Document b5RNRyXd3aywExN67MbNN0M6Z

IVlonseri-to I S A M E - L O C A T I O N -- P H O N E ) Dept, of Medicine & Environmental Health J- M. Kronenberg - G2WD, (4-8832) DATE S U B JE C T February 11, 1985 y JOURNAL CLUB' : J REFERENCE TO : Toxicology Group cc. F. ZJ t n ' j n \v;.: RGH i:RJ .U-AK TJL DPM R3N RO r-GR UJ3 i.iWS _ __ _ _ Johannsen This week's Journal Club meeting will be held on Thursday, February 14th at 12:30 p.m. I will be discussing the basic concepts (not the mathematics) proposed in the attached article on ADIs. JMK/pp Enclosure Joel IN-10-M {REV. 10/83) 003731 FUNDAMENTAL AND APPLIED TOXICOLOGY 4, 854-871 ( 1984) _ A New Method for Determining Allowable Daily Intakes,? Kenny S.Crump K. S. Crump and Company, Jnc.. 201 Gaines Street. Ruston. Louisiana 7270 A New Method for Determining Allowable Daily Intakes. Crump, IC S. (1984). Fundam. Appl. Toxicol 4, 854-871. The usual method for establishing allowable daily intake (ADI) for a chemical involves determining a no-observed-effcct level (NOEL) and applying a safety factor. Even though this method has been used for many years, there appear to be no general guidelines or rules for defining a NOEL. The determination of a NOEL is particularly uncertain for lesions which occur naturally in untreated animals. NOELs also have shortcomings in that smaller experiments tend to give larger values (this should be reversed because larger experiments can provide greater evidence of safety) and that the steepness of the dose response in the dose range where effects occur plays little or no role in the determination of a NOEL. This paper proposes and illustrates the use of a "benchmark dose'* (BD) as an alternative to a NOEL. A BD is a statistical lower confidence limit to a dose producing some predetermined increase in response rate such as 0.01 or 0.1. The BD is calculated using a mathematical dose-response model. This approach makes appropriate use of sample size and the shape of the dose-response curve. The BD normally will not depend strongly upon the mathematical model used because the method does not involve extrapolation far below the experimental range. Thus the method sidesteps much of the model dependency often associated with extrapolation of carcinogenicity data to low doses. The method can be applied to either "quantal" data in which only the presence or absence of an effect is recorded, or "continuous" data in which the severity of the effect is also noted. 1984 Society of Toxicology. , I I. INTRODUCTION A common approach to quantifying permissible human exposure to a toxic agent is to establish a no-effect level using experimental animal data and then to apply a safety factor-- or uncertainty factor, as it is sometimes called--to arrive at a permissible exposure level for humans. Allowable daily intakes for chemicals (ADIs), such as were employed by EPA in calculating water quality criteria (EPA, 1980), furnish one example of such calculations. Threshold limit values (TLVs), which are provided by the American Conference of Government and Industrial Hygienists (ACGIH) for many chemicals to which workers are exposed, are calculated in a similar fashion 1PupaitH for U.S. Environmental Promotion Agency, Environmental Criteria Assessment Office, Contract 6803-3H i, Work Assignment 15. (ACGIH, 1976). The term "daily intake level" (DIL) has also been employed. In this paper, we will refer to ADIs as a matter of convenience, although the discussion will apply to all such estimates. The calculation of an ADI by applying a safety factor to a no-effect level will be referred to as a NOEL-SF approach, In recent years, ADIs for carcinogens have sometimes been calculated by fitting mathematical models to experimental dose-response data. These models are used to estimate the dose corresponding to some specified small amount of additional risk. EPA (1980) used this approach to set water quality criteria for carcinogens, and used a NOEL-SF method for noncarcinogens. This dichotomy was based upon the supposition that carcinogens are un likely to have a threshold; consequently, a NOEL-SF approach would be inappropriate because it assumes the existence 0 f a threshold, The object of this paper is to present some 0272-0590/84 S3.00 Copynghi <019X4 by the Society of Toncdofy. Atl right* of reproduction in tny form rmerved. 854 ,es 1 o Fundam. ADI) for ty factor, uidelines <r lesions smaller ems can se range proposes BD is a response lei. T h is1 rve. The method idesteps data to fence or effect is ily intake level" I. In this paper, alter of conve rt will apply to tion of an ADI i no-effect level -SF approach, treinogens have *fitting mathel dose-response to estimate the specified small 'A (1980) used lity criteria for "L-SF method omy was based nogens are unmsequently, a inappropriate .of a threshold. ) present some DETERMINING ALLOWABLE DAILY INTAKES 855 mathematical and statistical approaches to calculating ADIs for effects other than cancer. In tlte next section some potential shortcom ings of the NOEL-SF approach are discussed. The following section describes some math ematical models and related statistical meth ods. These methods have two features that are somewhat novel. First, some of the models include the possibility of thresholds below which no effect will occur. Second, methods are suggested for application to "severity" or "continuous" data rather than just on inci dence data. In the next section, a recommen dation is made for replacing the NOEL in the NOEL-SF approach with a "benchmark dose." This benchmark represents a statistical lower confidence limit on the dose corre sponding to a small increase in effect over the background level. The amount of increase in effect used to define the benchmark is small enough so that the estimate of the benchmark dose will reflect the shape of the dose-response curve, and it is large enough so that the lower confidence limit will not depend critically upon the mathematical model used in its cal culation. A number of examples are presented illustrating the calculation of benchmark doses and comparing them with NOELS. II. DIFFICULTIES WITH THE NOELSAFETY FACTOR METHOD Definition of a NOEL considering effects which have nonzero back ground levels. Consider, for example,- liver weight; all animals have nonzero background levels of this "effect." Liver weights constitute a continuous measure (as opposed to "inci dence" or "quanta!" data) which can be ob tained for each animal. It might happen that the average liver weight in some, or even all, of the treated groups is above that of the con trol group. Since this could happen by chance, usually the NOEL is taken to be the largest dose for which the increase in liver weight is not statistically