Estimating upper confidence limits for extra risk in quantal multistage models

Multistage models are frequently applied in carcinogenic risk assessment. In their simplest form, these models relate the probability of tumor presence to some measure of dose. These models are then used to project the excess risk of tumor occurrence at doses frequently well below the lowest experimental dose. Upper confidence limits on the excess risk associated with exposures at these doses are then determined. A likelihood-based method is commonly used to determine these limits. We compare this method to two computationally intensive "bootstrap" methods for determining the 95% upper confidence limit on extra risk. The coverage probabilities and bias of likelihood-based and bootstrap estimates are examined in a simulation study of carcinogenicity experiments. The coverage probabilities of the nonparametric bootstrap method fell below 95% more frequently and by wider margins than the better-performing parametric bootstrap and likelihood-based methods. The relative bias of all estimators are seen to be affected by the amount of curvature in the true underlying dose-response function. In general, the likelihood-based method has the best coverage probability properties while the parametric bootstrap is less biased and less variable than the likelihood-based method. Ultimately, neither method is entirely satisfactory for highly curved dose-response patterns.     

The linearized multistage model and the future of quantitative risk assessment

The linearized multistage (LMS) model has for over 15 years been the default dose-response model used by the U.S. Environmental Protection Agency (USEPA) and other federal and state regulatory agencies in the United States for calculating quantitative estimates of low-dose carcinogenic risks from animal data. The LMS model is in essence a flexible statistical model that can describe both linear and non-linear dose-response patterns, and that produces an upper confidence bound on the linear low-dose slope of the dose-response curve. Unlike its namesake, the Armitage-Doll multistage model, the parameters of the LMS do not correspond to actual physiological phenomena. Thus the LMS is 'biological' only to the extent that the true biological dose response is linear at low dose and that low-dose slope is reflected in the experimental data. If the true dose response is non-linear the LMS upper bound may overestimate the true risk by many orders of magnitude. However, competing low-dose extrapolation models, including those derived from 'biologically-based models' that are capable of incorporating additional biological information, have not shown evidence to date of being able to produce quantitative estimates of low-dose risks that are any more accurate than those obtained from the LMS model. Further, even if these attempts were successful, the extent to which more accurate estimates of low-dose risks in a test animal species would translate into improved estimates of human risk is questionable. Thus, it does not appear possible at present to develop a quantitative approach that would be generally applicable and that would offer significant improvements upon the crude bounding estimates of the type provided by the LMS model. Draft USEPA guidelines for cancer risk assessment incorporate an approach similar to the LMS for carcinogens having a linear mode of action. However, under these guidelines quantitative estimates of low-dose risks would not be developed for carcinogens having a non-linear mode of action; instead dose-response modelling would be used in the experimental range to calculate an LED10* (a statistical lower bound on the dose corresponding to a 10% increase in risk), and safety factors would be applied to the LED10* to determine acceptable exposure levels for humans. This approach is very similar to the one presently used by USEPA for non-carcinogens. Rather than using one approach for carcinogens believed to have a linear mode of action and a different approach for all other health effects, it is suggested herein that it would be more appropriate to use an approach conceptually similar to the 'LED10*-safety factor' approach for all health effects, and not to routinely develop quantitative risk estimates from animal data.     

