 ## ISSN 2517-5718

### Bayo H Lawal*

Department of Statistics & Mathematical Sciences, Kwara State University, Malete, Kwara State, Nigeria

### CitationCitation COPIED

Lawal BH. On Modeling the Double and Multiplicative Binomial Models as Log-Linear Models. Adv Comput Sci.2018 Jan; 1(1):103.

### Abstract

In this paper we have fitted the double binomial and multiplicative binomial distributions as log-linear models using sufficient statistics. This approach is not new as several authors have employed this approach, most especially in the analysis of the Human sex ratio in . However, obtaining the estimated parameters of the distributions may be problematic, especially for the double binomial where the parameter estimate of π may not be readily available from the Log-Linear (LL) parameter estimates. Other issues associated with the LL approach is its implementation in the generalized linear model with covariates. The LL uses far more parameters than the procedure that employs conditional log-likelihoods functions where the marginal likelihood functions are minimized over the parameter space. This is the procedure employed in SAS PROC NLMIXED. The two procedures are essentially equivalent for frequency data. For models with covariates, the LL uses far more parameters and the marginal likelihood functions approach are employed here on three data set having covariates.

### Keywords

Double Binomial; Multiplicative Binomial; Log-Linear; Marginal Likelihood Functions

### Introduction

In the formulations of the multiplicative binomial distribution, in Altham and its corresponding double binomial distribution in Efron, both distributions were characterized with intractable normalizing constants c(n, ψ) and c(n, π) respectively [2,3]. Consequently, these models were implemented in by utilizing a generalized linear model with a Poisson distribution and log link to the frequency data. This approach has earlier being similarly employed in [1,4]. This approach which employs joint sufficient statistics in both distributions was earlier proposed in Lindley & Mersch .

Both distributions are fitted using a Poisson regression model having sufficient statistics from both distributions as explanatory variables with the frequencies being the mean dependent variables. For the Double binomial model (DBM), the sufficient statistics are y log(y) and (n − y) log(n − y). Similar joint sufficient statistics for the multiplicative binomial model (MBM) are y and y(n − y) with the offset being

$Z=log\left(\begin{array}{c}n\\ y\end{array}\right)$
for both models. For instance, for the DBM the model would be:

$log\left(n_{i}/z\right)=y+\theta$.....................................................................(1)

Where $\theta=\theta_{1}+\theta_{2}$

$\theta_{1}=\begin{cases}0 & ify = 0\\ylog\left(y\right) & otherwise\end{cases},\theta_{2}=\begin{cases}0& ify = n\\n-ylog\left(n-y\right) & otherwise\end{cases}$

Similarly, for the multiplicative binomial, the model is estimated by the log-linear model (or Poisson Model):

$log\left(\begin{array}{c}n_{i}\\ z\end{array}\right)=y+\delta$....................................................................(2)

Where, $\delta=\delta_{1}+\delta_{2}$ and

$\delta_{1}=\begin{cases}0 & ify = 0\\y & otherwise\end{cases},\delta_{2}=\begin{cases}0& ify = n\\y\left(n-y\right)& otherwise\end{cases}$

However, in recent times, both models have been fully formulated with the intractable normalizing constants fully formulated. These distributions are described later in this paper with the normalizing constants fully formulated. In this study, we compare fitting these two probability models to two example frequency data, two data set examples arising from teratology studies, and randomized complete block design example having binary outcome by the method of sufficient statistics described above and by the method of numerically maximizing the marginal likelihood function arising from engaging the problem as a mixed generalized linear model. The SAS PROC NLMIXED which performs the Maximum Likelihood estimation numerically by using the Adaptive Gaussian Quadrature and Newton-Raphson optimization algorithm. We shall designated the sufficient StatisticsPoisson regression approach as LL, while the marginal likelihood function maximization via PROCNLMIXED is designated MgL in this study. The sufficient statistics procedure uses a Poisson regression with an offset and is implemented in SAS PROC GENMOD.

### Models Under Consideration

We describe in the following section, the two probability distribution models employed in this paper.

The Multiplicative Binomial Model-MBM

[2,6,7] Lovinson proposed an alternative form of the twoparameter exponential family generalization of the binomial distribution first introduced by  which itself was based on the original Cox representation as:

$f\left(y\right)=\frac{\left(\begin{array}{c}n\\ y\end{array}\right)\psi^{y}\left(1-\psi\right)^{n-y}\omega^{y\left(n-y\right)}}{\sum_{j=0}^{n}\left(\begin{array}{c}n\\ j\end{array}\right)\psi^{j}\left(1-\psi\right)^{n-j}\omega^{j\left(n-j\right)}},y=0,1,........,n$.........................................(3)

where 0 < $\psi$ < 1 and $\omega$> 0. When $\omega$ = 1 the distribution reduces to the binomial with $\pi$ = $\psi$. If $\omega$ = 1, n → ∞ , and $\psi$→ 0, then n$\psi$ →$\mu$ and the MBD reduces to Poisson($\mu$).

