Proc. Natl Acad. Sci. USAVol. 78, No. 6, pp. 3779-3783, June 1981Genetics

Statistical analysis of the Ames Salmonella/microsome test(Ames test/mutagenesis/negative binomial model/statistical modeling/toxicity)

BARRY H. MARGOLIN*, NORMAN KAPLAN*, AND ERROL ZEIGERt*Biometry Branch and tLaboratory of Molecular Genetics, National Institute of Environmental Health Sciences, Research Triangle Park, North Carolina 27709

Communicated by Gerald N. Wogan, March 17, 1981

ABSTRACT A family of statistical models for analysis ofAmesSalmonella/microsome test data is constructed that considers mu-tation and toxicity as competing risks and allows hyper-Poissonvariability. These models have a parameter that can be employedas a mutagenic index because it approximates the slope at zero doseofa dose-response curve adjusted for toxicity. A second parameterquantifies plate-to-plate variability and lends itself to the study offactors affecting internal test reproducibility. The detection ofaberrant plate counts is also addressed. This methodology is il-lustrated with data from a Salmonella test.

The Ames Salmonella/microsome test (1) currently holds apreeminent position among the various tests available to thegenetic toxicologist for investigation of a chemical's mutagen-icity. Indeed, in many quarters, a positive Salmonella test resultis presumptive evidence for carcinogenicity. Although variousauthors (2-5) have proposed analyses ofdata from this test, eachtreatment has been deficient in some way: a Poisson distributionmodel for the sampling variability of the data is assumed, thepossibility of toxicity is disregarded or treated in an ad hoc man-ner, or the multigenerational aspect of the test is ignored.

There is substantial empirical evidence that the Poissonmodel, nearly universal in its adoption as the sampling distri-bution of revertants per plate from a Salmonella test, lacks theflexibility to adequately describe the variability in a set of platecounts. This can lead to false conclusions concerning the out-come of the assay. An alternative sampling distribution is pro-posed that incorporates a measure of experimental reproduc-ibility able to reflect dependence of internal variability amonga set of replicate plates on the specific laboratory, technician,strain, or any other factor influencing the results from a singletest.

Toxicity denotes any genetic or cellular phenomenon pre-cluding reversion to histidine prototrophy or expression of thehis' phenotype. Dose-response curves from a test frequentlyexhibit departures from monotonicity because of toxicity. Abiologically motivated family of mathematical models is con-structed to relate plate counts, on average, to their respectivedoses of the test chemical. This family considers both mutationand toxicity to be dose-dependent competing risks that confronteach Salmonella bacterium. These models are limited in theirabilities to resolve ambiguities caused by a limited understand-ing of the behavior ofa chemical once it is placed on a test plate.

A CRITIQUE OF THE POISSON MODELThe Poisson distribution has been employed extensively in sta-tistical modeling of microbial data. Its applicability to Salmo-nella test data appears to be a consequence of the following setof assumptions. (i) All microbes, whether on the same plate ornot, behave in a stochastic manner independently ofeach other.

(ii) Each microbe placed on a particular plate (and its lineage)experiences the same environment as any other microbe (andits lineage) plated on the same plate; this environment, denotedby A, is a function ofthe test compound and its dose, the amountof histidine present, and sundry other factors. (iii) The targetednumber ofmicrobes placed on a plate, denoted by No, is large-e.g., 108. (iv) The probability, P(6), that any single plated mi-crobe, through its descendants, gives rise to a visible revertantcolony is small, say 10-5 to 10-9, and is not altered by the num-ber of microbes plated.

These four assumptions imply that the number of revertantcolonies observed on a plate with environment f, denoted byX4, has approximately a Poisson distribution with mean param-eter N0P(g) (6)-i.e., for any nonnegative integer k, the prob-ability that X, equals k is given by

pr{X, = k} = [NoP()]kexp[-NoP(o)]/k!, [1]

in which exp[t] = et.For a set of r replicate plates with observed counts Xfj,X.2... .,X*, it is commonly assumed (e.g., refs. 2, 3, and 5) thatthese r counts are adequately described as a random samplefrom Eq. 1. This does not follow solely from the four assump-tions above but rests on an additional assumption: the r repli-cated plates share sufficiently similar environments and platednumbers of microbes, so that the realized mean parameters areessentially constant across replicates. If one accepts assump-tions i-iv, the validity of the additional assumption can be eval-uated empirically through a standard test of whether observedcounts XfX,.... .-,X. behave as a random sample from Eq. 1.To investigate this, four laboratories, without being told the