Statistical modeling of carcinogenic risks in dogs that inhaled 238PuO2

Combined analyses of data on 260 life-span beagle dogs that inhaled 238PuO2 at the Inhalation Toxicology Research Institute (ITRI) and at Pacific Northwest National Laboratory (PNNL) were conducted. The hazard functions (age-specific risks) for incidence of lung, bone and liver tumors were modeled as a function of cumulative radiation dose, and estimates of lifetime risks based on the combined data were developed. For lung tumors, linear-quadratic functions provided an adequate fit to the data from both laboratories, and linear functions provided an adequate fit when analyses were restricted to doses less than 20 Gy. The estimated risk coefficients for these functions were significantly larger when based on ITRI data compared to PNNL data, and dosimetry biases are a possible explanation for this difference. There was also evidence that the bone tumor response functions differed for the two laboratories, although these differences occurred primarily at high doses. These functions were clearly nonlinear (even when restricted to average skeletal doses less than 1 Gy), and evidence of radiation-induced bone tumors was found for doses less than 0.5 Gy in both laboratories. Liver tumor risks were similar for the two laboratories, and linear functions provided an adequate fit to these data. Lifetime risk estimates for lung and bone tumors derived from these data had wide confidence intervals, but were consistent with estimates currently used in radiation protection. The dog-based lifetime liver tumor risk estimate was an order of magnitude larger than that used in radiation protection, but the latter also carries large uncertainties. The application of common statistical methodology to data from two studies has allowed the identification of differences in these studies and has provided a basis for common risk estimates based on both data sets.     

Renal expression of interleukin-2 receptor mRNA in patients with crescentic glomerulonephritis

A microcolony assay was used in conjunction with fractionated gamma irradiation to determine the number of clonogens in murine intestinal crypts with varying doses of irradiation used in the determination. The experimental design allows direct comparison between two-dose methodologies, employing one and two (or two or four) equal dose fractions, and multiple-dose methodologies involving determination of the crypt survival curves for a number of fractionation regimens using equal doses per fraction. The two-dose methodology yielded estimates of clonogen number of between 3 and 4 at low delivered dose (single and double fractions each of 6.5-7.5 Gy), rising to around 40 at high biological doses (two and four fractions each of 5.75 or 6.5 Gy). The multifraction methodology yielded estimates of clonogen number which increased from 13 after a single fraction to values of 26 and 22 after three and four fractions. However, the latter values were reduced to 11 and 9, and showed little evidence of any dependence on fraction number, when data pertaining to high biologically effective doses were excluded. Hence it is concluded that the high values for clonogen number typically deduced from such multiple-dose protocols, compared with the generally lower (but dose-dependent) values obtained from two-dose protocols, may be explained at least partially by the higher biological doses generally employed in the multiple-dose protocols.     

A statistical test of compatibility of data sets to a common dose-response model

Quantitative estimates of cancer risk generally involve low-dose extrapolation based on an exponential dose-response model for dichotomous response data. Frequently more than one data set is available. If a careful analysis of the biological issues indicates that more than one of the available data sets could be used in the quantitative estimate of cancer risk, it is reasonable to think of combining the data. Before combining data, however, it would be prudent to test whether the data sets are compatible with a common dose-response model. If they are not, it could be concluded that an underlying biological factor is responsible. If they are statistically compatible, the decision to combine data sets based on biological issues would be reinforced. A statistical test based on the generalized likelihood ratio method is proposed for evaluating the compatibility of different data sets with a common dose-response model. This method of constructing a statistical test and the associated asymptotic theory is consistent with the approach used by GLOBAL86 (R. B. Howe, K. S. Crump, and C. Van Landingham, GLOBAL86: A Computer Program to Extrapolate Quantal Animal Toxicity Data to Low Doses, K. S. Crump & Co., Ruston, LA, 1986) for estimating the confidence limits that are used as a basis for quantitative estimates.     

Statistical issues in biological radiation dosimetry for risk assessment using stable chromosome aberrations

Biological dosimeters are useful for epidemiologic risk assessment in populations exposed to catastrophic nuclear events and as a means of validating physical dosimetry in radiation workers. Application requires knowledge of the magnitude of uncertainty in the biological dose estimates and an understanding of potential statistical pitfalls arising from their use. This paper describes the statistical aspects of biological dosimetry in general and presents a detailed analysis in the specific case of dosimetry for risk assessment using stable chromosome aberration frequency. Biological dose estimates may be obtained from a dose-response curve, but negative estimates can result and adjustment must be made for regression bias due to imprecise estimation when the estimates are used in regression analyses. Posterior-mean estimates, derived as the mean of the distribution of true doses compatible with a given value of the biological endpoint, have several desirable properties: they are nonnegative, less sensitive to extreme skewness in the true dose distribution, and implicitly adjusted to avoid regression bias. The methods necessitate approximating the true-dose distribution in the population in which biological dosimetry is being applied, which calls for careful consideration of this distribution through other information. An important question addressed here is to what extent the methods are robust to misspecification of this distribution, because in many applications of biological dosimetry it cannot be characterized well. The findings suggest that dosimetry based solely on stable chromosome aberration frequency may be useful for population-based risk assessment.     

