INTRODUCTION
The National Health Insurance Scheme (NHIS) was introduced in Nigeria with the promulgation of decree No. 35 of 1999. The broad objective of the scheme is to ensure that every Nigerian has access to good health care services at affordable costs. Participants are expected to pay capitation fees to licensed HMOs, which would allow the subscriber to have access to registered health care providers. Given the inefficiencies experienced with public hospitals and the rather expensive costs of private hospitals, it was expected that the populace would readily embrace the scheme. For some reasons this does not appear to be the reality. Worse still, it is unclear the level of participation. Thus, it is difficult to know what the scale should be of say, an enlightenment programme or some other strategy that could boost participation, which need to be put in place.
This study uses Bayesian methods of inference to estimate the rate of participation in the NHIS scheme based on the outcome of a survey conducted in an earlier study. It derives motivation from Rubin’s (1983) illustrations of the Bayesian method as an extremely powerful tool for the applied statistician especially in the way, it could provide sensible answers in a straightforward manner in problems where, sampling theories approaches appear awkward. Further support is derived from Link and Sauer (1996), who argued that Empirical Bayes methods provide alternative approaches that incorporate the structural advantages of Bayesian models while, requiring less stringent specification of prior knowledge.
MATERIALS AND METHODS
Bayes methods: Empirical Bayes methods have received considerable attention in the statistical study since, their introduction in the early part of the 20th century (Robbins, 1956). Morris (1983) argued that when considered in the context of a group of related parameters the Bayesian procedures yield improved estimates of individual parameters. Also in what appears to be a bold attempt at making a case for the adoption of the method Rubin (1983) compared the frequentist approach to the Bayesian method and concluded that the frequentist perspective appears to be devoid of easily followable principles that would lead to the construction of good inferences whereas, the Bayesian perspective naturally lead to the construction of such inferences.
Empirical Bayes methods have been applied in a number of contexts. Some recent
examples include applications to problems in forest science (Burk and Ek, 1982)
to monitoring of air pollution (Suggs and Curran, 1983) and in a variety of
medical applications (Stijnen and van Houwelingen, 1990). It would seem that
applications of empirical Bayes methodology in nonmedical biological settings
are few. In the ecological study where, some application had been recorded for
instance, Johnson (1989) noted that in spite of the theoretical justification
of empirical Bayes methods, their use has not been widespread. Within a period
of one decade since this apparent disinterest in the methods, there seems to
be an awakening. Recent applications include estimating population sizes from
survey data (Johnson, 1989), estimating numbers of species (Mingoti and Meeden,
1992), capturerecapture data (Smith, 1991), toxicity data (Piegorsch, 1994),
summary analysis of avian trends (Link and Sauer, 1995) and identification of
extremes in collections of parameter estimates (Link and Sauer, 1996).
Link and Sauer (1996) advanced that a Bayesian analysis can be thought of as a combination of existing knowledge with new knowledge, the two being synthesized in such a way as to account for the amount of confidence that is placed in each source of knowledge.
Usually, the existing knowledge base and updated knowledge base are summarized by probability distributions describing the likely range of values for each unknown parameter, which are referred to as the prior and posterior distributions, respectively. Informally, the prior distribution can be thought as an approximation to the histogram of the true, unknown values of the parameters under investigation.
Application of empirical Bayes analysis to NHIS data: In the original data, the respondents were structured along eight categories, viz.: educational qualification, marital status, number of children, number of children under 18 years, income per annum, occupation, awareness of NHIS and gender. Therefore, for a class of subscribers we let X denote the number of persons registered with HMO in a random sample of N subscribers. Given that the true proportion is P, X is assumed to be a binomial random variable with mean (NP). This second estimate is the observed proportion given as:
Observed proportion P has variance and standard error given, respectively as:
Then, for a single observation the marginal likelihood of the number of subscribers can be represented by the βBinomial distribution (Leonard and Hsu, 1999) thus:
Using the moments, the expectation and variance follow:
and
so,
We equate these to the empirical moments:
and
and taking n_{i} = t,
We now reparameterize the posterior distribution of P by setting:
and
The mean of the posterior distribution easily follows:
RESULTS AND DISCUSSION
The result of the reparametized hyper parameters from the NHIS data are shown in Table 1.
To obtain empirical Bayes estimates, we need only substitute estimates of the hyper parameters π and θ in the formula for the Bayes estimator. Following Link and Sauer (1995), we define an empirical Bayes estimate of a proportion P by:
We note also that the estimated hyper parameters can be used to estimate the posterior distribution for P, from which, we can construct confidence intervals for likely range of values for P. The true value for the formula in the posterior distribution is then obtained by substituting the hyper parameter estimates computed in Table 1. The results are shown in Table 2.
The empirical Bayes analysis of proportions of NHIS subscribers, assuming an underlying βbinomial model, show that the percentage of male subscribers ranges from about 8.810.2%, while that of female subscribers ranges from about 25.227.1%. For the occupational group, the result reveals that the proportion of Civil servants, who subscribed to the NHIS scheme as well as that of professionals is about 20%. The proportion of the
respondents who are aware of the benefits of subscribing to the NHIS scheme
is between 24 and 26%. For employees earning <100,000
annum^{1} the subscription rate is 7.610%. The respective subscription
proportions for those earning about 240,000
annum^{1} is 23.5% between 25.5%, for those earning between 500,0001,000,000
annum^{1}, it is 7.6%, for those earning between 1,000,0002,400,000
annum^{1}, it is between 6.4 and 9%, while for employees earning >2,400,000
annum^{1} the subscription is between 37 and 43%.
The proportion of subscribers among families that have only one child is between
16 and 20% that of families with 2 children is between 28 and 32%, families
with three children have a proportion between 19 and 22%. Families with 4 children
have a subscription proportion between 1012% while, the last category in this
group, families with 6 or more children record a proportion of between 5965%.
Table 1: 
Estimated reparameterised hyper parameters 


Table 2: 
Computed proportion of participation in NHIS 


When, a family has only one child who is under 18 years the proportion is between
1720%. For a family with two of the children <18 years, the proportion is
between 3539%; for a family with 3 children the proportion is between 710%.
The result for the level of education category shows that the proportion of
subscribers among holders of general certificate of education is between 21.223.9%
that for holders of OND is between 35% that for holders of HND or Bachelor
of Science is between 1416%; for 2nd degree, it is between 33 and 37%, while
the proportion for holders of Ph.D is between 21 and 23.9%. The proportion of
subscribers among single respondents is between 2 and 3%, while that of couples
is between 24 and 26%.
Overall, the computed confidence interval at the 95% level of significance provides evidence of a very narrow width (maximum width: 0.05 or 5%), thereby justifying the choice of Bayes method of estimation for the analysis.
CONCLUSION
The overaching objective of the study is to update previous knowledge about subscription to the NHIS scheme. As expected, the Bayesian analysis has saliently brought out the structure of the subscription to the NHIS scheme and inevitably underscore the need to raise the level of awareness of the populace about the NHIS scheme. It has presented, an opportunity to craft a marketing plan targeted at specific subgroup. For instance, since the proportion of women subscribers is about three times the proportion of male subscribers, it would suggest that more awareness campaign need to be done on men than women. Also, since the least subscribers to the scheme as indicated in the ‘rank’ column of Table 2 are the single individuals, efforts should be made at creating more awareness of the benefit of the scheme for this group. Similarly, since there does not appear to be much difference in the proportion of subscribers between civil servants and professionals, uniform marketing programmes can be mounted for them.
The study also, opens a vista for further research. Specifically, it raises questions about the cause of low subscription rates among some categories. This deserve to be investigated further.