intended purpose, were requested to produce 20 replicate platecounts under solvent control conditions with Salmonella strainTA100. This strain was selected because it is often cited as themost sensitive of the Salmonella strains. The number of plateswas chosen to be that round number nearest the total numberof plates involved in a single strain experiment consisting of azero dose or solvent control and five or six chemical doses, withthree replicate plates per dose; this experimental design wasa recent mutagenicity conference recommendation (7). The re-sultant counts, ordered by magnitude, together with varioussummary statistics, are in Table 1.The statistic Tf evaluated for each data set is

To= > (X_-)2/X,i=l

[2]

in which Xf = (l/r)fXX. The statistic T, is proportional to theratio ofsample variance to sample mean and is the usual statisticused to test whether a random sample of size r exhibits behaviorconsistent with Eq. 1. If the underlying distribution is suffi-ciently near a Poisson distribution, then the distribution of T,is well-approximated by a x2 distribution with (r - 1) degrees

The publication costs ofthis article were defrayed in part by page chargepayment. This article must therefore be hereby marked "advertise-ment" in accordance with 18 U. S. C. §1734 solely to indicate this fact.

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Table 1. Twenty replicated solvent control plate counts withSalmonella TA100 from four laboratories

Laboratory*1 2 3 4Ht 4Dt

81, 86 63, 64 74, 77 168, 171 132, 13493, 97 65, 68 81, 85 174, 175 144, 14599,104 69, 70 87, 88 185, 189 145, 145

105, 110 72, 73 89, 90 190, 191 157, 158112, 112 75, 80 93, 97 195, 197 158, 161113, 114 82, 83 98, 99 198, 198 162, 164115, 117 83, 84 102, 103 203, 205 166, 169118, 122 84, 85 105, 108 205, 207 174, 177131,135 90,91 108,110 210,214 201,208155, 183 t, t 111, 124 216, 218 208, 219

Mean115.10 76.72 96.45 195.45 166.35

Variance540.62 81.15 163.10 226.58 631.29

Te89.24 17.98 32.13 22.03 72.10

C (X 102)2.7 0.0 0.6 0.1 1.5

SE (e)(x102)1.1 0.4 0.5 0.2 0.7

* Revertant colonies per plate listed in ascending order within eachlaboratory and paired solely for conciseness of presentation.

t Laboratory 4 conducted this experiment twice, once with water (4H)and once with dimethylsulfoxide (4D) as the solvent. The same in-dividual did the two tests but on different days. All other laboratoriesused water.

t Contaminated plate, not counted.

of freedom (8). For 20 replicates, this would imply that Te ex-ceeds 30.1 once in 20, 36.2 once in 100, and 43.8 once in 1000.Table 1 clearly demonstrates that these data are not Poissondistributed in all laboratories at all times. Zeiger et al. (9) andVollmar (cited in ref. 10) have previously reported this finding.Although Vollmar stated that a Poisson distribution can be as-sumed up to about 150 revertants per plate, the data in Table1 contradict this conclusion.The main practical implication ofTable 1 is that the variability

of any statistic used to quantify or judge the evidence for mu-tagenicity may be substantially underestimated if one disre-gards the possibility of hyper-Poisson variability. To the extentthat the results in Table 1 typify laboratory functioning, an un-derestimate of the variance by a factor of 2-4 is not rare. This,in turn, would elevate the real false-positive rate, and possiblyeven the real false-negative rate, and would yield confidencelimits for parameters of interest whose coverage rates are over-stated. The degree of elevation of the two error rates is depen-dent on the statistical procedure used. A later example willdemonstrate how variance underestimation affects model pa-rameter estimates. For present illustrative purposes, considera one-tailed test for mutagenicity based on a statistic that is ap-proximately normally distributed with mean 0 and variance p9under an assumption of no mutagenicity. If this statistic istreated as if its variance is o2, then the real false-positive ratesfor this test when nominally 5% and 1% are respectively 12%and 5% ifp = 2 and 21%o and 12%0 ifp = 4. The possible presenceof hyper-Poisson variability underscores the need for replicateplates; only by this means can one properly quantify internal

plate-to-plate variability and, thereby, validly weigh the evi-dence for mutagenicity.

