22 Statistical methods for reproductive risk assessment

P. K. Sen and C. R. Rao, eds., Handbook of Statistics, Vol. 18 © 2000 Elsevier Science B.V. All rights reserved.

,~t.~

Statistical Methods For Reproductive Risk Assessment

Sati Mazumdar, Yikang Xu, Donald R. Mattison, Nancy B. Sussman and Vincent C. Arena

1. Introduction

Recently attention has been directed toward the normal processes which influence reproduction (Baird et al., 1999) as well as those risks which impair reproduction and development (Wilcox et al., 1988; Wilcox et al., 1990; Schardein, 1993; Samuels et al., 1995; Baird and Wilcox, 1985; Mattison, 1982; Weinberg et al., 1989; Ratcliffe et al., 1992). For example, exposures to unscavanged nitrous oxide (concentrations estimated between 100 and 1000 ppm) for five or more hours per week decreases fertility to less than half that observed in dental offices where the anesthetic gas is scavenged (Rowland et al., 1992). Several studies, including a recently concluded industry wide evaluation, have demonstrated an increased risk of spontaneous abortion among women working in the fabrication of semiconductors (Eskenazi et al., 1995a, b; Swan et al., 1995; Schenker et al., 1995). The US Environmental Protection Agency, in its reanalysis of the human health risk assessment of dioxin, has suggested that adverse reproductive and developmental effects may be the most critical and sensitive endpoints for this toxicants (Environmental Protection Agency, 1997). In order to assess the impact of an environmental exposure on health such as reproduction, alternately fertility, biomarkers are frequently used as surrogates for the endpoints in an analysis for determining an exposure risk (Baird et al., 1999). Biomarkers are measurable physical or chemical parameters affected by the exposure and associated with the health endpoint (National Research Council, 1989). Because reproduction/fertility requires multiple biological processes, more than one biomarker may be required for complete identification of the toxicity of the exposure (Baird et al., 1999). This article describes first reproductive toxicology, biomarkers of reproduction, and currently available statistical methods for characterizing reproductive toxicity. It then presents a recently developed method in the context of quantitative risk assessment - to utilize several biomarkers characterizing the same 649

650

S. Mazumdar, Y. J(u, D. R. Mattison, N. B. Sussman and V. C. Arena

hazard, and the possible recovery mechanisms of the affected biomarkers by extending the quantitative reproductive risk estimation (QRRE) method (Meistrich and Brown, 1983). Inferential procedures for this extension include nonparametric density estimation and multivariate mixed-effects models for longitudinal data. The methods are illustrated in the context of the reproductive effects of the pesticide 1,2,-dibromo-3-chloropropane for humans using data from a toxicologic inhalation fertility study for rabbits (Rao et al., 1980, 1982), a fertility study for humans (MacLeod et al., 1951a,b,c), and a health risk assessment study (Pease et al., 1991; Reed et al., 1987).

2. Reproductive toxicology and biomarkers of reproduction Reproductive toxicity includes any adverse effect on the male, female, or couple resulting from exposures which produce alterations in sexual behavior, reduced fertility, adverse pregnancy outcome, or modifications in other functions dependent on the reproductive system (Environmental Protection Agency, 1994). As suggested by this, successful reproduction entails events in men and women which precede recognized pregnancy, as well as critical events in the fetus and placenta. It is these interactions which must be considered in risk assessment of reproductive toxicity and for which biomarkers of reproduction and development are so valuable (National Research Council, 1989). A biomarker is a representative biological sign or signal of an event in a biological system or biological sample. The appropriate use of biomarkers should make it possible to determine the association between xenobiotic exposure and transient and permanent impairments of reproductive or developmental health (Mattison, 1991). Defining and validating a range of biomarkers from exposure to disease, (e.g., biomarkers of susceptibility, biomarkers of external dose, biomarkers of internal dose, biomarkers of the biological effective dose, biomarkers of early and late biological responses, biomarkers of altered function, and biomarkers of disease), may help establish the sequence of processes in the relationship between exposure and disease (Baird et al., 1999). Commonly used endpoints for reproduction are male fecundity, female fecundity and couple specific factors. Male fecundity is considered to be the measure of the ability of a male to fertilize, or the reproductive effectiveness of a male. Biomarkers which may be applicable for the characterization of male fecundity include ejaculate volume, ejaculate composition, sperm number in the ejaculate, sperm motility, sperm morphology, and anatomical or genetic factors. Female fecundity is a measure of the ability of a female to be fertilized following delivery of sperm to the reproductive tract, or the reproductive effectiveness of a female. Biomarkers that appear appropriate for characterizing female reproductive function include ovulatory frequency, hormones which describe follicular phase characteristics or luteal phase characteristics, parameters which describe endometrial function or the functional characteristics of the Fallopian tubes (Baird et al., in press). Couple-dependent factors may include behavioral parameters

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which are summarized by the frequency and timing of intercourse, male-female interactions, female-male interactions, and the repair of sperm and oocyte D N A damage by oocyte D N A repair enzymes. Once functions of biomarkers which describe male fecundity, female fecundity, couple-dependent factors and early pregnancy loss are defined and validated, it is possible to calculate the reproductive risk as some function of the individual and couple-dependent biomarker functions. This reproductive risk can be expressed by various measures of fertility, such as, cycle specific fertility rate or the percent of couples who fail to conceive after some interval of intercourse. Changes in these measures as a result of exposure to reproductive toxicants provide quantitative estimates of the reproductive risk.

3. Current statistical methods for characterizing reproductive toxicity Over the past decade, several statistical methods have been explored to characterize reproductive risks. We describe below three such methods that are of current use in the risk assessment arena.

No observed adverse effect level-safety factor (NOAEL-SF) method Initially the focus of risk assessment for reproductive toxicity was the identification of the no observed adverse effect level (NOAEL) and the application of a safety factor (SF), ranging from 1 to 1000, to the N O A E L to determine the allowable daily intake (ADI) (Crump, 1984). This approach assumes thresholds for reproductive toxicity. To determine the NOAEL, some investigators have simply identified the highest dose for which the response in the exposed group is not statistically different from that of the control. The N O A E L - S F method does not use all available data. Others have used Tukey's trend test procedure (Tukey et al., 1985) based on the notion of no-statistical-significance of trend (NOSTASOT) dose. The N O A E L identified in a study may be influenced by the sample size in the study, the selection of dose levels and their position on the dose-response curve. Moreover, a study's statistical power determines the potential rate of adverse response that may be associated with a NOAEL. Benchmark dose (BD) method The benchmark dose (BD) is defined as a statistical lower confidence limit on a dose associated with a predetermined change in the response from the control group. The BD is calculated from all the dose-response information. The BD is divided by a safety factor (SF), which can range from 1 to 1000, to determine the reference dose (Barnes and Dourson, 1988). It provides a starting point for the derivation of regulatory level which represents a known level of reproductive risk. While this approach has great appeal as it uses all of the data, it requires statistical analysis for the determination of the dose-response relationship. This analysis may be difficult as concerns exist about the minimum effect which can be

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S. Mazumdar, Y. Xu, D. R. Mattison, N. B. Sussman and V. C. Arena

detected with sufficient power in the presence of between and within individual variations.

