Linear Relative Risk Model Research Articles

Meta-analysis of published excess relative risk estimates.

A meta-analytic summary effect estimate often is calculated as an inverse-variance-weighted average of study-specific estimates of association. The variances of published estimates of association often are derived from their associated confidence intervals under assumptions typical of Wald-type statistics, such as normality of the parameter. However, in some research areas, such as radiation epidemiology, epidemiological results typically are obtained by fitting linear relative risk models, and associated likelihood-based confidence intervals are often asymmetric; consequently, reasonable estimates of variances associated with study-specific estimates of association may be difficult to infer from the standard approach based on the assumption of a Wald-type interval. Here, a novel method is described for meta-analysis of published results from linear relative risk models that uses a parametric transformation of published results to improve on the normal approximation used to assess confidence intervals. Using simulations, it is illustrated that the meta-analytic summary obtained using the proposed approach yields less biased summary estimates, with better confidence interval coverage, than the summary obtained using the more classical approach to meta-analysis. The proposed approach is illustrated using a previously published example of meta-analysis of epidemiological findings regarding circulatory disease following exposure to low-level ionizing radiation.

Issues in Interpreting Epidemiologic Studies of Populations Exposed to Low-Dose, High-Energy Photon Radiation.

This article addresses issues relevant to interpreting findings from 26 epidemiologic studies of persons exposed to low-dose radiation. We review the extensive data from both epidemiologic studies of persons exposed at moderate or high doses and from radiobiology that together have firmly established radiation as carcinogenic. We then discuss the use of the linear relative risk model that has been used to describe data from both low- and moderate- or high-dose studies. We consider the effects of dose measurement errors; these can reduce statistical power and lead to underestimation of risks but are very unlikely to bring about a spurious dose response. We estimate statistical power for the low-dose studies under the assumption that true risks of radiation-related cancers are those expected from studies of Japanese atomic bomb survivors. Finally, we discuss the interpretation of confidence intervals and statistical tests and the applicability of the Bradford Hill principles for a causal relationship.

Poisson excess relative risk models: New implementations and software

Two new implementations for fitting Poisson excess relative risk methods are proposed for assumed simple models. This allows for estimation of the excess relative risk associated with a unique exposure, where the background risk is modelled by a unique categorical variable, for example gender or attained age levels. Additionally, it is shown how to fit general Poisson linear relative risk models in R. Both simple methods and the R fitting are illustrated in three examples. The first two examples are from the radiation epidemiology literature. Data in the third example are randomly generated with the purpose of sharing it jointly with the R scripts.

Risk Estimation with Epidemiologic Data When Response Attenuates at High-Exposure Levels

BackgroundIn occupational studies, which are commonly used for risk assessment for environmental settings, estimated exposure–response relationships often attenuate at high exposures. Relative risk (RR) models with transformed (e.g., log- or square root–transformed) exposures can provide a good fit to such data, but resulting exposure–response curves that are supralinear in the low-dose region may overestimate low-dose risks. Conversely, a model of untransformed (linear) exposure may underestimate risks attributable to exposures in the low-dose region.MethodsWe examined several models, seeking simple parametric models that fit attenuating exposure–response data well. We have illustrated the use of both log-linear and linear RR models using cohort study data on breast cancer and exposure to ethylene oxide.ResultsLinear RR models fit the data better than do corresponding log-linear models. Among linear RR models, linear (untransformed), log-transformed, square root–transformed, linear-exponential, and two-piece linear exposure models all fit the data reasonably well. However, the slopes of the predicted exposure–response relations were very different in the low-exposure range, which resulted in different estimates of the exposure concentration associated with a 1% lifetime excess risk (0.0400, 0.00005, 0.0016, 0.0113, and 0.0100 ppm, respectively). The linear (in exposure) model underestimated the categorical exposure–response in the low-dose region, whereas log-transformed and square root–transformed exposure models overestimated it.ConclusionAlthough a number of models may fit attenuating data well, models that assume linear or nearly linear exposure–response relations in the low-dose region of interest may be preferred by risk assessors, because they do not depend on the choice of a point of departure for linear low-dose extrapolation and are relatively easy to interpret.

