Statistical methods for the beta-binomial model in teratology.

The beta-binomial model is widely used for analyzing teratological data involving littermates. Recent developments in statistical analyses of teratological data are briefly reviewed with emphasis on the model. For statistical inference of the parameters in the beta-binomial distribution, separation of the likelihood introduces an likelihood inference. This leads to reducing biases of estimators and also to improving accuracy of empirical significance levels of tests. Separate inference of the parameters can be conducted in a unified way.


Introduction
Because teratological data include observations on fetuses from the same litter, binary responses have litter effects that cause overdispersion against the binomial model. By taking account the litter structure, several statistical models have been introduced, and many of their inference procedures have been proposed and improved. Reviews of this subject were presented in Haseman and Kupper (1) and in Krewski et al. (2). In the next section, we give a brief review ofrecent developments for statistical inference of the semiparametric model and the parametric model in the teratological data analysis and especially that ofthe (3-binomial model. Then we review our recent work on modifications for the moment estimators of the parameters in the model. Recent developments for likelihood inference emphasize advantages of separation of the likelihood (3). We will apply separate likelihood inference for the (3-binomial population in the third section ofthis paper for expectation of improvement ofthe usual likelihood inference. A simulation study is conducted for examining performance of the applied inference procedures. In the final section, we discuss unsolved problems and future studies on the (3-binomial model.

Review of Teratological Data Analysis General View
For the test of the difference between prevalence rates in two samples, Gladen (4) proposed the jackknife method. On the assumption ofthe first two moments modeled on the litter structure, Williams (S) proposed the quasi-likelihood method for the dose-response regression analysis. On the other hand, the binomial sampling error model was generalized for litter effects as the following parametric models. Williams (6) introduced the (3-binomial model in the teratological data analysis. He assumed a (3 distribution between prevalence rates of litters. Kupper and Haseman (7) introduced the correlated binomial model by considering the correlation between two binary responses within the same litter. A different approach was used by Ochi and Prentice (8). In their model, binary responses within the same litter are defined according to whether the corresponding components of a multivariate normal variate with common mean, variance, and correlation exceed a common threshold. The usual likelihood methods for inference of the parameters have been used in the above models.
Among the existing models, the most important one is the (3-binomial model. This model has been used widely in the analysis of teratological data and has been studied by many biostatisticians (9,10). Recent topics for the (3-binomial model are concerned with the regression analysis and the incorporation of historical control data. Kupper et al. (11) considered the fitting of a logistic doseresponse curve to litter proportions in a (3-binomial sampling error model. They showed from their simulation study that the maximum likelihood estimates (MLEs) of regression coefficients are seriously biased ifthe intralitter correlation is falsely assumed to be homogeneous across all dose groups. A simple explanation of the source of these large biases was given by Williams (12), and the theoretical aspect was discussed by Yamamoto and Yanagimoto (13).
Incorporating historical controls to a current toxicological experiment is another attractive topic. Throne (14) assumed that the prevalence rate of the current control varies according to a (3 distribution. Hoel and Yanagawa (15) constructed a conditional test given the fixed number of responses in the current control group. In applications to actual data, estimates ofthe parameters in the ( distribution are necessary, which might be obtained from the historical control data distributed in a (3-binomial distribution. Recently, Prentice et al. (16) conducted a non-Bayes approach to incorporating the historical control data. They assumed that the historical controls follow a (3-binomial distribution and the current experiment data follow a binomial or a (3-binomial distribution. Inference of the parameters in an applied model is based on thejoint likelihood ofthe historical and the current data.
For simplicity we assume that the number of fetuses, ni, is common among litters. Because ofthe above unfavorable properties, the two traditional estimation methods, the method of moments and the maximum likelihood method, have been used routinely.
A few attempts to improve estimators have been made. Kleinman (20) claimed the superiority ofthe moment estimator of ir with proper weights. Tamura and Young (21) proposed the use of a stabilizer for the usual moment estimator of 4. Crowder (22) suggested good performance ofthe conditional MLE of fixed the sample mean, xi, where he approximated the distribution of xi by a normal distribution. For the test ofthe null hypothesis = 0, the traditional asymptotic likelihood ratio test (LRT) theory has been falsely applied in spite of the fact that = 0 is the marginal point of the parameter range. Paul etal. (23) claimed that the LRT statistic is asymptotically distributed in the 50:50 mixture of the degenerate distribution at zero and the chi-square distribution with 1 degree of freedom (df) under the null hypothesis. The C(a)-test proposed by Tarone (24) and recommended by Paul et al. (23) is able to test this marginal point without any difficulty. Accuracy ofthe empirical significance level can be improved by using an alternative asymptotic distribution of the test statistic under the assumption of large litter sizes proposed by Kim and Margolin (25). As Prentice (17) noted, in the extended (3-binomial distribution, the point zero can be treated as an inner point of the parameter range so that the traditional LRT theory can be applied correctly.