significant. However, such a decision can seem rather arbitrary when there is a smooth dose-response trend which over laps the region where the increase is not sta tistically significant. A NOEL must be one of the experimental doses.2 This constraint can appear unneces sarily restrictive in some cases. Consider, for example, an experiment to detect liver effects which involves three dose levels. Suppose at the highest dose level there are very severe effects, at the middle there are barely discern ible effects, and at the low dose no effects at all are seen. Then the low dose likely will be designated the no-effect level even if the doseresponse from the middle to high dose indi cates that a much higher dose (one slightly less than the middle dose) would have had no discernible effect. Furthermore, if the data at the lowest dose had not been available, this experiment could not be used at all to define a no-effect level. The first problem one faces with the NOEL concept is one of definition: Just what con stitutes a NOEL? For effects which are un ambiguous because they do not occur in un treated animals, such as acute toxicity or the occurrence of rare tumors, determination of a NOEL can be reasonably straight-forward; if no effect is seen in any animal a NOEL is determined--otherwise a'NOEL is not deter mined. For less well-defined effects, such as necrosis or cloudy swelling of the liver, de termination of a NOEL requires the use of judgment. This problem is compounded when Effect of Sample Size It would be appropriate for larger studies to tend to produce larger ADIs because they 1A NOEL is not an inherent property of the animal system but depends upon the experimental design and outcome. Thus it represents, in statistical terms, a statistic or an estimate of a "true no-effect level." This latter terra refers to the highest dose which is absolutely safe and thus is an inherent property of the animal system, or. in sta tistical terms, a parameter. For an effect for which no threshold exists, the "true no-effect level" is zero. 856 KENNY S. CRUMP involve less random variation. However, the NOEL approach has the opposite tendency. A larger study has a better change of showing a statistically significant result and thus will, on average, produce a smaller ADI. As an illustration, suppose at the .control and one treated dose in a study involving 100 rats per dose, the resulting mean liv^rTal.per animal was 15.1 g in the control groupiand 18.4 g in the treatment group with a standard deviation of 10.0 g in each group. Thencthe t statistic for a difference between the twagroups is 3.3, which is significant at the l%tlavel. However, If the identical results came from a study in volving only 25 rats perrgroup, tfhe t statistic is 1.14, which is not significant at the 10% level. Thus, a NOEL wduld p&ssibly be esti mated for the smaller study biitmot the larger. d-8 Ojb d** Dose Fig. I. Hypothetical responses with 95% confidence limits for two experiments. Therefore, rather than encouraging larger studies to demonstrate greater evidence of which is less than the NOEL d2n for Experi safety, the NOEL-SF insfeadSpenalized pro ment B. The dose-response methods to be ponents of chemicals -fon conducting large discussed in later sections are capable of fuller studies. This topsy-turvy state? of affairs has utilization of dose-response trends. made it necessary for[regulatory agencies to set minimally acceptable sample sizes. Quite naturally, many studies uset these minimal values. The NOEL-SF Approach Can Entail Unnec essary Restrictions and Expense Consider the following scenario; A company Utilization of Dose Response' wishing to market a new product implements a thorough toxicological testing program as A NOEL is determined?'solely by infor required by the regulatory agency involved. mation relating to whether.or not an effect Included in this program is a 2-year chronic was observed; the magnitude of positive effects toxicity and carcinogenesis bioassay, and a and relationships among the effects at the var two-generation reproduction and teratology ious doses (i.e., the dose-rresponse trend) is study. Each study involves three treatment and largely ignored. Consider fo'r.example the hy one control group with doses and sample sizes pothetical data in Fig. 1. Experiment A shows approved by the agency. The reproduction and a sharply increasing dose lEesponse. Experi teratology study is negative. In fact, the treated ment B shows a much flatter dose response, animals reproduce better than the control an which is, in fact, consistent with a linear re imals. This is apparently related to the fact sponse through the origins. Itiappears, because that the control animals are obese. The 2-year of the sharp decrease ircaresponse with de study likewise shows no effects of treatment creasing dose in Experiment A, as opposed to except for a dose-related weight reduction Experiment B, that the NOEL for A should which is apparently due to the fact that the be larger than the NOEL for B. However, be animals were fed the chemical in such high cause the response at doserrfjA was barely sig concentrations that their food was distasteful. nificant, the NOEL for E)5eriment A is As illustrated in Fig. 2, this weight loss follows L Fig. dence a cle statis The been for c requ stud; high in th is al< lustr dose thee loss is mi at li cont dose ADI fact< mar F unn spoi spot ADI and with 95% confidence - d2B for Experimethods to be capable of fuller trends. 2n Entail Unnecxpense lario: A company iuct implements ting program as igency involved, a 2-year chronic bioassay, and a and teratology ee treatment and and sample sizes eproduction and i fact, the treated n the control anated to the fact bese. The 2-year cts of treatment feight reduction :he fact that the ;al in such high 1was distasteful. :ight loss follows DETERMINING ALLOWABLE DAILY INTAKES 857 Arbitrariness o f Safety Factor The NAS Safe Drinking Water Committee made the following recommendations for un certainty factors (safety factors): 1. Valid experimental results from studies on prolonged ingestion by man with no indication of carcinogenicity. Uncertainty Factor = 0. Dose Fig. 2. Hypothetical average weights with 95% confi dence limits from two studies (see text for description). 2. Experimental results of studies of human ingestion not available or scanty (e.g., acute ex posure only). Valid results of long-term feeding studies on experimental animals or in the ab sence of human studies, valid animal studies on one or more species. No indication of carci nogenicity. Uncertainty Factor = 100. 