Estimation of the Relative Excess Risk Due to Interaction and Associated Confidence Bounds

The relative excess risk due to interaction (RERI) provides a useful metric of departure from additivity of effects on a relative risk scale. In this paper, the authors show that RERI is identical to the product term in a linear odds ratio or a linear relative risk model. SAS and STATA codes are provided for fitting a linear odds ratio model that directly parameterizes RERI. In addition, this paper presents a method for obtaining likelihood-based 95% confidence bound estimates for RERI. The authors show that likelihood-based confidence intervals may differ substantially from the asymptotic confidence interval estimates advocated by previous authors. The approach presented in this paper should facilitate estimation of RERI and associated likelihood-based confidence bounds, by using standard statistical packages.

Follow-up study of chrysotile textile workers: cohort mortality and exposure-response

Objectives: This report provides an update of the mortality experience of a cohort of South Carolina asbestos textile workers. Methods: A cohort of 3072 workers exposed to chrysotile in a...

Prognostic significance of six-month estimated glomerular filtration rate in cadaveric renal transplantation.

We hypothesized a linear relative-risk model for graft survival of cadaveric renal transplantation, with a 6-month estimated glomerular filtration rate (GFR) employed as the linear parameter quantifying functioning renal mass and the detrimental effects of early chronic allograft nephropathy. A retrospective study was conducted in a 15-year series of cadaveric renal transplantations (n = 227) with cyclosporin-based immunosuppression in a single transplant center. The Cockcroft-Gault formula was used for estimation of GFR, with a correction factor of 0.85 used for women. Stepwise Cox's regression analyses were applied to examine the prognostic significance of the 6-month estimated GFR. The Kaplan-Meier 5- and 10-year graft survival rates for this study were 80.67% and 59.43%, respectively. From univariate analysis, recipient gender, acute rejection and 6-month estimated GFR were significantly associated with graft survival (p < 0.05). Acute rejection became less significant (p = 0.0033) than 6-month estimated GFR (p = 0.0001) when 6-month estimated GFR was introduced in the stepwise regression procedures. We demonstrated that 6-month estimated GFR is a significant prognostic indicator in cadaveric renal transplantation and is an essential parameter in the regression modeling of long-term graft survival.

The bystander effect model of Brenner and Sachs fitted to lung cancer data in 11 cohorts ofunderground miners, and equivalence of fit of a linear relative risk model with adjustmentfor attained age and age at exposure

Bystander effects following exposure toα-particles have been observed in many experimental systems, and imply that linearlyextrapolating low dose risks from high dose data might materially underestimate risk.Brenner and Sachs (2002 Int. J. Radiat. Biol. 78 593–604; 2003 Health Phys. 85 103–8) haverecently proposed a model of the bystander effect which they use to explain the inversedose rate effect observed for lung cancer in underground miners exposed to radondaughters. In this paper we fit the model of the bystander effect proposed by Brenner andSachs to 11 cohorts of underground miners, taking account of the covariance structure ofthe data and the period of latency between the development of the first pre-malignant celland clinically overt cancer. We also fitted a simple linear relative risk model, withadjustment for age at exposure and attained age. The methods that we use for fitting bothmodels are different from those used by Brenner and Sachs, in particular taking account ofthe covariance structure, which they did not, and omitting certain unjustifiableadjustments to the miner data. The fit of the original model of Brenner and Sachs (with 0 yperiod of latency) is generally poor, although it is much improved by assuming a5 or 6 y period of latency from the first appearance of a pre-malignant cell tocancer. The fit of this latter model is equivalent to that of a linear relative riskmodel with adjustment for age at exposure and attained age. In particular, bothmodels are capable of describing the observed inverse dose rate effect in this dataset.