AN ALTERNATIVE SAMPLING DISTRIBUTIONIt is our conjecture that laboratories occasionally exhibiting hy-per-Poisson variability in a particular experiment are unablethat day to maintain near constancy across replicates ofthe plateenvironment because of variability in pipetting the microbes,the agar, the S9 liver homogenate, and the test compound. Foreach plate, the Poisson mean would then be a stochastic quan-tity, having its own sampling distribution G. It follows that aset of replicate plate counts are a random sample from a mixtureof Poissons represented by

pr{X= k} = (k!)-lf Tke-TdG(T). [3]

The distribution G must be that of a nonnegative randomvariable and is probably unimodal, the targeted mean deter-mining the mode if no systematic bias is involved. The distri-bution G that affords greatest mathematical tractability is thatassociated with a gamma density:

g(t;pC) = [(pc)C-[r(C-L)]Y-tc-l-lexp[-t/pcl] [4]in which ,uL > 0, c > 0. With this choice, Eq. 3 reduces to anegative binomial distribution:

pr{X, = kIA,c} = (k + c - [,/(,u + c-1)]k

x [C-'/(p + c-')]-l [5]in which k is a nonnegative integer. This distribution has meang and variance ,u(l + cpu), which is greater than the mean. Thenext section discusses modeling ,u as a function of applied dose.The negative binomial can be evaluated at c = 0 through a

limit argument to obtain the Poisson distribution; hence, largevalues of c are evidence for departures from Poisson samplingbehavior. Aside from unimodality and tractability, the gammadensity assumption rests on the historical role of the resultantnegative binomial distribution as an important alternative to thePoisson model in microbiology; its use for this purpose datesback at least to Gossett [Student] (11), who was concerned withcounting yeast cells using a haemacytometer.

The parameter c reflects the intrinsic precision of an exper-imenter in producing replicated plate environments on a givenday, despite the varying test chemical dose. Perfect reproduc-ibility of replicated plate environments is represented by c =0. Variability in replicated plate environments is representedby c > 0: the greater the variability, the larger the value of c.Table 1 includes the maximum likelihood estimate c of c andits estimated standard error (12) for each of the five data sets.

Because the variance of a negative binomial random variableis a function of both u and c, one is unable to estimate the stan-dard error of an observation y solely from its model-fitted orpredicted value y, something that is obtainable for Poisson dis-tributed data simply as y. If c is nearly constant within an ex-periment, then the standard error for y can be estimated by[y(1 + Cy)]'f2* Thus, an individual observation may be checkedempirically for aberrance by computing its absolute standard-ized residual:

IY - Yl/[Y(1 + VA1 [6]

An illustration of this diagnostic check occurs later in the text.

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MODELING DOSE RESPONSEThe lack of universal Poisson sampling behavior for Salmonellatest data is one complication of the statistical analysis of suchexperiments. Additional complexity results from the frequentlynonmonotonic relationship ofrevertants per plate to test chem-ical dose, a relationship that increases to a peak and then de-creases, sometimes to a value below the solvent control valuesand occasionally to zero. This departure from a nondecreasingresponse is commonly ascribed to microbial toxicity, both ge-netic and cellular, caused by the test chemical. Were the cur-rently recommended Salmonella test procedure to include aparallel assay that accurately measured the number ofmicrobessuccumbing to toxicity on the treated plates, one would thenbe able to modify statistical analyses to adjust for the effects oftoxicity. However, this dual experiment is not common practicebecause of cost and convenience.An effective way to disentangle the competing risks of mu-

tation and toxicity in order to draw inferences regarding theformer is to model A as NOPD, where NO is the average numberof microbes placed on a plate, and PD is the probability that asingle plated microbe yields a revertant colony when exposedto dose D of a test chemical. Haynes and Eckardt (13) have re-cently come to a similar conclusion and have proposed modelsfor PD incorporating toxicity; their models are essentially of theform:

PD = {1 - exp[-HM(D)]}exp[-H7<D)], [7]

in which HM(D) and HT(D) are typically low-order polynomialsand represent respectively the average number of mutagenicand toxic "hits" per microbe when dose D is applied to a plate.The simplest and most common case of Eq. 7 is the "single hit"or linear model, where

PD = {1 - exp[-(a + /3D)]}exp[-yD],a>0,(830, y-0. [8]

Models of the form in Eq. 7 have been investigated and dis-cussed extensively in the field ofradiation mutagenesis but maybe inappropriate for chemical mutagenesis. Radiation is an in-stantaneous process; in response to a single burst, microbeseither succumb to toxicity, mutate with regard to a specifiedgenetic trait, or survive unchanged with regard to that trait. Ifa plated microbe escapes the toxic effects of radiation but mu-tates with respect to the specified trait, it is essentially assuredof giving rise to a mutant colony. For .this process, the platedmicrobes are the only ones to run the twin risks ofmutation andtoxicity. In the Salmonella plate assay, however, various alter-natives are possible depending upon the test chemical. Ifit losesits mutagenic and toxic capabilities rapidly, say in a single mi-crobial generation, then- the process closely resembles that ofsingle-burst radiation; on the other hand, if it is active for anextended period of time, then the plated microbes and theirdescendants are at risk of both mutation and toxicity. Becausethe activity of a chemical on a plate is not generally known, thisuncertainty argues for broadening the class of models in Eq. 7.To this end, in addition to assumptions i-iv and the negativebinomial sampling assumption, the following are assumed. (v)The probability of plating preexisting mutants or of experienc-ing forward mutation from prototrophy to auxotrophy is negli-gible. (vi) Expression occurs within the first generation follow-ing mutagenesis. (vii) Sufficient histidine is placed on each plateto permit auxotrophic growth through k generations, where kis typically three to six. (viii) Mutation and toxicity are stochast-ically independent processes. A microbe on a plate with doseD of the test chemical has probability pM(D) of mutating from

his- to his' if it is a member of the first m generations afterplating, probability of PM(O) of mutating if it is a member of thenext k-m generations, and a zero probability thereafter. Bothauxotrophs and prototrophs have probability p7(D) ofsuccumb-ing to toxicity ifthey are members ofthe first t generations sinceplating and zero probability thereafter.The step-function behavior assumed in viii is intended as a

parsimonious approximation to probabilities that vary in a com-plex fashion with generation time. Because measurable muta-tion effectively ceases with histidine depletion (ifnot before) andbecause measurable toxicity (if present) is not necessarily sorestricted, it has been assumed in viii that m ' k, and m ' t' oo. In fact, for the single-hit assumptions subsequentlyadopted, setting m = k introduces negligible error in the pa-rameter estimates. Here t = oo is intended to indicate long-last-ing toxicity (e.g., the entire test time of 48 or 72 hr). The as-sumptions above permit the following modeling.

Let PD(m,t) denote the probability that a plated microbe ex-posed to dose D of the test chemical gives rise to a revertantcolony when mutation is present for m generations and toxicityfor t generations. Similarly define QD(S) to be the probabilitythat any microbe and its descendents exposed to doseD becomeextinct because of toxicity within s generations, counting theplated microbe as generation one. Then upon consideration ofthe possible first generation events, one obtains the followingrecursion:

PD(m,t) = pM(D)[1 - p7(D)][1 - QD (t - 1)] + [1 - pM(D)]x [1 - p7(D)]{l - [1 - PD(m-l,t-1)]2}. [9]

The quantity QD(s) also satisfies a recursion. If 1 < s < t, then

QD(S) = p7<D) + [1 -p7<D)]Q2(S_1);otherwise QD(S) = 0 ifs = 0, or QD(S) = QD(t) if s > t. Usefulvalues of QD(s) are QD(1) = p7(D) for all t . 1,

QD(2) = p7AD) + [1 - p7(D)]pR(D) for all t 2,and

QD(O) = min{l, p7gD)/[I-p7(D)]} for t = 00.