The normalizing constant is

$c\left(n,\psi\right)=\sum_{j=0}^{n}\left(\begin{array}{c}n\\ j\end{array}\right)\psi^{j}\left(1-\psi\right)^{n-j}\omega^{j\left(n-j\right)}$

the denominator expression in (3) in this case.  presented an elegant characteristics of the multiplicative binomial distribution including its four central moments. His treatment includes generation of random data from the distribution as well as the likelihood profiles and several examples-some of which are similarly employed in this presentation. Following  the probability $\pi$ of success for the Bernoulli trial, that is, P(Y = 1) can be computed from the following expression in (4) as:

$p_{i}=\psi_{i}\frac{K_{n-i}\left(\psi,\omega\right)}{K_{n}\left(\psi,\omega\right)},for i=1$............................................................................(4)

Where:

$k_{n-a}\left(\psi,\omega\right)=\sum_{j=0}^{n-a}\left(\begin{array}{c}n-a\\ y\end{array}\right)\psi^{y}\left(1-\psi\right)^{n-a-y}\omega^{\left(y+a\right)\left(n-a-y\right)};a=1,2,.......,n$............(5)

with p defined as in (4), $\psi$ therefore can be defined as the probability of success weighted by the intra-units association measure $\omega$ which measures the dependence among the binary responses of the n units. Thus if $\omega$ = 1, then p = $\psi$ and we have independence among the units. However, if $\omega$ = 1, then, p = $\psi$ and the units are not independent.

The mean and variance of the LMPD are given respectively as:

$E\left(Y\right)=np_{1}$......................................................................................................(6a)

$Var\left(Y\right)=np_{1}+n\left(n-1\right)p_{2}-np_1^2$.....................................................................(6b)

### The Double Binomial (DBM) Model

The Double Binomial (DBM) Model

In Feirer et al. , the double binomial distribution was presented, having the pdf form:

$f\left(y;\pi,\phi\right)=\frac{\left(\begin{array}{c}n\\ y\end{array}\right)\left[y^{y}\left(n-y\right)^{n-y}\right]^{1-\phi}\left[\pi/\left(1-\pi\right)\right]^{y\phi}}{\sum_{j=0}^{n}\left(\begin{array}{c}n\\ j\end{array}\right)\left[j^{j}\left(n-j\right)^{n-j}\right]^{1-\phi}\left[\pi/\left(1-\pi\right)\right]^{j\phi}},y=0,1,........,n$...........(7)

Again, the normalizing constant in this case is the denominator expression given by

$c\left(n,\psi\right)={\sum_{j=0}^{n}\left(\begin{array}{c}n\\ j\end{array}\right)\left[j^{j}\left(n-j\right)^{n-j}\right]^{1-\phi}\left[\pi/\left(1-\pi\right)\right]^{j\phi}}$

### Applications

We apply the models discussed above to two frequency data and to teratology data sets having four and two treatment groups. We first present the analyses for the two frequency data sets in Tables 1 through 5. The estimation of the parameters under each model for the MgL approach uses SAS PROC NLMIXED, using the following log-likelihoods for the MBM (LL1) and DBM (LL2) respectively. The procedure was discussed earlier in the paper.

$LL1=\log\left(\begin{array}{c}n\\ y\end{array}\right)+y\log\left(\psi\right)+y\left(n-y\right)\log\omega-\log\left[ \sum_{j=0}^{n}\left(\begin{array}{c}n\\ j\end{array}\right)[\psi^{j}\left(1-\psi\right)^{n-j} \omega^{j\left(n-j\right)}\right]$

$LL2=\log\left(\begin{array}{c}n\\ y\end{array}\right)+\left(1-\phi\right)\left[y\log\left(y\right)+\left(n-y\right)\log\left(n-y\right)\right]+y\phi\log\left(\frac{\pi}{1-\pi}\right)-\log\left[\sum_{j=0}^{n}\left(\begin{array}{c}n\\ j\end{array}\right)\left({j^{j}\left(n-j\right)}^{n-j}\right)^{1-\phi}{\left(\frac{\pi}{1-\pi}\right)}^{j\phi} \right]$

Example Data Set I-Geissler Data

The Distribution of males in 6115 families with 12 children in Saxony, previously analyzed in Sokal & Rohlf  is presented in Table 1. The data is originally from Geissler  and had similarly been analyzed in . Here Y ~ binomial (12,$\pi$ ). The frequencies are presented as counts having a total sum of 6115. The observed mean for the data is $\overline{y}$ = 6.2306 and the corresponding variance is $s^{2}$ = 3.4898. Under the binomial model, the estimated mean is 6.2304 and estimated variance being 2.9956. Hence the estimated dispersion parameter DP= $\overline{y}/s^{2}$ = 2.07 indicating over-dispersion in the data. The estimated probability of occurrence under the binomial model is $\pi$= 0.5192. The binomial does not fit the data ($X^{2}$ = 110.5051 on 11 d.f., p-value=0.0000) because the variance of the data is grossly under estimated by the model. The results of the application of the double binomial and the multiplicative models to this data are presented in Table 2. Further, the mixed model approach is based on one more degree of freedom as it estimates one parameter less than the LL model approach. The Mixed model approach gives the parameter estimates of the distribution. We can obtain the equivalent parameters estimates from the Log-linear (LL) approach for the multiplicative models as follows:

$\omega=e\times p\left(\hat{\delta}\right)=e\times p\left(- 0.02615\right)=0.9742,\psi=1/\left[1+e\times p\left\{-\left(\hat{\delta}+{\hat{y}}\right)\right\}\right]=0.5165$..........(8)
 Y 0 1 2 3 4 5 6 7 8 9 10 11 12 Count 3 24 104 286 670 1033 1343 1112 829 478 181 45 7

Table 1: Distribution of Males in 6115 families with 12 children

 Log-Linear Marginal Likelihood MBM DBM MBM DBM MLE Int=0.853840 Int=-3.096918 $\hat{\theta}$=0.5165 $\pi$=0.5192 $\hat{y}$=0.092157 $\hat{y}$=0.065977 $\omega$=0.9742 $\hat{\phi}$=0.8598 $\hat{\delta}$=−0.026150 $\hat{\theta}$=0.140205 $\pi$=0.5192 -2LL 104.5372 103.1298 24985.8 24984.3 AIC 110.5372 109.1297 24990 24988.3 $X^{2}$ 14.5354 13.042 1 14.5354 13.0421 $G^{2}$ 14.4686 13.0616 14.4686 13.0616 d.f 10 10 11 11