Quantitative reproductive risk estimation (QRRE) method In contrast to the above two methods, the quantitative reproductive risk estimation (QRRE) method (Meistrich and Brown, 1983) calculates the increased incidence of infertility in a human population resulting from exposure of males to a toxic agent. The steps in the Q R R E method follow the standard protocol for health risk assessment (National Research Council, 1983) with two exceptions. The standard protocol requires: (1) hazard identification; (2) hazard characterization (doseresponse assessment); (3) exposure assessment; and (4) risk characterization. In the Q R R E method the step requiring the identification of the hazard is assumed to have already taken place and an additional step consisting of the application of the interspecies extrapolation factor, which would otherwise be grouped with the uncertainty factor applied in the dose-response assessment, is added. The first step is to obtain from experimental animal data a dose-response curve for the toxicant induced alterations of a reproductive biomarker. The second step involves extrapolating the dose between species using the Interspecies Extrapolation Factor (IEF) defined as: IEF -

Dose necessary to produce a given change in test animal biomarker Dose necessary to produce equivalent change in human biomarker "

The third step requires that the level of human exposure to the reproductive toxicant be determined by some accepted technique (National Research Council, 1983; National Research Council, 1989). The human exposure assessment conducted in this step provides information such as: (1) the number of exposed individuals; (2) the number of men at risk; and (3) dose per unit of body weight or body surface area. The second and third steps are used together to calculate the alterations in the human biomarker in the exposed population. The final step uses this information and the two-distribution model (Meistrich and Brown, 1983) to calculate the increase incidence of infertility in the exposed population. This method is described below in detail. The two-distribution model and the calculation of incidence of infertility: In its simplest form, the distribution of a single reproductive biomarker X in a population can be expressed as a weighted sum of the corresponding distributions in the infertile and fertile populations. If we denote by a(x) and b(x) the distributions of the biomarker X in the fertile and the infertile populations, respectively, then the distribution of X in the entire population can be written as

c(x) ---- (1 - p)a(x) + pb(x)

(3.1)

where, p is the proportion of infertile couples in the general population. The value o f p is usually taken as 0.15, because about 15 % of unsterilized married couples in

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which the woman is 15 to 44 years old are unable to conceive within 1 year of unprotected intercourse (Mosher 1985; Mosher et al., 1985). The risk, f(x), that a man with a biomarker value x is from an infertile couple, which is the probability that he falls within the infertile group, is given by

f(x) = pb(x)/c(x) .

(3.2)

The incidence of infertility in a population is given by the expected value o f f ( x ) . This expected value can be calculated by integrating (over x) the frequency of occurrence of each value of x times f(x). If the population is not exposed, the expected value o f f ( x ) is p as noted above. If the exposure to the toxicant alters the distribution of X, the altered distribution is used for the calculation of this expected value of f(x). Denoting by c* (x) this altered distribution, this expected value i.e., the new incidence following exposure, is given by p* = E~(x)] = f f ( x ) c * ( x ) e ~

.

(3.3)

The distributions a(x) and b(x) are the key components of the two-distribution model. As data providing the empirical distributions, a(x) and b(x), are mostly found to be sparse, smoothing of these empirical distributions are needed to obtain these distributions over the entire range of x. To determine these smooth representations of a(x) and b(x), continuous distributions can be fitted to available data. Once c(x) is obtained from a(x), and b(x) and p, c*(x) can be derived either analytically or numerically. The derivation of c* (x) is simpler ifx values for all individuals in the population are reduced by a reduction factor s, i.e., x* = x/s. In that case, the incidence of infertility (p*) can be calculated as a function of s (Meistrich, 1992). Instead of using c* (x), an alternative approach uses individual sperm measure in a group of exposed men and a sum of discrete values instead of an integral as in Eq. 3.3. If the sperm measures of n exposed men are xi, i = 1,2,... ,n, then the incidence of infertility in the exposed population is

p* = ~ f ( x i ) / n

,

(3.4)

where, the sum is over all i = 1 , 2 , . . . , n. In both cases, the difference between p and p* measures the increase in the incidence of infertility due to the toxicant.

Semi-parametric mixture models in fertility studies Statistical models for fertility studies using female biomarkers have two major components. These are menstrual cycle length and timing of intercourse. The utility of ovarian cycle length as a biomarker of female fertility in experimental animals has recently been confirmed by analysis of a large number of chemicals in the continuous breeding protocol (Chapin et al., 1997; Chapin and Sloane, 1997). Also, there must be intercourse near the time of ovulation (Wilcox et al., 1995). It has been recognized that some menstrual cycles are "viable" and some are not,

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S. Mazurndar, Y. Xu, D. R. Mattison, N. B. Sussman and V. C. Arena

where "viability" is determined by whether hormonal, uterine, and gameterelated factors are favorable to gestation. The statistical methods consists of modeling the probability of conception accounting simultaneously for cycle viability, timing of intercourse relative to ovulation and effects of extraneous factors such as cigarette smoking and toxic exposures. The basic model for conception specifies that the probability of conception for a given menstrual cycle ij is Pr(Yij = 1/ {Xijk } ) = Aij[1 - (1 -- Pijk )Xijk] ,

(3.s)

where Aij is the probability that the cycle is viable and pijk can be interpreted as the conditional probability that conception would occur if there were sexual intercourse only on day k, given that the cycle is viable. Recently, there has been some methodological advancement in modeling the probability of conception by the introduction of semi-parametric mixture distributions to account for the variability associated with the different components of a fertility model (Zhou and Weinberg, 1999). The mixing distribution, the component that introduces the heterogeneity among the menstrual cycles that come from different couples, is characterized nonparametrically by a finite number of moments. The second component, the intercourse-related probability is modeled parametrically to assess the possible covariate effects (Zhou and Weinberg, 1999). The parametric model can be incorporated by a suitable link function between the covariates and Pijk. The paper discussed an EM algorithm based estimating procedure that incorporates the natural order in the moments for inference. The semi-parametric mixture model has robustness properties and has overcome the criticisms of an earlier work (Zhou et al., 1996) where the mixing distribution for the cycle viability was modeled as a beta distribution. Though the application of such a parametric assumption was seen (Sheps and Menken, 1973), the choice of the beta distribution was mainly for mathematical convenience.

