Measuring total health inequality: adding individual variation to grouplevel differences
 Emmanuela Gakidou^{1}Email author and
 Gary King^{2}
DOI: 10.1186/1475927613
© Gakidou and King; licensee BioMed Central Ltd. 2002
Received: 2 February 2002
Accepted: 12 August 2002
Published: 12 August 2002
Abstract
Background
Studies have revealed large variations in average health status across social, economic, and other groups. No study exists on the distribution of the risk of illhealth across individuals, either within groups or across all people in a society, and as such a crucial piece of total health inequality has been overlooked. Some of the reason for this neglect has been that the risk of death, which forms the basis for most measures, is impossible to observe directly and difficult to estimate.
Methods
We develop a measure of total health inequality – encompassing all inequalities among people in a society, including variation between and within groups – by adapting a betabinomial regression model. We apply it to children under age two in 50 low and middleincome countries. Our method has been adopted by the World Health Organization and is being implemented in surveys around the world; preliminary estimates have appeared in the World Health Report (2000).
Results
Countries with similar average child mortality differ considerably in total health inequality. Liberia and Mozambique have the largest inequalities in child survival, while Colombia, the Philippines and Kazakhstan have the lowest levels among the countries measured.
Conclusions
Total health inequality estimates should be routinely reported alongside average levels of health in populations and groups, as they reveal important policyrelated information not otherwise knowable. This approach enables meaningful comparisons of inequality across countries and future analyses of the determinants of inequality.
Keywords
Health inequality risk of death child mortality extended betabinomial modelBackground
The distribution of health, or health inequality, has become prominent on global policy agendas as researchers have come to regard average health status as an inadequate summary of a country's health performance [1, 2]. Almost all health inequality studies have in fact documented differences in average health status across groups of people. Those with an economic focus have measured differences in average health status across income groups [3, 4]. Researchers with a sociological focus have examined inequalities in average health status among social classes [5, 6]. and those with a political focus have looked at how political structure is related to differences in the average level of health [7]. Other scholars have focused on differences in average health status among racial or ethnic groups or by educational attainment or occupation [8–10]. And most researchers consider differences across political entities such as countries or local governments. Similarly, demographers have also long studied differences in average health status, particularly in children, across age, sex, education and racial groups [11–13]. In low and middleincome countries there exists a rich demographic literature on levels and trends in child mortality and causes associated with them [14–16].
In this paper, we define the concept of total health inequality, and demonstrate how to measure it by the variation in health status across individuals (within a country as a whole or any subgroup within a country). This approach complements the existing grouplevel approaches, a fact that can even be demonstrated mathematically. That is, the standard analysis of variance identity applies to variations in health status just as it does to all other coherent variables:
"Total" = "Between Group" + "Within Group"
Existing literature has focused exclusively on the "between group" component. In this paper, the missing "withingroup" component is added to the existing measures to arrive at total health inequality. With total health inequality, no individual variation in health status is ignored. With this measure added to existing reporting standards, public health policy can be targeted at reducing inequalities across individuals, in addition to its existing goal of reducing disparities in average health status across countries and groups in society.
We would like to emphasize that total health inequality complements group level measures; it does not replace them. After all, if average health attainment is the same across a given set of groups, total health inequality could still be unacceptably high (because of intragroup variation across individuals), whereas if total health inequality is small, then the differences among any set of groups, albeit potentially systematic, must also be small. In our view, between, within, and total levels of health inequality should be reported henceforth.
Preferably, measures of inequality in healthy life expectancy (the number of years in full health an individual born today can expect to live [17]) would be computed, but this paper focuses on a preliminary step for which data are more readily available – developing methods for the measurement of total inequality in the probability of child survival. Survival from birth to two years of age is only one aspect of health, but it is a useful place to start since it is a critical part of health status, particularly in developing countries [4, 18].
