Mutation Research, 128 (1984) 195-206 195 Elsevier

MTR 03908

A comparison of alternative distributions of postimplantation death in the dominant lethal assay

David M. Smith and Derek A. James * Departments of Scientific Computing and Statistics and Toxicology, The Wellcome Research Laboratories, Langley Court, Beckenham,

Kent BR3 3BS (Great Britain)

(Received 31 January 1984) (Revision received 4 May 1984)

(Accepted 10 May 1984)

Summary

Statistical analysis of dominant lethal assay data has been the subject of much discussion, a large part of which has been concerned with the distributional form displayed by postimplantation embryonic deaths. The purpose of this study was to compare the fit of the frequently suggested distributions and some others, and to comment on the effect of distribution upon interpretation of results, using a particular set of dominant lethal assay data. The data set used included experiments involving a known mutagen (cyclophosphamide). For these experiments the parameters of the best fitting distributions were reviewed to consider how a dominant lethal effect is displayed.

The dominant lethal assay is one of several tests for mutagenicity currently in use. It is an in vivo mammalian a s s a y considered pertinent to the fundamental question of genetic toxicology, i.e. is a chemical likely to produce genetic change that can be transferred to the offspring. The usual species used for this assay is the mouse. The experimental procedure is to administer the test compound to males of proven fertility, then mate them to one or more undosed females each week for a period of weeks sufficient to cover all stages of spermatogenesis. At about mid-gestation the females are killed and the number of implanta- tions and dead embryos recorded. The main effect looked for is a dose-related increase in the inci- dence of post implantation embryonic death. Haseman and Kupper (1979) review alternative analysis methods for this variable. One method

* Present address: Apoloco Ltd., 90 King Street, Newcastle- under-Lyme, Staffordshire ST5 1JB Great Britain.

considers the data have a specific distributional form. However, this approach has several unre- solved aspects, such as whether to take number or proportion of dead implants as the variable of analysis. Anderson et al. (1981) and Epstein et al. (1972) have looked at this question by considering the relationship between number of dead implants and number of implants. They concluded that number of deaths was better. The distributions considered here include some where the number of dead implants is the variable and others using the proportion of dead implants. Even when the choice of variable has been made it remains to be de- termined which is the best distribution to use. Various authors (Paul, 1982; Williams, 1975; Aeschbacher et al., 1977; Kupper and Haseman, 1978; Vuataz and Sotek, 1978) have made recom- mendations. Kupper and Haseman (1978) suggest use of a correlated binomial for proportion of dead implants. The others recommend either a beta binomial for proportion of dead implants or a

0027-5107/84/$03.00 © 1984 Elsevier Science Publishers B.V.

196

negative binomial for number of dead implants. These recommendations are generally based on comparisons against alternatives of binomial for proportion of deaths and Poisson for number of deaths.

The distributions considered here for propor- tion of deaths are the binomial, beta binomial, correlated binomial and a mixture of two binomi- als. For number of deaths the distributions consid- ered are Poisson, truncated Poisson and negative binomial. To our knowledge comparisons of beta binomial, correlated binomial and mixture of two binomials for proportion of deaths, and truncated Poisson against negative binomial for number of deaths have not been reported before.

For all experiments in the data set, 15 males per group and a mating ratio of 2 were used. Since it is controversial whether the female or the male should be the unit of analysis when a mating ratio greater than 1 is involved, distributions were fitted doing both. When analysing by male, only males who made both females pregnant were used.

The distributions were fitted to the individual treatment g roup /week results and their compara- tive fit assessed. Because of the small numbers of results available (a maximum of 30 taking females as the unit, or 15 taking the males as the unit) the usual goodness of fit tests such as chi-square or Kolmogorov-Smirnov could not be performed. Therefore goodness of fit was assessed by comparison of statistics originating from statistical information theory involving likelihood and the number of parameters of the distribution.

Materials and methods

Data set used Results from 17 dominant lethal assays were

used. Sixteen of these were performed as part of an Association of the British Pharmaceutical In- dustry (ABPI) collaborative study. An analysis of the data from this study has been reported by James and Smith (1982). All 17 assays were per- formed according to a standard protocol using the same or similar strains of mice with 3 treatment groups (control, low dose and high dose), 15 dosed males per group, a mating ratio of 2 and a time period of 8 weeks. Four of these assays used cyclophosphamide, a known mutagen. Four laboratories were involved.

Distributions fitted The 7 distributions were fitted to the individual

t rea tment /week results, taking separately the male and the female as the unit of analysis. Every experiment contributed 24 results (3 treatment groups × 8 weeks). Details of the distributions fitted and the methods of fitting follow. All itera- tive maximum likelihood procedures had the same convergence criterion.

Binomial distribution This is a commonly used distribution and de-

tails of it can be found in any basic statistics textbook. It assumes a constant probability of death p (the parameter of the distribution) for all animals.

Beta binomial distribution This is a 2-parameter distribution which ex-

tends the binomial by allowing the probability of death to vary from animal to animal. Relevant references are Kleinman (1973), Williams (1975), Aeschbacher et al. (1977) and Vuataz and Sotek (1978). The two parameters of the distribution are /z and 0 which can be considered a measure of average probability of death and a measure of spread of probability of death across animals re- spectively. The maximum likelihood estimation procedure used is detailed in Smith (1983).