3. No long-term or acute human data. Scanty results on experimental animals. No indication of carcinogenicity. Uncertainty Factor = 1000. a clear dose-response trend and the effect is statistically significant at the lowest dose tested. The agency rules that a no-effect level has not been determined and thus there, is no basis for calculating a ADI. The company is then required to conduct another 2-year study. This study also uses three treatment levels, the highest of which coincides with the lowest used in the previous study. A dose-response trend is also obtained in the follow-up study as il lustrated in Fig. 2. The data at the highest dose reproduce almost exactly the results in the earlier study at that dose level. The weight loss at the middle dose in the follow-up study is marginally significant and the average weight at the low dose is comparable to that of the control group. The agency rules that the low dose of the follow-up study is a NOEL; an ADI is then calculated by applying a safety factor, and the company is finally allowed to market its product Figure 2 shows that the follow-up study was unnecessary and only verified the dose-re sponse trend of the initial study. If dose-re sponse methods had been used for determining ADIs, the extra expense of the follow-up study and the 2-year delay could have been avoided. These uncertainty factors are used in every case as a divisor of the highest reported long term dose which is observed not to produce any adverse effect. (NAS, 1977) The application of a 100-fold safety factor to a results from long-term animal studies is a long standing practice. It has been interpreted as resulting from the product of two 10-fold safety factors: one factor to account for animalto-animal variation, and another to translate results from animal to man (Weil, 1972). However,-the use of the 100-fold safety factor probably developed simply because some op erational basis for setting allowable exposures was needed and a factor of 100 seemed "rea sonable" or "prudent." The fact that humans have 10 fingers undoubtedly played a role in the specific factor selected. When a safety factor is applied it is im plicitly assumed that a threshold exists and the resulting ADI is below the threshold and hence safe. However, whether a threshold ex ists for a specific effect and, if so, whether the ' ADI is below that threshold are open to ques tion. Also, as economic costs of regulations become more critical, there is increasing need for balancing the level of safety provided with wa t* V1 KENNY S. CRUMP 858 the costs involved. The NOEL-SF approach does not lend itself to cost-benefit analyses. III. FITTING DOSE RESPONSE MODELS TO TOXICOLOGICAL DATA Quantitative toxicological data are basically of two types: quantal and continuous. Quantal or incidence data specify the number of an imals affected, but not the degree of harm. The numbers of animals with tumors or some genetic anomaly are examples of quantal data. On the other hand, with continuous data the level of harm is specified for each animal. Or gan weights, triglyceride levels in liver, and serum measurements are examples of effects that are usually recorded as continuous data. these methods could be applied are caused by diverse mechanisms, most of which are poorly understood. Therefore, it seems that it would not be fruitful to attempt to develop doseresponse models from detailed assumptions regarding these mechanisms. Instead, we pro pose the use of relatively simple generic mod els. For illustrative purposes we shall consider the following models: the quantal linear regression (QLR) model P{d) = c + (1 - c){ 1 - expi-<?i(i* - d0)l} for d s* do = c for d < do (1) where 0 * c < 1, do 0, q, 5s 0; the quantal polynomial regression (QPR) model P{d) = c + (1 - c){ 1 - e x p l - ? ^ - dQ) Dose Response Methods for Quantal Data Quantal data from a toxicological experi ment can be represented as a collection of triplets (Nh X it dt)-- one triplet for each treat ment or control group--where Ni is the num ber of animals in the tth group, Xiis the num ber of affected animals, and d, is the dose. Let P(d) represent a dose-response model appli cable to quantal data, where P(d) is the prob ability that an effect will occur in an animal subject to a dose d. The parameters of the model can be estimated by fitting the model to quantal dose-response data using maxi mum likelihood procedures. A number of dose-response models have been suggested for use with cancer data. Some of these, such as the one-hit, multistage, multihit, and Weibull models can be derived from detailed assump tions about carcinogenic mechanisms. Other models, such as the/probit or logit models, can be thought of as representing the distri bution of individual tolerances in a large pop ulation (Krewski and Van Ryzin, 1981). In the next section some ways are suggested for applying mathematical models to non cancer data. The toxic endpoints to which - . . . - gk(d - 4)*]} for d 5* do - c for d < do (2) where 0 ^ c ^ I, do > 0, q{ > 0 for i = 1, . . . , k\ the quantal Weibull (QW) model t P{d) = c + (1 - c)[l - exp(-atf*)] (i) where 0 * c ^ 1, a 5* 0, and k 5* 1; and the log-normal (LN) model P(d) = c + (1 - c)N(a + b log d) (4) where 0 < c < 1, b 5* 1 and N is the standard normal distribution function. Readers familiar with the carcinogenesis dose-response literature will recognize (1) and (2) as slightly modified versions of, respec tively, the one-hit and multistage models as applied to carcinogenesis data (Krewski and Van Ryzin, 1981). Each has been modified here to include a threshold dose do: doses be low this threshold produce no effect. Thus these models allow for the possibility that thresholds could exist for some effects; how ever, the models could be applied with the threshold fixed to 0. Although we have not done so, thresholds could also be included in the Weibull and log-normal models. V $ . ed are caused by which are poorly ms that it would > develop doseled assumptions Instead, we pro zie generic modve shall consider quantal linear Q\(d --do)J} for d > do 0) * 0; the quantal *model -qt( d - do) *]} for d ^ d o (2) > 0 for i = I, QW) model tpi-arf*)] (3) k > l; and the + b log d) (4) ' is the standard carcinogenesis .