The normative principles involved in choosing a measure of inequality are discussed briefly. Instead of making an arbitrary choice, the inequality measure selected is consistent with the results of a survey of normative preferences of over 1000 health professionals conducted by WHO and used in the World Health Report 2000 [19]. Comparisons with applications of other popular measures of income inequality to health are also presented.
Methods
DHS survey year and sample size
Country  Year  No. of households interviewed 

Bangladesh  1997  9,127 
Benin  1996  5,491 
Bolivia  1994  8,603 
Brazil  1996  12,612 
Burkina Faso  1993  6,354 
Burundi  1987  3,970 
Cameroon  1997  5,501 
Central African Republic  1995  5,884 
Colombia  1995  11,140 
Comoros  1996  3,050 
Cote d'Ivoire  1994  8,099 
Dominican Republic  1996  8,422 
Ecuador  1987  4,713 
Egypt  1995  14,779 
Ghana  1994  4,562 
Guatemala  1995  12,403 
Haiti  1995  5,356 
India  1993  89,777 
Indonesia  1994  28,168 
Kazakhstan  1995  3,771 
Kenya  1993  7,540 
Liberia  1986  5,239 
Madagascar  1997  7,060 
Malawi  1992  4,849 
Mali  1996  9,704 
Mexico  1987  9,310 
Morocco  1992  9,256 
Mozambique  1997  8,779 
Namibia  1992  5,421 
Nepal  1996  8,429 
Nicaragua  1998  13,634 
Niger  1995  7,577 
Nigeria  1990  8,781 
Pakistan  1991  6,611 
Paraguay  1990  5,827 
Peru  1996  28,951 
Philippines  1998  13,983 
Rwanda  1992  6,551 
Senegal  1997  8,593 
Sudan  1990  5,860 
Thailand  1987  6,775 
Togo  1998  8,569 
Trinidad and Tobago  1987  3,806 
Tunisia  1988  4,184 
Uganda  1995  7,070 
United Republic of Tanzania  1996  8,120 
Uzbekistan  1996  4,415 
Yemen  1992  6,010 
Zambia  1996  8,021 
Zimbabwe  1994  6,128 
The DHS are conducted through inperson interviews. The samples, which are all above 3,000 households in the countries analyzed in this study, are the result of a multistage stratified sampling design [26]. The DHS sampling weights are used to produce nationally representative estimates.
For each country we used the latest year of available data from a nationally representative DHS, ranging from 1987 to 1997. For each mother surveyed the number of children born and the number survived to age 2 was calculated. A tenyear observation period was used ending two years prior to the interview year, to avoid censoring effects. This period is a compromise between providing recent estimates and ensuring enough births to reduce the effects of sampling error. Measuring survival to (or death by) age 5, would involve a longer censoring period, produce older estimates of inequality, and not differ much from the under 2 mortality because on average, 80% of under 5 deaths occur in the first two years of life [26, 27].
To provide a partial but independent validation of the DHSbased results, mortality data by municipality in Mexico [28] and Brazil [29] from different data sources were analyzed. Data on socioeconomic variables [30] and on the political system [31] of each country were also collected to help us explore possible causes of differences in inequality. The socioeconomic variables were collected for the year the survey was conducted in each country.
The population of interest includes all children born alive in a country in a given time period. Ideally, one would measure the length of time each child is expected to live from birth to two years and then use a measure of inequality to summarize the distribution of these survival expectations. Making the inference from the dichotomous data on child survival to health inequality requires several methodological steps.
The first step is to estimate the distribution of the probability of death across children in each national sample. The chief methodological difficulty here is that for any one child, only the dichotomous variable of survival to two years is measured, while the probability of dying for each child is not observed. These probabilities are estimated using the extended betabinomial model [32–34]. This model has been widely applied in biomedical research, most commonly for modeling animal littermate survival probabilities, and in political science to model voting statistics [32, 34–38]. In this application, we model the number of child deaths within a family with a binomial distribution with equal risk of dying per child, and then allow the risks to vary across families according to a beta distribution [35]. (See Additional file 1 for more details on the specification of the model.)
