Quantalization of Continuous Data for Benchmark Dose Estimation

REGULATORY TOXICOLOGY AND PHARMACOLOGY ARTICLE NO.

24, 246–250 (1996)

0137

Quantalization of Continuous Data for Benchmark Dose Estimation DAVID W. GAYLOR National Center for Toxicological Research, U.S. Food and Drug Administration, Jefferson, Arkansas 72079 Received March 18, 1996

Benchmark doses corresponding to low levels of noncancer disease risk have been proposed to replace the no-observed-adverse-effect level for establishing allowable daily intakes or reference doses. For quantal data each animal is classified with or without a disease. The proportion of animals with an adverse effect (risk) is observed as a function of dose of a toxic substance. The calculation of a benchmark dose is relatively straightforward. For continuous data a somewhat more complicated designation of risk is required. Because of the more direct procedures with quantal data, consideration could be given to converting continuous data to quantal data before estimating benchmark doses. The purpose of this paper is to compare the precision of the two approaches (use of continuous or quantalized data) for a number of sublinear dose–response curves ranging from low to high probabilities of risk at the highest dose. In these studies, five animals per dose were generally satisfactory to estimate the benchmark dose for continuous data, whereas the corresponding quantalized data generally do not perform as well even with 10 to 20 animals per dose. For quantalized data, the lower 95% confidence limits on the estimates of the benchmark dose were generally a factor of 3 to 4 below the true benchmark dose, whereas the confidence limits using the continuous data were generally within a factor of 2 of the true benchmark dose. Although the use of quantalized data for the estimation of risk is more direct, estimates of benchmark doses using the continuous data were more precise. Based on this study, converting continuous data to quantal data is not recommended. q 1996 Academic Press, Inc.

1. INTRODUCTION

Substances that are potentially toxic to humans are regulated by establishing a ‘‘safe’’ dose or allowable daily intake (ADI). For noncancer end points, the ADI or reference dose (RfD) is established by dividing the no-observed-adverse-effect level (NOAEL) observed in humans and/or animal bioassays by appropriate safety (uncertainty) factors. A description of this process is summarized by Barnes and Dourson (1). Various au-

246

0273-2300/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

AID

RTP 1067

/

6e0d$$$$41

thors have discussed difficulties associated with the NOAEL [e.g., Crump (2), Kimmel and Gaylor (3), and Gaylor (4)]. Notably, the NOAEL is limited to one of the experimental doses employed, does not make full use of the dose–response data, and is highly dependent on the number of animals used. On the latter point, the NOAEL has the undesirable property that the fewer animals used the lower the statistical power to detect effects, resulting in higher NOAEL values. Hence, poorer experiments are unjustly rewarded with higher ADI or RfD values. To avoid the shortcomings associated with the NOAEL, Crump (2) proposed a ‘‘benchmark’’ dose to replace the NOAEL. The benchmark dose (BMD) is chosen to correspond to a dose associated with a low level of risk, e.g., 1 to 10%. To account for variability in experimental data, a lower confidence limit on the BMD is used to replace the NOAEL in the determination of an ADI or RfD. Use of the lower confidence encourages better experiments to obtain tighter confidence limits resulting in higher ADI or RfD values. For quantal data, each animal is classified as normal or possessing a deleterious health effect such as a birth defect. The identification of an adverse effect for a quantal effect is straightforward. The proportion of abnormal animals as a function of dose provides a direct procedure for determining a BMD. Various dose–response models using quantal data have been compared by Allen et al. (5) for reproductive and developmental data. For continuous data, such as measurements of clinical chemistry, somewhat more indirect procedures are needed to estimate risk and obtain BMD values. The estimation of risk for continuous data requires knowledge of the distribution of values. If the values are normally distributed (or lognormal), estimates of the mean and standard deviation provide all the needed information. In the absence of a biologically defined abnormal level, a range can be defined for normal and abnormal values. For example, values below the first percentile or above the 99th percentile could be designated as abnormal. In this case, for the normal distribution, values that deviate by more than 2.33 standard deviations from the mean would be considered abnor-