Method of Moments
The moment estimator of wr is i = xi In, which is unbiased and has a potential efficiency (20). The moment estimator of is denoted by ns2 (n-(n-1)(n -a) where = 2(x,-x )2/(m-1), which is known to have an asymptotic positive bias. The estimator ),,0 is given as the root of the estimating equation (26) 9b(x; k) = ns2 _ -(n--)-(n-1)i(n-x) = 0, which is not unbiased, that is, E[gb(x; 4) ] * 0. This comes from the fact thatix (nxi ) is not an unbiased estimator ofn2w (1-4v). Yanagimoto and Yamamoto (27) claimed that removing the bias from an estimating equation for a usual moment estimator leads to better performances in many examples appearing in the actual statistical analyses. An unbiased estimating equation for 4 g(x; )=(mn -1)s2 Mi(ni) -n -)(mi(n -) + -a') = OX gives a moment estimator (1) This treatment reduces the bias and the mean square error. Yamamoto and Yanagimoto (28) made an extensive comparison of performance of 4 and )mo. Until now the method ofmoments focuses only on the estimation procedure and consequently does not attract our attention to the test procedure. By using an unbiased moment estimating equation g( x ;0) = 0. we can produce a following test statistic for the null hypothesis 0 = 0o defined by where 0 is the unbiased moment estimator given by g(x ;0) = 0 and the denominator is an unbiased moment estimator for the expectation of l g(x ;0)d0. Note that the test statistic T,m,, can be explained as a signal-noise ratio under the hypothesis. Applying this test procedure to the (3-binomial case, the test statistic for the null hypothesis ir = -ro is given by i0 -n7ro)2 TM = s2/M which is the square ofthe well known t-test statistic. The test for 4 needs complicated calculations of the third and the fourth moments, so we do not pursue the moment test procedure for 4 any further.

Innovation in Likelihood Inference
Outline of Principle Recent developments for likelihood inference ofthe mean, IL, and the dispersion parameter, 0, put emphasis on the advantage of separation of the likelihood (3), which is based on factorization of the density function of a sample x; f(X; Ad ) = fm(t;Ii,0)fc(Z,0I1t), (2) where t is the sample sum or the sample mean. The marginal density is used as the likelihood for inference ofA, and the conditional density is used as the likelihood for 0.  (29).

Inference Procedures
Though the (3-binomial distribution cannot be factored into the form of Equation 2, we can expect that the application of separation of the likelihood leads to improving usual likelihood inference. Inference of w is based on the marginal density of the sample sum (t = E xi) and that of 4 is based on the conditional density fixed t. Following the principle outlined above, thejoint density ofx is separated in f(X; Ir, d) = fm(t; Ad9!)fc(X; 7rq St) Unfortunately, the marginal densityfm is of a complicated form, and also ir remains in the conditional densityfi. Therefore, we will evaluate the former by the following approximation: The first and second moments of t are given by where N = mn and (N -1)4)' = (n -1)0. Crowder (22) applied a normal distribution for a candidate of the approximated distribution of t. We propose here the use of a (3-binomial distribution as a more reasonable candidate, because the distribution of t is skewed. Especially the (3-binomial approximation has merits such that the sample distribution is discrete and closed in the (3-binomial family. Fixing the first two moments oft, the above two approximated distributions have the following probability density  The results are shown in Tables 1 and 2. Noteworthy findings 0.04. The squared t-test overstates all the parameter values and in Table 1 are that the median biases of the two conditional MLEs has stable empirical levels when r is large. The approximations are about half those of the MLEs. The unbiased moment by the normal distribution and the (-binomial distribution to the estimator decreases by 1 when w is large. Table 2 indicates that marginal distribution of t have resulted in the similar perforthe MSEs of the four estimators are comparable to each other. mance for the estimation of 4), but T, shows differences from Summarizing results of the simulations, we conclude that the TmB. The empirical levels of Tuare unstable, and their range is conditional MLEs perform the best, and the unbiased moment wider than that of the usual LRT. estimator is superior to the MLE.
On the other hand, TB has smaller empirical levels than the usual LRT for all the parameter values. The understatement Table 4. Empirical significance levels of tests for of Tm8 can be improved by using the chi-square distribution in the parameter wwith 1% nominal level place of the F distribution to yield the critical values when an (litter size = 10, number of animals = 20, number of iterations = 10,000). estimate X is 0. In the right-hand columns of Tables 3 and 4,  w  4  LRT  t2-Test  TmN  TmB  TmB  modified empirical levels are  Note that the one-sided U test can be produced by the signed LRT (31) as

5.20
'Values in parentheses were obtained using a one-sided test. 4.72 alternative leads to better accuracy of empirical levels for the 4.88 two conditional LRTs than the usual LRT. 4.69 In conclusion, the simulation study has shown that separate 4.83 likelihood inference has the ability to innovate in statistical inference of the parameters (T,O) in the B-binomial distribution. aValues in parentheses were obtained using a one-sided test.

Further Problems
In this paper we do not consider the heterogeneous litter-size case, two-sample problems, and the regression analysis. Notice that the inference procedures proposed above are derived from separation ofthe likelihood. We expect that this principle can be applied to these statistical problems successfully. For example, in two-sample problems, the estimation of a common 4 may be conducted by maximizing the conditional likelihood Pim(t, *1, ' )P2(Yt, #2,0) Pim(tl 7 l 1)P2m(t2) '2 i0) where t1= = Xi, t2 = E y,, f#= X /n,, f2= 5FM/f2, and the marginal distributions of t, and t2 are adjusted to (B-binomial distributions, respectively. The t-test for the difference between two incidence rates with a common dispersion may be constructed by the signed marginal LRT 21PM(ti X *1 X MP2m(t2 X *-2 X v sgn(i -9) n dgn(X y'v Pl m(t) ' *a O)P2m(t2l a ¢) where WC = (mif#I + m2z2)/(m, + m2) and + is the above conditional MLE. For regression problems, the construction of inference procedures for regression coefficients looks more difficult but is worth future study.