4. Extensions of the QRRE method

This article presents two recent extensions of the QRRE method (Xu, 1996). The first extension allows the QRRE method to use several biomarkers characterizing the same endpoints simultaneously and calculates the infertility risk from their joint change due to exposure. The second extension addresses the possible recovery mechanisms of biomarkers and provides an approach to calculate the reproductive risks at selected points of time during the observation period encompassing both exposure and post-exposure periods thus allowing the estimation of both acute and chronic effects. In the mathematical formulation of the first extension of the QRRE method, the biomarkers are used as a vector denoted by X. Hence, the calculation of the risk (Eq. 3.3) requires multiple integration. The analytical and computational difficulties lie in obtaining the smooth forms for a(x), and b(x), the joint distri-

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butions of the biomarkers for the fertile and infertile couples. These can be obtained from available data, and by determining c*(x), the joint distribution of the altered biomarkers in the population. Nonparametric density estimation methods (Scott, 1992) are considered to be practical and feasible to obtain smoothed forms of the distributions of fertile, a(x), and infertile, b(x), males. As the available data for the estimation of these distributions are most likely to be sparse, choosing and fitting multivariate continuous functions are usually difficult. Once these two joint distributions are determined over a grid of values of the biomarkers, the distribution for the population, c(x), can be obtained numerically, as a weighted sum of a(x) and b(x). Then f(x), the risk of infertility, is calculated as: f(x) = pb(x)/c(x) .

(4.1)

If the post-exposure biomarkers x* can be expressed as a function of the preexposure biomarkers x, i.e., if x* = h(x), where h(x) is differentiable function, then c* (x) can be obtained from c(x) using the usual method of transformation of variables. When several biomarkers are considered simultaneously, even under linear alterations in all the individual biomarkers, the effects of the exposure can vary from one biomarker to the other altering them differently. In addition, the IEF values for the different biomarkers may very well be different. Under these circumstances, the usual method of transformation of variables is computationally difficult. We have developed a computationally feasible method for the derivation of c*(x) for the above mentioned situations (Xu, 1996). This method requires that the altered biomarker vector X* be expressed as a sum of X (the pre-exposure values) and the changes in the biomarkers due to the exposure and random errors. The distribution of X* can then be obtained as the convolution of the distribution of these components. The multivariate dose-response relationship relating preexposure, exposure, and post-exposure animal biomarkers and the IEF values together provide estimates of these changes and their joint distributions.

4.1. Derivation of the joint distribution of the toxicant exposure altered biomarkers using cross-sectional data and the calculation of the incidence of infertility Let Yk and Yk* the pre-exposure and post-exposure values for the kth animal biomarker, respectively, d is the dose of the toxicant, /~k is the dose regression coefficient and ek is the random error (k = 1 , . . . , m). Then a multivariate, linear, dose-response model can be written as Y* - Y = p d + e

(4.2)

where, V

=

(Y1,Y2,...,

' . = . (Yi,. Y ;. , ' ' ' , E ) , ' Ym),Y

!I = (ill, f12,'", tim)' and e = (el, e 2 , . . . , em)' •

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S. Mazumdar, Y. Xu, D. R. Mattison, 3/. B. Sussman and V. C. Arena

The above model can be fitted using methods of standard multivariate regression analysis providing estimates b of p. It should be noted that Y and Y* may represent the transformed biomarkers and other factors, e.g., age, can be included in the model. Moreover, instead of modeling the deviations from the pre-exposure values, the post-exposure biomarker values can be used as the response variable with an intercept term added to the model. The quantity bd provides the estimated changes in the biomarker vector from dose d. The distribution of bd is obtained from the distribution of e and the assumptions about II (fixed or random). If we denote by IEFk the interspecies extrapolation factor for the kth biomarker, then for a dose of amount d in the humans, the extrapolated dose to produce equivalent change in the kth animal biomarker will be IEFkd (k = 1 , . . . , m). These m extrapolated doses are denoted by a matrix D = I E F d where, IEF1 0

0 IEF2

•.

0

• '

0

IEF = 0

0

••

IEFm

/

(4.7)

We further assume that the forms of the m dose- response models and the m error distributions are same for the humans and the animals. This assumption is reasonable if the biomarkers for the humans and the animals are of the same quantitative type, such as sperm count. Denoting by X, the pre-exposure h u m a n biomarker vector, the exposure altered biomarker vector X*, from a dose d of the toxicant can be estimated as X* = X + D b + e

(4.8)

where, e is a random error term. Hence e* (x) can be derived as the convolution of the distributions of X and Db + e. The distribution of X is e(x) and the distribution of Db + e is obtained from the distribution of e and the model assumptions as noted earlier. It is usually assumed that e has a multivariate normal distribution with zero mean and covariance matrix W. The parameter 11 is also assumed to have a random-effects component. The formulation permits either or both of I E F and d to be r a n d o m to account for the uncertainties associated with them. Once e*(x) is calculated, p*, the incidence of infertility, is calculated as p* = Elf(x)] = [ f(x)c*(x)dx . J

(4.9)

4.2• Derivation of the joint distribution of the toxicant exposure altered biomarkers using longitudinal data and the calculation of the incidence of infertility I f the biological processes measured by the biomarkers possess recovery mechanisms, i.e., if the exposure altered biomarker values can improve after the ces-

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sation of exposure, time-dependent models should be considered to fully describe the dose-response relationships. When several biomarkers are considered simultaneously, we propose multivariate, mixed-effects, time-dependent, dose-response models. These models account for the correlations between longitudinal observations within the individuals and the correlations between the biomarkers and possible individual specific effects of dose, time, and other factors (Longford, 1993). An m-variate mixed-effects model is given by

Y~ = (Ti @ Ira)l] + (Z/@ Im)7i + ~i,

i = 1,2,... ,N ,

(4.10)

where, Y~ is a m n i × 1 column vector for the ith animal consisting of ni subvectors each with m biomarker values from the exposure and post-exposure period. Ti is a n i x p design matrix, Ii is a m p x 1 vector of fixed-effects parameters, Zi is a n~ x q design matrix, yi is a m q x 1 vector of random-effects parameters, and e; is a mn~ × 1 vector of errors. The T~ and Z~ matrices are assumed to be of same structures for all i excepting the row dimensions that depend on missing observations. Moreover, we may assume that the values in Y* are deviations from the corresponding pre-exposure values, which allows the exclusion of an intercept term in Ti. We need the distributions of the random-effects parameters and the errors to fit this model. The design matrices T~ (i = 1 , 2 , . . . , N), includes the time variables, dose, and other factors. The columns of Z~ are subsets of the columns of Ti. Denoting by Tt a row of the fixed-effects design matrix at the time point t during the observation period, we define Dt = IEF(Tt ® Ira)