Chernobyl accident: reconstruction of thyroid dose for inhabitants of the Republic of Belarus

The Chernobyl accident in April 1986 resulted in widespread contamination of the environment with radioactive materials, including (131)I and other radioiodines. This environmental contamination led to substantial radiation doses in the thyroids of many inhabitants of the Republic of Belarus. The reconstruction of thyroid doses received by Belarussians is based primarily on exposure rates measured against the neck of more than 200,000 people in the more contaminated territories; these measurements were carried out within a few weeks after the accident and before the decay of (131)I to negligible levels. Preliminary estimates of thyroid dose have been divided into 3 classes: Class 1 ("measured" doses), Class 2 (doses "derived by affinity"), and Class 3 ("empirically-derived" doses). Class 1 doses are estimated directly from the measured thyroidal (131)I content of the person considered, plus information on lifestyle and dietary habits. Such estimates are available for about 130,000 individuals from the contaminated areas of the Gomel and Mogilev Oblasts and from the city of Minsk. Maximum individual doses are estimated to range up to about 60 Gy. For every village with a sufficient number of residents with Class 1 doses, individual thyroid dose distributions are determined for several age groups and levels of milk consumption. These data are used to derive Class 2 thyroid dose estimates for unmeasured inhabitants of these villages. For any village where the number of residents with Class 1 thyroid doses is small or equal to zero, individual thyroid doses of Class 3 are derived from the relationship obtained between the mean adult thyroid dose and the deposition density of (131)I or 137Cs in villages with Class 2 thyroid doses presenting characteristics similar to those of the village considered. In order to improve the reliability of the Class 3 thyroid doses, an extensive program of measurement of (129)I in soils is envisaged.     

Risk assessment of the nuclear medicine patient

Two types of risk are identified following the administration of a radiopharmaceutical to a patient: the risk to the patient, and the risk to critical groups exposed to the patient. The method for quantifying the risk to the patient is described in terms of estimating the effective dose. The main limitations in these estimates for adult and paediatric patients are uncertainties in the biokinetic data, and the assumption of a uniform distribution of activity in each organ. Effective doses from most nuclear medicine procedures will not exceed twice the annual dose from natural background radiation in the UK. Lack of human placental transfer data is now the main limitation to estimating fetal doses. The characteristics of two methods which can be used to derive the dose to critical groups exposed to nuclear medicine patients are reviewed. It is shown that studies using either method have indicated that the current recommendations in the UK for restricting the exposure of these groups and the recommendation recently proposed for restricting the exposure of pregnant members of staff are not appropriate. Revised recommendations for restricting the behaviour of patients administered iodine-131 should await the results of a current multicentre trial. The method to estimate the dose to a breast-fed infant from a mother administered a radiopharmaceutical is outlined, and the recently revised guidance for interrupting breast feeding is summarised. When a recommendation for controlling risk is to be derived from dosimetry data obtained from a number of individuals, an outstanding issue to be resolved is the value (e.g. 95% upper confidence limit) on which it should be based.     