Potential confounders, including mother's age, number of children, level of education, and average birth interval, were controlled for [13]. This procedure relaxes the assumptions of the model, making it more flexible. However, the basic model fits the data well, and controlling for these variables does not materially affect the estimates of health inequality. When the covariates have no effect, the beta distributed random effect portion of the model ensures that the level of variability is not underestimated.
For Mexico and Brazil, the extended betabinomial model was also applied to the municipalitylevel mortality data sets to validate the model. The underlying assumption is that small geographical areas (which are treated analogously to families) include mostly homogeneous populations for which the risk of death is similar. In both countries, the estimates of inequality from the extended betabinomial model did not materially differ between the two data sets used.
The second step is to transform the estimated probability of death between birth and age two for each child (_{2}q_{0} in demographic notation) to the expected survival time in the first two years of life, S. Although the results do not change materially, we opted to measure inequality in survival time, instead of probability of survival, as it is analogous to inequality in health expectancy and is more interpretable. Expected survival time can be calculated as
where S is expected survival time, and _{2}m_{0} is the mortality rate in the first two years of life [39]. _{2}m_{0} can, in turn, be calculated from the probability of dying in the first two years of life, [39].
Finally, since printing fifty plots like Figure 1 would be unwieldy, we give numerical summaries of health inequality. To do this, several normative criteria have to be addressed. At least three general normative dimensions are relevant [17]. First, measures of inequality range from absolute to relative. Absolute measures are independent of mean survival time, whereas relative measures adjust for the mean. If one believes that more variation in health states is acceptable when average survival time is higher, then a measure close to the relative end of the continuum would reflect that choice; on the other hand, if one believes that a given discrepancy in expected survival across people should be considered in the same way, irrespective of the mean survival time in that population, then an absolute measure of inequality would be appropriate. The second normative dimension is the weight given to outliers. One might believe that the majority of children is what measures should be based on, or one might instead want to focus primarily on the worst and best off. The final dimension is whether individuals should be compared to the average of their communities or to each of the individuals within their communities separately.
A range of measures of inequality that reflect many different normative positions were developed, including measures used in quantifying income inequality (such as the Gini index), variance measures, and many that have not been previously considered [17]. Although it need not have turned out this way, in the present analysis these measures all gave substantively consistent empirical results. For empirical analyses, the inequality index (II) used was derived from a survey of the normative preferences of over 1,000 health professionals and other individuals with an interest in health systems [19]. The index is defined as
where s_{ i } is the expected survival time between birth and age two of individual i, and s is the average expected survival time in the first two years of life in the population. This index of inequality (II) is logically between a relative and an absolute measure, so the average survival time is included in the denominator. The index is based on comparing each child with every other child in the population (thus the sum of the differences in the numerator), and gives a large weight to the best and worst off (the differences are raised to the power of three). Larger values of II indicate more individual level inequality in child survival. The health inequality point estimates and uncertainty bounds are mean posterior estimates and 95% credible intervals, respectively, computed from the extended betabinomial model with flat priors and the traditionally used asymptotic normal approximations (e.g. [40]).
Results
Child survival inequality index for 50 countries, estimates and 95% confidence intervals.