Correlated binomial distribution Both the binomial and beta binomial distribu-

tions make the assumption that individual im- plants within a litter are independent. The corre- lated binomial extends the binomial by assuming that the results for the individual implants within a litter are correlated. This distribution has been proposed by Kupper and Haseman (1978) as an alternative to the beta binomial. This distribution has 2 parameters/x and 0 where again /~ can be considered a measure of average probability of death and 0 as the covariance between implants within the same litter. These parameters were estimated by iterative maximum likelihood estima- tion with initial estimates of # and 0 the binomial parameter p and 0 respectively. It assumes con- stant parameters across animals.

Mixture of two binomials This is a 3-parameter distribution which hy-

pothesizes that the results arise from mixing two binomial distributions. This distribution was em- ployed because a survey of the available data highlighted a number of cases where multi-modal- ity existed. The binomial, beta binomial and corre- lated binomial are uni-modal distributions and it was felt that a simple distribution capable of hav- ing more than one mode was worth consideration. The 3 'parameters involved are pl, p2 and h where pl and p2 are the probabilities of death for the two distributions and h is the mixing ratio. Maximum likelihood estimation of mixture model parameters' is not straightforward and so a method known as the E-M algorithm (which converges to a maxi- mum likelihood solution) was used. Details of this method are given in Aitkin and Tunnicliffe Wilson (1980). Initial moment estimates of the parameters were obtained by the methods given in Blischke (1962, 1964).

Poisson distribution All the previous distributions described relate

to the proportion of deaths. Some authors (Ander- son et al., 1981; Epstein et al., 1972; Salsburg, 1973) have proposed the number of deaths as a more meaningful variable for analysis. This is a commonly used distribution and details of it can be found in any basic statistics textbook. It as- sumed a constant death rate ~, (the parameter of the distribution).

Truncated Poisson distribution This distribution is a Poisson distribution with

the number of dead implants for an animal re- stricted to being less than or equal to the number of implants. This distribution was employed be- cause it is a simple way of involving number of implants when analysing number of deaths. The one parameter h (the death rate) was obtained by the maximum likelihood estimation procedure given in Cohen (1954).

Negative binomial distribution This is a 2-parameter distribution whose rela-

tionship to the Poisson is like that of the beta binomial to the binomial in that it extends the Poisson by allowing the death rate to vary from animal to animal. McCaughran and Arnold (1976) have recommended its use. The iterative maximum

197

likelihood estimation procedure used is detailed in Bliss (1953) and Fisher (1953). The two parame- ters of the distribution are M and K, where M is a measure of mean death and K is the exponent involved in the distribution. It is better to use 1 /K because it can be considered a measure of spread of results and because it has advantages when iterating. A value for 1 /K of zero is equivalent to fitting a Poisson distribution. It is possible for 1 /K to take the value infinity ( K equal to zero). This is equivalent to fitting a logarithmic series distribution which assigns a zero probability to all units of value zero. This distribution assumes that death rate varies across animals.

Other possible distributions Two other distributions which extend the bi-

nomial are described by Altham (1978). They were not considered because their fit closely matches that of the beta binomial. Other extensions of the Poisson distribution of possible interest are de- scribed in Hinde (1982), and Katti and Gurland (1961, 1962).

Statistic of goodness of fit Two information-theory-based statistics of

goodness of fit were considered. Both use the value of the log likelihood and a function of the number of parameters estimated. The first measure known as AIC (Akaike's information criterion) is:

- log(likelihood) + (number of parameters)

An alternative measure known as SIC (Schwarz's information criterion), is:

- log(likelihood) + 0.5 x log N x (number of parameters)

where N is the number of observations. Papers relevant to these two measures are Akaike (1973), Atkinson (1980), Schwarz (1978) and Stone (1979). For both measures, when comparing distributions, that giving the lower value is considered the better fit. For each experiment the t rea tment /week re- sults were summed.

This summation implicitly assumes indepen- dence of results across weeks. Because the same males are used this assumption is unlikely to be

198

T A B L E 1

R E S U L T S ( p values) O B T A I N E D F R O M A N A L Y S E S B A S E D O N D I F F E R E N T D I S T R I B U T I O N S

Uni t o f analysis females .