Treatment of intrahepatic cancers with radiation doses based on a normal tissue complication probability model

PURPOSE: To attempt to safely escalate the dose of radiation for patients with intrahepatic cancer, we designed a protocol in which each patient received the maximum possible dose while being subjected to a 10% risk of radiation-induced liver disease (RILD, or radiation hepatitis) based on a normal tissue complication probability (NTCP) model. We had two hypotheses: H1; with this approach, we could safely deliver higher doses of radiation than we would have prescribed based on our previous protocol, and H2; the model would predict the observed complication probability (10%). PATIENTS AND METHODS: Patients with either primary hepatobiliary cancer or colorectal cancer metastatic to the liver and normal liver function were eligible. We used an NTCP model with parameters calculated from our previous patient data to prescribe a dose that subjected each patient to a 10% complication risk within the model. Treatment was delivered with concurrent hepatic arterial fluorodeoxyuridine (HA FUdR). Patients were evaluated for RILD 2 and 4 months after the completion of treatment. RESULTS: Twenty-one patients completed treatment and were followed up for at least 3 months. The mean dose delivered by the current protocol was 56.6 +/- 2.31 Gy (range, 40.5 to 81 Gy). This dose was significantly greater than the dose that would have been prescribed by the previous protocol (46.0 +/- 1.65 Gy; range, 33 to 66 Gy; P < .01). These data are consistent with H1. One of 21 patients developed RILD. The complication rate of 4.8% (95% confidence interval, 0% to 23.8%) did not differ significantly from the predicted 8.8% NTCP (based on dose delivered) and excluded a 25% true incidence rate (P < .05). This finding supports H2. CONCLUSION: Our results suggest that an NTCP model can be used prospectively to safely deliver far greater doses of radiation for patients with intrahepatic cancer than with previous approaches. Although the observed complication probability is within the confidence intervals of our model, it is possible that this model overestimates the risk of complication and that further dose escalation will be possible. Additional follow-up and accrual will be required to determine if these higher doses produce further improvements in response and survival.     

1-[Carbon-11]-glucose radiation dosimetry and distribution in human imaging studies

1-[Carbon-11]-D-glucose ([11C]-glucose) is an important imaging agent for PET studies that have been used to study the normal brain, encephalitis, epilepsy, manic-depressive disorder, schizophrenia and brain tumors. METHODS: Dosimetry estimates were calculated in subjects undergoing imaging studies to help define the radiation risk of [11C]-glucose PET imaging. Time-dependent radioactivity concentrations in normal tissues in 33 subjects after intravenous injection of [11C]-glucose were obtained by PET imaging. Radiation absorbed doses were calculated according to the procedures of the Medical Internal Radiation Dose (MIRD) committee along with the variation in dose based on the calculated standard deviation of activity distribution seen in the individual patients. RESULTS: Total body exposure was a median of 3.0 microGy/MBq in men and 3.8 microGy/MBq in women. The effective dose equivalent was 3.8 microGy/ MBq in men and 4.8 microGy/MBq in women. The critical organs were those that typically take up the most glucose (brain, heart wall and liver). CONCLUSION: The organ doses reported here are small and comparable to those associated with other commonly performed nuclear medicine tests and indicate that potential radiation risks associated with this radiotracer are within generally accepted limits.     

High dose baclofen is neuroprotective but also causes intracerebral hemorrhage: a quantal bioassay study using the intraluminal suture occlusion method

Agonists of the GABA-A receptor are neuroprotective after experimental stroke, but studies of GABA-B agonists have contradicted each other. To further investigate whether GABA-B agonists may be neuroprotective, we devised a quantal bioassay using the intraluminal occlusion method of inducing reversible cerebral ischemia. Subjects underwent middle cerebral artery occlusion for varying amounts of time, ranging from 5 to 90 min. Behavioral outcome was measured 48 h later with a quantal observational scale: score of abnormal given for any one of asymmetric forepaw flexion on tail lift, asymmetric grip, circling, reduced exploration, seizures, or death. To the grouped response data the logistic equation was used to find the ED50, the duration of occlusion that caused one-half of the subjects to be abnormal. To find the potency ratio for each drug, we divided the ED50 for treatment by that for vehicle. We administered baclofen, a GABA-B agonist, intraperitoneally 5 min after the onset ofischemia. Baclofen (20 mg/kg) was neuroprotective (potency ratio of 3.0, P < 0.05), but a lower dose (10 mg/kg) was not. However, both doses of baclofen caused significantly more intracerebral hemorrhages than control. In awake animals, both baclofen doses caused significant increases in mean arterial pressure, but no changes in other cardiorespiratory variables. The glutamate antagonist MK-801, the GABA-A agonist muscimol, and hypothermia were all protective using the bioassay (potency ratios ranging from 1.5 to 3.0). We conclude that although baclofen (20 mg/kg) may be neuroprotective, its utility is complicated by postischemic hypertension and cerebral hemorrhages.     