Country  Inequality Index  95% CI  Country  Inequality Index  95% CI 

Liberia  .75  .56 – .91  Comoros  .36  .17 – .53 
Mozambique  .73  .59 – .87  Egypt  .35  .29 – .42 
Central African Republic  .69  .53 – .85  Uganda  .34  .23 – .48 
Nigeria  .66  .55 – .77  Burkina Faso  .34  .21 – .47 
Malawi  .62  .44 – .78  Kenya  .34  .24 – .44 
Rwanda  .56  .43 – .68  Ecuador  .32  .18 – .44 
Niger  .54  .42 – .66  Benin  .31  .19 – .45 
Pakistan  .54  .43 – .64  Bangladesh  .30  .20 – .41 
Côte d'Ivoire  .52  .40 – .64  Bolivia  .27  .17 – .37 
Mali  .51  .41 – .60  Tunisia  .25  .14 – .35 
Namibia  .47  .31 – .61  Morocco  .25  .15 – .34 
United Republic of Tanzania  .47  .35 – .59  Brazil  .23  .14 – .33 
Togo  .46  .35 – .57  Guatemala  .23  .16 – .30 
Zambia  .46  .32 – .59  Senegal  .22  .14 – .32 
Madagascar  .45  .33 – .58  Peru  .22  .17 – .26 
Yemen  .44  .34 – .53  Zimbabwe  .21  .11 – .31 
Nepal  .41  .29 – .52  Dominican Republic  .21  .11 – .30 
Cameroon  .40  .25 – .54  Nicaragua  .20  .13 – .27 
Sudan  .40  .29 – .51  Trinidad and Tobago  .15  .04 – .25 
Burundi  .40  .24 – .55  Thailand  .15  .05 – .24 
Indonesia  .40  .34 – .45  Mexico  .14  .06 – .21 
India  .39  .36 – .43  Paraguay  .12  .05 – .20 
Haiti  .39  .22 – .55  Kazakhstan  .11  .01 – .21 
Ghana  .39  .23 – .53  Philippines  .10  .05 – .16 
Uzbekistan  .36  .21 – .52  Colombia  .08  .03 – .15 
Relative ranks of child survival inequality by four measures of inequality. Rank 1 refers to the most unequal.
Country  II  Std. deviation  Coefficient of variation  Gini coefficient 

Liberia  1  2  1  1 
Mozambique  2  1  2  2 
Central African Republic  3  3  5  6 
Nigeria  4  4  7  7 
Malawi  5  5  3  3 
Rwanda  6  6  8  9 
Nigeria  7  9  4  4 
Pakistan  8  7  13  17 
Cote d'ivoire  9  8  10  10 
Mali  10  10  6  5 
Namibia  11  14  22  26 
Tanzania  12  11  12  12 
Togo  13  12  16  18 
Zambia  14  13  9  8 
Madagascar  15  15  11  11 
Yemen  16  16  14  13 
Nepal  17  17  15  14 
Cameroon  18  19  23  23 
Sudan  19  18  20  21 
Burundi  20  20  17  15 
Indonesia  21  24  30  31 
India  22  21  26  25 
Haiti  23  22  19  19 
Ghana  24  23  25  24 
Uzbekistan  25  30  35  39 
Comoros  26  25  27  27 
Egypt  27  26  28  29 
Uganda  28  27  24  22 
Burkina Faso  29  28  18  16 
Kenya  30  29  32  32 
Ecuador  31  32  33  33 
Benin  32  31  21  20 
Bangladesh  33  33  29  28 
Bolivia  34  34  31  30 
Tunisia  35  36  38  37 
Morocco  36  35  36  35 
Brazil  37  39  40  40 
Guatemala  38  37  37  36 
Senegal  39  38  34  34 
Peru  40  40  39  38 
Zimbabwe  41  41  41  41 
Dominican Republic  42  42  42  42 
Nicaragua  43  43  43  43 
Trinidad & Tobago  44  46  47  48 
Thailand  45  45  45  45 
Mexico  46  44  44  44 
Paraguay  47  47  46  46 
Kazakhstan  48  50  50  50 
Philippines  49  48  48  47 
Colombia  50  49  49  49 
Additionally, inequality in childhood survival is positively related to the mean probability of death (2q0), but at a given level of mortality there is significant variation in inequality. This confirms the expected relationship and also reflects the fact that traditionally reported measures of average levels of health are insufficient for summarizing the health experience of a population. Finally, each point in each graph also codes the type of political system. The graphs seem to indicate that full democracies (represented as diamonds) tend to have lower values of inequality than partial democracies (squares) or autocracies (triangles), as would be expected. (Partial democracies include countries that have adopted some democratic practices, such as popular elections to legislatures with limited powers, but most have not completed the transition from autocratic practices.) However, and perhaps surprisingly, health inequality is otherwise unrelated to the type of political system either directly or in interaction with any of the five potential explanatory variables.