D r u g / E x p t . No. Dis t r i - W e e k b

bu t ion a 1 2 3 4 5 6 7 8

C y c l o p h o s p h a m i d e

1

E thano l

5

G luco se

8

N B < 0.001 * < 0.001 * 0.260 0.139 0.491 0.891

T P < 0.001 * < 0.00l * 0.111 0.115 0.221 0.695

MB < 0.001 * < 0.001 * 0.576 0.816 0.705 0.804

CB < 0.001 * < 0.001 * 0.213 0.147 0.530 0.779

BB < 0.001 * < 0.001 * 0.235 0.051 0.409 0.776

N B <: 0.001 * < 0.001 * < 0.001 * 0.637 0.970 0.331

T P < 0.001 * < 0.001 * < 0.001 * 0.933 0.563 0.415

M B < 0.001 * < 0.001 * < 0.001 * 0.800 0.583 0.172

CB < 0.001 * < 0.001 * < 0.001 * 0.835 0.996 0.234

BB < 0.001 * < 0.001 * < 0.001 * 0.699 0.875 0.261

N B < 0.001 * < 0.00l * 0.257 0.464 0.616 0.903

T P < 0.001 * < 0.001 * 0.043 * 0.504 0.253 0.781

MB < 0.001 * < 0.001 * 0.002 * 0.476 0.114 0.406

CB < 0.001 * < 0.001 * 0.105 0.180 0.475 0.787

BB < 0.001 * < 0.001 * 0.065 0.294 0.301 0.777

N B < 0.001 * < 0.001 * 0.994 0.638 0.111 0.850

T P < 0.001 * < 0.001 * 0.620 0.430 0.003 * 0.281

M B < 0.001 * < 0.001 * < 0.001 * 0.978 0.044 * 0.722

CB < 0.001 * < 0.001 * 0.585 0.879 0.009 * 0.256

BB < 0.001 * < 0.001 * 0.952 0.818 0.034 * 0.559

N B 0.487 0.026 * 0.045 * < 0.001 * 0.030 * 0.702

TB 0.067 0.006 * 0.008 * 0.345 < 0.001 * 0.339

M B 0.059 0.083 0.056 - 0.013 * 0.527

CB 0.070 < 0.001 * 0.044 * 0.687 < 0.1301 * 0.506

BB 0.116 0.005 * 0.041 * 0.824 0.002 * 0.659

N B 0.404 0.066 0.394 0.691 0.598 0.953

T P 0.595 0.154 0.281 0.267 0.124 0.601

M B 0.603 0.314 0.447 0.069 0.287 0.958

CB 0.197 0.090 0.154 0.219 0.073 0.727

BB 0.516 0.021 * 0.438 0.688 0.362 0.849

N B 0.465 0.513 0.091 0.942 0.788 0.062

T P 0.068 0.512 0.005 * 0.425 0.603 < 0.001 *

M B 0.814 0.324 0.122 0.895 0.464 0.097

CB 0.312 0.922 0.014 * 0.877 0.129 0.035 *

BB 0.543 0.532 0.058 0.866 0.880 0.204

N B 0.428 0.680 0.042 * 0.862 0.420 0.351

T P 0.118 0.394 0.008 * 0.556 0.962 0.175

M B 0.700 0.914 0.063 0.531 0.500 0.296

CB 0.522 0.292 0.010 * 0.448 0.254 0.188

BB 0.468 0.348 0.019 * 0.751 0.274 0.262

N B 0.593 0.419 0.671 0.021 * 0.712 0.783

T P 0.355 0.124 0.591 0.089 0.613 0.382

M B 0.790 0.624 0.904 0.063 0.996 0.805

CB 0.422 0.393 0.927 0.026 * 0.908 0.694

BB 0.475 0.451 0.837 0.007 * 0.858 0.724

0.880

0.604

0.950

0.791

0.816

0.230

0.001 *

0.307

0.044 *

0.283

0.203

0.132

0.404

0.117

0.157

0.316

0.050

0.024 *

0.242

0.310

0.873

0.968

0.748

0.945

0.952

0.495

0.186

0.099

0.085

0.390

0.372

0.006 *

0.202

0.063

0.497

0.405

0.652

0.197

0.267

0.192

0.164

0.909

0.406

0.293

0.230

0.222

0.225

0.253

0.180

0.747

0.619

0.080

0.070

0.285

0.276

0.211

0.279

0.113

0.193

0.812

0.148

0.893

0.600

0.831

0.793

0.955

0.977

0.850

0.723

0.153

0.306

0.669

0.146

0.108

0.388

0.009 *

0.042 *

0.011 *

0.802

0.174

0.118

0.471

0.013 *

0.340

0.702

0.590

0.825

0.201

0.742

T A B L E 1 (cont inued)

199

D r u g / E x p t , No. Distri- Week b

but ion a 1 2 3 4 5 6 7 8

Glucose

10

11

Penicillin G 12

13

14

5-Fluorouraci l

15

16

C o m p o u n d X 17

NB 0.416 0.258 0.551 0.662 0.193 0.738 0.189 0.042 *

T P 0.089 0.192 0.552 0.986 0.342 0.172 0.129 0.009 *

MB 0.529 0.243 0.608 0.981 - 0.011 * 0.298 0.332

CB 0.306 0.023 * 0.332 0.600 0.067 0.247 0.245 0.019 *

BB 0.345 0.098 0.286 0.479 0.112 0.286 0.118 0.034 "