`cognize (1) and .ions of, respec>tage models as a (Krewski and been modified >se do; doses be10 effect. Thus possibility that ne effects; howpplied with the ;h we have not i be included in nodels. DETERMINING ALLOWABLE DAILY INTAKES 859 Note that the restrictions k > 1 and b > 1 are assumed for the Weibull and log-normal models, respectively. Some restrictions of this nature seem necessary with these models; oth erwise these models can exhibit very extreme and biologically implausible behavior. The re striction k > 1 was selected for the Weibull model because k < 1 corresponds to a supralinear curve shape which is implausible for any biological effect (Crump, 1984). Although the restriction b 5= 1 for the slope parameter in the log-normal model does not have a strong theoretical basis, it does have a precedent, as it was recommended by Mantel et al. (1975) for cancer data. Dose-Response Methodsfor Continuous Data Continuous dose-response data consist of the dose level and the response level for each animal. With most continuous effects there will be variation about a nonzero value in the control group. There has been little experience in applying dose-response models to such data. It is possible to convert continuous data to quantal data by considering all animals with responses beyond a particular value as "af fected" and all others as "unaffected." How ever, this procedure entails a considerable loss of information as well as requiring the arbi trary choice of a cut-off value. The following method makes more complete use of the data. The method will be based upon the* sup position that the responses in an animal group subject to a dose dt are normally distributed with mean m(dt) and variance aj. There is both a theoretical reason and a pragmatic rea son for assuming the normal distribution. First, by the Central Limit Theorem of prob ability theory (Loeve, 1963) the sample means will be approximately normally distributed for large samples regardless of the form of the underlying distribution. Second, with the nor mal distribution, maximum likelihood meth ods can be applied knowing only the doses du - , dg, the numbers of animals at each dose nt, . . . , nf , and the corresponding sample means and standard errors (jci, S\). (x2, s2), . . . , (jcf, sg). If a non-normal distribution were assumed, maximum likelihood methods would require knowledge of the individual animal responses, which usually are not readily available. The choice of the normal distri bution is not a critical decision as this distri bution only determines the error structure, and not the dose response. The mean function m(d), which represents the average response at a dose d, determines the dose response. We do not require any assumptions regarding the variances (other than that they are finite); it is not necessary to assume, for example, that the variances in the different dose groups are all equal. For illustrative purposes, we will consider the following forms for m(d): the continuous linear regression (LCR) model m(d) = c + Q\(d --do) for-d > do = c for d < do (5) where do > 0, but c and qx are unrestricted; the continuous polynomial regression (CPR) model m(d) - c + qx(d - do) + * + qd - d0)k for d 2* do = c for d < do (6) where do > 0 and the q,'s are restricted to be either all positive (increasing dose response) or all negative (decreasing dose response); and the continuous power (CP) model m(d) = c + qx{d - dot. (7) These models are analogous to (1), (2), and (3), which were suggested for use with quantal data. With both quantal and continuous data, in addition to the selection of a dose-response model, the proper use of statistical confidence limits is also of critical importance. Often, different confidence limit procedures yield dif ferent results; this makes it important to use the same procedures when comparing dose- ; ( me 860 ICENNY S. CRUMP response models. A standard method for computing confidence limits is to base them upon the asymptotic normal distribution of maximum likelihood estimates. However, these confidence limits have been shown to behave poorly in a low dose extrapolation set ting (Crump and Masterman, 1979; Krewski and Van Ryzin, 1981; Crump and Howe. 1983); the upper and lower limits are often too close together to be believable. Further, these limits are not invariant under parameter transformations, and different transformations applied to the same model at low doses can yield vastly different confidence bounds. Cox and Lindley (1974) noted these difficulties in a more general context, and argued that con fidence limits based upon the asymptotic dis tribution of the likelihood ratio statistic "can be expected to behave much more sensibly" than those based upon the asymptotic nor mality of maximum likelihood estimates. Crump and Howe (1983) reviewed confidence limit procedures for use in dose-response evaluations and recommended limits based upon the distribution of the likelihood ratio statistic as the method of choice. This method for constructing confidence limits is outlined in the Appendix and will be used exclusively throughout this paper. IV. THE BENCHMARK-SAFETY FACTOR METHOD FOR COMPUTING ADIs In this section we examine the implications of modifying the NOEL-SF method for cal culating ADIs by replacing the NOEL by a "benchmark dose" (BD) calculated using the methods described in the last section. A BD is defined as a lower statistical confidence limit for the dose corresponding to a specified in crease in level of health effect over the back ground level. The increased level of effect upon which the BD is based would be near the lower limit of the experimental range; i.e,, near the lower limit of increases in health effects which can be measured with reasonable accuracy in toxicological studies. This value is estimated to be something on the order of a 10% change from background at typical sample sizes. Benchmark doses calculated in this fashion will have several advantages/ over NOELs. They will reflect the dose-response pattern to a much greater degree than NOELs. They will also make more reasonable use of sample size (larger experiments will tend to produce larger BDs. which is not true of NOELs). It will not be necessary to define a NOEL in order to determine an ADI. Because these BDs cor respond to risks in the experimental range, their value will not depend strongly upon the particular dose-response model used in their calculation. For quantal data we define the BD to be a dose d which corresponds to a specified value for the extra risk ( W ) - P(0)]/[l - P(0)1. Extra risk can be interpreted as the probability of an effect at dose d given that no effect would have occurred in the absence of the dose. This interpretation is valid irrespective of whether there is independent action between the backj ground and the stimulus. Extra risk places greater weight upon the same increase in rate for a common lesion than for a rare lesion. For example, it takes an increase of 9% above the background level for a lesion with a 10% background rate to attain a 10% extra risk compared to only a 5% increase for a lesion with a 50% background rate. Because of this property, some may prefer using the additional risk P{d) -- P{0) to extra risk. For continuous data we define the BD to be a dose d which corresponds to a specified amount of absolute