11-29-96 14:51:32

rtpa

AP: Reg Tox Pharm

BENCHMARK DOSE FOR QUANTALIZED DATA

mal. Then, the proportion (risk) of animals having values in the abnormal range can be estimated as a function of dose. Gaylor and Slikker (6, 7), Chen and Gaylor (8), Kodell and West (9), and West and Kodell (10) describe procedures for estimating BMD values for continuous data. Because of the more direct procedures for estimating risk with quantal data, it is worthwhile to consider converting continuous data to quantal data for calculating BMD values. Instead of assuming normality and estimating the standard deviation for each dose group, the values initially could be designated as abnormal or not and a quantal analysis conducted. However, in the absence of a biological definition of abnormal, this would still require statistically defining an abnormal range. In this paper, the effect of initially dichotomizing continuous data and performing a quantal analysis to calculate BMD values will be compared using the original continuous data. 2. METHODS

To illustrate the effect of converting continuous measurements to quantal responses for estimating the BMD, a simple nonlinear dose response is chosen that mimics the upward curvature observed at low doses for many biological effects, ln y Å b0 / b1d

247

or the average background level can be thought of as 100% of the typical level in unexposed animals. It is assumed that the values of ln y in a dose group are normally distributed with a standard deviation denoted by s. The effects of dichotomizing continuous data for estimating the BMD are investigated for values of b1 Å ks with k Å 1, 2, and 4, representing small (1 standard deviation) to large changes (4 standard deviations) in the mean value of y at the largest dose. All nine combinations of n Å 5, 10, and 20 and k Å 1, 2, and 4 are investigated. Since adverse levels for continuous measurements often are not available for experimental animals, an abnormal cutoff will be chosen so that the proportion of abnormal animals in the control group is 0.01 or 0.05. The levels of excess risk at the BMD are investigated for 0.05 and 0.10, resulting in 2 1 2 Å 4 different BMD values for each of the nine cases described above. The abnormal values of ln y corresponding to proportions of 0.01 and 0.05 are ln y Å Zas, where Za Å 2.327 and 1.645, respectively, are the standard normal deviates. Suppose a high value of y is considered an adverse biological effect. If the abnormal cutoff is chosen so that 1% of the control animals are considered abnormal, values of ln y greater than 2.327s are designated as abnormal. For example, the BMD corresponding to an excess risk of 0.05 occurs when 6% of the measurements are above 2.327s. This occurs when the mean value of ln y is Zb Å 1.555 standard deviations below 2.327s. That is, ln y Å (2.327 0 1.555)s Å 0.772s. The true BMD is the dose d where

or ln y Å b1d Å ksd Å 0.772s y Å e(b0/b1d) giving where y is the biological effect, d is dose, and b0 and b1 are parameters estimated from bioassay data. For the above log-linear model, the rate of change in y as a function of dose is proportional to the level of y. That is, the derivative is Ìy/Ìd Å b1y. The log-linear model was used by Gaylor and Slikker (6) to describe neurotoxic effects. Further biological effects frequently are log-normally distributed (6). If so, then ln y(logey) is normally distributed. Common experimental bioassay designs are considered with control animals and three doses spaced logarithmically. Without loss of generality, the highest dose is set equal to 1. For the design investigated, the doses are 0, 14, 12, and 1. Three different sample sizes are used: n Å 5, 10, and 20 animals per dose. The smaller sample sizes are typical of subchronic toxicological bioassays and the larger sample sizes are typical of reproductive and developmental bioassays. The total number of animals is N Å 4n. For the underlying dose–response curve, b0 is set equal to 0. Hence, at d Å 0, the value of y Å 1. This is equivalent to a relative risk of 1 for the control animals

AID

RTP 1067

/

6e0d$$$$41

11-29-96 14:51:32

BMD Å 0.772/k. The other three BMD values corresponding to the various combinations of abnormal proportions and excess risk can be calculated in a similar manner. Confidence Limits for the BMD Using Continuous Data For the dose–response model ln y Å b0 / b1d, the BMD occurs at the dose d where (Za 0 Zb)s Å b0 / b1BMD Å b0 / ksBMD where Za and Zb are the normal deviates chosen to give the designated proportion of abnormal control animals

rtpa

AP: Reg Tox Pharm

248

DAVID W. GAYLOR

and the designated excess risk level at the BMD. Replacing b0 by b*0s and setting Z Å (Za 0 Zb) gives

i.e., better relative precision. For continuous data with the four-dose experimental design,

BMD Å (Z 0 b*0)/k.