Because PD(r,v) is very small for all r - m and v ' t, Eq. 9 maybe approximated byPD(m,t) pM(D)[- pDD)][1- QD (t- 1)]

+ 2[1 - pM(D)][1 - pAD)]PD(m-1,t-1)-Ifm = k, Eq. 10 admits the following solution:

m-1

PD(m,t) = pM(D)[l - p7(D)]{Z (2[1lpM(D)]j=1

x [1 - p7#D)]y X [I Q2(t-l-j)] .

[10]

[11]

Four models are boundary members of the family of modelsin Eq. 11 subject to m c min (k,t); because they straddle thefamily, these models give good evidence ofthe varied behaviorspossible in Eq. 11:

I. For m = k = 1 = t: PD(1,1) = pM(D)[l - p7(D)].This is Eq. 7 with pM(D) = 1 - exp[-HM(D)] and p7(D)= 1 - exp[-HT(D)].

II. Form= k= 1, t= o:

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PD(1,oo) = PM(D)[1-p(D)I[1 Q2 (00)] = PM(D\[1-2p[ D)]+' ~[1-pA~D)]

i.e., mutation for one generation but long-lasting toxicity;[x]+ = max(0,x).

III. For m = k = t > 1: PD(k,k) in Eq. 11 admits no furthersimplification; both mutagenicity and toxicity are histidine-limited processes.

IV. Form = k > 1, t = 00 PD(k,) = (2[1-PM(D)][1 -pT(D)])i. Here mutation is histidine-limited, but tox-icity, if present, is long lasting.

For the remainder of the discussion, simplifying single-hitassumptions for mutation and toxicity are adopted, and m is setto k. These plus earlier assumptions lead to the following: pM(D)= 1 - exp[-(a + /3D)], a > 0, P3 2 0, and p7<D) = 1 -

exp[- yD], y 0. The key parameter in this case is /, becausethe chemical under study is mutagenic if and only if ,3> 0; itis the only index of mutagenicity to date that both derives frommechanistic modeling of the Ames test and is adjusted fortoxicity.One distinction among the models is immediately apparent.

The duration of toxicity on a plate can have a marked impacton the form ofPD and, hence, ,u. For example, model I describessurvival by the exponential decay function exp[- yD], whereasmodel II depicts it as the "shoulder-shaped" function [2 -

exp(yD)]+, reaching zero at a finite dose D = y-'ln 2 for Y>0. The visible effects oftoxicity are usually apparent only at highdoses. If the positive doses are selected to be equispaced on a

logarithmic scale, then there may be little information regardingthe form of the survival function. As the example in the nextsection illustrates, models I and II may fit the data equally well,as judged by the evaluated logarithmic likelihoods, yet the dif-ferences in their estimates of the parameter P can frequentlyexceed that which is explainable by sampling variability. Ofgreater concern are differences in estimates of /3 that modelsII and IV can exhibit because the duration of mutagenicity on

a plate is unknown. The example below illustrates this point.

APPLICATION AND IMPLICATIONSOF THE MODELS

Quinoline was studied by W. Speck (unpublished) as part oftheEnvironmental Mutagenesis Test Development Program at theNational Institute of Environmental Health Sciences. The datain Table 2 were obtained from a test with Salmonella strain TA98and induced rat liver homogenate S9. Dimethyl sulfoxide was

the solvent control, and each dose was replicated three times.Models I-IV, together with the single-hit and negative binomialassumptions, were fitted to these data by maximum likelihoodtechniques (14). Parameter estimates of a, I, y, and c and theirestimated standard errors are given in Table 3. A fifth model(V) that assumes m = k = 6 and t = is included as a furthercheck on model sensitivity.

The fitted values of these five models (Table 4) show sub-stantial agreement in fitting the data despite the differences in

Table 2. Quinoline data (TA98, rat liver homogenate S9)Revertant colonies with increasing quinoline*

0 10 33 100 333 100015 16 16 27 33 2021 18 26 41 38 2729 21 33 60 41 42

* Revertant colonies per plate are listed in ascending order within eachdose group (dIg of quinoline per plate). Three plates were tested perdosage;

Table 3. Parameter estimates from negative binomial single-hitmodels I-V for quinoline data

ModelI II III IV V

(m = 1, (m = 1, (m = 3, (m = 3, (m = 6,t= 1) t= m0) t= 3) t= 00) t= 0c)