The individuallevel approach to conceptualizing and measuring health inequality appears to complement the grouplevel approaches. To show that the total health inequality measures offered here are at least sometimes distinct from grouplevel analyses, the results of the present analysis are compared to those of Wagstaff [4] and Brockerhoff and Hewett [16]. Wagstaff calculated inequalities among income groups in 7 countries, measured by a concentration index. Brockerhoff and Hewett measured ethnic differences in 11 countries via odds ratios. Brockerhoff and Hewett used subsets of the same DHS datasets as used in this analysis, while Wagstaff used mostly data from the Living Standards Measurement Surveys.
Conclusions
This paper presents the first measures of total health inequality of a population. Such measures could serve as an important complement to existing grouplevel approaches in the literature on health inequalities among groups. Including individuallevel variation, as done here, produces estimates of inequality that capture the entire distribution of risk of death in the population and that are directly comparable across countries.
At the same average level of health status, countries can achieve widely varying levels of health inequality. Since measuring and communicating this type of information seems essential to making informed public policy, we believe inequality should be measured and reported together with average levels of health status.
Estimating the underlying distribution of risk is useful for understanding the nature and possibly the causes of health inequality using observed dichotomous outcomes such as survival and death. This or a related approach should prove useful for examining the risk of illhealth for all age groups, such as in measures of inequality in health expectancy.
Considerable future research needs to be conducted into health inequality. For one area, efforts should continue to measure inequalities in child survival outside of the fifty countries analyzed here. For another, the normative underpinnings of popular measures of health inequality should be further clarified. Similarly, other measures that formalize richer normative principles should be developed. Further efforts need to be made to measure what types of people, policymakers, and democratic electorates prefer one normative position rather than another. Third, new databases need to be created and statistical methods developed that enable researchers to expand measures of inequality in child survival in the first two years of life to inequality in health expectancy in general. Fourth, we should seek further external validation of these results, along the lines of the vital registrationbased analysis conducted for Mexico and Brazil. Finally, and most importantly for influencing health policy globally, scholars should pursue an understanding of the determinants of inequality. We need to understand not only how average levels of health status of populations can be raised but also how health inequalities can be reduced.
There are several limitations to this study. The ranking of countries is influenced by the year the data were collected and particularly for those most affected by the HIV/AIDS epidemic, the estimate of the inequality index might change if more recent data were available. Since women of reproductive age are the basic sampling units in these surveys, their premature death (from maternal or other causes) excludes their children from the studies. Such children often have an elevated mortality risk and their exclusion may bias estimates child mortality (both level and inequality) downward. This bias is likely to be greater in countries with higher maternal mortality and HIV/AIDS epidemics. Our preliminary explorations of this issue indicate that the estimate of the inequality index changes very little, and not enough to result in a change of rankings across countries.
Some of the potential implications of this article include a research program devoted to developing and improving measures of health inequality, a substantial change in data collection efforts by public health authorities internationally, and even ongoing changes in national and international public policy as a result. All this possible activity takes nothing away from the important existing focus on differences in average health levels across groups, but measuring and reporting individual health inequality adds an important new perspective as well.
List of abbreviations
 DHS:

Demographic and Health Surveys
 II:

Inequality index
 WHO:

World Health Organization
Declarations
Acknowledgements
The authors thank Christopher Murray, Julio Frenk, Alan Lopez, Brad Palmquist, Joshua Salomon, Lana Tomaskovic, and Brodie Ferguson for valuable comments. Our thanks go to the World Health Organization, the U.S. National Science Foundation (SBR9729884) and the National Institutes on Aging (PO1 AG1762501) for research support.