Dietary no-effect level of a dithiocarbamate fungicide, thiram, evaluated from measurement data on rats. I. Choice of the model of the dose-response relationship

Rats were fed diets containing various amounts of added thiram, a dithiocarbamate fungicide. As thiram feeding resulted in decreased appetite, control rats not receiving thiram were pair-fed to the experimental ones. On d 30 of the experiment the animals were weighed and sacrificed, and the following organs were weighed: liver, kidneys, heart, epididymal and perirenal fat pads, testes, seminal vesicles, tibia, adrenals, and thyroid. Liver concentrations of lactate, pyruvate, beta-hydroxybutyrate, acetoacetate, ATP, and ADP were determined by enzymatic-spectrofluorimetric assay. For each parameter studied and each thiram dosage, values for treated rats were compared to those for control rats and the probability under the null hypothesis was computed. These probabilities were transformed into probits, logits, or "Weibull transforms" and plotted against the logarithms of the respective doses. Models were fitted to the data by linear regression techniques. Finally, the dose inducing the least significant difference (LSD dose), and the dose considered "safe" at P = 0.95, 0.99, and 0.999 were calculated. Significant pesticide-induced changes in the following parameters were found: food intake; weights of the whole body, kidneys, epididymal and perirenal fat pads, testes, and seminal vesicles; and liver beta-hydroxybutyrate/acetoacetate and lactate/pyruvate ratios. As the models did not differ in fit to the experimental data or in computed LSD doses, they were discriminated on the grounds of their underlying theoretical assumptions and their prediction of safe doses in a long-term study. The log-probit model was rejected for the former reason, and it was shown that the Weibull model foresees a nonnegligible risk of change, with thiram feeding at low doses, for too many parameters. The analysis resulted in the selection of the log-probit model for further use. Weight of fatty tissues was the most sensitive parameter and, using the log-probit model, the predicted no-effect dose at the 95 percent confidence level was 38 ppm thiram in the diet.     

A bootstrap approach to confidence regions for genetic parameters from Method R estimates

Confidence regions (CR) for heritability (h2) and fraction of variance accounted for by permanent environmental effects (c2) from Method R estimates were obtained from simulated data using a univariate, repeated measures, full animal model, with 50% subsampling. Bootstrapping techniques were explored to assess the optimum number of subsamples needed to compute Method R estimates of h2 and c2 with properties similar to those of exact estimators. One thousand estimates of each parameter set were used to obtain 90, 95, and 99% CR in four data sets including 2,500 animals with four measurements each. Two approaches were explored to assess CR accuracy: a parametric approach assuming bivariate normality of h2 and c2 and a nonparametric approach based on the sum of squared rank deviations. Accuracy of CR was assessed by the average loss of confidence (LOSS) by number of estimates sampled (NUMEST). For NUMEST = 5, bootstrap estimates of h2 and c2 were within 10(-3) of the asymptotic ones. The same degree of convergence in the estimates of SE was achieved with NUMEST = 20. Correlation between estimates of h2 and c2 ranged from -.83 to -.98. At NUMEST < 10, the nonparametric CR were more accurate than parametric CR. However, with the parametric CR, LOSS approached zero at rate NUMEST(-1). This rate was an order of magnitude larger for the nonparametric CR. These results suggested that when the computational burden of estimating genetic parameters limits the number of Method R estimates that can be obtained to, say, 10 or 20, reliable CR can still be obtained by processing Method R estimates through bootstrapping techniques.     

