Inference for functions of parameters in discrete distributions based on fiducial approach: Binomial and Poisson cases

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Abstract

In this article, we propose a simple method of constructing confidence intervals for a function of binomial success probabilities and for a function of Poisson means. The method involves finding an approximate fiducial quantity (FQ) for the parameters of interest. A FQ for a function of several parameters can be obtained by substitution. For the binomial case, the fiducial approach is illustrated for constructing confidence intervals for the relative risk and the ratio of odds. Fiducial inferential procedures are also provided for estimating functions of several Poisson parameters. In particular, fiducial inferential approach is illustrated for interval estimating the ratio of two Poisson means and for a weighted sum of several Poisson means. Simple approximations to the distributions of the FQs are also given for some problems. The merits of the procedures are evaluated by comparing them with those of existing asymptotic methods with respect to coverage probabilities, and in some cases, expected widths. Comparison studies indicate that the fiducial confidence intervals are very satisfactory, and they are comparable or better than some available asymptotic methods. The fiducial method is easy to use and is applicable to find confidence intervals for many commonly used summary indices. Some examples are used to illustrate and compare the results of fiducial approach with those of other available asymptotic methods.

Introduction

Many summary indices in comparative clinical trials can be expressed as a function of independent binomial success probabilities or of Poisson means. For instance, if pe denotes the probability of an event (such as death or adverse symptoms) in the exposed group of individuals, and pc denotes the same in the control group, then the ratio pe/pc is a measure of relative risk for the exposed group. The ratio of odds is defined by pe(1pc)/(pc(1pe)) which represents the relative odds of event in the exposed group compared to that in the control group. Another measure of importance is the relative difference defined by (p2p1)/(1p1), where p1 denotes the proportion of patients who “improve” when treatment 1 is administered and p2 is the corresponding proportion of patients treated with treatment 2 (Fleiss, 1973). There are other applications where it is of interest to estimate the product of two or more independent binomial probabilities (Buehler, 1957, Harris, 1971). Another well-known problem of interest is comparison of two proportions via the difference p1p2. In their recent article, Brown and Li (2005) compared several interval estimation procedures for p1p2 with respect to coverage probabilities and recommended some for practical applications.

The problem of estimating a function of several Poisson parameters arises in many applications. For instance, the ratio of two independent Poisson means is used to compare the incident rates of a disease in a treatment group and a control group, where the incident rate is defined as the number of events observed divided by the time at risk during the observed period. A weighted sum of independent Poisson parameters is commonly used to assess the standardized mortality rates (Dobson et al., 1991). Confidence intervals (CIs) for a product of Poisson parameters are also used to estimate the reliability of a parallel system (Harris, 1971).

Wald method and the asymptotic method based on the likelihood approach are commonly used for many of the aforementioned problems. The Wald confidence intervals (CIs) that use the standard errors obtained with delta method often work poorly for small to moderate sample size (often conservative). Closed form formulas for score or likelihood CIs are not available, and some iterative methods are required to evaluate them. Even though the likelihood ratio or score CIs are computationally more complex than Wald intervals, these intervals often perform better than Wald CIs. For estimating the odds ratio, one can use the exact conditional method which is based on the power function of Fisher's exact test for testing equality of two binomial proportions (Thomas and Gart, 1977). This type of conditional confidence limits are usually too conservative yielding CIs that are unnecessarily wide.

In this article, we propose simple inferential procedures for some of the aforementioned problems based on Fisher's fiducial argument. The idea of fiducial probability and fiducial inference was introduced by Fisher, 1930, Fisher, 1935. As mentioned in Zabell (1992) fiducial inference has been a subject of severe criticisms concerning the interpretation of fiducial distribution. Efron (1998) remarked in Section 8 of his paper that fiducial distribution is generally considered to be Fisher's biggest blunder; however, he concluded the section by stating “Maybe Fisher's biggest blunder will become a big hit in the 21st century!” We observed from these two review articles just cited that many criticisms about the fiducial approach are philosophical than practical. In particular, fiducial approach is a useful tool to find solutions to many complex problems with satisfactory frequentist properties. In fact, fiducial inference in many situations are now well accepted. For example, Clopper and Pearson's (1934) fiducial limits for a binomial proportion and Garwood's (1936) fiducial limits for a Poisson mean are now commonly referred to as the exact (in the frequentist sense) confidence intervals. Furthermore, the exact conditional CI for the ratio of two Poisson means by Chapman (1952), and the exact CI for the correlation coefficient of a bivariate normal distribution (see Anderson, 1984, Section 4.2) are also fiducial intervals. For other situations where fiducial inference led to exact CIs, see Dawid and Stone (1982).