(4.11)

This matrix Dt consists of time-dependent extrapolated doses for the m biomarkers. Hence, the altered h u m a n biomarker X~ at a time t (after the start of exposure) can be expressed as X; = X + Dtb + e

(4.12)

where, X is the pre-exposure biomarkers vector, b is the fixed-effects parameter estimates and e is the error vector. Writing Vt = Dtb + e, the distribution of X~, denoted by e~ (x), can be obtained as the convolution of the distributions of X and Vt at selected times. In fitting the model in (Eq. 4.10) it is usually assumed that the random-effects have a multivariate normal distribution with zero mean and covariance matrix B, the within subject errors have a multivariate normal distribution with zero mean and covariance matrix W, and the random-effects and within subject errors are independently distributed. Hence, the distribution of e is multivariate normal with zero mean and covariance matrix (Z' ® Im)l~(Zt ® Im) + ~¢

(4.13)

where, 1~ and W are the estimates of B and W, and Z t is the row of the randomeffects design matrix at time t.

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Once e; (x) is calculated for selected points of time, p~, the incidence of infertility at time t, can be calculated by Pt =

E[f(x)] =

f(x)et(x)dx .

(4.14)

5. Illustrative applications The illustrative applications use data from three studies to estimate the impact of DBCP on human male reproductive performance (Warren et al., 1984; Whorton et al., 1984; Whorton and Milby, 1980). The results from the applications of the NOAEL-SF, the BD method and the Q R R E method with cross-sectional data and a single biomarker are summarized with appropriate references for the computational details. The application of the Q R R E method with longitudinal data and two biomarkers is detailed to illustrate the methodological extensions by the present authors. 5.1. D a t a

Animal data were obtained from the inhalation fertility study in rabbits (Rao et al., 1980, 1982). In this study, groups of 10 male New Zealand white rabbits were exposed to DBCP vapor in individual animal chambers at 4 different dose levels: 0, 0.1, 1.0, 10 ppm respectively. The pre-exposure data collection period was 2 weeks. DBCP exposure lasted for 14 weeks with a 5 days/week, 6 hrs/day schedule with the exception of rabbits in the 10 ppm group that were exposed for only 8 weeks due to a high incidence of mortality. The duration of exposure and recovery periods were designed to encompass the length of the spermatogenic cycle in the rabbits. Semen specimens were collected from the rabbits, prior to exposure, each week during the 14-week exposure period, and at periodic intervals during recovery period. Due to sacrifices and high mortality rate in the high dose group, only 29 rabbits survived after week 14. A number of sperm parameters, i.e., biomarkers, were measured in each specimen. Sperm concentration (sperm count per ml. of ejaculate) and sperm viability (percentage of normal cells in the sperm specimen) are the biomarkers used in the present applications. Tables 1 and 2 present the weekly means and standard deviations of these sperm measures for the control and the DBCP exposed rabbits. These tables also include the number of observations on which these summary measures are based. Substantial amounts of variability are seen in these measures for individual animals over time, between animals within each dose group, and across dose groups. Following the cessation of exposure, reversible trends are seen (Figures 1 and 2). The human sperm biomarker distribution data used in this example are taken from the study of male factor in fertility and infertility analyzing several aspects of semen quality in 1000 men of known fertility with those in 1000 men whose

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Table 1 Sperm concentration a from control and DBCP exposed rabbits Week of study

p p m DBCP 0

Millions of sperm/ml of Pre-exposure -2 835 -1 694 Exposure 1 728 2 671 3 657 4 575 5 475 6 675 7 641 8 426 9 509 10 473 11 458 12 716 13 758 14 602 Post-exposure 16 604 19 396 24 540 26 452 27 701 28 327 30 728 32 357 34 520 36 374 38 667 40 637 42 429 44 497 46 637

0.1

1.0

l0 b

semen, mean + SD (N) + 237 (9) ± 452 (9)

576 -k 263 (8) 673 ± 416 (8)

630 ± 549 (9) 551 + 312 (6)

579 -4- 468 (8) 546 + I98 (6)

± ± ± ± ± + ± ± ± ± ± ± ± ±

280 342 181 275 205 294 501 167 255 206 132 234 502 282

(10) (10) (9) (10) (10) (10) (9) (9) (7) (8) (8) (7) (9) (8)

494 521 835 618 391 576 531 423 419 459 360 382 738 498

± 281 ± 135 ± 488 ± 290 ± 160 ± 440 ± 571 ± 232 ± 122 + 284 ± 201 ± 208 -4- 376 ± 499

538 373 465 467 353 473 386 301 510 376 204 356 248 109

± ± + ± ± ± ± ± ± ± ± -± ±

276 244 231 126 186 165 251 167 122 288 131 210 217 160

(9) (10) (9) (10) (10) (9) (9) (10) (7) (10) (7) (7) (9) (7)

503 ± 347 ± 392 ± 548 ± 467 ± 455 ± 277 ± 124 t 124 + 7.2 ± 5.4 ± 2.4 ± 3.7 ± 1.9 ±

± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

273 220 560 133 501 213 678 202 467 197 689 433 284 239 350

(6) (6) (6) (5) (6) (5) (5) (6) (6) (6) (6) (5) (5) (5) (5)

448 500 691 629 415 409 526 627 535 485 497 616 231 522 583

± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

292 199 532 328 359 542 411 394 544 471 382 488 559 395 430

± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

210 221 411 180 189 261 327 299 301 155 442 544 587 313 107

(5) (4) (5) (5) (5) (5) (5) (5) (4) (5) (5) (4) (4) (3) (3)

(10) (7) (10) (10) (10) (10) (10) (9) (6) (10) (10) (9) (10) (9)

157 (6) 190 (5) 121 (6) 115 (4) 259 (6) 308 (6) 244 (6) 24 (4) 235 (6) 239 (6) 196 (6) 341 (6) 181 (6) 196 (6) 276 (6)

0.9 0.3 0.4 1.8 7.7 5.1 4.9 6.0 7.5 1.7 39.6 28.7 2.0 8.4 27.2

± + ± ± ± ± ± ± ± ± ± ± ± ± ±

255 (9) 141 (7) 28t (10) 272 (10) 272 (10) 250 (9) 170 (7) 254 (6) 275 (6) 7.03 (5) 2.0 (4) 3.4 (4) 7.2 (5) 2.4 (5) 0.8 (5) 0.2 (5) 0.5 (5) 3.5 (5) 10.9 (5) 9.6 (4) 5.9 (5) 7.0 (5) 11.9 (5) 2.0 (5) 73.0 (5) 44.1 (5) 2.5 (5) 8.6 (5) 37.2 (5)

a Data from Rao et al. (1980). All measures from animal 34 in the 10 p p m group were discarded. Observations are missing due to death, sacrifice and insufficient sample or low sperm concentration. b The exposure period for rabbits in the 10 p p m group ends with week 8 and the post-exposure period begins in week 9.

marriages

were infertile (MacLeod

et al., 1951a,b,c).

sperm viability distributions

in these unexposed

present

distributions

applications.