A file with all data and information necessary to replicate the results in this paper is available from the authors
Authors’ Affiliations
References
 World Health Organization: Health Systems: improving performance. World Health Report 2000. Geneva, Switzerland. 2000Google Scholar
 Acheson D: Independent inquiry into inequalities in health. 1998. London, UK, The Stationery Office. 1998Google Scholar
 World Bank: Country Reports on Health, Nutrition, Population, and Poverty. Washington, DC, USA. 2002, [http://www.worldbank.org/poverty/health/data/intro.htm]Google Scholar
 Wagstaff A: Socioeconomic inequalities in child mortality: comparisons across nine developing countries. Bulletin of the World Health Organization. 2000, 78: 1929.PubMedPubMed CentralGoogle Scholar
 Marmot MG, Smith GD, Stansfeld S, Patel C, North F, Head J, White I, Brunner E, Feeney A: Health inequalities among British civil servants: the Whitehall II study [see comments]. Lancet. 1991, 337: 13871393. 10.1016/01406736(91)93068K.View ArticlePubMedGoogle Scholar
 Kawachi I, Marshall S, Pearce N: Social class inequalities in the decline of coronary heart disease among New Zealand men, 1975–1977 to 1985–1987. Int J Epidemiol. 1991, 20: 393398.View ArticlePubMedGoogle Scholar
 Navarro V: Health and equity in the world in the era of "globalization". Int J Health Serv. 1999, 29: 215225.View ArticlePubMedGoogle Scholar
 Kunst AE, Geurts JJ, van den Berg J: International variation in socioeconomic inequalities in self reported health. J Epidemiol Community Health. 1995, 49: 117123.View ArticlePubMedPubMed CentralGoogle Scholar
 Kunst AE, Groenhof F, Mackenbach JP, Health EW: Occupational class and cause specific mortality in middle aged men in 11 European countries: comparison of population based studies. EU Working Group on Socioeconomic Inequalities in Health [see comments]. BMJ. 1998, 316: 16361642.View ArticlePubMedPubMed CentralGoogle Scholar
 Mackenbach JP, Kunst AE: Measuring the magnitude of socioeconomic inequalities in health: an overview of available measures illustrated with two examples from Europe. Soc Sci Med. 1997, 44: 757771. 10.1016/S02779536(96)000731.View ArticlePubMedGoogle Scholar
 Preston SH, Haines MR: Fatal years: child mortality in late nineteenthcentury America. Princeton, New Jersey, Princeton University Press. 1991, 266:Google Scholar
 Caldwell JC, McDonald P: Influence of maternal education on infant and child mortality: levels and causes. In: International Population Conference, Manila. 1981, 2: 7996.Google Scholar
 Bicego G, Ahmad O: Infant and Child Mortality #20. Calverton, Macro International, Inc. 1996Google Scholar
 Hill K, Pande R, Mahy M, Jones G: Trends in child mortality in the developing world: 1960 to 1996. New York, NY: UNICEF. 1999Google Scholar
 Hill K, Pande R: The recent evolution of child mortality in the developing world. Arlington, VA, Partnership for Child Health Care, Basic Support for Institutionalizing Child Survival [BASICS]. Current Issues in Child Survival Series. 1997Google Scholar
 Brockerhoff M, Hewett P: Inequality of child mortality among ethnic groups in subSaharan Africa. [Review] [36 refs]. Bulletin of the World Health Organization. 2000, 78: 3041.PubMedPubMed CentralGoogle Scholar
 Gakidou E, Murray CJL, Frenk J: Defining and measuring health inequality: an approach based on the distribution of health expectancy. Bulletin of WHO. 2000, 78: 4254.Google Scholar
 Zaba B, David P: Fertility and the distribution of child mortality risk among women: an illustrative analysis. Popul Stud. 1996, 50: 263278. 10.1080/0032472031000149346.View ArticleGoogle Scholar