Table 2: Parameter estimates under the five Models

For the DB, $\hat{\phi}$ can equivalently be obtained as $1-\hat{\phi}=1-$ 0.140205 = 0.8598, but the estimated probability $\pi$ seems intractable in this case and no equivalent solution is available in this case. We may note here that the estimate $\psi$ = 0.5165 under the multiplicative model is not an estimate of the success probability$\pi$ . For this data, we must use the expressions in (4) and (5) to obtain this estimate. Here, $k_\left({n-1}\right)$ = 0.42723 and $k_\left({n-1}\right)$ = 0.42499.

Consequently, $\pi=\psi\left(\frac{k_\left({n-1}\right)}{k_{n}}\right)=0.5165\left(\frac{0.42723}{0.42499}\right)=0.5192$

The mean and Variance can therefore be computed respectively from (6a) and (6b). Alternatively, the means and variances can be empirically obtained from the fitted models using the elementary principles of

$E\left(Y\right)=\sum_{i=0}^{n}y_{i}p_{i}$
and Var of Y being $E\left(y^{2}\right)-\left[E\left(y\right)\right]^{2}$ . These distributions are displayed in Table 3.

 Y Count Double Binomial Multiplicative Binomial $p_{i}$ $p_{i}$ $\sum y_{i}p_{i}$ $V_{1}$ $V_{1}$ $p_{i}$ $\sum p_{i}$ $\sum y_i^2p_{i}$ $\sum y_i^2p_{i}$ $V_{2}$ 0 3 0.0005 0.0005 0.0000 0.0000 0.0000 0.0004 0.0004 0.0000 0.0000 0.0000 1 24 0.0038 0.0043 0.0038 0.0038 0.0038 0.0037 0.0041 0.0037 0.0037 0.0037 2 104 0.0171 0.0214 0.0379 0.0721 0.0706 0.0171 0.0212 0.0380 0.0723 0.0708 3 286 0.0503 0.0717 0.1889 0.5250 0.4893 0.0508 0.0721 0.1905 0.5298 0.4936 4 670 0.1068 0.1785 0.6160 2.2333 1.8538 0.1072 0.1793 0.6194 2.2454 1.8617 5 1033 0.1698 0.3483 1.4652 6.4792 4.3325 0.1694 0.3487 1.4665 6.4812 4.3304 6 1343 0.2067 0.5550 2.7056 13.9220 6.6015 0.2057 0.5544 2.7008 13.8867 6.5924 7 1112 0.1938 0.7488 4.0622 23.4178 6.9164 0.1933 0.7478 4.0542 23.3605 6.9239 8 829 0.1390 0.8878 5.1743 32.3146 5.5414 0.1396 0.8874 5.1712 32.2961 5.5553 9 478 0.0748 0.9626 5.8472 38.3710 4.1810 0.0755 0.9629 5.8511 38.4154 4.1803 10 181 0.0289 0.9915 6.1364 41.2628 3.6074 0.0291 0.9920 6.1418 41.3227 3.6009 11 45 0.0074 0.9989 6.2178 42.1579 3.4972 0.0071 0.9992 6.2204 42.1873 3.4939 12 7 0.0011 1.0000 6.2306 42.3116 3.4915 0.0008 1.0000 6.2306 42.3094 3.4893

Table 3: Empirical Means and Variances for both the DBM and MBM

In the above Table, Some of the columns are self explanatory. The columns labeled $V_{1}$ and $V_{2}$ are cumulative values of

for both models respectively. Thus, the mean is the value at y = 12. The variance for the DBM for instance, is computed as 42.3116− $\left(6.2306\right)^{2}$ = 3.4915. In Table 4 are presented the expected values under both models for the two approaches (LL & MgL), both approaches give exact results as expected. The Table also displays the mean of the distributions under both approaches as well as the empirical variances, designated here as var. We recall that for the observed data in Table 1, $\overline{y}$= 6.2306 and $s^{2}$ = 3.4898.We see from Table 4, that while the two models estimate the mean of the data well, the estimated variance under the binomial model of 12(0.5192)(1−0.5192) = 2.9956 underestimates the observed variance of the data, and this explains the poor fit to the data by the binomial model. On the other hand, for the two models, the variance of the observed data are reasonably well estimated, because of the extra parameter in the model (dispersion parameter) of $\phi$ and $\omega$ for the DBM and MBM respectively.

Table 4 also displays the corresponding Pearson‘s $X^{2}$ and the corresponding degrees of freedom (d.f.). Clearly, for this data set, both the double binomial and the multiplicative models fit the data well with the Double binomial being slightly providing a better fit. Although the expected values generated are the same for both fitting approaches, we see that the marginal likelihood (MgL) approach gives a more parsimonious model because it is based on one more degree of freedom.