Fiducial inference appears to have made a resurgence recently under the label of generalized inference by Tsui and Weerahandi (1989) and Weerahandi (1993). Hannig et al. (2006) have noted that the generalized variable procedures are a special case of fiducial inference procedures, and are asymptotically exact in many situations. For more details and applications of the generalized inference, see the books by Weerahandi, 1995, Weerahandi, 2004. For the continuous case, the fiducial approach has been used successfully to estimate or to test a function of parameters where ordinary pivotal quantities are available for individual parameters (e.g., lognormal mean, normal quantiles and quantiles in one-way random model). We note that no ordinary pivotal quantity is available to make inference on a binomial success probability or for a Poisson mean, and so the methods for the continuous case cannot be extended to these discrete distributions. Thus, we shall explore an alterative approximate approach on the basis of our observation of the method by Cox (1953) for constructing confidence limits for the ratio of two independent Poisson means.

The rest of the article is organized as follows. In the next section we describe fiducial quantities for a binomial success probability and for a Poisson mean. In Section 3, we provide approximate fiducial CIs for the relative risk and the ratio of odds. For each of the problems, we evaluate the coverage probabilities of the fiducial CI and compare them with those of popular CIs based on asymptotic methods, and provide illustrative examples. In Section 4, we provide fiducial approaches for estimating the ratio of two Poisson means and for estimating a weighted sum of Poisson means. The validity of the fiducial approach is evaluated using Monte Carlo simulation, and the results are illustrated using some examples. A test procedure based on a fiducial quantity is outlined in Section 5. Applications of the fiducial approach to other problems are given in Section 6. Some concluding remarks and limitations of the fiducial approach are given in Section 7.

Section snippets

A fiducial quantity for a binomial p

Let Xbinomial(n,p), and let Ba,b denote the beta random variable with shape parameters a and b. It is well-known that, for an observed value k of X, P(Xk|n,p)=P(Bk,nk+1p) and P(Xk|n,p)=P(Bk+1,nkp). On the basis of this relation, Stevens (1950) pointed out that there is a pair of fiducial distributions for p, namely, Bk,nk+1 for setting lower limit for p and Bk+1,nk for setting upper limit for p. The Clopper–Pearson (1934) CI based on this pair of fiducial variables is given by (BX,nX+1

Inference for binomial distributions

In the following we shall provide fiducial CIs for the relative risk and the ratio of odds. Throughout the section, it is assumed that X1 binomial(n1,p1) independently of X2 binomial(n2,p2), and (k1,k2) is an observed value of (X1,X2).

CIs for the ratio of two Poisson means

Let Yi be the number of random occurrences of an event over a period of time ti (or from a sample of ni units) with mean rate λi, so that YiPoisson(tiλi), i=1,2. The problem of interest here is to find confidence intervals for the ratio PR=λ1/λ2. As confidence interval for the ratio can be obtained from the CI for λ1t1/λ2t2, we can ignore t1 and t2 while developing CI for the ratio of means.

Binomial-score CI: This CI is based on the result that the conditional distribution of Y1 given Y1+Y2=M>0

Fiducial tests

In general, following the generalized variable approach by Tsui and Weerahandi (1989), a fiducial test variable for testing a parameter is obtained as the fiducial quantity minus the parameter. For example, the fiducial test variable for testing a binomial success probability p is given by Qt=Bk+1/2,nk+1/2p. Because, for a given k, Qt is stochastically decreasing in p, the fiducial p-value for testing H0:pp0vs.Ha:p>p0 is given by supH0P(Qt0)=P(Bk+1/2,nk+1/2p00). The fiducial test rejects

Application to other problems

We shall now briefly explain other problems where the fiducial approach produces satisfactory results. More details with examples and coverage studies are given in Lee (2010).

As mentioned in the introduction, Fleiss (1973) introduced the relative difference, defined by RD=(p2p1)/(1p1), as the index of benefit when patients who respond positively to treatment 1 are also expected respond positively to treatment 2. The FQ for the RD can be obtained by replacing the parameters by their fiducial

Concluding remarks

In this article we showed that the fiducial approach is not only simple but is also useful to obtain satisfactory solutions to several important problems involving estimation of a function of binomial or of Poisson parameters. In many situations, the fiducial approach produced results that are comparable to or better than those of some existing methods. In addition, the accuracy of the fiducial approach for a specific problem can be evaluated using Monte Carlo simulation. Even though we made no

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