Estimation of a common risk ratio in stratified case-cohort studies

The case-cohort study design is a useful modification of the case-control design, which allows direct estimation of the risk ratio without the rare-disease assumption. While several risk ratio estimation procedures have been proposed under large-strata settings, only the Mantel-Haenszel point estimator is available in sparse stratifications. This paper provides simple confidence limits methods, based on the large-sample distribution of the Mantel-Haenszel risk ratio, that apply to both large-strata and sparse-data situations. For the Tarone risk ratio, I give a new large-strata variance estimate.     

The effect of selective reporting on estimates of weaning weight parameters in beef cattle

The effect of selective reporting on estimates of weaning weight parameters in beef cattle was evaluated by comparing REML estimates from unaltered and altered simulated data. Selective reporting reduced estimates of weaning weight direct (WWD), maternal milk (MAT), and error variances. However, heritability estimates were not greatly affected because the reductions in variance estimates were relatively proportionate. When the true value for the direct-maternal (DM) correlation was zero or negative, selective reporting caused estimates of DM to be less positive or more negative in 50 of 62 comparisons, with an average change of -.136. When the true value for DM was positive, selective reporting increased the positive magnitude of DM estimates in 12 of 20 comparisons, with an average change of +.040. In BLUP of unaltered data with a true DM value of -.09, using a -.28 and a zero DM correlation reduced the correlation of MAT EPD with true values .065 and .041, respectively. These results suggest that the reliability of parameter estimates (and BLUPs) would be improved by estimating parameters from representative subsets of data free of reporting bias.     

Immunohistochemical assessment of prostaglandin H-synthase in bovine endometrial biopsy samples collected throughout the oestrous cycle

In the Zhytomyr region, about 52,000 measurements of the 131I activity in thyroids were performed. On the basis of these measurements, individual doses have been assessed for the people monitored and age-dependent average doses have been estimated for those settlements with more than 11 direct measurements. In order to estimate the pattern of thyroid exposure in the Zhytomyr region, these doses have been interpolated or extrapolated to population groups who were not monitored during May-June 1986. For this purpose, a model has been developed based on a correlation between thyroid dose estimates with the 137Cs deposition and the co-ordinates of the settlements relative to Chernobyl. Collective doses of people who were born in the years 1968 to 1986 were calculated. The radiation-induced thyroid cancer incidence in the period 1991 to 1995 was assessed by subtracting the spontaneous incidence from the observed incidence. The result is considerably lower than that observed in longer periods after external exposures. Possible reasons for this difference are discussed.     

Low- versus high-dose aspirin in prevention of ischemic stroke

Aspirin is the most extensively studied drug for the prevention of ischemic vascular disease. Meta-analyses confirm that aspirin is effective in prevention of ischemic events, including stroke. Recently, there has been considerable discussion about the best dose of aspirin to prevent stroke. Several studies tested aspirin in a daily dose of 975 mg or more alone or in combination with another drug, most commonly dipyridamole, and noted that aspirin was effective. Successively lower doses of aspirin were tested and recent studies demonstrate that low doses (< 100 mg/day) are effective. Only one study, enrolling patients with transient ischemic attack or minor stroke, has examined aspirin in a daily dose of approximately 325 mg. Side effects of aspirin are dose related; gastrointestinal bleeding and epigastric pain are less with low doses. Available data cannot confirm that low doses (< 100 mg/day) of aspirin are either more or less effective than larger (975 mg/day) doses. A direct comparison of the usefulness of low doses (< 100 mg/day) or large doses (approximately 1,000 mg/day) in patients at high risk of stroke is needed. Until the results of such a study are known, the better safety profile of low doses favors aspirin in a daily dose of 100 mg or less.     