change in the mean value relative to the mean value in the absence of the dose--i.e., the dose d corresponding to a specific value for the "extra response" m(d) --m(0) m (0 ) Other terms, such as the standard error of the responses in control group, could be used in the denominator in place of m{0) to normalize this expression. fa 10% change sample sizes, in this fashion over NOELs. onse pattern to >ELs. They will : of sample size produce larger iLs). It will not IL in order to hese BDs corimental range, jngly upon the ;1 used in their the BD to be a specified value TO)]. the probability no effect would fthe dose. This ive of whether ween the backtra risk places ncrease in rate ' a rare lesion, se of 9% above on with a 10% .0% extra risk se for a lesion lecause of this ; the additional 'tine the BD to s to a specified .he mean value the absence of sponding to a sponse" xd error of the uld be used in J) to normalize DETERMINING ALLOWABLE DAILY INTAKES TABLE l - Quantal Data Used to Illustrate Quantitative Dose-R esponse Methodology Ethyleneihiourea (ETU) (Khera, 1973) Fetal anomalies in rats Doses (mg/kg) No. affected/total No. 0 0/167 5 0/132 10 1/138 20 14/81 2,3.1.8~Tetrachlorodibenzo-p-dio.xin (TCDD) (Khera and Ruddick, 973) intestinal anomalies in rat fetuses Doses (jig/kg) No. afTected/total No. 0 0/24 0.125 0/38 2.3,7.8-Telrachlorodibenzo-p-dioxin (TCDD) (Murray et ai. 1979) Rats dead at birth 0.25 1/33 Doses (jig/kg/day) No. affected/total No. 0 22/318 0.001 16/224 Hexachlorobenzene (HCB) (Khera. 1974) 14th rib anomaly in rat fetuses Doses (mg/kg) 0 10 20 No. affected/total No. 0/80 4/79 8/91 Botulinum toxin-- Type A (Food Research Inst.. Univ, of Wisconsin) (FSC. 1978) Death due to Botulinum Doses (ng) No. dead/total No. Doses (ng) No. dead/total No. .01 0/30 .034 11/30 .015 0/30 .037 10/30 .020 0/30 .040 16/30 .024 0/30 .045 26/30 40 142/178 0-5 3/31 40 15/87 .027 0/30 .050 26/30 80 24/24 i,o 3/10 0.01 17/100 60 25/96 .030 4/30 Examples of Benchmark Doses Calculated from Quantal Data To illustrate application of the benchmark approach to quantal data, we have applied it to five sets of quantal dose-response data, in volving exposures to ethyleneihiourea (ETU); 2.3.7,8-tetrachIorodibenzo-p-dioxin (TCDD) (two data sets); hexachlorobenzene (HCB); and botulinum toxin--Type A (BT-A). The data are listed in Table 1. A summary of the fits of the four models to these data is given in Table 2. Graphs of the data, along with the fit of the QPR model (which was the only model that fit all five data sets adequately) are given in Figs. 3-7. Doses corresponding to 'various levels of extra risk are furnished in Tables 3-7. Doass (mg/kg) Fig. 3. Probability of fetal anomaly in rats (Khera et at, 1977) from exposure to ETU with 90% confidence bars and best-fitting polynomial regression model. 862 KENNY S. CRUMP TABLE 2 Summary of Fits of Models to Q uantal Data in Table 1 Data Model' X1 d f p value ETU QLR 17 3 0.0007 QPR 0.0 2 1.0 QW 1.3 2 0.73 LN 0.46 2 0.93 TCDD (Khera and Ruddick. 1973) QLR QPR QW LN 0.17 2 0.92 0.014 1 0.91 0.32 3 0.85 0.23 3 0.89 TCDD (Murray et ai. 1979) HCB QLR QPR QW LN QLR QPR QW LN 00 00 00 00 0.11 2 0.95 0.09 1 0.76 0.11 2 0.95 0.31 2 0.86 D m i (ug/kg) F(G. 4. Probability o f intestinal anomaly in rat fetuses from exposure to TCDD (Khera et ai. 1973), with 90% confidence bars and best-fitting polynomial regression model. Botulinum toxin QLR QPR QW LN 7.0 4.4 162 159 8 0.54 7 0.73 8 0.00001 Y 8 0.00001 / a Code: QLR = quanta! linear regression, QPR = quantal polynomial regression. QW = quantal Weibull, LN = log-normal. data (Figs. 3 and 4). As a result, all four of the models fit these data quite well (Table 2). Predictions of the four models also agree closely down to extra risks of 0.01 but differ considerably at extra risks of 10-6. The ETU data involve a sizable number of animals and are characterized by a NOEL at 5 mg/kg followed by a steeply rising dose re sponse that reaches 100% response at 80 ppm. Each of the models except the QLR model fits these data quite adequately. Also, except for the QLR model, MLEs and lower confi dence limits for doses corresponding to various levels of increased risk computed using the various models agree rather closely for extra risks of 0.1, 0.05, and 0.01. However, doses corresponding to extra risks of 10"6 differ by larger amounts. The Khera and Ruddick (1973) TCDD data on intestinal anomalies involve a dose-related increase in response for doses larger than a NOEL of 0.125 Mg/kg. However, the confi dence intervals on responses at the experi mental doses are wider than those for the ETU O OOI at Dos [ug/kg/doy) Fig. 5. Probability of fetal death in rats from exposure to TCDD (Murray et aL. 1979). with 90% confidence bars and best-fitting polynomial regression model. y in rat fetuses )73), with 90% nial regression , all four of ;11 (Table 2). also agree ) 1 but differ s from exposure confidence bars odel. DETERMINING ALLOWABLE DAILY INTAKES 863 TABLE 3 Doses Corresponding to G iven Levels of Extra Risk for Quantal ETU Data" Dose (mg/kg) Dot*i (mg/kg) Fig. 6. Probability of 14th rib anomaly from exposure to HCB (Khera, 1974), with 90% confidence bars and best-fitting polynomial regression model. Both the Murray et al. (1979) TCDD data on postnatal survival and the Khera (1974) HCB data on rib anomalies are nearly linear and are described well by all four of the mod els. Doses calculated from the models corre sponding to extra risks of 0.1, 0.05. and 0.01 are also in close agreement. The data on probability of death after ex posure to Botulinum Toxin (FSC, 1978) ex- Model QLR/ QPR QW LN Extra risk 0.1 MLE 12.2 16.4 17.9 17.1 QLR:QPR QW LN 0.05 11.0 13.2 14.5 14.8 QLR/ QPR QW LN 0.01 10.1 10.2 8.9 11.2 QLR > QPR QW LN 1 X 10"* 9.9 9.4 5.9-1 4.2 " Source. Khera, 1977. * 1.5-3 means 1.5 X I0~\ 95% lower 11.8 13.9 15.7 15.3 99% lower 11.6 13.2 14.7 14.6 10.6 10.4 11.6 11.0 12.2 11.3 13.0 12.2 9.7 9.5 7.2 6.0 7.0 6.2 9.5 8.8 9.5 9.2 4.0-1 . 1.5-3* 2.9-1 2.1-1 3.1 4.7 TABLE 4 Doses Corresponding to G iven Levels of Extra Risk for Quantal TCDD Data" Dose (mg/kg) Fig. 7. Probability of death from exposure to botulinura toxin--Type A (Food Research Inst., Univ. of Wisconsin, see F&C, 1978), with 90% confidence bars and best-fitting polynomial regression model. Model Extra risk MLE 95% lower QLR QPR QW LN 0.1 4.6-1* 5.0-1 5.2-1 4.9-1 3.2-1 3.2-1 3.6-1 3.5-1 QLR 0.05 3.1-1 2.1-1 QPR 3.3-1 2.2-1 QW 3.6-1 2.0-1 LN 3.5-1 2.1-1 QLR 0.01 2.0-1 5.7-2 QPR 1.7-1 4.9-2 QW 1.6-1 4.3-2 LN 1.8-1 6.4-2 QLR QPR QW LN 1 x IO-* 1.7-1 1.3-1 1.6-3 2.0-2 7.8-3 4.9-6 4.3-6 5.4-4 - aSource. Khera and Ruddick, 1973. 