&(X10-8) 20.3 22.0 2.99 3.04 0.325SE(&)(x 10-9) 26.1 26.4 3.73 3.74 0.4133(x10-10) 21.5 10.7 2.38 2.14 0.318SE(,4)(x10"-) 62.7 32.4 8.17 5.90 0.898ifx 10-4) 21.2 5.72 5.71 5.14 3.79SE(j4(x 10-5) 35.3 4.01 15.2 5.07 5.61cF 10-2) 5.36 6.53 5.45 6.02 5.49SE()(x 10-2) 2.98 3.37 3.01 3.20 3.02

model form and parameter estimates. The five logarithmic like-lihoods obtained were -63.20, -64.36, -63.73, -63.86, and-63.34, respectively. Estimates of the mutagenic index /3 vary

from 2.15 x 10-9 to 3.18 x 10-11. Models I and II differ by a

factor of 2 in this estimate, which is solely attributable to dif-ferences in their modeling of toxicity. The wide variation inestimates of 3 among models II, IV, and V, all of~which assumelong-lasting toxicity, can be elucidated. If toxicity is essentiallynegligible, then because pM(D) is generally very small one can

closely approximate Eq. 11 by:

PD(m, t) = (2m - 1)pM(D)

= (2m -1){1 - exp[-(a + /3D)]}. [12]

Consequently, it is virtually impossible to distinguish empiri-cally between the models associated with PD(mt) and PD(1,t),since their fits to the data will be nearly the same. However,their parameter estimates for a and /3, denoted by (&(m), frm))

and (&'(), fin)), respectively, will differ by nearly a factor of (2'1)-i.e., &° (2m - 1)&(m) and #i) _ (2m 1)1m). The re-

spective standard errors of these estimates will differ by thesame factor. The estimates of in Table 3 exhibit comparablemultipliers in the presence of mild toxicity and moderate sam-

pling variability-e.g., P'1I _5 PIV and I" 34 3V.Therefore, in the presence ofnegligible toxicity and sufficient

histidine for more than one generation, if it is assumed thatmutagenic activity persists for m* generations but in reality itpersists for m generations, then the estimate 83 actually esti-mates approximately (2m- 1)/(2m -1) times the real P. If suf-ficient histidine for four or five generations is supplied to eachplate, then the uncertainty regarding the mutagenic index /3 can

be nearly an order of magnitude. This is far greater than theuncertainty evidenced by the estimated standard errors in Table3, which must be interpreted as uncertainties conditional on thevalidity of the corresponding models. This uncertainty will af-fect any quantitative analysis that attempts to measure muta-genic "potency" because the number of microbes at risk of mu-

Table 4. Fitted values for quinoline data

Quinoline, Model Averaged platettg per plate I II III IV V counts

0 20.3 22.0 20.9 21.3 20.5 21.710 22.0 22.9 22.3 22.5 22.1 18.333 25.6 25.0 25.2 25.1 25.4 25.0100 33.9 30.8 32.6 . 31.9 33.4 42.7333 45.5 45.5 46.4 45.7 45.5 37.31000 28.4 29.4 28.9 29.1 28.6 29.7

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tation is not known. Nevertheless, it is conceivable that forcertain purposes one can accept an order of magnitude of un-certainty in estimating P.3. Alternatively, one can theoreticallyreduce this uncertainty by supplying only enough histidine fora small number of generations. For identifying mutagens, asopposed to the potentially more useful quantification of thestrength of mutagenicity, the natural one-tailed statistical testthat treats S = U/SES as a standard normal random variable inthe absence of mutagenicity is much less affected by the un-certainty of the number of microbes at risk of mutation. Thevalues of S for the five models considered here are 3.43, 3.30,2.91, 3.62, and 3.54, all of which indicate that quinoline ismutagenic.The distinction between the negative binomial and Poisson

sampling models can be inferred from Table 3 and the followingvalues offP ± SEW obtained for models I-V, respectively, underPoisson distribution sampling: (2.04 ± 0.363) X 10-9, (9.67± 1.69) X 10-10, (2.24 ± 0.466) x 10-"°, (1.97 ± 0.321) x10-10, and (2.99 ± 0.512) X 10-11. The estimates of 3 for thequinoline data under the Poisson and negative binomial modelsexhibit small differences for any one of the five models consid-ered, but their respective standard errors under the negativebinomial model are approximately 1.75 times larger than underthe Poisson model. This is intuitively reasonable because thetwo sampling models as used here differ not in expected valuesbut in variances.