 Y Count Double Binomial Multiplicative Binomial MgL LL MgL LL 0 3 2.956 2.956 2.3486 2.3487 1 24 23.3861 23.386 22.5809 22.581 2 104 104.3138 104.3137 104.8482 104.8484 3 286 307.7531 307.7531 310.8921 310.8923 4 670 652.8854 652.8858 655.6551 655.6551 5 1033 1038.546 1038.546 1036.077 1036.077 6 1343 1264.242 1264.242 1257.907 1257.907 7 1112 1185.039 1185.04 1182.293 1182.293 8 829 850.0634 850.0639 853.7711 853.7724 9 478 457.2186 457.2188 461.9646 461.9659 10 181 176.8358 176.8359 177.7841 177.785 11 45 45.2369 45.2369 43.6925 43.6928 12 7 6.5246 6.5246 5.1858 5.1858 $G^{2}$ 13.0421 13.0421 14.4686 14.4686 $X^{2}$ 13.0612 13.0612 14.5354 14.5354 d.f. 10 9 10 9 P-value 0.2213 0.1607 0.1527 0.1066 $\overline{y}$ 6.2306 6.2306 6.2306 6.2306 6.2306 var 3.4898 3.4893 3.4915 3.4915 3.4915

Table 4: Expected values under the two models and Approaches with corresponding Pearson’s $X^{2}$ Statistic values

Data Example II

This example is taken from Nelder & Mead  and relates to the number of candidates having an “alpha”, i.e. at least 15 scores out of a total 20 points from each of nine questions employed in assessing the final class of candidates in an examination. There were a total of 209 candidates for the exam and Table 5 gives the distribution of these scores for the 209 candidates.

 Y Count Double Binomial Multiplicative Binomial MgL LL MgL LL 0 3 2.956 2.956 2.3486 2.3487 1 24 23.3861 23.386 22.5809 22.581 2 104 104.3138 104.3137 104.8482 104.8484 3 286 307.7531 307.7531 310.8921 310.8923 4 670 652.8854 652.8858 655.6551 655.6551 5 1033 1038.546 1038.546 1036.077 1036.077 6 1343 1264.242 1264.242 1257.907 1257.907 7 1112 1185.039 1185.04 1182.293 1182.293 8 829 850.0634 850.0639 853.7711 853.7724 9 478 457.2186 457.2188 461.9646 461.9659 $\overline{y}$ 1.5598 1.5598 -1.5598 1.5598 1.5598 $s^{2}$ 2.8245 2.6428 2.6428 20.811 20.811 $\pi$=0.1537 Int= -7.7413 $\psi$= 0.3630 Int= 4.1887 0.3928 −0.6701 0.8051 −0.3457 0.6072 $\pi$=0.1733 −0.2168 -2LL 713 51.4985 703.1 41.6374 AIC 717 57.4985 707.1 47.6374 $X^{2}$ 13.9366 13.9366 2.6948 2.6948 $G^{2}$ 12.9164 12.9164 3.0554 3.0554 d.f 7 6 7 6

Table 5: Expected Values under the two models and approaches with corresponding Pearson’s $X^{2}$ Statistic Values

Results

The results of applying both models DBM and MBM to the data using both approaches (LL and MM) are presented in Table 5. Again, both approaches lead to the same results in terms of expected values. However, the MgL models have one more degree of freedom under both models than the LL approach. Again, to get equivalent parameter estimates from the LL model, we have, for the multiplicative model,$\omega=exp\left(\hat{\delta}\right)=exp\left(−0.2168\right)=0.8051,\psi=1/\left[1+exp\left\{-\left(\hat{\delta}+\hat{y}\right)\right\}\right]=0.3630$. For the double binomial, an equivalent estimate for $\phi$ is $\phi$=1-$\hat{\theta}$= 1−0.6072 = 0.3928. As discussed earlier, the corresponding estimate for $\pi$ is not readily available. For this model, the multiplicative model is the most parsimonious and fits the data very well.

### Regression Model Formulations

When there are covariates in our data, the sufficient statistic approach-here into referred to as Log-linear (LL) does not lend itself to easier formulation and implementation. Lindsey and Altham  employed this approach to fitting amongst others, the two models considered in this study to the distribution of males in families in Saxony during 1885-1976 (the human sex ratio data). This approach employs far too many parameters, 13 to be precise when the same group of models can be implemented with only four parameters with the same results. Further, the implementations under this approach are not readily available. Thus in this study, we will employ the alternative MgL procedure that utilizes PROC NLMIXED in SAS. One advantage of this is that it will based on more degrees of freedom than the log-linear model. We see for the frequency data in Tables 1 for instance, that the LL approach is based on 1 d.f. more than the MgL model based.

Example I: Teratology-Ossification on the Phalanges

Teratology is the study of abnormalities of physiological development. The offspring of animals that were exposed to a toxin during pregnancy are studied for malformation. The number of malformed offspring in a litter of size n is not typically distributed binomial because the responses of the offspring from the same litter are not independent, hence their sum does not constitute a binomial r.v. Thus, data in teratological studies exhibit over-dispersion because of the correlation among responses from off springs in the same litter.

 report data from a completely randomized design that studies the teratogenicity of phenytoin in 81 pregnant mice. The treatment structure of the experiment is an augmented factorial. In addition to an untreated control, mice received 60 mg/kg of phenytoin (PHT), 100 mg/kg of trichloropropene oxide (TCPO), and their combination. The design was augmented with a control group that was treated with water. As in , the two control groups are combined here into a single group. The presence or absence of ossification in the phalanges on both the right and left forepaws on each of the fetuses is considered a measure of the teratogenic effect. The data is presented below. For the control for instance, there are 35 pair of observations designated as (n , r). Thus, the numbers of rats in each group are respectively {35,19,16,11}.