Radiation-induced cancers: state of the art in 1997

Scientists now have available a large amount of data dealing with radiation-induced neoplasms. These data went back to anecdotal observations which were made in the very first years of utilization of X-rays and radioactive elements. In fact, it is essentially the strict follow-up of the Japanese populations irradiated by the Hiroshima and Nagasaki bombing which allowed a more precise evaluation of the carcinogenicity of ionizing radiations. Further refinements came from therapeutical irradiations: it is now possible to study large cohorts of patients given well-known doses in well-defined volumes and followed for more than 20 years. Last but not least, a significant increase in the incidence and mortality of thyroid cancer has been detected in children contaminated by iodine radioisotopes after the Tchernobyl accident. Recently, some data suggested the emergence of "clusters" of leukemias close to some nuclear facilities, but this question remains highly polemical, both in France and in the UK. Other questions are still waiting for a precise answer; of course, the extrapolation of our available data to very low doses delivered at very low dose rates, but also the carcinogenic risk at high doses. For these "high" doses (about 30 to 70 Gy), a competition between mutagenesis and cell killing was expected, so that these dose levels were expected to be less carcinogenic than lower (a few sieverts) doses. Actually, recent data suggest that the carcinogenic risk goes on increasing up to relatively important doses. In addition, carcinogenic factors, such as tabacco, anticancer chemotherapy and individual susceptibility, are found more and more to be closely intricated with ionizing radiation in the genesis of a given cancer. Even if a number of questions are still pending, the already available data allow specialists, both in medicine and radioprotection, to edict strict rules which can be reasonably expected to have significantly reduced the risk of radiation-induced neoplasms in most situations.     

Risk of benzene-induced leukemia: a sensitivity analysis of the pliofilm cohort with additional follow-up and new exposure estimates

This report updates the risk assessment by Crump and Allen (1984) for benzene-induced leukemia that was used by OSHA (1987) to support its reduction of the permissible exposure limit (PEL) to 1 ppm and that also was the basis for EPA's (1985) interim "unit risk" for benzene. The present study derives new risk estimates using data from follow-up through 1987 (whereas the earlier assessment only had follow-up available through 1978), and using new exposure estimates for this cohort developed by Paustenbach et al. (1992) that account for a number of factors that were unknown or not fully evaluated in earlier exposure assessments. There was a significant excess of acute myelocytic or acute monocytic leukemia (AMML, the only forms of acute nonlymphatic leukemia observed) in this cohort, and this end point also exhibited a strong dose-response trend. AMML was the only hematopoietic or lymphatic cancer that was clearly linked to benzene exposure. However, quantitative estimates of risk based on modeling either AMML or all leukemia differed by only 20%. Differences between the two Pliofilm plant locations in the occurrence of AMML were not statistically significant (.12 < or = p < or = .21) after differences in levels of benzene exposure were taken into account. The Paustenbach et al. exposures predicted a quadratic dose response, based on a measure of exposure that weighted intensity of exposure more heavily than duration of exposure. The best-fitting quadratic models predicted an additional lifetime risk of a benzene-related death from 45 yr of exposure to 1 ppm of between 0.020 and 0.036 per thousand. Statistical confidence intervals (90%) on these estimates were barely wide enough to include risk estimates based on linear dose response models. These linear models predicted risks of between 1.6 and 3.1 per thousand.     

Modeling hazard functions in families

A genetic frailty model is presented for censored age of onset data in nuclear families where individuals carrying a genetic susceptibility gene have an increased risk of becoming affected. We use maximum likelihood via the EM algorithm to estimate the genetic relative risk and the allele frequency under a dominant susceptibility type and a proportional hazards model. When sampling is from a disease registry, likelihood corrections are necessary for reducing bias in the parameter estimates. In these biased samples, the full conditional likelihood is approximated by a likelihood conditional on the proband's age of onset. For unbiased samples, simulations show the distributions of the estimates are similar under both a semiparametric and the correctly specified parametric likelihoods. For biased samples, simulations under the approximate conditional likelihood show the median estimates of the allele frequency and genetic relative risk tend to under- and overestimate, respectively, the true values; however, the approximation is better for rarer allele frequencies (0.0033 vs. 0.01). In practice, large samples or more complex ascertainment corrections are recommended. Using the approximate conditional likelihood on familial breast cancer onset data collected as part of a case-control study at the Fred Hutchinson Cancer Research Center in Seattle, Washington, we estimate an allele frequency of 0.0009 (approximate 95% CI 0.0006-0.002) and a genetic relative risk of 104 (approximate 95% CI 55-181).