N B 0.558 0.043 * < 0.001 * 0.042 * 0.049 * 0.395 0.244 0.361

T P 0.324 0.011 * 0.011 * 0.007 * 0.001 * 0.019 * 0.537 0.016 *

MB 0.814 0.010 * 0.041 * 0.087 0.112 0.521 0.792 0.271

CB 0.802 0.369 0.012 * 0.043 * 0.023 * 0.549 0.126 0.116

BB 0.724 0.125 0.126 0.021 * 0.064 0.417 0.307 0.322

NB 0.513 0.728 0.232 0.950 0.911 0.063 < 0.001 * < 0.001 *

TP 0.409 0.357 0.412 0.684 0.634 0.007 * 0.030 * 0.521

MB 0.776 0.530 0.664 0.978 0.375 0.425 -

CB 0.442 0.257 0.427 0.742 0.896 0.040 * 0.014 * 0.013 *

BB 0.622 0.489 0.200 0.924 0.840 0.045 * 0.059 0.169

NB 0.236 0.817 0.913 0.889 0.023 * 0.537 0.492 0.670

TP 0.234 0.686 0.212 0.799 0.008 * 0.228 0.388 0.866

MB 0.841 0.926 0.999 0.958 0.063 0.470 0.569 0.615

CB 0.112 0.839 0.992 0.302 0.005 * 0.311 0.165 0.858

BB 0.263 0.932 0.862 0.716 0.007 * 0.463 0.149 0.603

N B 0.156 0.268 0.001 * 0.088 0.724 0.065 0.026 * 0.014 *

T P 0.009 * 0.112 < 0.001 * 0.005 * 0.094 < 0.001 * 0.004 * < 0.001 *

MB 0.115 0.486 0.026 * 0.247 0.965 0.259 0.062 0.019 *

CB 0.506 0.063 < 0.001 * 0.018 * 0.091 0.054 0.023 * < 0.001 *

BB 0.160 0.288 0.005 * 0.072 0.360 0.074 0.035 * 0.006 *

N B 0.037 * 0.197 0.540 0.067 0.007 * 0.855 0.637 0.959

T P 0.007 * 0.200 0.188 0.103 < 0.001 * 0.559 0.473 0.752

MB 0.117 0.290 0.406 - 0.119 0.618 0.128 0.997

CB 0.013 * 0.041 * 0.328 0.035 * < 0.001 * 0.833 0.405 0.827

BB 0.025 * 0.091 0.574 0.071 0.002 * 0.833 0.726 0.968

NB 0.726 0.753 < 0.001 * 0.048 * 0.810 0.915 0.032 * 0.249

T P 0.251 0.662 0.004 * 0.038 * 0.375 0.465 0.007 * 0.004 *

MB 0.589 0.833 0.893 0.039 * 0.792 0.008 * 0.439 -

CB 0.149 0.451 0.073 0.008 * 0.590 0.992 < 0.001 * 0.007 *

BB 0.446 0.744 0.207 0.015 * 0.569 0.896 0.033 * 0.208

NB 0.307 0.460 0.064 0.330 0.494 0.174 0.048 * 0.149

T P 0.370 0.908 < 0.001 * 0.431 0.289 0.033 * 0.010 * 0.471

MB 0.481 0.807 0.026 * 0.179 0.745 0.242 0.481 0.535

CB 0.296 0.900 < 0.001 * 0.174 0.353 0.035 * 0.014 * 0.128

BB 0.042 * 0.538 0.003 * 0.083 0.499 0.052 0.011 * 0.012 *

a NB, l ikelihood rat io test based on negat ive binomial ; TP, l ikelihood rat io test based on t runca ted Poisson; MB, l ikelihood rat io test

based on mix tu re of two binomials ; CB, likelihood rat io test based on corre la ted b inomial ; BB, l ikelihood rat io test based on be ta b inomial .

b . indicates a statistically significant result ( p < 0.05); - indicates test cannot be p e r fo rmed (see Discussion).

2OO

T A B L E 2

R E S U L T S ( p values) O B T A I N E D F R O M A N A L Y S E S B A S E D O N D I F F E R E N T D I S T R I B U T I O N S

Uni t of analysis males.