control 35 8 8 9 9 7 9 0 5 3 3 5 8 9 10 5 8 5 8 1 6 0 5

8 8 9 10 5 5 4 7 9 10 6 6 3 5 8 9 7 10 10 10

1 6 6 6 1 9 8 9 6 7 5 5 7 9 2 5 5 6 2 8 1 8

0 2 7 8 5 7

PHT 19 1 9 4 9 3 7 4 7 0 7 0 4 1 8 1 7 2 7 2 8 1 7

0 2 3 10 3 7 2 7 0 8 0 8 1 10 1 1

TCPO 16 0 5 7 10 4 4 8 11 6 10 6 9 3 4 2 8 0 6 0 9

3 6 2 9 7 9 1 10 8 8 6 9

PHT2 11 2 2 0 7 1 8 7 8 0 10 0 4 0 6 0 7 6 6 1 6 1 7

Suppose $Y_{ij}$ denote the number of deaths in litter i. Further, let $p_{ij}$ be the probability of a fetus in litter i dying. $Y_{ij}$ has the overdispersed binomial distribution with mean $n_{i}p_{ij}$  and variance $n_{i}p_{ij}\left(1-p_{ij}\right)\phi$ , with $\phi$ characterizing the correlation between any two fetuses within the same litter.

The probability of fetal death is modeled with the logit link viz:

$\log\left(\frac{p_{ij}}{1-p_{ij}}\right)=\beta_{0}+\beta_{2}z_{2i}+\beta_{3}z_{3i}+\beta_{4}z_{4i}$...........................................(9)

We have assumed here that β0 is similar across litters, and that,

$z_{2}=\begin{cases}1 & if PHP\\0 & otherwise\end{cases},z_{3}=\begin{cases}1 & if TCPO\\0 & otherwise\end{cases},z_{4}=\begin{cases}1 & if PHT_{2}\\0 & otherwise\end{cases}$

• The model that assumes $p_{0}=p_{1}=p_{2}=p_{3}$ with a common dispersion parameter $\phi$ or $\omega$ for the double binomial and the multiplicative models respectively, and, where $p_{0}$ and $p_{1}$, $p_{3}$, $p_{3}$ refer respectively to the corresponding probabilities in the control and other treatment groups.
• Here, $p_{ij}=\frac{1}{\left[1+e\times p\left(-\beta_{0}\right)\right]},\phi=e^{a0}$

$\omega=exp\left(c_{0}\right)$ with $a_{0}\neq e^{ao}$ . These ensure that the dispersion parameters are positive.

• The model here has pi = pj i i ≠ j and the dispersion parameters are functions of the covariates. That is, φ = exp (a0 + a2 z2 + a3 z3 + a4 z4 ) and ω = exp(c0 +c2 z2 + c3 z3 + c4 z4 ).
• The model here has pi ≠ pj with the ps modeled as in (9) and the dispersion parameters are modeled as functions of the covariates as in the preceding case.

Results

From the results in Table 6, the two cases (II & III) with variable dispersion parameters fit better than the model in case I, where the dispersion is uniform across the four groups. Of the models in Cases II and III, the models in case III fits much better than those in case II. Case II models assume that the four groups have a common estimated probability $\pi$, which are estimated respectively as 0.2125 and 0.2158 in both the DBM and MBM. However, the models in III which assume heterogeneous success probabilities across the four groups and variable dispersion parameters (that are functions of the covariates) fit better than those in case II. The DBM here is based on $X^{2}$ = 115.1333 on 72 d.f. The estimated $\pi$ s under the MBM are functions of n, hence these values are different for different n in the final model (Case III). We may note here that the Ψs should not be mistaken for the success probabilities.

 Case I                        Case II               Case III Parameters DBM MBM DBM MBM DBM MBM $\pi$= 0.4808 $\psi$= 0.4977 $\pi$= 0.2125 $\psi$= 0.4977 $\psi_{0}$= 0.7956 $\psi_{0}$= 0.5566 $\pi$= 0.4912 - $\pi$= 0.2158 $\pi{1}$= 0.2158 $\psi_{2}$= 0.2814 - - $\psi_{2}$= 0.4949 $\psi_{2}$= 0.4988 - - $\pi{3}$= 0.0000 $\psi_{3}$= 0.4484 $\hat{\phi}$= 0.1224 $\omega$= 0.7463 $\phi_{0}$= 0.0000 $\phi_{0}$= 0.7342 $\phi_{0}$= 0.1732 $\omega_{0}$== 0.7506 $\phi_{1}$= 0.6830 $\omega_{1}$= 0.7658 $\phi_{1}$= 0.6856 $\omega_{1}$= 0.9120 $\omega_{2}$= 0.0574 $\omega_{2}$= 0.7981 $\phi_{2}$= 0.2219 $\omega_{2}$= 0.7981 $\phi_{3}$= 0.1054 $\phi_{3}$= 0.6028 $\phi_{3}$= 0.0004 $\omega_{3}$= 0.6145 -2LL 329.3874 340.1538 309.7278 328.3978 291.8264 295.9996 AIC 333.3874 344.1538 319.7278 338.3978 307.8264 311.9996 $X^{2}$ 161.85 158.9242 144.0727 159.0039 115.1333 118.7587 d.f 78 78 75 75 72 72

Table 6: Parameter estimates for the Models in all the Cases

Data Example II-Trout Egg Data

The data in Table 7 from Manly  relate to the number of surviving eggs from boxes of eggs that were buried at five different locations in a stream and at four different times a box from a location was sampled. The data is presented as y/n where y is the number surviving and n is the number of eggs in the box.