L/BMD Å 1 0

Note that under the above formulation the BMD does not depend upon the standard deviation; s is simply a scalar. In fitting the model to experimental data, the BMD is estimated by BMO D Å (Z 0 bO *0)/kO where P denotes an estimate. To satisfy the condition of no excess risk for the control animals, the true response at d Å 0 is taken to be bO *0. Thus, the uncertainty in the estimate of the BMD is dependent only on the estimate of k. In calculating a reference (allowable) dose, a lower confidence limit (Lˆ) for the BMD is used. One choice is LO Å BMO D 0 tVO 1/2, ˆ is the estimated variance of BM ˆ D and t is where V Student’s t corresponding to the selected confidence level with (N 0 2) degrees of freedom. ˆ D is approximated by The variance of BM VÅ

S

D

d BMD 2 V(kO ), dk

where d denotes the derivative and V(kˆ ) is the variance of kˆ . Hence, VÅ

S

D

S D

Z 0 b*0 2 BMD 2 O V(k ) Å V(kO ). k2 k

For normally distributed ln y with constant s and the dose response passing through the mean of the control animals,

/

6e0d$$$$41

k 1.3125n

.

Confidence Limits for the BMD Using Dichotomized Data For each of the 3 k values 1 3 n values 1 2 abnormal levels 1 2 risk levels Å 36 combinations of conditions under investigation, the proportions of animals with abnormal values of ln y can be estimated for each dose level. When the data are used in dichotomized form, a value is considered abnormal if it exceeds Zas, where Za is the standard normal deviate corresponding to the proportion of control animals defined to be abnormal. At a dose d, the mean value is b1d Å ksd. The probability that an animal exceeds the abnormal cutoff at dose d is the probability that the standard normal deviate exceeds (Zas 0 ksd)/s Å (Za 0 kd). These proportions range from 0.01 in control animals up to 0.99 at the highest dose for the steepest slope (k Å 4). The expected proportions (P) for a given set of conditions can be fit to a polynomial exponential model, P Å 1 0 exp[0(q0 / q1d / q2d2 / q3d3)], using the computer program GLOBAL82 (11). The integer closest to nP is used for the number of abnormal animals. From this analysis, the lower 95% confidence (L) based on the expected tumor proportions can be calculated from which the relative precision, L/BMD, can be obtained for dichotomized data for each of the 36 conditions. 3. RESULTS

where kˆ is the least-squares estimate of k. For the experimental design with d Å 0, 14, 12, and 1, ( d2 Å 1.3125n. The expected lower 95% confidence limit for BMD is L Å (BMD 0 tV1/2), where t Å 1.73, 1.69, and 1.66 for n Å 5, 10, and 20 animals per dose, respectively. The ratio of L/BMD provides a measure of the relative precision of the estimate of the BMD. The larger the ratio, the closer the confidence limit is to the BMD,

RTP 1067

t

The relative precision increases as the slope of the dose response increases, i.e., as k increases and as the number of animals per dose, n, increases. It is interesting to note for continuous data that the relative precision is not dependent on the choice of the cutoff for the abnormal level, i.e., on the proportion of abnormal animals among the controls or on the excess risk at the BMD.