D r u g / E x p t . No . Distri- Week b

but ion a 1 2 3 4 5 6 7 8

Cyc lophosphamide

1

Ethanol

5

Glucose

8

NB 0.002 *

T P < 0.001 *

MB < 0.001 *

CB < 0.001 *

BB < 0.001 *

NB 0.002 *

TP < 0.001 *

MB < 0.001 *

CB < 0.001 *

BB < 0.001 *

N B

T P < 0.001 *

MB < 0.001 *

CB < 0.001 *

BB < 0.001 *

NB < 0.001 *

T P < 0.001 *

MB < 0.001 *

CB < 0.001 *

BB < 0.001 *

0.038 * 0.551 0.193 0.411 0.632 0.488 0.599

< 0.001 * 0.228 0.279 0.021 * 0.362 0.212 0.151

< 0.001 * 0.626 0.702 0.768 0.918 0.636 0.294

< 0.001 * 0.621 0.001 * 0.110 0.435 0.217 0.294

0.006 * 0.631 0.144 0.288 0.409 0.347 0.693

- 0.001 * 0.466 0.983 0.369 0.702 0.637

< 0.001 * < 0.001 * 0.745 0.972 0.716 0.195 0.541

< 0.001 * < 0.001 * 0.443 0.249 0.226 0.217 0.490

< 0.001 * < 0.001 * - 0.846 0.290 0.950 0.047 *

< 0.001 * < 0.001 * 0.436 0.924 0.203 0.616 0.388

< 0.001 * 0.522 0.426 0.950 0.610 0.306 0.415

< 0.001 * 0.053 0.437 0.611 0.314 0.117 0.226

< 0.001 * 0.126 0.401 0.889 0.150 0.640 0.297

< 0.001 * 0.155 0.031 * 0.312 0.118 0.164 0.296

< 0.001 * 0.232 0.180 0.919 0.405 0.206 0.256

< 0.001 * 0.714 0.680 < 0.001 * 0.436 0.423 0.730

< 0.001 * 0.305 0.478 < 0.001 * 0.418 0.088 0.150

< 0.001 * 0.020 * 0.921 0.123 0.689 0.072 0.568

< 0.001 * 0.570 0.710 0.014 * 0.371 0.123 0.336

< 0.0131 * 0.397 0.747 0.015 * 0.441 0.390 0.782

NB 0.380 0.030 * 0.026 * 0.758 0.106 0.263 < 0.001

T P 0.102 0.002 * 0.007 * 0.683 < 0.001 * 0.195 0.464

MB 0.158 0.348 0.426 0.981 0.035 * 0.717 0.085

CB 0.182 < 0.001 * 0.001 * 0.302 0.002 * 0.323 0.386

BB 0.154 0.011 * 0.008 * 0.642 0.062 0.263 0.846

NB 0.224 0.602 0.647 0.552 0.323 0.964 0.194

T P 0.103 0.161 0.303 0.485 0.102 0.617 0.247

MB 0.274 0.332 0.560 0.002 * 0.228 0.659 0.669

CB 0.056 0.211 0.439 0.236 0.005 * 0.624 0.210

BB 0.218 0.295 0.797 0.532 0.290 0.866 0.201

N B 0.434 0.854 0.057 0.969 0.624 0.157 0.365

T P 0.029 * 0.673 0.006 * 0.577 0.483 0.004 * 0.006

MB 0.899 0.461 0.132 0.946 0.298 0.496 0.219

CB 0.038 0.379 0.008 * 0.713 0.210 0.017 * -

BB 0.352 0.684 0.091 0.936 0.859 0.233 0.470

N B 0.041 * 0.688 0.054 0.854 0.868 0.156 0.189

T P 0.012 * 0.436 0.007 * 0.619 0.926 0.172 0.246

MB 0.830 0.719 0.314 - 0.341 0.068 0.367

CB 0.032 * 0.287 0.025 * 0.519 0.355 0.149 0.017 *

BB 0.061 0.439 0.045 * 0.779 0.690 0.119 0.094

N B 0.527 0.119 0.639 0.176 0.733 0.666 0.426

T P 0.242 0.118 0.371 0.229 0.576 0.304 0.128

MB 0.632 0.307 - 0.138 0.642 0.926 0.989

CB 0.401 0.128 - 0.118 0.800 0.554 0.616

BB 0.498 0.137 0.602 0.120 0.534 0.546 0.675

* < 0.001 *

0.888

0.250

0.546

0.623

0.260

0.499

0.626

0.172

0.188

0.284

* 0.014 *

0.550

0.078

0.621

0.076

0.077

0.325

0.183

0.088

0.816

0.554

0.999

0.952

0.951

201

TABLE 2 (continued)

Drug/Expt. No. Distri- Week b

bution a 1 2 3 4 5 6 7 8

11

Penicillin G

12

13

14 NB TP MB

CB BB

5-Fluorouracil 15

Glucose

10 NB 0.560 0.077 0.877 0.274

TP 0.173 0.037 * 0.548 0.915

NB 0.591 0.286 0.970 0.136

CB 0.452 0.008 * 0.720 0.057

BB 0.485 0.050 0.774 0.279

0.185 0.375

0.050 0.069

0.422 0.302 0.073

0.181 0.121 0.008 *

0.869 0.591 0.063

0.140 0.002 * 0.056

0.316 0.300 0.049 *

NB 0.509 0.021 * 0.171 0.172 0.154 0.225 0.515 0.319

TP 0.518 0.031 * 0.008 * 0.046 * 0.005 * 0.074 0.291 0.015 *

MB 0.378 0.001* 0.430 0.968 0.203 0.182 0.666 0.224

CB 0.339 0.320 0.283 0.154 0.094 _ 0.110 0.039 +

BB 0.624 0.110 0.237 0.224 0.252 0.228 0.326 0.232

NB 0.301 0.263 0.622 0.825 0.997

TP 0.274 0.020 * 0.394 0.507 0.918

MB 0.437 0.750 0.707 0.753 0.986

CB 0.402 0.053 0.364 0.479 0.946

BB 0.465 0.130 0.678 0.770 0.946

NB 0.149 0.646 0.927 0.818 0.041*

TP 0.150 0.539 0.368 0.912 0.017 *

MB 0.986 0.733 0.826 0.141 0.014 * CB 0.120 0.652 0.782 0.624 0.024 * BB 0.155 0.635 0.908 0.655 0.037 *

0.149 0.007 *

0.378 0.185

0.436 0.004 * 0.149 0.200 0.117 c 0.001* 0.005 * 0.234 0.011 * 0.038 * 0.571 0.101

0.026 * 0.003 * 0.104 0.035 * 0.388 0.022 * 0.140 0.058

0.118 < 0.001 * < 0.001 * 0.004 * 0.025 * 0.448 0.419 0.412 1.000

0.129 0.016 * 0.363 0.148 0.393 0.653

0.497 0.518 0.712 0.273 0.303 0.775

0.688 - 0.879 0.019 * 0.407 0.907 0.510 0.590 0.824

0.112 0.028 * 0.054 < 0.001 * < 0.001* < 0.001*

0.047 * 0.351 0.002 *

0.002 * 0.010 * 0.006 * 0.091 0.031 * 0.015 *

NB 0.040 * 0.108 0.298 0.188 0.014 * 0.719 0.466 0.954 TP 0.006 * 0.166 0.125 0.056 0.001* 0.446 0.209 0.606 MB 0.218 0.245 0.502 0.645 0.052 0.977 0.055 0.902 CB 0.018 * 0.055 0.009 * 0.132 -z : 0.001* 0.620 0.163 0.737 BB 0.024 * 0.056 0.283 0.199 0.005 * 0.655 0.476 0.942

16 NB 0.600 0.698 0.255 0.770 0.690 0.807 0.118 TP 0.187 0.452 0.007 * 0.341 0.340 0.509 0.008 * MB 0.667 0.964 0.293 0.265 0.723 0.817 0.218 CB 0.026 * 0.976 0.020 * 0.643 0.001* 0.802 0.046 * BB 0.437 0.743 0.341 0.477 0.443 0.633 0.141

Compound X 17 NB 0.894 0.479 0.080

TP 0.663 0.959 < 0.001* MB 0.605 0.975 0.045 * CB 0.939 0.241 -= 0.001* BB 0.947 0.516 0.016 *

0.572 0.965 0.112 0.143 0.059 0.815 0.769 0.008 l 0.013 * 0.663 0.409 0.723 0.379 0.631 0.676 0.305 0.873 0.005 * 0.017 l 0.104 0.327 0.887 0.046 * 0.202 0.015 *

’ NB likelihood ratio test based on negative binomial; TP likelihood ratio test based on truncated Poisson; MB likelihood ratio test based on mixture of two binomials; CB likelihood ratio test based on correlated binomial; BB likelihood ratio test based on beta binomial.

b * indicates statistically significant result (p -z 0.05); - indicates test cannot be performed (see Discussion).

