for each (i , j ) th cell is sufficiently large. Under the Poisson approximation, the X ij ' s are independent Poisson random variables with the parameter N ij λij , so that the likelihood of the sample is

[

= ∑ ∑ − N ijθ i φ j + X ij ln N ij + ln θ i + ln φ j − ln( X ij ) i

]

(4)

j

From (4), we obtain the MLE’s of θi and φ j as ^

θi =

X i. ^

∑ N ij φ j

,

j

X.j

^

φi =

∑N

^

ij

θi

i

^

which again have to be solved iteratively. Convenient initial values may be φ j = X . j / N . j ^

and θ i = X i . / N i . Bayesian Model Assume that each observed count X ij follows a Poisson distribution with mean E ij λij , where Eij is the expected number of Qatari deaths due to CHF in the (i , j ) th cell based upon the

associated incidence. The expected values are estimated using indirect standardisation based on employing age specific death rates of CHF as standard. i.e. E ij = N ij R j , where N ij is the incidence associated with the (i , j ) th cell and R j is the standard rate of CHF in the j th age group. The relative risk λ ij , SMRij , can be estimated by assuming a random effect model for λij . This allows for over-dispersion in the Poisson model caused by unobserved confounding factors (Mollie 1995; Clayton & Bernardinelli, 1996). The most common method for estimating the random effect λij is by using Bayesian approach. A gamma prior distribution for the relative risk λij combines conveniently with the Poisson likelihood to give a gamma posterior distribution.

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That is λ ij ~ gamma ( ai , bi ) as the prior and λij ( X ij , ai , bi ) ~ gamma ( ai + X ij , bi + E ij ) as posterior distribution (Lawson et al 2003, Bener and Abdalla 2004). The posterior mean of the relative risk λ ij for the (i , j ) th cell is weighted average of the SMRij for the (i , j ) th cell and the relative risk associated with the i th gender and is given by E[ λij | X ij , ai , bi ] =

ai + X ij bi + E ij

= ω ij SMRij + (1 − ω ij )

ai , bi

where

ω ij =

E ij bij + E ij

For rare diseases and small counts ω ij is small, thus the posterior mean is shrunk towards a global (gender) mean of (

ai ). In the cells with large counts the posterior mean of the relative risk will bi

be slightly different from the original relative risk (

X ij E ij

)

By indirectly simulating observations from the gamma posterior distribution using Markov Chain and Monte Carlo methods estimates and Bayesian confidence limits (95% confidence intervals of the relative risk in the (i , j ) th cell, SMRij , can be obtained together with estimates of gender relative risk SMRi , and age specific CHF death rates (see Gilks et al, 1996). Lorenz curve and Gini index Lorenz curve is a widely used and practical method for modelling mortality rates of a disease (Lee 1996, 1997, 1999, Llorca and Delgado, Rodriguez 2002, Bener and Abdalla 2004). The Lorenz curve is constructed by plotting the empirical cumulative distribution of CHF deaths against the cumulative distribution of incidence, provided that exposure levels (age groups) are rearranged such that the first level will have the lowest fatality risk and the last one would have the highest. If the risk of CHF at different exposure levels is equal to each other, the Lorenz curve will be a straight diagonal line for that exposure (that is, age is not a risk factor that influence the distribution of CHF incidence). The greater the inequality of CHF incidence over exposure levels is, the greater the discrepancy between the Lorenz curve and the line of perfect equality.

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The variation depicted using a Lorenz curve can be summarized using the Gini index (Lee, 1997). The value of the index is between 0 and 1, with larger va lues reflecting greater variability in CHF incidence over exposure levels (age groups) and smaller values reflect greater uniformity. One realization of the index is expressed mathematically as k xi −1 Gini = ∑ x i =1

i

yi −1

yi = ∑ ( xi −1 yi − xi yi −1 )

where, ( x i , y i ) represent the coordinates of the points in the Lorenz curve, x i = ∑ j ≤i n i / N and yi = ∑ j ≤i ai / A . n i denotes the incidence in the ith (age group) exposure level (i = 1,K , k ) and

N denotes the total incidence ( N = ∑i ni ). A and ai denote, respectively, the total number of CHF deaths and number of deaths in the i th exposure level, A = ∑iai .

Case Study – An Application Background: Although CHD mortality rates have been halved in many developed countries in recent decades, some studies have pointed out significant differences regarding time-related tendencies of mortality between the sexes. With this recognition, recently considerable interest has been focused on the study of sex-based differences in the outcome of patients with CHF in the Western world. However, the vast majority of these studies being performed in Western population that may not necessarily apply to other ethnic groups such as middle-eastern population and overall data on the outcome of women after CHF in the developing world are lacking.

Subjects and Statistical Analysis: This study was conducted in the State of Qatar, the Gulf. The estimated population of State of Qatar for the yea r 2005 was about 840,000. This study was based at Hamad Medical Corporation, Doha. The hospital provides comprehensive tertiary health care services for the residents in Qatar , hence it is an ideal centre for population-based studies. All patients with acute myocardial infarction, angina and congestive heart failure requiring hospitalisation in Qatar are treated at this hospital. Our study focused only on Qatari patients because it is a stable population

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and avoids the bias in the fluctuation of expatriate population in the country, which varies with time. Hamad Medical Corporation, Coronary Care Unit database used for this study, prospectively collected data on all patients admitted to the cardiology department at Hamad General Hospital. The patient’s physicians collected data from the clinical records at the time of the patient’s hospital discharge according to predefined criteria for each data point. These records have been coded and registered at the cardiology department since January 1992. With the described database all patients presenting with CHF in the ten-year period between 1991 and 2002 were identified. Congestive Heart Failure was defined for this study according to World Health Organization criteria, International Classification Diseases for which Version 10 (ICD-10) was used. The age of presentation, gender, ethnicity, and cardiovascular risk factor profiles status were analysed. We also studied the trends of in-hospital mortality, and morbidity.

Results Table 1 shows the comparison between CHF female and male patients. Of the total 3712 patients admitted with CHF, 1236 patients were female and 2476 were males. When compared to male patients, female patients were older; the mean±(SD) age of women was 62.3±12.5 yrs, whereas that of the men was 60.9±12.3 yrs. Global in-hospital mortality among patients was 9.3% which did not show any significant difference according to gender, males 9.6%, females 8.8%. We have implemented Bayesian methods in particular, Gibbs sampling MCMC methods in the software package WinBUGS. This environment permits great flexibility in model specification, which allows non-standard model specification to be fitted in an undemanding manner. Using WinBUGS it was possible to treat missing observation as unknown quantities to be estimated by the model. This maximised the amount of information available to estimate and validate the model. As our data had no deaths reported in the age group