64.6-1 means 4.6 X 10-1, 99% lower 2.8-1 2.8-1 3.0-1 2.9-1 1.5-1 1.5-1 1.5-1 1.4-1 3.0-2 3.0-2 2.9-2 3.0-2 3.0-6 3.0-6 2.9-6 1.1-4 864 KENNY S. CRUMP TABLE 5 Doses Corresponding to G iven Levels of Extra Risk for Quantal Murray Postnatal Survival Data* TABLE 6 Doses Corresponding to G iven Levels of Extra Risk for Quantal HCB Data" Dose (mg/kg) Dose (jjg/kg/day) Model Entra risk MLE 95% lower .99% lower QLR QPR QW LN QLR QPR QW LN QLR QPR QW LN QLR QPR QW LN 0.1 0.05 0.01 1 x 10~* 9.3-3* 9.6-3 9.3-3 4.9-3 6.2-3 5.5-3 1.6-3 2.3-3 2.0-3 8.0-4 9.6-6 6.0-5 5.3-3 Same as QLR 5.3-3 4.6-3 2.6-3 Same as QLR 2.6-3 6.M Same as QLR 5.1-4 4.1-4 5.1-8 Same as QLR 5.0-8 1.5-6 4.4-3 4.4-3 3.5-3 2.1-3 2.1-3 1.5-3 4.2-4 4.2-4 3.2-4 4.2-8 4.2-8 1.2-6 ` Source. Murray et al.. 1979. *9.3-3 means 9.3 X 10_. Model Extra risk MLE QLR QPR QW LN 0.1 21.7 22.3 22.0 21.0 QLR QPR QW LN 0.05 10.6 11.0 10.8 11.1 QLR QPR QW LN 0.01 2.1 2.2 2.2 3.4 QLR QPR QW LN 1 X IO"4 2.1-4* 2.2-4 2.6-4 5.0-2 " Source. Khera, 1974. *2.1-4 means 2.1 X 10"4. 95% lower 17.4 17.4 17.4 14.2 8.5 8.5 8.5 6.1 1.7 1.7 1.7 1.3 1.7*4 1.74 1.7-4 4.8-3 99% lower 16.0 16.0 16.0 12.2 7.8 7.8 7.8 5.3 1.5 1.5 1.5 M 1.5^4 1.5*4 1.5-4 4.1-3 TABLE 7 Doses Corresponding to G iven Levels of Extra Risk for Quantal Botulinum Toxin Data" hibit a very rapid rise in response for doses larger than the NOEL of 27 ng. Neither of the nonthreshold models--QW or LN--fit these data. However, both the QLR and QPR models fit quite adequately (Table 2). All four of the models give comparable doses corre sponding to given extra risk levels, even down to levels of extra risk of 10~6. Table 8 compares NOELs with BDs cor responding to three levels of extra risk. With the exception of the HCB data, the NOELs generally correspond to the BDs for extra risks between 0.01 and 0.05. However, these were all reasonably large studies and involved effects not seen in control animals; for smaller studies or for effects which can occur spontaneously, NOELs are liable to be larger relative to the BDs. The data for HCB illustrate a particular ad vantage the benchmark approach has over the Model Extra risk MLE QLR QPR QW LN 0.1 3.0-2* 3.0-2 3.3-2 3.3-2 QLR 0.05 2.9-2 QPR 2.8-2 QW 3.0-2 LN 3.1-2 QLR 0.01 2.9-2 QPR 2.7-2 QW 2.4-2 LN 2.9-2 QLR QPR QW LN 1 x 10~* 2.9-2 2.7-2 7.3-3 2.0-2 0Source. FSC. 1978. *3.0-2 means 3.0 X 10_. Dose (ng) 95% lower 2.8-2 2.9-2 2.7-2 3.1-2 2.8-2 2.9-2 2.2-2 2.9-2 2.7-2 2.5-2 1.6-2 2.5-2 2.7-2 2.3-2 1.8-3 1.6-2 99% lower 2.8-2 2.8-2 2.6-2 3.0-2 2.7*2 2.6-2 2.2-2 2.8-2 2.7-2 2.4-2 1.5-2 2.4-2 2.6-2 2.2-2 1.5-3 1.4-2 ; ls of Extra ta" mg/kg) % 99% ver lower .4 16.0 '.4 16.0 '.4 16.0 t.2 12.2 ;.5 7.8 :.5 7.8 :.5 7.8 >.l 5.3 .7 1.5 .7 1.5 .7 1.5 .3 l.l 7-4 1.5-4 7-4 1.5-4 7-4 1.5-4 3-3 4.1-3 els of Extra xiN Data" (ng) % ver 99% lower i-2 2.8-2 >-2 2.8-2 7-2 2.6-2 1-2 3.0-2 8-2 2.7-2 .9-2 2.6-2 2-2 2.2-2 9-2 2.8-2 7-2 2.7-2 5-2 2.4-2 5-2 1.5-2 5-2 2.4-2 7-2 2.6-2 3-2 2.2-2 8-3 1.5-3 6-2 1.4-2 DETERMINING ALLOWABLE DAILY INTAKES TABLE 8 Comparison of Benchmark Doses with NOELS for Quantal Data 865 Benchmark doses" corresponding to % extra risk Data set Dose units NOEL ETU (Khera. 1973) TCDD (Khera and Ruddick, 1973) TCDD (Murray et at. 1979) HCB (Khera. 1974) Botulinum toxin (Food Research Institute. Univ. of Wisconsin) mg/kg **8/kg Mg/kg/day mg/kg ng 5 0.125 1.0-3 ND* 0.027 a Benchmark doses = 95% lower limits derived from QPR model. * ND = not determined. 10% 13.9 0.32 5.3-3 17.4 0.029 5% 11.6 0.22 2.6-3 8.5 0.027 1% 7.2 0.049 5.1-4 1.7 0.025 NOEL approach. Since a NOEL was not de termined, the NOEL-SF method can not be used with these data to determine an ADI. However, these data would present no diffi culty in determining an ADI from a BD. Examples o f Benchmark Doses Calculated from Continuous Data Table 9 contains dose-response data on liver fat in rats after exposure to carbon tet rachloride (Alumot et ai, 1976), mean body weights in rats after exposure to hexachlorobutadiene (HCBD) (Kociba et ai, 1977), and thymus weights in rats after exposure to TCDD (Murray et ai, 1979). Figures 8-10 contain graphs of the responses and 90% con fidence intervals, along with the dose-response curve obtained by fitting the continuous poly nomial regression (CPR) model to the data. In the Kociba et ai data numbers of animals were not provided and the total number on TABLE 9 Continuous Data Used to Illustrate Quantitative Dose- R esponse Methodology Carbon tetrachloride (Alumot et ai. 976} Average liver fat in male rats Dose (ppm in diet) Ave. SE (mg/g) No. of animals 0 61.0 6.6 6 150 71.0 6.0 6 275 136 21 6 520 229 49 6 Hexachlorobmadiene (HCBD) (Kociba et ai. 1977) Mean body weight of male rats Dose (mg/kg/day) Ave. SE (gra) No. of animals 0 586 43 90 0.2 568 53 40 2.0 557 52 40 20.0 494 15 40 2.3.7,8-Tetrachlorodibenzo-p-dioxin (TCDD) (Murray et al.. 1979) Thymus weights of male offspring, f 3 generation Dose (fig/kg/day) , Ave. S E (g) No. of animals 0 0.19 0.01 5 0.001 0.19 0.06 5 0.01 0.08 0.02 4 866 KENNY S. CRUMP Carbon te et ai.. 1 HCBD(K mean b TCDD (fi, Do m i (ppm in dial) Fig. 8. Mean liver fat in rats exposed to carbon tet rachloride (Alumot et at.. 1976), with 90% confidence bars and best-fitting continuous linear regression model. Fig. 10. Mean thymus weights of male offspring, F3 generation (Murray ei aL 1979), with 90% confidence bars and best-fitting continuous linear regression model. test was assumed. Also, values reported by Kociba el ai as "s.d." were assumed to mean "s.e." As Table 10 shows, all of the models fit each of these data sets adequately. Tables 11-13 show that the estimates of doses corresponding to given levels of extra response calculated using the four models are quite similar. In fact, corresponding lower confidence limits are almost identical in Tables 12-14. In Table 11 the 95% lower limits differ by as much as a factor of 2 for an extra re sponse of 0.01 and by larger amounts for smaller values of extra response. Table 14 compares BDs with NOELs for the continuous data. Question marks are in cluded beside the NOELs because it is not clear when a NOEL has been determined. For example, although for the data for carbon tet rachloride the average liver fat in 150-ppm animals is not statistically different from that of control animals, there is an increase at 150 ppm that appears to be part of a dose-response trend (Fig. 8). For these three data sets, the BD corresponding to an extra response of 1% are roughly comparable to the NOELs. Code: * NS = to a spf 10%. It the tra< Doses Responsi Data" Mo CLR CPR CP CP (no tl CLR CPR CP CP (no tl V. DISCUSSION Fig. 9. Mean body weights in rats exposed to HCBD (Kociba et at,, 1977), with 90% confidence bars and best fitting continuous linear regression model. In this paper we have examined an alter native to the NOEL-SF approach which in volves fitting a mathematical model to toxi cological dose-response data. The model is used to define a BD, which represents a sta tistical lower limit on the dose corresponding CLR CPR CP CP (no tl CLR CPR CP CP (no i Souri ! bring, Fj mfidencc n model. ; lower Tables ts differ ttra re nts for ELs for are in is not ed. For x>n tet?0-ppm )m that : at 150 esponse >ets, the -e of l% _s. n alterlich in to toxitodel is s a sta n d in g DETERMINING ALLOWABLE DAILY INTAKES 867 TABLE 10 Summary of Fits to Models to Continuous Data in Table 9 Data Carbon tetrachloride (Alumot et dl,, 1976) Model" CLR CPR CP CP (no threshold) F statistic 0.29 0.29 0.29 1.25 df (1.20) 0 .2 0 ) (1. 