Joint

fertile and the infertile couples

Sperm

populations

of these two

are presented

in Table

concentration

sperm 3.

and

are necessary for the measures

for the

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S. Mazumdar, Y. J(u, D. R. Matt±son, N. B. Sussman and V. C. Arena

Table 2 Sperm viability from control and DBCP exposed rabbits Week of study

ppm DBCP 0

% Live sperm, mean :t: SD (N) Pre-exposure -2 87 ± 9 (10) -1 93 ± 3 (8) Exposure 1 88 ~ 15 (10) 2 92 :t: 5 (10) 3 76 ± 21 (10) 4 88 ± 6 (10) 5 78 -4- 12 (10) 6 89 -4- 5 (10) 7 89 ± 5 (5) 8 90 + 4 (8) 9 92 -4- 4 (5) 10 83 ± 18 (6) 11 88 -t= 5 (6) 12 74 ± 15 (2) 13 88 + 15 (8) 14 68 ± 35 (6) Post-exposure 16 89 ± 7 (4) 19 80 -4- 27 (4) 24 82 ± 7 (3) 26 62 t 54 (3) 27 87 ± 16 (3) 28 98 ± 1 (2) 30 90 -4- 3 (2) 32 84 ± 10 (4) 34 79 + 12 (4) 36 87 ± 11 (4) 38 58 ± 39 (4) 40 89 4- 8 (3) 42 54 ~_ 13 (3) 44 88 ± 6 (3) 46 88 -4- 5 (3)

0.1

1.0

l0 b

88 ± 6 (9) 90 ± 8 (6)

92 ± 6 (8) 90 -- 7 (5)

87 + 7 (7) 87 ± 5 (5)

71 86 84 80 81 86 87 84 86 90 84 91 84 87

+ ± ± + ± ± -± ± ± ± ± + ±

81 82 79 83 80 80 86 73 84 82 78 73 7I 79

± 25 (10) ± 20 (7) ± 15 (8) ± 5 (10) -4- 15 (10) + 14 (8) ± 10 (9) ± 25 (10) ± 9 (6) ± 11 (10) ± 16 (5) ± 24 (6) ± 12 (8) -4- 25 (3)

90 90 86 75 78 82 81 43 46 44 41 _a -

92 92 85 96 46 72 79 88 76 75 54 87 84 91 79

± 5 (4) -4- 0 (2) ± 8 (4) ± 3 (3) ± 25 (2) :t: 30 (5) -4- 15 (5) ± 9 (2) ± 29 (4) ± 26 (4) -4- 21 (4) ± 5 (5) ± 6 (3) ± 4 (5) ± 5 (5)

89 91 83 92 94 89 79 86 53 64 76 84 81 92 28

~: 1 (3) :t: 9 (3) ± 16 (4) ± 6 (4) -4- 5 (4) ± 13 (4) ± 18 (4) ± 13 (4) ± 30 (3) ± 43 (4) -4- 36 (4) ± 5 (4) ± 3 (4) -- 4 (2) ± 10 (2)

-

25 (10) 9 (7) 12 (10) 14 (10) 5 (10) 6 (10) 10 (10) 8 (8) 12 (5) 7 (9)6 19 (9) 6(6) 9 (8) 6 (6)

36 51 47 92 47 59 53 68 71 73 78

± 5 (7) + 6 (6) -4- I0 (8) ± 20 (9) ± 8 (10) J: 8 (9) ± 11 (6) i 25 (5) ± 54 (2) -4- 17 (3) ± (1)

(1) (1) (1) (1) ± 17 (2) ± 30 (2) -4- 16 (2) ± 8 (2) (1) (1) ± 1 (2)

a Insufficient number of sperm for determination of the percentage of live sperm. u The exposure period for rabbits in the 10 ppm group ends with week 8 and the post-exposure period begins in week 9.

The human exposure data are taken from a study of health risk assessment o f D B C P i n C a l i f o r n i a ( R e e d e t al., 1987; M c k o n e , 1987). I t w a s e s t i m a t e d that approximately 220,000 Californians were exposed to DBCP in water provided by large public systems in the Central Valley. The pathways considered in assessing total personal exposure to DBCP from domestic water

Statistical methods for reproductive risk assessment

661

Dose = 0 ppm

14001200-" 10002 800 ;

BOO-" 4002O0O2

-~" ' ~ " ' ; "

'1;'

'1:* " iT

'2'£ " '2;' ' 'a'o" ' ~'

Week

Dose =

1400-1

~'12ooq

TIT T

/

~

E ~oolzI I~,xTI ITF'H'

i

t

2oo-] ot

~

'4T '

'4;

ppm

T

~H ~ TI000 60B]OO / ~1~ ( " E

0.1

'

~

-I

i

til

[~ I-F!

±l

l ltlV -

l

xx

-

C: 0

-2

2

6

10 14 18 22 26 30 34 38 42 46 Week

E

Dose

= 1 ppm

~ 140012001000800CL 600400-

200O-W'

'~"

' ;"

'1'0' ' 'G"

1400]

I'W

Dose

2T

Week

'2'6 " 'a'o' ' 'a'4' ' '3'8' ' 'g2' '

'26

= 10 ppm

4

1200~

,oo1

1°°°1'~

T

6O0 4OO

-~' ' ' ~ ' ' ~ "

'1'o' ' iT'

'1'8' ' '2'2" ' '2'6 ' '3'o ' '~"

Week

'a'W' '4'2' ' '4'6

Fig. 1. Means and standard deviations (error bars) of weekly sperm concentration (millions per ml of ejaculate) from control and DBCP exposed rabbits.

are: ingestion o f D B C P - c o n t a m i n a t e d water, inhalation o f D B C P volatilized into i n d o o r air from daily water use, and dermal absorption from bathing. The average dose per exposed person ranges between 2 and 4 x 10 s mg/kgday.