V(kO ) Å 1/( d2,

AID

q

11-29-96 14:51:32

ˆ D is BMD Å (Za Recall that the expected value of BM 0 Zb)/k. The values of Za and Zb for the four conditions are provided below. Proportion abnormal

Excess risk at BMD

Za

Zb

(Za 0 Zb)

0.01 0.01 0.05

0.05 0.10 0.05

2.327 2.327 1.645

1.555 1.227 1.282

0.772 1.100 0.363

rtpa

AP: Reg Tox Pharm

249

BENCHMARK DOSE FOR QUANTALIZED DATA

TABLE 1 Relative Precision: Expected Lower 95% Confidence Limits for an Estimated BMD Divided by the BMD, Where the Proportion of Abnormal Animals in the Controls Was Selected to Be 0.01 or 0.05 and the Excess Risk at the BMD Is 0.05 or 0.10 Dichotomized data Proportion abnormal Å 0.01 k

n

Continuous data

1

5 10 20 5 10 20 5 10 20

0.32 0.53 0.68 0.66 0.77 0.84 0.83 0.88 0.92

2

4

0.05

0.10

1.645

1.036

Risk Å 0.05

Risk Å 0.10

Risk Å 0.05

Risk Å 0.10

0.28 0.39 0.42 0.25 0.31 0.36 0.23 0.25 0.34

0.40 0.55 0.59 0.36 0.45 0.51 0.32 0.36 0.48

0.21 0.29 0.34 0.20 0.24 0.32 0.16 0.23 0.31

0.25 0.30 0.41 0.24 0.29 0.40 0.20 0.29 0.38

0.609

For continuous data, the relative precision (lower confidence limit of the estimate of the BMD divided by the BMD) does not depend upon the cut-off level selected to identify abnormal control animals or on the excess risk at the BMD, L/BMD Å 1 0

q

t

k 1.3125n

,

where d Å 0, 41, 21, and 1. These values are displayed in Table 1. For example, a shallow dose–response slope of 1 standard deviation (k Å 1) across the dose range and n Å 5 animals per group results in t Å 1.73, L/BMD Å 0.32 (Table 1). For this case, the lower 95% confidence limit is a factor of 1/0.32 Å 3.1 below the BMD. For the dichotomized data, the proportion of animals exceeding the abnormal cutoff for each dose is used to obtain a relationship between risk and dose. From this dose–response relationship the BMD is estimated and a lower confidence limit is obtained (Howe and Crump, 1982) for the estimate of the BMD. The probability that an animal exceeds the abnormal cutoff is given by the probability of exceeding the standard normal deviate z Å (Za 0 kd). For the case where 1% of the control animals are defined to be abnormal, Za Å 2.327. When k Å 1, the values of Z are 2.327, 2.077, 1.827, and 1.327 at d Å 0, 14, 12, and 1, respectively. The probabilities of exceeding these values are 0.010, 0.019, 0.034, and 0.092, respectively. The expected number of abnormal animals is obtained by multiplying by the sample size n. For use in curve fitting with GLOBAL82, the expected number of abnormal animals is rounded off to the nearest integer and the denominator adjusted to give the correct proportion. For example, with n Å 20, the expected numbers of abnormal animals are 20 1

AID

RTP 1067

/

6e0d$$$$41

Proportion abnormal Å 0.05

11-29-96 14:51:32

.010 Å 0.20, 20 1 0.019 Å 0.38, 20 1 0.034 Å 0.68, and 20 1 0.092 Å 1.84 at d Å 0, 14, 12, and 1, respectively. Rounding these to the nearest integer, the numbers of abnormal animals at each dose are 0, 0, 1, and 2. The proportions used for fitting the polynomial–exponential model are 0/20, 0/20, 1/29.4 Å 0.034 and 2/21.7 Å 0.092 for d Å 0, 14, 12, and 1, respectively. For this case (k Å 1, n Å 20, proportion abnormal Å 0.01), L/BMD Å 0.42 and 0.59 for excess risks of 0.05 and 0.10 at the BMD, respectively (Table 1). The sample sizes are not all equal to 20, but maintained to be as near 20 as possible with integer values for the numbers of abnormal animals. 4. DISCUSSION