As mentioned earlier, one can use Eq. 6 to check for evidenceof aberrant observations that might unduly influence the esti-mated (8. For the data on quinoline, modelsI-V all point to theobserved 60 revertant colonies at a dose of 100 ug per plate asthe candidate for designation as a rogue observation. The valuesof Eq. 6 for the reading of 60 under each of the five negativebinomial models are 2.67, 3.03, 2.88, 2.91, and 2.73, respec-tively; there is marginal-to-moderate evidence for aberrance inthe reading of 60, depending upon the model employed in theanalysis. The implication of this finding can be investigated byrefitting each model to the quinoline data set with the obser-vation of 60 deleted. The new values of A plus or minus theirstandard errors for the five models are (1.55 ± 0.447) x 10-9,(8.58 ± 2.27) X 10-10, (1.78 ± 0.584) x 10-1", (1.67 + 0.427)X 10'l, and (2.35 ± 0.653) X 10-11, respectively, which doesnot alter the conclusion that quinoline is mutagenic. The majorimpact of omitting the 60-revertants reading is to reduce c,which reflects lack of fit and lack of reproducibility, both ofwhich are reduced when the most discrepant observation isomitted from the modeling. The values ofc upon deletion oftheobservation of 60 are 0.0212, 0.0249, 0.0215, 0.0231, and

0.0215, respectively. Each model still leads to the conclusionthat the sampling distribution is not Poisson.

DISCUSSIONThe models constructed and discussed here are approximationsto the biochemistry and genetics of the Ames Salmonella testthat ignore possible complexities of pharmacokinetics, thresh-old phenomena, and expression delay. This family of modelsprovides a methodology for analysis of mutagenicity data thatdoes not presuppose or exclude either Poisson variability ortoxicity or rely on subjective judgments of density of the back-ground lawn to determine whether, toxicity is present. The pa-rameters in these models serve different purposes. The param-eter a can be employed to check the credibility of solventcontrol data vis-a-vis historical values. The parameter 83 rep-resents a mutagenic index that estimates the instantaneousslope at zero dose of the observed dose response adjusted fortoxicity. Finally, the parameter c serves to quantify plate-to-plate variability within a test and can be used as a means to im-plement quality control for a test protocol that is rapidly ap-proaching a level of mass production in many laboratories.

The assistance of K. Risko with all numerical computations is grate-fully acknowledged.

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2. Katz, A. J. (1979) Mutat. Res. 64, 61-77.3. Weinstein, D. & Lewinson, T. M. (1978) Mutat. Res. 51, 433-

434.4. Venitt, S. & Crofton-Sleigh, C. (1979) Mutat. Res. 68, 107-116.5. Stead, A. G., Hasselblad, V., Creason, J. P..& Claxton, L. (1981)

Mutat. Res. 85, 13-27.6. Feller, W. (1967) An Introduction to Probability Theory and Its

Applications (Wiley, New York), 3rd. Ed., Vol. 1.7. de Serres, F. J. & Shelby, M. D. (1979) Environ. Mutag. 1, 87-

92.8. Snedecor, G. W. & Cochran, W. G. (1967) Statistical Methods

(Iowa State Univ. Press, Ames, IO), 6th Ed.9. Zeiger, E., Chhabra, R. S. & Margolin, B. H. (1979) Mutat. Res.

64, 379-389.10. Seiler, J. P., Mattern, I. E., Green, M. H. L. & Anderson, D.

(1980) Mutat. Res. 74, 71-75.11. Gossett, W. S. [Student] (1907) Biometrika 5, 351-360.12. Fisher, R. A. (1941) Ann. Eugenics 11, 182-187.13. Haynes, R. H. & Eckardt, F. (1980) in Chemical Mutagens, Prin-

ciples and Methods for Their Detection, eds. Hollaender, A. &de Serres, F. J. (Plenum, New York), Vol. 6, pp. 271-307.

14. Rao, C. R. (1973) Linear Statistical Inference and Its Applica-tions (Wiley, New York), 2nd Ed.

Genetics: Margolin et al.

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