 Gakidou E, Murray CJL, Frenk J: Measuring preferences for health systems performance assessment. GPE Discussion Paper 20. 2000. Geneva, World Health Organization. 2000Google Scholar
 USAID & Macro International: Demographic and Health Surveys. 1984, [http://www.measuredhs.com]Google Scholar
 Stanton C, Abderrahim N, Hill K: An assessment of DHS maternal mortality indicators. Studies in Family Planning. 2000, 31: 111123.View ArticlePubMedGoogle Scholar
 Sullivan J, Rutstein S, Bicego G: Infant and Child Mortality #15. Calverton, Macro International, Inc. 1994Google Scholar
 Curtis SL: Assessment of the quality of data used for direct estimation of infant and child mortality in DHSII surveys. Calverton, Maryland, Macro International. Demographic and Health Surveys. 1995Google Scholar
 Boerma JT, Sommerfelt AE: Demographic and Health Surveys (DHS): contributions and limitations. World Health Statistics Quarterly. 1993, 46: 222226.Google Scholar
 Hill K, Pande R, Mahy M, Jones G: Trends in child mortality in the developing world: 1960–1996. New York: UNICEF;. 1999Google Scholar
 Rutstein S: Factors associated with trends in infant and child mortality in developing countries during the 1990s. Bull World Health Organ. 2000, 78: 12561270.PubMedPubMed CentralGoogle Scholar
 Macro International Inc.: An assessment of the quality of health data in DHS surveys #2. Calverton, Maryland, Macro International Inc. 1993Google Scholar
 Fundacion Mexicana para la Salud, Instituto Nacional de Estadística Geografía e Informática. Deaths and population by municipality in Mexico. 1998
 Instituto Brasilero de Geografía e Estadística (IBGE). Sistema de Informacao sobre Mortalidade. 1994, http://www.datasus.gov.br, http://www.ibge.gov.br
 World Health Organization: GPE Socioeconomic database. WHO. 2000Google Scholar
 Gurr TR, Jaggers K: Polity III: Regime Change and Political Authority, 1800–1994. Computer file, Interuniversity Consortium for Political and Social Research, Ann Arbor, MI. 1996Google Scholar
 Prentice RL: Binary regression using an extended betabinomial distribution, with discussion of correlation induced by covariate measurement errors. Journal of the American Statistical Association. 1986, 81: 321327.View ArticleGoogle Scholar
 King G: Unifying political methodology: the likelihood theory of statistical inference. Cambridge: Cambridge University Press. 1989Google Scholar
 Palmquist B: Analysis of proportions data. College Station, Texas.
 Brooks SP, Morgan BJ, Ridout MS, Pack SE: Finite mixture models for proportions. Biometrics. 1997, 53: 10971115.View ArticlePubMedGoogle Scholar
 Yamamoto E, Yanagimoto T: Statistical methods for the betabinomial model in teratology. Environmental Health Perspective. 1994, 102S: 2531.View ArticleGoogle Scholar
 Liang KY, McCullagh P: Case studies in binary dispersion. Biometrics. 1993, 49: 623630.View ArticlePubMedGoogle Scholar
 Griffiths D: Maximum likelihood estimation for the betabinomial distribution application to the household distribution of the total number of cases of a disease. Biometrics. 1973, 29: 637648.View ArticlePubMedGoogle Scholar
 Preston SH, Keyfitz N, Schoen R: Causes of death: Life tables for national populations. New York and London: Seminar Press. 1972Google Scholar
 King G, Tomz M, Wittenberg J: Making the most of statistical analyses: improving interpretation and presentation. American Journal of Political Science. 2000, 44: 341355.View ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article: verbatim copying and redistribution of this article are permitted in all media for any purpose, provided this notice is preserved along with the article's original URL.