The gain in relative precision from increasing the number of animals per dose from 5 to 20 is generally between a factor of 1.5 and 2, except for continuous data with k § 2 where the relative precision is already high. There is little gain in the precision from doubling the numbers of animals per dose. The relative precision is up to a factor of 1.5 better when the proportion of abnormal animals for the controls is set equal to 0.01 rather than 0.05 when the data are dichotomized. The relative precision is better when the excess risk at the BMD is 10% rather than 5% for dichotomized data. This would be akin to a low-observed-adverseeffect level (LOAEL) and would require an additional uncertainty (safety) factor for setting a reference (allowable) dose. As noted before, when the continuous data are used, the relative precision is not affected by the level chosen for the proportion of abnormal animals or the excess risk at the BMD.

rtpa

AP: Reg Tox Pharm

250

DAVID W. GAYLOR

A comparison of the ratios of L/BMD based on continuous data and quantalized data indicates a clear advantage for using continuous data. Samples sizes of 5 animals per dose perform reasonably well with continuous data, whereas sample sizes using the corresponding quantal data generally do not perform as well with n Å 10 or even n Å 20 animals per dose. That is, confidence limits for BMD are tighter for continuous data, resulting in larger limits (L values) and hence larger acceptable daily intakes or reference doses for regulation generally on the order of two- to threefold, where k Å 2 or 4. When k Å 1, using the continuous data generally results in larger L values, up to a factor of 2 higher than those based on the dichotomized data. Part of the poorer performance of the dichotomized data could be due to the choice of the multistage model. When the logistic model is used for the dichotomized data, the results (not shown) are similar. In general the L values for the dichotomized data are higher for the logistic model than for the multistage model. Hence, the differences from the continuous data are not quite as great with the logistic model. When k Å 2 or 4, the L values are generally on the order of 1.5 to 3 times higher using the continuous data. When k Å 1, use of the continuous data generally results in L values up to a factor of 1.4 higher than those based on dichotomized data using the logistic model. Although the use of quantal data for the estimation of risk is more direct, the estimates of BMD based on the continuous data are generally much more precise. Since continuous data contain more information than their quantal counterpart, converting continuous data to quantal data is not recommended.

AID

RTP 1067

/

6e0d$$$$41

11-29-96 14:51:32

ACKNOWLEDGMENT The author appreciates the assistance of Susan Taylor in performing the calculations for Table 1.

REFERENCES 1. Barnes, D. G., and Dourson, M. (1988). Reference dose (RfD): Description and use in health risk assessment. Regul. Toxicol. Pharmacol. 8, 471–486. 2. Crump, K. S. (1984). A new method for determining allowable daily intakes. Fundam. Appl. Toxicol. 4, 854–871. 3. Kimmel, C. A., and Gaylor, D. W. (1988). Issues in qualitative and quantitative risk analysis for developmental toxicology. Risk Anal. 8, 15–20. 4. Gaylor, D. W. (1992). Incidence of developmental defects at the no observed adverse effect level (NOAEL). Regul. Toxicol. Pharmacol. 15, 151–160. 5. Allen, B. C., Kavlock, R. J., Kimmel, C. A., and Faustman, E. M. (1994). Dose–response assessment for developmental toxicity: III. Statistical models. Fundam. Appl. Toxicol. 23, 496–509. 6. Gaylor, D. W., and Slikker, W., Jr. (1990). Risk assessment for neurotoxic effects. NeuroToxicology 11, 211–218. 7. Gaylor, D. W., and Slikker, W., Jr. (1994). Modeling for risk assessment of neurotoxic effects. Risk Anal. 14, 333–338. 8. Chen, J. J., and Gaylor, D. W. (1992). Dose–response modeling of quantitative response data for risk assessment. Commun. Stat. Theory Methods 21, 2367–2381. 9. Kodell, R. L., and West, R. W. (1993). Upper confidence limits on excess risk for quantitative responses. Risk Anal. 13, 177–182. 10. West, R. W., and Kodell, R. L. (1993). Statistical method of risk assessment for continuous variables. Commun. Stat. Theory Methods 22, 3363–3376. 11. Howe, R. B., and Crump, K. S. (1982). GLOBAL82: A Computer Program to Extrapolate Quantal Animal Toxicity Data to Low Doses. ICF Clement Assoc., Crump Division, Ruston, LA.

rtpa

AP: Reg Tox Pharm