 Location in stream Survival Period (weeks) 4 7 8 11 1 89/94 94/98 77/86 141/155 2 106/108 91/106 87/96 104/122 3 119/123 100/130 88/119 91/125 4 104/104 80/97 67/99 111/132 5 49/93 11/113 18/88 0/138

Table 7: Number of Surviving eggs against number of eggs in a box

The model of interest here is:

$\log it\left(p_{ij}\right)=\beta_{0}+\sum_{k=1}^{4}\beta_{k}z_{k}+\sum_{l=1}^{3}\beta_{1}x_{1}$......................................(10)

where $z_{k}$ are four dummy variables for location effects, and $X_{l}$ are three dummy variables representing the Time effects. The structure here is that of a randomized block design having locations as blocks and Survival times as treatments. Thus, the structure of the Pearson’s $X^{2}$  would be for Location (L) and Survival time (T):

 Source d.f. L|T 4 T|L 3 Residual* 12

The degree of freedom of 12 refers only to the binomial model. For all other distributions, the d.f. must account for the additional dispersion parameter estimates. Under the Binomial model $X^{2}$ = 63.9639 on12 d.f, giving an estimated dispersion parameter of 5.3303 > 1, indicating that the data is highly overdispersed.

Because of the overdispersion in the data, we now apply our models, DBM and the MBM to the data, giving the results in Table 8.

 Models A Models B Source BIN DBM MBM MBM $\hat{\phi}$= 0.3116 $\omega$= 0.9884 - Residual ($X^{2}$) 63.9639 27.2582 18.5493 5.0120 d.f. 12 11 11 8 -2LL 141.0292 120.4564 125.7706 112.7608 AIC 157.0292 138.4564 143.7706 136.7608

Table 8: Results of Analysis of Data in Table 7.

Models in (A) fit both the double binomial and the multiplicative binomial with constant dispersion parameter. For this group of models, the multiplicative binomial performs much better with constant dispersion parameter of 0.9884, very close to 1, indicating there is partial independence in the data ignoring the effects of locations. Models in B, have variable dispersion parameters that are functions of the covariates (Time), that is, $\phi=exp\left(a_{0}+a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}\right)$ and $\omega_{i}=exp\left(c_{0}+c_{1}x_{1}+c_{2}x_{2}+c_{3}x_{3}\right)$. Under this formulation, the double binomial computation does not converge, but that of the multiplicative binomial converged. This model gives a Pearson $X^{2}$ of 5.0120 on d.f. The results of this final model are presented in Table 9. Note that for the multiplicative, the estimated probabilities of success $\pi$ which are not the same as the $\phi$ in the model formulation in (3) are computed using expressions in (4) and (5). Note that $\psi=\pi$ . The column labeled $\sum X^{2}$ gives the cumulative contributions of observation towards $X^{2}$. The value 5.0120 is the sum of all 20 contributions towards $X^{2}$. Under the final multiplicative model, the estimated average probabilities of surviving in the first 4, 7, 8 and 11 weeks are respectively {0.8854, 0.6999, 0.6831, 0.6656}.

 # n y $\psi$ $\pi_{1}$ $\pi_{2}$ $m_{i}$ $\omega_{i}$ $s^{2}$ $\sum X^{2}$ 1 94 89 0.9895 0.9863 0.9727 92.7093 1.003 1.2635 0.1484 2 98 94 0.9042 0.9059 0.8207 88.7817 0.9997 8.3863 0.4551 3 86 77 0.9226 0.8714 0.7592 74.9446 1.009 8.2376 0.5115 4 155 141 0.7915 0.9327 0.87 144.5644 0.9903 12.016 0.5994 5 108 106 0.9868 0.9821 0.9645 106.0685 1.003 1.8759 0.5994 6 106 91 0.8823 0.8844 0.7822 93.7495 0.9997 10.894 0.6801 7 96 87 0.9044 0.8413 0.7076 80.7662 1.009 10.4533 1.1612 8 122 104 0.7509 0.8805 0.7755 107.4158 0.9903 17.1125 1.2698 9 123 119 0.9736 0.9633 0.928 118.4912 1.003 4.2347 1.272 10 130 100 0.787 0.7902 0.6244 102.7271 0.9997 21.7874 1.3444 11 119 88 0.8235 0.7383 0.5447 87.8609 1.009 16.3371 1.3446 12 125 91 0.5977 0.7114 0.5077 88.9207 0.9903 50.801 1.3933 13 104 104 0.9782 0.9711 0.943 100.9938 1.003 2.8698 1.4827 14 97 80 0.8183 0.8206 0.6734 79.5995 0.9997 14.3821 1.4848 15 99 67 0.8504 0.7776 0.6042 76.9785 1.009 13.1452 2.7782 16 132 111 0.6442 0.7911 0.6268 104.4282 0.9903 37.8028 3.1918 17 93 49 0.5274 0.5241 0.2743 48.7372 1.003 20.3948 3.1932 18 113 11 0.1006 0.0986 0.0097 11.1422 0.9997 10.0944 3.195 19 88 18 0.1238 0.1869 0.0346 16.4498 1.009 10.8223 3.3411 20 138 0 0.0431 0.0121 0.0001 1.6708 0.9903 1.7055 5.012

Table 9: Parameter estimates under the multiplicative model with variable dispersion parameter

Example III-Teratology

The data below is from an unpublished toxicological studies on pregnant mice, Kupper & Haseman (1978). The study is concerned with the effect of compounds on fetal death or the occurrence of some abnormalities in physiological development. Ten pregnant female mice in each of two groups (one group is the control and the other is the treated group) are employed in the study. The data is presented below for (y/n). y is the number of off springs dead in n litters.

control 10 0/5, 2/6, 0/7, 0/7, 0/8,

0/8, 0/8, 1/9, 2/9, 1/10

TRT 10 0/5, 2/5, 1/7, 0/8, 2/8

3/8, 0/9, 4/9, 1/10, 6/10.