662

S. Mazumdar, Y. Xu, D. R. Mattison, N. B. Sussman and V. C. Arena

Dose

= 0 ppm

10090 2 80 70 ~ 60 50' 40" 30" 2010" O" -2''

,....,, "0 U~ +1 ,,,-

v

O

"8 O

'2''

'6''

'I'0" ' 'I~I' ' '18' ' '22" ' '216' ' Week

Dose

= 0.1

"3'0' ' '314' " '318' '

'412' ' '4'0

ppm

80706050403020-

lo:

c-

-~" '~' ' ';"

"6

'1'0' ' '1:*' " W'

~) 131

Dose

E

'2~" '2'o" ' 'a'o' ' "3'4' ' '3'o' ' '4'*' ' '4'6

Week

= 1 ppm

@

,o

o 8o~ 70-

6o>

.E

5040 30 ' 20-

,o!

¢) Q.

''

o)

'2'"

';''

1~0' '

'1~1' " ' 1 ; '

Dose

' "2~2' ' Week

'216 ' '

= 10

ppm

'3()' " '314 ' '

'318 ' '

'412 ' ' '4'0

90 2 80 7O 2 6O 50 40 3O 2O 100-

-~,' ' ' ~ " ' ~ "

1'0' ' '1~.' ' '1;"

'2'2' ' '2'0' ' '3'0' ' '3'4' ' '3'8' ' '&'

' '4'0

Week

Fig. 2. Means and standard deviations (error bars) of weekly sperm viability (% of live sperm) from control and DBCP exposed rabbits. 5.2. Regulatory levels and risk estimation using a single biomarker The N O A E L - S F method Based on the a n i m a l s p e r m c o u n t data, N O A E L has been identified as 0.1 p p m (Pease et al., 1991). This value o f N O A E L is f o u n d to be in a g r e e m e n t o f the earlier conclusions c o n c e r n i n g D B C P risk ( R a o et al., 1980, 1982).

663

Statistical methods for reproductive risk assessment

Table 3 Joint distributions of sperm concentration and sperm viability Sperm viability (% of live sperms) Fertile group < 60 60-79 80-89 90-99 Total Infertile group < 60 60-79 80-89 90-99 Total

Sperm concentration (millions per c.c. of semen) <20

20-39

40-79

>80

Total

9 15 14 5 43

26 49 31 10 116

32 121 79 32 264

23 136 249 158 566

90 321 373 205 989

66 67 21 6 160

38 54 30 4 126

57 90 62 21 230

41 131 181 100 453

202 342 294 131 969

The a d m i n i s t e r e d dose c o r r e s p o n d i n g to this N O A E L is derived to be 0.054 m g / k g - d a y a n d the e s t i m a t e d a b s o r b e d dose is 0.027 m g / k g - d a y ( R e e d et al., 1987). T h e reference dose ( R f D ) is c a l c u l a t e d as follows: N O A E L b a s e d R f D = ( N O A E L x reference b o d y w e i g h t ) / S F = (0.027 m g / k g - d a y x 70 k g ) / 1 0 0 0 = 1.9 ~ g / d a y .

The B D method T h e b e n c h m a r k dose is c a l c u l a t e d as the statistical 9 5 % l o w e r confidence limit on the dose associated with 10% o f the c h a n g e in the s p e r m c o u n t from the pree x p o s u r e values (Pease et al., 1991). In these calculations, the s p e r m counts are p o o l e d over the time interval d u r i n g which the effects o f D B C P first b e c a m e significant in the 1 p p m dose g r o u p (week 11-14 in the experiment). The high dose g r o u p o f 10 p p m is e l i m i n a t e d f r o m the m o d e l i n g because D B C P ' s syst e m a t i c toxicity interfered with expressions o f its g o n a d a l toxicity effects a n d m o r e o v e r , h a l f o f the test a n i m a l s died in the high dose g r o u p even t h o u g h e x p o s u r e was t e r m i n a t e d after eight weeks. Since D B C P acts as an a l k y l a t i n g agent a n d kills stem a n d o t h e r s p e r m a t o g e n i a b y D N A d a m a g e , s p e r m counts following exposure, N, are a s s u m e d to follow an e x p o n e n t i a l d o s e - r e s p o n s e curve. This e x p o n e n t i a l d o s e - r e s p o n s e curve takes a linear forn'l b y t r a n s f o r m i n g s p e r m c o u n t s l o g a r i t h m i c a l l y . A linear m o d e l is fitted to these p o o l e d , cross-sectional, l o g a r i t h m i c a l l y - t r a n s f o r m e d , s p e r m c o u n t data. T h e m a x i m u m l i k e l i h o o d estim a t e s o f the intercept a n d the dose are o b t a i n e d u n d e r the a s s u m p t i o n o f the n o r m a l l y d i s t r i b u t e d errors a n d the c o n s t r a i n t that the intercept is positive. T h e

664

S. Mazumdar, Y. Xu, D. R. Mattison, N. B. Sussman and V. C. Arena

BD is calculated as 0.015 mg/kg-day. Using the 1000 fold Safety Factor (SF) to reflect the uncertainty in the estimation, the reproductive RfD is calculated as follows: BD based RfD = (BD x reference body weight)/SF = (0.015 x 70 kg)/1000 = 1.1 gg/day .

The Q R R E method Meistrich (Meistrich, 1992) presents the details of the calculations of h u m a n infertility resulting from exposures of men to D B C P using the Q R R E method. The four steps of this calculations are: Dose-response assessment: The dose-response model is given by N/No = e -D/32 where, N is the sperm counts following exposure, No is the pre-exposure count and D(mg/m2-day) is the daily absorbed dose per body surface area Interspecies Extrapolation Factor (IEF): I E F is chosen as 19 based on the results for reduction of sperm count data at short times following irradiation (same mechanism of action as DBCP) and expressing dose in the unit of body surface area H u m a n exposure assessment: The estimated number of men at risk is 17,500 (based on a total of 220,000 exposed individuals, and assuming that 50% of exposed individuals are w o m a n of which 50% are of reproductive age (15-44), 52% of such women are currently married, and that 39% of such married women or their spouses have been surgically sterilized). The population-weighted average absorption from drinking water in Central Valley of California (per body surface area) is D(human) = 1.4

x 10 -3

mg/m2-day .

The dose to produce equivalent sperm count decrease in animals is D(animal) = D(human) × 19 = 1.4 × 10 .3 mg/m2-day x 19 = 2.6 x 10 2 mg/m2_day . The sperm count reduction factor for this dose is s = 1~(N/No) = 1/e -0"026/3'2 = 1.0083 .

Statistieal methods for reproductive risk assessment

665

Calculation of human infertility: p* = 0.15 + 0.0334 × ln(1.0083) = 0.15 + 0.00027 , where, 0.15 is the intercept (representing the incidence of infertility in the unexposed population) and 0.0334 is the slope of the line relating p* and log(s). The number of additional cases of infertility = ( p * - 0 . 1 5 ) x 17,500 = 4.8.