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exactly true. However, Aeschbacher et al. (1977) state that for their data females mated with the same male behaved as if independent. Ryt tman (1976) suggests a multivariate approach to analysis but a test for complete independence (Morrison, 1967) performed on the covariance matrix given did not reject that hypothesis. Therefore assump- tion of independence was not felt to be unreasona- ble.

Tables of AIC and SIC values are not given. Copies can be obtained from the authors.

Results

Assessment of AIC and SIC values showed the fits of the beta binomial and negative binomial generally superior to the alternatives for their re- spective variables. For proportion of deaths, females the unit of analysis, the beta binomial always fitted better than the correlated binomial, but for 2 experiments the mixture of two binomi- als fitted better than the beta binomial. However, in these experiments the differences in AIC were small. For the same variable, males the unit of analysis, for 4 experiments the correlated binomial fitted better than the beta binomial (with the differences in AIC again small), but the mixture of two binomials was always a worse fit. For 2 ex- periments with females the unit of analysis and 3 experiments with males the unit of analysis the truncated Poisson fitted better than the negative binomial to number of deaths.

The results from the majority (16 out of 17) of the experiments of this data set have been reported by James and Smith (1982), who concluded that of the compounds used only cyclophosphamide produced a dominant lethal effect. This effect occurred in the first 3 weeks of dosing only. The compound used in the other experiment included here was considered to be a non-mutagen.

Tables 1 and 2 show the p values obtained from an analysis of the data set on a per compound, per experiment, per week basis. These analyses consisted of likelihood ratio tests of the equality of distributions, i.e. twice the difference in log likeli- hood between fitting separate distributions to each treatment group and one distribution to the combined groups was compared to a chi-square distribution with appropriate degrees of freedom.

T A B L E 3

P E R C E N T A G E O F F A L S E P O S I T I V E R E S U L T S (out o f

124) F O R T H E D I F F E R E N T D I S T R I B U T I O N S A N D

U N I T S O F A N A L Y S I S

D i s t r i bu t i on U n i t o f ana lys i s

Fema le s Ma les

False N o t Fa lse N o t

pos i t ives ca l cu lab le posi t ives Ca lcu lab le (~) (~) (~) (~)

N B 18.5 - 11.3 -

T P 28.2 - 29.8 -

M B 9.7 7.5 7.3 4.0

C B 24.2 - 28.2 2.4

BB 17.7 11.3

Defining a false positive result as a statistically significant result ( p < 0.05) not occurring within the first 3 weeks of cyclophosphamide Table 3 gives the percentage of false positive results for the different distributions and units of analysis. With females the unit of analysis, the mixture of two binomials has the lowest percentage (note the per- centage of occasions on which a test statistic was not calculable), followed by the beta and negative binomials (approximately equal), correlated bi- nornial and truncated Poisson. For males the unit of analysis, the order is: mixture of two binomials, negative and beta binomials equal, correlated bi- nomial and truncated Poisson. There are two differences between the percentages for the differ- ent units of analysis. Firstly, with males the unit, using a correlated binomial distribution, some- times a test statistic was not calculable. Secondly, for the negative and beta binomial distributions the percentages were much lower with males the unit of analysis.

Tables 4 and 5 give the mean (and S.E.) of the parameters of the fitted negative and beta bi- nomial distributions for the different doses, weeks and units of analysis averaged over the 4 cyclophosphamide experiments. The results in Ta- bles 4 and 5 show how the dominant lethal effect is manifested.

Discussion

The beta binomial distribution is flexible and able to handle a range of distributional shapes.

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TABLE 4

MEAN (and S.E) OF THE PARAMETERS OF THE NEGATIVE AND BETA BINOMIAL DISTRIBUTIONS FROM 4 Expts. WITH CYCLOPHOSPHAMIDE

Unit of analysis females

Dose Negative binomial

(mg/kg) Week

1 2 3 4 5 6 7 8

M Control 1.20(0.29) 1.25(0.51) 1.38(0,57) 1.21(0.31) 1.16(0.34) 1.32(0.30) 1.22(0.39) 20 3.17(1.18) 3.14(0.97) 1.82(0,50) 1.20(0.22) 1.60(0.57) 1.57(0.44) 1.17(0.39) 80 5.89(0.32) 5.40(1.32) 2.51(1,33) 1.25(0.41) 1.28(0.54) 1.35(0.34) 1.53(0.80)

1/K Control 1.08(1.18) 1.16(0.59) 0.41(0.30) 0.52(0.71) 0.54(0.45) 0.34(0.29) 0.54(0.56) 0,69(0.13) 20 0,06(0.06) 0.12(0.16) 0.47(0.32) 0.44(0.35) 0.77(0.34) 0.60(0.49) 0.36(0.38) 0,41(0.40) 80 0,05(0.04) 0.12(0.13) 0.32(0.32) 0.50(0.41) 0.58(0.37) 0.74(0.72) 0.51(0.40) 0.34(0.43)