Methods for Detecting and Estimating Population Threshold Concentrations for Air Pollution‐Related Mortality with Exposure Measurement Error

The association between daily fluctuations in ambient particulate matter and daily variations in nonaccidental mortality have been extensively investigated. Although it is now widely recognized that such an association exists, the form of the concentration‐response model is still in question. Linear, no threshold and linear threshold models have been most commonly examined. In this paper we considered methods to detect and estimate threshold concentrations using time series data of daily mortality rates and air pollution concentrations. Because exposure is measured with error, we also considered the influence of measurement error in distinguishing between these two completing model specifications. The methods were illustrated on a 15‐year daily time series of nonaccidental mortality and particulate air pollution data in Toronto, Canada. Nonparametric smoothed representations of the association between mortality and air pollution were adequate to graphically distinguish between these two forms. Weighted nonlinear regression methods for relative risk models were adequate to give nearly unbiased estimates of threshold concentrations even under conditions of extreme exposure measurement error. The uncertainty in the threshold estimates increased with the degree of exposure error. Regression models incorporating threshold concentrations could be clearly distinguished from linear relative risk models in the presence of exposure measurement error. The assumption of a linear model given that a threshold model was the correct form usually resulted in overestimates in the number of averted premature deaths, except for low threshold concentrations and large measurement error.

Methods for detecting and estimating population threshold concentrations for air pollution-related mortality with exposure measurement error.

The association between daily fluctuations in ambient particulate matter and daily variations in nonaccidental mortality have been extensively investigated. Although it is now widely recognized that such an association exists, the form of the concentration-response model is still in question. Linear, no threshold and linear threshold models have been most commonly examined. In this paper we considered methods to detect and estimate threshold concentrations using time series data of daily mortality rates and air pollution concentrations. Because exposure is measured with error, we also considered the influence of measurement error in distinguishing between these two completing model specifications. The methods were illustrated on a 15-year daily time series of nonaccidental mortality and particulate air pollution data in Toronto, Canada. Nonparametric smoothed representations of the association between mortality and air pollution were adequate to graphically distinguish between these two forms. Weighted nonlinear regression methods for relative risk models were adequate to give nearly unbiased estimates of threshold concentrations even under conditions of extreme exposure measurement error. The uncertainty in the threshold estimates increased with the degree of exposure error. Regression models incorporating threshold concentrations could be clearly distinguished from linear relative risk models in the presence of exposure measurement error. The assumption of a linear model given that a threshold model was the correct form usually resulted in overestimates in the number of averted premature deaths, except for low threshold concentrations and large measurement error.

Curvature in the cancer mortality dose response in Japanese atomic bomb survivors: absence of evidence of threshold

Purposes : To investigate the evidence for a threshold in the cancer dose-response curve. Materials and methods : Japanese atomic bomb survivor cancer mortality data, based on follow-up to 1990, was used, taking account of random errors in DS86 dose estimates. Results : For all solid cancers analysed together, there is a significant positive dose response (two-sided p 0.05) if all survivors who received 0.5Sv are considered, but the significance vanishes if doses of 0.2 Sv are considered; the same is also true for leukaemia. For solid cancer mortality there is no indication of curvilinearity in the dose response: no statistically significant improvement in fit to a linear relative risk model is provided by addition of quadratic or threshold dose terms. If a relative risk model with a threshold (the dose response is assumed linear above the threshold) is fitted to solid cancer mortality data, the best estimate of the threshold is 0.00 Sv (95% CI 0.00-0.13). If a linearquadratic-threshold model is used the best estimate of the threshold is 0.00 Sv (95% CI 0.00-0.15). For leukaemia mortality there is highly statistically significant upward curvature in the dose response. In particular, if a relative risk model with a threshold (the dose response is assumed linear above the threshold) is fitted to the leukaemia data, the best estimate of the threshold is 0.16Sv (95% CI 0.05-0.40) (twosided p 0.001 for test of departure of threshold from 0). However, there is no evidence for a threshold effect (two-sided p 0.16) when a quadratic term is included in the dose response: the best estimate of threshold in this case is 0.09Sv (95% CI 0.00-0.29). Moreover, addition of a quadratic term improves the fit of a linear-threshold model at borderline levels of statistical significance (two-sided p 0.07). Therefore, the most parsimonious description of the leukaemia dose response is provided by a linear-quadratic function of dose. Conclusions : There is no evidence of threshold-type departures from the linear-quadratic dose response either for solid tumours or for leukaemia in the Japanese atomic bomb survivor mortality data.