If we let $\pi_{ij}$ ij denote the probability of death for fetus j in litter i. Then, we would model this probability for both models with the logit link, viz:

$\log it\left(\pi_{ij}\right)=\beta_{0}+\beta_{1}trt$.................................(11)

where (trt=1 if treatment group and 0, otherwise). Again here, we fit three competing models (17) viz:

1. The model that assumes $\pi_{1}=\pi_{1}$ with a common dispersion parameter $\phi$ or $\omega$ for the double binomial and the multiplicative models respectively, and, where $\pi_{0}$ and $\pi_{1}$refer respectively to the corresponding probabilities in the control and treatment groups. Here, $\log it\left(\pi_{ij}\right)=\beta_{0}$ , $\phi=a_{0}$ and $\omega=c_{0}$ with $a_{0}\neq c_{0}$
2. The model here has  $\pi_{0}=\pi_{2}$ and the dispersion parameters are functions of the covariate. That is, $\phi=a_{0}+a_{1}$ trt and $\omega=c_{0}+c_{1}trt$
3. The model here has$\pi_{0}=\pi_{1}$  with the$\pi$ S modeled as in (11) and the dispersion parameters are modeled as functions of the covariates as in preceding case.

The results of these models are presented in Table 10.

 Case I Case II Case III Parameters DBM MBM DBM MBM DBM MBM MLE Est. $\pi$= 0.1269 $\psi$= 0.3033 $\pi_{0}$= 0.0853 $\pi_{0}$= 0.3019 $\pi_{0}$= 0.0703 $\psi_{0}$= 0.0624 $\psi_{1}$= 0.2293 $\psi_{1}$= 0.3566 $\hat{\phi}$= 0.3648 $\omega$= 0.8314 $\phi_{0}$= 0.8035 $\omega_{0}$= 0.7590 $\omega_{0}$= 0.7180 $\omega_{0}$= 1.0412 $\phi_{1}$= 0.1772 $\omega_{1}$= 0.8964 $\omega_{1}$= 0.4445 $\omega_{1}$= 0.8514 -2LL 60.3121 63.5982 57.7621 59.4377 55.6644 57.1084 AIC 64.3121 67.5982 63.7621 65.4377 63.6644 65.1084 $X^{2}$ 28.6119 41.6813 22.8846 27.5236 22.366 28.159 $G^{2}$ 11.2724 39.9362 9.6845 32.2834 9.4538 30.5746 d.f 17 17 16 16 15 15

Table 10: Parameter estimates under the three cases for the two probability models.

### Appendix A. SAS Program for the Example

options nodate nonumber ls=85 ps=66;

data ex1;

do L=1 to 5;

do T=1 to 4;

input y n @@;

output;

end; end;

datalines;

89 94 94 98 77 86 141 155

106 108 91 106 87 96 104 122

119 123 100 130 88 119 91 125

104 104 80 97 67 99 111 132

49 93 11 113 18 88 0 138 ;

run;

proc print;

run;

/*generate indicator variables for Location*/; data w1;

set ex1;

array x(5) z1-z5;

do j=1 to 5;

if j=L then x(j)=1;

else x(j)=0;

end;

drop j;

run;

/*generate indicator variables for Time*/;

data w2;

set ex1;

array d(4) x1-x4;

do k=1 to 4;

if k=T then d(k)=1;

else d(k)=0;

end;

drop k;

run;

data new;

merge w1 w2;

run;

proc sort data=new;

by T;

run;

proc nlmixed data=new tech=newrap maxit=2000;

parms b0=-0.1 b1=1.1 b2=0.4 b3=.1 b4=0.2 s1-s3=0.0 a0=0 a1=0 a2=0 a3=0;

lp=b0+b1*z1+b2*z2+b3*z3+b4*z4+s1*x1+s2*x2+s3*x3;

lr=a0+a1*x1+a2*x2+a3*x3; omega=exp(lr);

p=1/(1+exp(-lp)); sum=0.0;

do j=0 to n;

z1=lgamma(n+1)-lgamma(j+1)-lgamma(n-j+1);

u=z1+ j*log(p) + (n-j)*log(1-p) + j*(n-j)*log(omega);

sum=sum+exp(u);

end;

keep sum;

z2=lgamma(n+1)-lgamma(y+1)-lgamma(n-y+1);

LL=z2+ y*log(p) + (n-y)*log(1-p) + y*(n-y)*log(omega)-log(sum);

model y~general(LL);

predict p out=aa;

predict omega out=bb;

run; Ods rtf close;

data q1;

set aa;

psi=pred;

run;

data q2;

set bb;

omega=pred;

run;

data qq4;

merge q1 q2;

suma=0;

do j=0 to n;

zz1=lgamma(n+1)-lgamma(j+1)-lgamma(n-j+1);

u1=zz1+ j*log(psi) + (n-j)*log(1-psi) + j*(n-j)*log(omega);

suma=suma+exp(u1);

end;

sumb=0;

do k=0 to n-1;

zz2=lgamma(n)-lgamma(k+1)-lgamma(n-k);

u2=zz2+ k*log(psi) + (n-k-1)*log(1-psi) + (k+1)*(n-k1)*log(omega);

sumb=sumb+exp(u2);

end;

sumc=0;

do t=0 to n-2;

zz3=lgamma(n-1)-lgamma(t+1)-lgamma(n-1-t);

u3=zz3+t*log(psi)+(n-t-2)*log(1-psi)+(t+2)*(n-t-2)*log(omega);

sumc=sumc+exp(u3);

end;