5.3. Risk estimation using the Q R R E method with two biomarkers and longitudinal data The two biomarkers chosen for this application are sperm counts per c.c. of ejaculate and sperm viability as % of live sperm. The sperm count data are transformed logarithmically and the sperm viability data are transformed by a logit transformation. The logarithmic transformation for the sperm counts is justified earlier in this paper from a biological point of view. The logit transformation is commonly used with bioassay data that are expressed as percentages. Our exploratory analysis showed less variability and better approximations to normal distributions in these transformed data. Sharp decreases in the weekly rates of change of the transformed sperm measures suggest that the effect of the toxicant accumulates over time and remains constant after the cessation of exposure. To accommodate this behavior, we transform the dose to a different metric and call it cumulative dose. Denoting by to the time when the dosing stops, the cumulative dose at time t for an animal exposed at a constant dose level d is defined as

~l = td

ift_
~l=tod

ift>t0

.

(5.1)

The cumulative dose increases during the exposure period, and after the cessation of exposure, it remains constant at its last attained value. We assume a bivariate, mixed-effects, third degree polynomial, dose- response model suggested by the Figures 1 and 2. Using the notations of (Eq. 4.10) the model is given by: Y* = (Ti @ I2)~ + (Zi @ I2)7i + e i i

= 1,2,... ,N

where,

Y,* = (Yil (til) y i 2 ( t i l ) . . . Y i l

(lin,) Yi2( t ini)) : ,

e; = ( , , (t,) u a ( < ) . . .

1,i

Ti= ni

~lini tini

~ 2 dinitin'

N 3 di"'ti"'

J

ILgi.,

(5.2)

666

S. Mazumdar, Y. Xu, D. R. Mattison, N. B. Sussman and V. C. Arena = (flllfl12f121f122f131f132f141f142) t, ~i = (]]il~)i2) t,

i= 1,2,...,N; j=

1,2,...,hi

.

Here, yik(tij)(k = 1,2) is the deviation of the kth b i o m a r k e r of the ith a n i m a l at time tij, from the c o r r e s p o n d i n g pre-exposure value, [tij is the c u m u l a t i v e dose at time tij o f the ith animal, Ii is the fixed-effects p a r a m e t e r vector, ~i is the r a n d o m effects p a r a m e t e r vector for the ith a n i m a l a n d the u i k ( @ ' s are the within a n i m a l errors. A FORTRAN program BIGROWTH (Tan, 1993) is used to fit this bivariate mixed-effects model. The p r o g r a m a c c o m m o d a t e s incomplete a n d u n e q u a l l y spaced data. The random-effects p a r a m e t e r s for each subject are assumed to have a m u l t i v a r i a t e n o r m a l d i s t r i b u t i o n with zero m e a n s a n d u n k n o w n covariance matrix B. The within subject errors are modeled as a c o n t i n u o u s time first-order autoregressive [CAR(l)] process with p a r a m e t e r s ,co a n d o-2. The K a l m a n filter is used to calculate the exact likelihood of the data. The cubic p o l y n o m i a l model (Eq. 5.2) a n d its reduced versions, b o t h linear a n d quadratic, are fitted. U s i n g A k a i k e ' s I n f o r m a t i o n Criterion (AIC), the cubic p o l y n o m i a l s are f o u n d to be the best fit models. I n each case, the a s s u m p t i o n of autoregressive error structure is retained as p r o v i d i n g a better fit t h a n the indep e n d e n t error structures as indicated by the likelihood ratio tests. Weekly changes calculated from the fitted models are f o u n d to be c o m p a r a b l e with the empirical changes i n d i c a t i n g the suitability of the assumed dose-response m o d e l a n d the c u m u l a t i v e dose metric (Xu, 1996). The estimates o f the linear parameters a n d the variance c o m p o n e n t s are given in Table 4.

Table 4 Parameter estimates, standard errors and t-values from the linear bivariate mixed-effects time-dependent dose-response model Variables in the model Responsea: logarithm of sperm concentration Dose Week (Week)2 (Week)3 Responsea: logit of sperm viability Dose Week (Week)2 (Week)3 Variance components: (p 4.568 × 10-s e2 0.0237 B 3.24 x 10.8

Estimates

Standard errors

t-values

-4.497 x 10-1 4.561 x 10 2 -1.485 x 10 . 3 1.538 x 10.5

0.79 × 10 t 0.24 x 10.2 0.31 × 10 - 3 0.20 x 10-s

-5.69* 19.1" -4.79" 7.69*

-3.798 x 10 1 3.99 x 10 . 2 -1.32 x 10 - 3 1.38 x 10 5

0.65 x 10-I 0.18 × 10 . 2 0.27 × 10 . 3 0.16 x 10 5

-5.84" 18.1" -4.88* 8.62*

Responses are deviations from the pre-exposure values. * Significantat 0.05 level of significance.

Statistical methods for reproductive risk assessment

667

The joint distributions of the human sperm count and sperm viability data for the fertile and infertile populations (Table 3) are smoothed, after transforming to logarithmic and logit scales, respectively, by using the kernel density estimation method (Scott, 1992; Xu, 1996). These smooth distributions are used to find e(x) and the final altered biomarkers' joint distribution, e*(x), is derived for selected time points. We note here that the univariate, mixed-effects, time-dependent, dose-response models are also fitted separately to these two transformed biomarkers using the same B I G R O W T H F O R T R A N program. The parameter estimates are found to be similar to those that are given in Table 4 with somewhat larger standard errors. The univariate distributions of the human sperm count and sperm viability are also smoothed appropriately by fitting continuous functions and the corresponding two altered biomarker distributions are obtained (Meistrich and Brown, 1983). Using the DBCP dose of 1.4 x 10-3 mg/m2-day and IEF of 19 for both the biomarkers, the reproductive risks are calculated (Figure 3). Finally the incidences of infertility are calculated (Eq. 4.9) at week 14, the end of the period of exposure, and at week 40 following the cessation of exposure and a period of recovery of reproductive function (Table 5). This table also includes the results when these two biomarkers are considered singly. The short-term effects on fertility observed at week 14 are about 3-fold greater than the permanent effects observed at week 40 which remain after cessation of treatment and recovery of testicular function. It can also be noted that the risk estimate

1.o1 0.9" 0.8 0.7

0.6" ~-~

0.5

,~

0.4

!

i

0.3 0.2

/o " ~

0.1 0.0

I

2

r~Concentration) Fig. 3. Risk of infertility as a function of logarithm of sperm concentration and logit of sperm viability.