20) (2. 20) HCBD (Kociba et ai. 1977), mean body weights CLR 0.14 (2, 206) CPR 0.14 (2, 206) CP 0.14 (2, 206) CP (no threshold) 0.14 (2. 206) p value NS* NS NS NS NS NS NS NS TCDD (Murray et ai, 1979) CLR CPR CP (no threshold) 0 0 0 NS NS NS a Code: CLR = continuous linear regression, CPR = continuous polynomial regression. CP = continuous power. 6NS = not significant (p value greater than 0.1). to a specific increase in risk between 1 and proach mitigates several of the problems raised 10%. It is suggested that such a BD replace in Section II concerning the NOEL-SF the traditional NOEL. We believe this ap- method. TABLE It Doses Corresponding to G iven Levels of Extra Response for Continuous Carbon T etrachloride Data" Model Extra response Dose (ppm) MLE 95% lower CLR CPR CP CP (no threshold) CLR CPR CP CP (no threshold) CLR CPR CP CP (no threshold) CLR CPR CP CP (no threshold) 0.1 0.05 0.01 0.001 141 141 141 95.6 134 134 134 68.0 129 129 129 30.8 127 127 127 9.90 102 63 67.2 47.1 94.1 37.6 44.7 29.2 87.9 9.48 17.3 9.53 86.5 1.03 4.4 1.89 e Source. Alumot et a/.. 1976. TABLE 12 Doses Corresponding to G iven Levels of Extra Response for Continuous HCBD Data on Mean Body Weights'1 Model Extra response Doses (mg/kg/day) MLE 95% lower CLR CPR CP CP (no threshold) 0.1 14.1 14.1 14.1 14.1 CLR CPR CP CP (no threshold) 0.05 7.03 7.03 7.03 7.03 CLR CPR CP CP (no threshold) 0.01 1.41 1.40 1.40 1.41 CLR CPR CP CP (no threshold) 0.001 1.40-1 1.41-1 1.41-1 1.41-1 "Source. Kociba et at., 1977. 69.14-1 means 9.14 x 10"` = 0.914. 9.14 9.14 9.14 9.14 4.57 4.57 4.57 4.57 9.14-1 * 9.14-1 9.14-1 9.14-1 9.14-2 9.14-2 9.14-2 9.14-2 86 8 KENNY S. CRUMP TABLE 13 Doses Corresponding to G iven Levels of Extra Response for Continuous TCDD Data4 Model Extra response Dose (Mg/kg/day) MLE 95% lower CLR CPR CP (no threshold) CLR CPR CP (no threshold) CLR CPR CP (no threshold) CLR CPR CP (no threshold) 0.1 0.05 0.01 0.001 2.55-3 2.55-3 6.37-3 1.78-3 1.78-3 5.53-3 1.16-3 1.16-3 1.21-3 1.02-3 1.02-3 1.96-3 1.32-3 1.32-3 1.32-3 6.61-4 6.61-4 6.61-4 1.32-4 1.32-4 1,32-4 1.32-5 1.32-5 1.32-5 0Source. Murray et at.. 1979. b2.55-3 means 2.55 X I0"J = .00255. A BD is calculated using a mathematical dose-response curve estimated from all of the dose-response data. Thus the benchmark should better reflect the shape of the dose re sponse than the NOEL. Because a benchmark represents a statistical lower limit, larger ex periments will tend on average to give larger benchmarks, thus rewarding good experimen tation. As we pointed out, NOELs have the opposite tendency. With the NOEL approach, ADIs cannot be determined until a NOEL has been established. An otherwise well-conducted experiment may therefore be considered in appropriate for calculating an ADt if no NOEL is established. In such a case, determining an ADI could require an additional'experiment, resulting in considerable additional costs and delays. On the other hand, the original ex periment might be quite acceptable for cal culating a BD. This situation is illustrated by the quantal data for HCB (Fig. 6). A BM-SF approach to setting ADIs would allow proponents of chemicals more leeway in the design of experiments than is possible under the NOEL-SF approach. With the latter method minimum sample sizes must be spec ified by the regulatory agency in order to ensure that NOELs are established to the agency's satisfaction. With a BM-SF approach, the agency would still in some cases need to specify methods for choosing the maximum dose and the sample size to be used at this dose, because otherwise important effects might not be de tected at all. Beyond this requirement, however, proponents of a chemical could be given wide latitude in selecting dose levels and sam ple sizes. Of course, the larger a study and better designed it is to estimate the BD, the higher the benchmark is liable to be. If an accurate benchmark is considered critical, the experimentors may wish to conduct a large study and consider carefully the placement of the experimental doses; otherwise, a smaller TABLE 14 Comparison of Benchmark Doses with NOELS for Continuous Data Benchmark doses4 corresponding to % extra risk Data set Dose units NOEL 10% 5% 1% Carbon tetrachloride (Alumot er ai.. 1976) HCBD (Kociba et at.. 1977) mean body weights TCDD (Murray er at.. 1979) ppm mg/kg/day jig/kg/day 150? 2.0? 0.001? 141 14.1 .0026 *Benchmark doses = 95% lower limits derived from QPR model. b ? indicates that it is doubtful whether a NOEL has been established. 134 7.0 .0012 129 1.4 .0010 stu inf< cur exp cor che che mo geu safe A con sup tha pro ben Th< wot this ben tha: mei ! wot ; con ben per ? * wer 1 of kett S ratii ben AD cuk mei cui; star fror mir the sam the fer ; A mat non elirr sibli NOEL has -conducted .sidered inf no NOEL rminingan xperiment, il costs and mginal exble for calustrated by i. vDIs would lore leeway i is possible th the latter ust be spec ter to ensure be agency's aroach, the _`d to specify m dose and )se, because ; not be de ment, howaid be given Is and sami study and .he BD. the o be. If an critical, the luct a large lacement of e, a smaller ponding to 1*% 129 1.4 .0010 DETERMINING ALLOWABLE DAILY INTAKES 869 study may be considered adequate. Any prior luctance to recommend this application stems information on the shape of the dose-response from the uncertainty as to the shape of the curve could be used in optimally designing an dose-response curves at low doses for toxic experiment Such prior information might effects iij general. Dose-response curves which , come from pilot studies or studies of similar are linear at low doses have been used to set `chemicals. Given such choices, proponents of upper bounds for low dose cancer risks (EPA, chemicals should be able to design studies 1980), This approach has been justified on the more in keeping with their needs and bud grounds that cancer mechanisms that would getary constraints without compromising safety. As an example of how experimental design considerations could be put to effective use, suppose a company knows the smallest ADI