/* Generate p1,p2, expected values and variances*/;

p1=psi*(sumb/suma);

p2=(psi*psi)*(sumc/suma);

exp=n*p1;

var=n*p1+(n*(n-1)*p2)-(n*n*p1*p1);

/* Generate Wald, LRT and Pearson’s GOFs */;

wald+((y-exp)**2)/var;

if y=0 then lrt+0;

else lrt+2*y*log(y/exp);

XX+((y-exp)**2)/exp;

run;

proc print data=qq4;

var n y psi p1 p2 exp omega var XX LRT Wald;

format psi p1 p2 exp omega var xx LRT Wald 10.4;

run;

Results

Results from Table 10 show that for cases I to III, the model for case II is the most parsimonious. The difference between the Likelihood-test statistic, $G^{2}$ between models II and III being 0.2307 on 1 d.f (p-value=0.6310), which is not significant. We have used the $G^{2}$ rather than the Wald or Pearson’s $X^{2}$ because only the $G^{2}$ statistic has the partitioning property, (see, ). However, while this model seems the best, it does not tell us much about the probability of success ($\pi_{i}$ , i = 0, 1) for each group. The model assumes a common success probability for both groups. Our results further indicate that we probably do not need variable dispersion parameters for both probability models, that is, a common dispersion parameter would be adequate since neither the $\phi$or $\omega$ associated with the treatment groups are significant in model III. Thus, a reduced model of case III which models the probability of success separately for the treatments but assumes a common dispersion parameter. The models are based on 16 d.f. Here, under the double binomial model, the estimated probabilities of fetal deaths for the control and experimental groups are respectively 0.0552 and 0.2332 and these estimated probabilities are constant across the treatment levels. The corresponding goodness-of-fit values are $G^{2}$ = 9.1437 and $X^{2}$= 22.9671 with common dispersion parameter estimate being $\hat{\phi}$ = 0.4900. We notice a considerable discrepancy between the values of $G^{2}$ and $X^{2}$ for this data here. This is because, of the twenty observations in the data, nine of them have zeros for the values of Y. Consequently, these observations do not contribute to the overall $G^{2}$ and this accounts for the lower values of $G^{2}$ compared to their corresponding $X^{2}$ .

For the MBM, while the estimated $\psi$ are specific to each treatment and constant across each treatment, the estimated probabilities $\pi_{1}$ of successes vary by the number of litters n as outlined in expression (4). Thus for n = 8, $\pi_{1}$  equals 0.0769 and 0.2469 respectively for the control and treatment groups. We present in Table 11 the estimated probabilities and other variables under the multiplicative model for this case.

 # TRT n y $\psi$ $\pi_{1}$ $s^{2}$ $\sum G^{2}$ $\sum X^{2}$ $\sum$Wald 1 0 5 0 0.1565 0.1086 0.5467 0 0.543 0.5392 2 0 6 2 0.1565 0.0974 0.6076 4.9215 3.9724 3.8374 3 0 7 0 0.1565 0.0868 0.6485 4.9215 4.5799 4.4064 4 0 7 0 0.1565 0.0868 0.6485 4.9215 5.1873 4.9754 5 0 8 0 0.1565 0.0769 0.6702 4.9215 5.8023 5.5396 6 0 8 0 0.1565 0.0769 0.6702 4.9215 6.4172 6.1039 7 0 8 0 0.1565 0.0769 0.6702 4.9215 7.0321 6.6681 8 0 9 1 0.1565 0.0677 0.6743 5.9118 7.2823 6.8942 9 0 9 2 0.1565 0.0677 0.6743 10.6648 10.4546 9.7616 10 0 10 1 0.1565 0.0594 0.6637 11.7067 10.7322 10.0101 11 1 5 0 0.3409 0.2949 1.3354 11.7067 12.2067 11.6382 12 1 5 2 0.3409 0.2949 1.3354 12.926 12.394 11.845 13 1 7 1 0.3409 0.2643 1.9864 11.6956 12.7845 12.2088 14 1 8 0 0.3409 0.2469 2.3069 11.6956 14.7597 13.8999 15 1 8 2 0.3409 0.2469 2.3069 11.7456 14.76 13.9002 16 1 8 3 0.3409 0.2469 2.3069 14.2534 15.2918 14.3554 17 1 9 0 0.3409 0.2282 2.6004 14.2534 17.3455 15.9774 18 1 9 4 0.3409 0.2282 2.6004 19.5864 19.1899 17.4341 19 1 10 1 0.3409 0.2084 2.8434 18.1179 19.7537 17.8473 20 1 10 6 0.3409 0.2084 2.8434 30.8076 27.1123 23.2406

Table 11: Parameter estimates under the multiplicative model with constant dispersion parameter.

We may note here that for this data, we have also computed the Wald’s test Statistic and it seems to give the lowest value of 23.2406. The GOF values are cumulated so that the last values give the sums over all observations.

### Conclusion

Results presented in the preceding sections showed that while it is relatively easier to fit both the double binomial and the multiplicative binomial with joint sufficient statistics employing Poisson regression for frequency data, this approach cannot easily be implemented with data having co-variates. Further, the sufficient statistics approach is based on more degrees of freedom than the MgL method, which makes the MgL method more parsimonious in all cases. We would encourage the use of the MgL methods in applications of these models to binary count data. Of the two binomial models, the multiplicative binomial seems more consistent and fits much better than the double binomial. Further, it does not have much convergence problems than the DBM.

The SAS programs for implementing all the models discussed in this paper are readily available from the author. Meanwhile, we have attached a typical program in the appendix for implementing the MGM for the Manly data discussed in section 5.3.