1.27(0.33) 1.37(0.50) 1.35(0.74)

Beta binomial

Control 0.11(0.02) 0.10(0.03) 0.11(0.04) 0.09(0.02) 0.09(0.03) 0.11(0.03) 0.09(0.03) 0.10(0.02) 20 0.27(0.10) 0.28(0.07) 0.15(0.04) 0.10(0.03) 0.14(0.04) 0.13(0.03) 0.10(0.03) 0.10(0.03) 80 0.71(0.03) 0.62(0.16) 0.25(0.16) 0.10(0.03) 0.11(0.04) 0.12(0.04) 0.12(0.06) 0.11(0.05)

Control 0.15(0.17) 0.17(0.09) 0.08(0.08) 0.08(0.08) 0.08(0.08) 0.07(0.07) 0.07(0.07) 0.11(0.04) 20 0.06(0.04) 0.06(0.06) 0.13(0.10) 0.10(0.11) 0.16(0.06) 0.14(0.10) 0.05(0.04) 0.06(0.06) 80 0.17(0.20) 0.10(0.09) 0.14(0.12) 0.07(0.05) 0.09(0.09) 0.10(0.10) 0.08(0.08) 0.07(0.09)

TABLE 5

MEAN (and S.E.) OF THE PARAMETERS OF THE NEGATIVE AND BETA BINOMIAL DISTRIBUTIONS FROM 4 Expts. WITH CYCLOPHOSPHAMIDE

Unit of analysis males.

Dose Negative binomial

(mg/kg) Week

1 2 3 4 5 6 7 8

M Control 2.26(1.24) 2.68(0.90) 2.87(1.22) 2.47(0.63) 2.24(0.56) 2.78(0.77) 2.38(0.57) 20 6.35(2.62) 5.87(1.75) 3.84(1.42) 2.35(0.46) 3.39(1.26) 3.09(0.72) 2.46(0.66) 80 11.98(0.82) 11.30(2.36) 5.06(2.90) 2.63(0.73) 2.58(1.46) 2.80(0.83) 2.83(1.48)

1/K Control 0.22(0.32) 0.84(0.63) 0.14(0.11) 0.14(0,18) 0.44(0.38) 0.10(0.13) 0.29(0.36) 20 0.05(0.06) 0.00(0.01) 0.17(0.17) 0.44(0,46) 0.41(0.26) 0.30(0.25) 0.10(0.12) 80 0.05(0.06) 0.03(0.07) 0.15(0.06) 0.16(0,23) 0.31(0.25) 0.74(0.86) 0.24(0.18)

2.57(0.74) 2.80(0.97) 2.94(1.69)

0.40(0.17) 0.17(0.21) 0.11(0.08)

Beta binomial

Control 0.10(0.05) 0.10(0,04) 0.11(0.04) 0.09(0.02) 0.09(0.02) 0.11(0.03) 0.09(0.02) 0.10(0.03) 20 0.28(0.11) 0.26(0,07) 0.15(0.04) 0.09(0.02) 0.14(0.05) 0.13(0.03) 0.10(0.02) 0.10(0.03) 80 0.70(0.06) 0.64(0,21) 0.26(0.17) 0.11(0.03) 0.11(0.05) 0.11(0.04) 0.12(0.06) 0.12(0.06)

Control 0.05(0.05) 0.13(0,13) 0.03(0.04) 0.02(0.02) 0.05(0.06) 0.02(0.03) 0.03(0.03) 0.06(0.03) 20 0.07(0.04) 0.01(0,01) 0.03(0.03) 0.07(0.07) 0.07(0.04) 0.06(0.04) 0.02(0.02) 0.03(0.03) 80 0.05(0.09) 0.05(0.06) 0.07(0.07) 0.03(0.04) 0.07(0.09) 0.11(0.16) 0.04(0.03) 0.02(0.02)

H o w e v e r , a l l the s h a p e s have o n l y one m o d e . O n a n u m b e r of o c c a s i o n s in th is d a t a set a m u l t i - m o d a l f o r m was d i s p l a y e d . In such c i r c u m s t a n c e s the

b e t a b i n o m i a l f i t t ed the d a t a r e l a t ive ly p o o r l y a n d the m i x t u r e o f b i n o m i a l s p r o v i d e d a m u c h b e t t e r fit. Th i s is n o t u n e x p e c t e d b e c a u s e the m i x t u r e of

204

binomials is capable of handling multi-modality. The multi-modality was usually caused by isolated proportions distant from the main body of data (outliers). An improvement in fit could be achieved if an alternative to the beta distribution for proba- bility of death across animals could be found with similar flexibility but able to produce multi-modal shapes.

The negative binomial distribution uses only the number of dead implants and somehow incor- porating the total number of implants could bring about an improvement in fit. One possible way is to use the number of implants as a truncation value. This has the advantage of not involving any extra parameters. However, estimation of the parameters of a truncated negative binomial with variable non-zero truncation values is difficult, which is why a truncated Poisson (with truncation at number of implants) was included. However, this distribution only fitted marginally better than the Poisson, suggesting that use of a truncated negative binomial might not be worthwhile. In- cluding the number of implants in some simple functional form involving only one or two parame- ters, might be more productive.