668

S. Mazumdar, Y. Xu, D. R. Mattison, N. B. Sussman and V. C. Arena

Table 5 Increased risk a of infertility and the increased number of cases of infertility among 17,500 males exposed to DBCP at an average dose of 1.4 × 10-3 mg/m2-day in drinking water Biomarker

Increased number of cases of infertility b per 17,500 males

Increased risk

Week 14

Week 40

Single biomarker characterization of reproductive risk Sperm count 2.3 × 10-4 0.8 × 10-4 Sperm viability 2.4 × 10-4 0.8 × 10-4 Multiple biomarkers characterization of reproductive risk Sperm count and viability 4.1 × 10-4 1.3 × 10-4

Week 14

Week 40

4.0 4.2

1.4 1.4

7.2

2.3

a Calculated from the dose response data from Rao et al. (1980) and an IEF 19. b Increased risk of infertility x number of men at risk (17,500).

for infertility increases when two biomarkers of reproductive function are combined.

6. Discussion

This article describes reproductive toxicology, biomarkers of reproduction, and currently available statistical methods for characterizing reproductive toxicity. It then focuses on the Q R R E method for the quantitative estimation of reproductive risks. The reproductive risk is defined as the increased incidence of infertility in a male population based on alterations in their semen characteristics, the biomarkers for reproductive toxicity. Our extensions of the Q R R E method are based on the following two premises: (1) when a toxicant independently disrupts multiple processes needed for reproductive success, it is necessary to include all of those factors in the risk assessment calculations to avoid the underestimation of the actual risk; and (2) because of the adaptive nature of the reproductive system, reproductive risk calculations should include the characterization of both reversible and permanent effects on reproductive functions. The illustrative applications show that while some recovery may be possible after the cessation of exposure permanent damage may also remain and combining two biomarkers of reproductive function substantially increases the estimate of infertility risk. More andrologic research is necessary to gather appropriate data. At present the risks are calculated based on the fertility status of the couple, rather than that of the man, because only semen characteristics of men from infertile couples (as opposed to infertile men) are available. Issues related to time-dependence of IEF and uncertainties of the dose should be addressed. More computational procedures and computer software are required for density estimations from sparse data, and to fit mixed effects, multivariate, dose-time-response models accommodating irregularly spaced and missing data.

Statistical methods for reproductive risk assessment

669

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Rao, K. S., J. D. Burek, F. J. Murray, J. A. John, B. A. Schwetz, J. E. Beyer and C. M. Parker (1982). Toxicologic and reproductive effects of inhaled 1,2-dibromo-3-chloropropane in male rabbits. Fundam. Appl. Toxicol. 2, 241-251. Rao, K. S., J. D. Burek, J. S. Murray, J. A. John, B. A. Schwetz, F. J. Murray, A. A. Crawford, M. M. Deacon, W. J. Potts, B. N. Sutter, D. A. Dittenber, T. J. Bell, J. E. Beyer, R. R. Albee, J. E. Battjes and C. M. Parker (1980). 1,2-Dibromo-3-Chloropropane: Inhalation fertility study in rats and rabbits. Toxicol. Res. 27 245. Ratcliffe, J. M., B. C. Gladen, A. J. Wilcox and A. L. Herbst (1992). Does early exposure to maternal smoking affect future fertility in adult males? Reprod. Toxicol. 6, 297 307. Reed, N. R., H. E. Olsen, M. Marty, L. M. Beltran, T. Mckone, K. T. Bogen, N. L. Tablante and D. P. H. Hsieh (1987). Health Risk assessment of 1,2-Dibromochloropropane (DBCP) in California Drinking Water. Department of Environmental Toxicology, University of California, Davis, CA. Rowland, A. S., D. D. Baird, C. R. Weiuberg, D. L. Shore, C. M. She and A. J. Wilcox (1992). Reduced fertility among women employed as dental assistants exposed to high levels of nitrous oxide. New Eng. J. Med. 327, 993 997. Sanmels, S. J., S. A. McCurdy, D. Pocekay, S. K. Hammond, L. Missell and M. B. Schenker (1995). Fertility history of currently employed male semiconductor workers. Am. J. Ind. Med. 28, 873-882. Schardein, S. J. (1993). Chemically induced birth defects. Second Edition, Revised and Expanded. Mercel Dekker, Inc. New York. Schenker, M. B., E. B. Gold, J. J. Beaumont, B. Eskenazi, S. K. Hammond, B. L. Lasley, S. A. McCurdy, S. J. Samuels, C. L. Saiki and S. H. Swan (1995). Association of spontaneous abortion and other reproductive effects with work in the semiconductor industry. Am. J. Ind. Med. 28, 639 659. Scott, D. W. (1992). Multivariate Density Estimation, Theory Practice and Visualization. John Wiley and Sons, New York. Swan, S. H., J. J. Beaumont, S. K. Hammond, J. VonBehren, R. S. Green, M. F. Hallock, S. R. Woskie, C. J. Hines and M. B. Schenker (1995). Historical cohort study of spontaneous abortion among fabrication workers in the Semiconductor Health Study: agent-level analysis. Am. J. Ind. Med. 28, 751 769. Tan, L. (1993). A multivariate growth model with random effects and CAR (1) errors. Ph.D. Dissertation, Department of Biostatistics, Graduate School of Public Health, University of Pittsburgh. Tukey, J. W., J. L. Ciminera and J. F. Heyse (1985). Testing of statistical certainty of a response to increasing doses of a drug. Biometrics, 41, 295-301. Warren, D. W., J. R. J. Wisner and N. Ahmad (1984). Effects of 1,2- dibromo-3-chloropropane on male reproductive function in the rat. Biol. Reprod. 31, 454-463. Weinberg, C. R., A. J. Wilcox and D. D. Baird (1989). Reduced fecundability in women with prenatal exposure to cigarette smoking. Am. J. Epidemiol. 129, 1072-1078. Whorton, D., T. H. Milby, R. M. Krauss and H. A. Stubbs (1979). Testicular function in DBCP exposed pesticide workers. J. Occup. Med. 21, 161-166. Whorton, M. D. and T. H. Milby (1980). Recovery of testicular function among dbcp workers. J. Occup. Med. 22(3), 177 179. Wilcox, A. J., C. R. Weinberg and D. D. Baird (1995). Timing of sexual intercourse in relation to ovulation - Effects on the probability of conception, survival of the pregnancy, and sex of the baby. New Engl. J. Med. 333, 1517-1521. Wilcox, A. J., C. R. Weinberg and D. D. Baird (1990). Risk factors for early pregnancy loss. Epidemiol. 1, 38~385. Wiieox, A. J., C. R. Weinberg, J. F. O'Connor, D. D. Baird, J. P. Schlatterer, R. E. Canfield, E. G. Armstrong and B. C. Nisula (1988). Incidence of early loss of pregnancy. New Engl. J. Med. 319, 189-194.

Statistical methods for reproductive risk assessment

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