that would permit the marketing of their product. It would be simple to calculate the benchmark that would produce this ADI. They could then design an experiment that would be optimal under the assumption that this needed benchmark is in fact the true benchmark. If the true benchmark were lower produce linear dose responses at low doses appear quite plausible and those that would produce supralinear responses seem highly implausible. The low dose linearity concept could be used to determine upper limits of risks of noncarcinogenic effects as well. How ever, many of these effects appear threshold like. The assumption of a linear response could greatly overestimate risk in cases where a threshold exists. The threshold models dis cussed in this paper might be used to deter mine risks at low doses for effects which appear than what they were hoping for, the statistical methods used in calculating the benchmark would insure that human safety would not be compromised. On the other hand, if the to be threshold-like. However, we have not recommended this in this paper because of both the uncertainty as to the existence of a threshold and because these threshold esti benchmark were near that for which the ex mates are apt to differ widely depending upon periment was designed, the extra care that the specific model used. went into the design might allow the marketing The model-fitting techniques proposed here of a product that could not have been mar have fairly minimal data requirements. When keted if a less optimal design had been used. quantal data are used, the basic needs are the Since safety factors are largely arbitrary, one doses, number of animals in each group, and rationale for choosing safety factors to use with the number of these animals which are af benchmarks would be to make the resulting fected. With continuous data one needs the ADIs comparable, on average, to those cal-` doses,' number of animals in each group, the culated previously using the NOEL-SF average response in each group, and the stan method. This could be accomplished by cal dard errors of these responses. Some effects, culating benchmarks for a number of sub such as cloudy swelling of the liver, are in stances for which ADIs have been developed herently difficult to quantify and are normally from the NOEL-SF method, and then deter classified qualitatively, such as by present/ab- mining the safety factor that, when applied to sent or mild/severe. Even for effects which are the benchmarks, would on average yield the quantifiable, the data needed to apply dose- same ADI. Of course, ADIs calculated using response methods are frequently not reported the NOEL-SF and BM-SF methods could dif in the literature. Thus, it will not be possible fer appreciably in specific cases. to apply these methods universally. However, Although we have not discussed the use of the introduction of these methods would en mathematical models for extrapolation of courage more complete presentation of data, noncarcinogenesis data to low dose and thus as well as generally encouraging the use of eliminating safety factors, this is another pos quantitative methods in toxicology. sible application of these methods. Our re It should be kept in mind that determining 870 KENNY S. CRUMP ADIs does not involve purely statistical meth ods. Toxicological evaluation of data on nu merous species and biological endpoints may be required. Included in the many consider ations should be differences in species sensi tivities to various chemicals and the need for affording different levels of protection for dif ferent toxicological effects. The statistical methods proposed in this paper should be useful in this process but they should not sup plant a careful toxicological evaluation of all the data. APPENDIX Description o f Maximum Likelihood Procedures Likelihoodfor Quantal Data Consider an experiment with g dose levels dl t . . . , dg, and let V, and X, be, respectively, the number of animals tested and the number of animals affected at the ith dose level. Let P{d) be the probability of a response at a dose d. Assuming that Xi has a binomial distri bution with parameter and P(di}t the like lihood of the data can be written as L= n /-I - P(dA*-x, Likelihoodfor Continuous Data Consider an experiment with g dose levels dlt . . . , d%; let W(-be the number of animals in the Ah dose group, and let x i}yj = 1, . . . , N i f i - 1 , . . . . g represent the response of the j th animal in the ith dose group. It is assumed that x u has a normal distribution with mean m(dj) and variance a). The parameters in the model consist of those involved in the defi nition of m (d\ plus 8\.........ffg. Let jc*be the sample mean in the Ah dose group, i.e., Si Xi = Z XfJNi. i-1 and s? the sample variance, i.e., St sj = Z (Xij - x,)2/(Ni ~ l). Then the likelihood of the data can be_written as g L = (2ir)-*'2 n "7' e x p [-W - l)s! -1 - Nt(x - m(d,))2]/tt}. Estimation and Confidence intervals The parameters are estimated as the values which maximize the appropriate likelihood. The "likelihood method" (Cox and Lindley, 1974; Crump and Howe, 1983) is used to cal culate confidence limits. For example, when using quantal data the lower 95% limit on the dose d corresponding to an extra risk of P(d) - P(0) _ 1 - P(0) is calculated as the smallest d which satisfies P(d) - P{0) = 0.1 I ~ P(0) and 2 logtZ-tM*/!.) = (1.645)2 where ma* is the maximum value of the like lihood L. When using continuous data the same approach is followed except the formula for extra response replaces the one for extra risk. Computer Programs These methods require iterative numerical calculations. We have developed computer programs to perform these calculations and intend to have them available for the general public in the near future. ACKNOWLEDGMENTS The authors acknowledge the input received at a number of workshops conducted by the EPA Environmental Cri- teria Asses utarly the Dourson. [ ton Nelsoi Stan. Alth been funds through Ci the Agency does not n no official ACGIH (I Confcrei Ohio. ALbMOT. 3 Bondi. , and acct the rat c Cox. D. 3 Stannic Crump, K sponse n (P. F. Fi glewood Crump, k ment of ! vironme Papers. nology * Crump, K ods for trapolat Clayson Press, C EPA098C Fed. Ri 79379. ; written 0>2]/*?. e values elihood. Lindley, d to cale, when it on the k of satisfies the likedata the formula or extra imen cal imputer ons and : general t a number rental Cri- DETERMINING ALLOWABLE DAILY INTAKES 871- teria Assessment Office in Cincinnati, including partic ularly the comments of Dr. Roy Albert. Dr. Michael Dourson, Dr. Rick Hertzberg. Dr. Rolf Homing, Dr. Nor ton Nelson. Dr. Marvin Schneiderraan, and Dr. Jerry Stara. 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