Tables 1 and 2 show good general agreement between the p values obtained by analyses based on the different distributions but there are a num- ber of instances of important differences reflected in Table 3. The false positive percentages are all greater than the nominal value of 5%, suggesting a general problem with a distributional model analy- sis, although the sample size (124) is small. These inflated percentages appear to have 3 main causes, i.e. isolated outlying values (outliers), the inaccu- racy of the chi-square approximation to the distri- bution of the likelihood ratio statistic, and estima- tion problems with certain distributions. Where the negative-binomial-based analysis displays an exceptionally small p value it appears that one of the fits has produced a value for 1/K of infinity i.e. a logarithmic series distribution has resulted. When this occurs in a separate group fit a large chi-square value and hence a small p value results (e.g Table 1: Expt. 16, week 3). On two occasions in Table 2 (Expt. 2, week 2 and Expt. 3, week 1) the test was not performable because only one or two observations were available. The exceptionally small p values in the truncated Poisson analysis

are caused by the inability of this distribution to adequately represent the spread of number of deaths. For both the mixture of two binomials and the correlated binomial on occasions a test statistic was not calculable because the combined group fit was better than the summed individual groups. Almost always this was caused by one or more of the parameters reaching a boundary value. Boundary value problems also appeared to be the cause of many of the exceptionally small p values for these distributions. Again for both distribu- tions, at times it appeared that the estimation procedures suffered from problems of false max- ima. Where the analysis based on the beta bi- nomial shows an exceptionally small p value the cause is usually odd outlying proportions distant from the main body of the data. Such outlying proportions appreciably affect the parameter estimation and hence the likelihood ratio test, a point made by Kupper and Haseman (1978), and James and Smith (1982).

Although the false positive percentages are im- portant the false negative percentages (i.e. the percentage of times an effect is not identified) are more so. There is insufficient data here for meaningful estimates. However, it is interesting that on two occasions for females the unit of analysis, and on one for males the unit of analysis, the mixture of two binomials produced a signifi- cant result for cyclophosphamide in week 3 where none of the other distributions did.

Non-parametric methods have been considered (Haseman and Kupper, 1979; Mitchell et al., 1981). Generally these are less powerful than those based on distributional models. Shirley and Hickling (1981) suggest that analysis based on the beta binomial is actually less powerful than equivalent non-parametric techniques. It is felt that this indi- cates the theoretical statistical problems associated with the relevant significance tests. A distribu- tional model approach makes possible use of powerful model programs such as MLP (Ross, 1980) and G L I M (Baker and Nelder, 1978) to fit complex functional models relating time and dose to a measure of death, enabling analysis of the experiment as a single entity. Williams (1982) and Hinde (1982) indicate how such models could be implemented.

As the negative and beta binomial distributions

205

were generally the best for their respective varia- bles Tables 4 and 5 give the mean (and S.E.) of the pa rame te r s of these dis tr ibut ions for the cyclophosphamide experiments, for females and males the unit of analysis respectively. These ta- bles show how the dominant lethal effect for cyclophosphamide is displayed. James and Smith (1982) concluded that there was a definite effect in weeks 1 and 2 with a probable tail over effect into week 3. Assuming this, it can be seen from Tables 4 and 5 how the parameters of the two distribu- tions change. For the negative binomial, M (aver- age number of deaths) shows a rise from 1.2 for control to values of over 3 for both low and high doses of the compound in the first 2 weeks and lower values in week 3. The parameter 1/K shows a fall towards zero with increasing M. For the beta binomial the parameter /~ (average proportion of deaths) shows a rise from 0.11 for controls to over 0.20 for both doses in weeks 1 and 2 with lower values in week 3. Unlike the negative binomial there appears to be no change in the spread parameter O. The results in Tables 4 and 5 are very similar. The negative binomial parameter M in Table 5 is approximately twice its value in Table 4 because it represents the number of deaths for pairs of females mated by the same male.

The false positive percentages for the negative and beta binomial distributions are markedly less for males than females the unit of analysis. How- ever, if the odd outlying values were due to females (note proven males and virgin females were used) this would not be unexpected since the summing over 2 animals would reduce the outlier's in- fluence. Analysis taking females as the unit of analysis has advantages. If a mating ratio greater than 1 is used an empirical assessment of whether high outlying values represent a mutagenic effect is possible (James and Smith, 1982). Also using females as the unit of analysis does not involve the problematic handling of non-pregnant females which is necessary when analysing number of dead implants per male. For strong dominant lethal effects choice of unit of analysis appears not to be of major importance. Where there is doubt about an effect it is advisable to perform analyses with both units.

Conclusions

(1) Among the distributions considered for their respective variables the negative binomial (number of dead implants) and beta binomial (proportion of dead implants) were generally the best.

(2) The correlated binomial was a worse fit than the beta binomial and also had a higher percentage of false positive results. This result contrasts with that of Kupper and Haseman (1978) who found little difference between the two distri- butions. Its use on dominant lethal data is not recommended.

(3) Where strong dominant lethal effects exist choice of unit of analysis as male or female is not felt to be of major importance. Where there is doubt analysis using both is advised.

(4) An analysis based on the negative binomial distribution can have problems when values for 1/K of infinity occur.

(5) An analysis based on the beta binomial distribution can have problems with odd outlying values. Consideration should be given in the analy- sis to the effects of such values.

Acknowledgements

The authors wish to express their appreciation for comments made on this work by Mr. J. Wood (Wellcome) and Dr. R.A. Ferguson (ICI Phar- maceuticals). Thanks are also due to Ms. V. Pettit for typing the manuscript.

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