Estimation of the temporal distribution and dose dependency of lung cancers among workers of nuclear fuel reprocessing plants.

Lung cancer mortality among 4,279 workers at the Mayak nuclear complex who were exposed to chronic irradiation both externally and internally from incorporation of plutonium was analyzed in terms of a linear relative risk model. It is estimated that excess lung cancer risk is about 1.9 Sv(-1), with an average latent period of 24 y.

Compensating lung cancer patients occupationally exposed to coal tar pitch volatiles.

To investigate the process of deciding on compensation claims by lung cancer patients exposed occupationally to coal tar pitch volatiles. For each case of lung cancer the probability that it was caused (probability of causation (PC)) by coal tar pitch volatiles was expressed as an increasing function of cumulative exposure to benzo-a-pyrene-years. This was assessed from several exposure-response models fitted to data from a large epidemiological study of aluminum production workers. For some models, PC depended also on the smoking habit of the cancer patient. Estimation of relative risk by exposure group indicated that over 50% of lung cancers were attributable to coal tar pitch volatiles (PC > 50%) at exposures above 100 micrograms/m3-years benzo(a)pyrene. A linear relative risk model indicated that 50% PC was first achieved at 342.2 micrograms/m3-years benzo(a)pyrene, or 190.1 micrograms/m3-years benzo(a)pyrene according to the upper 95% confidence limit for risk increment. Corresponding figures for a power curve model were 210.3 and 45.9. With these five figures as compensation criteria compensation would have resulted in 31.4%, 2.7%, 19.2%, 15.7%, and 39.2% of cancers studied, compared with an estimated total proportion of cancers studied attributable to coal tar pitch volatiles of 15%-26%. If risks due to coal tar pitch volatiles and smoking multiply, PC does not depend on the amount smoked. If the two risks are additive, however, PC depends on the amount smoked according to a formula, with the figures mentioned applying to an average smoking history (24.4 pack-years). Because of its simplicity and because it falls within the range of criteria based on several more sophisticated approaches, we prefer the criterion of 100 micrograms/m3-years, based on the relative risks by exposure group. However, the compensation board of the Canadian province of Quebec, on consideration of these alternatives, has proposed as a criterion that the upper 95% confidence limit of PC for the patient be at least 50%, assuming an additive relative risk model and allowing for their smoking habit.

Tests of Loglinear and Linear Relative Risks for Cox's Model

SUMMARY A score test statistic is derived to test for the loglinear relative risk function in the Cox proportional hazards model for failure time data. The loglinear and linear relative risk functions are embedded within a power family transformation of the linear predictor. In two simple cases some empirical results are obtained for the score test statistic using expected information and for the likelihood ratio statistic. Both statistics have distributions well approximated by the chi-squared distribution. Problems of identifiability and problems associated with the linear relative risk model are also discussed. A real example is given.

Lung cancer in radiochemical industry workers

The frequency of lung cancer was investigated among 2346 workers in the radiochemical plant ‘Mayak’ who were exposed to radiation, linear both externally and internally from incorporation of plutonium. The results from the follow-up were analysed in terms of a linear relative risk model, accounting for the protracted exposures. On the basis of the observed data, it is deduced that the life-span incidence of radiation-induced lung cancer is several times larger than the value 0.0085 Sv −1 that is recommended in ICRP Publication 60 for the risk to this organ.

Noncancer Disease Incidence in the Atomic Bomb Survivors: 1958-1986

Using the longitudinal data of the Adult Health Study (AHS) cohort collected during 1958-1986, we examined for the first time the relationship between exposure to ionizing radiation and the incidence of 19 nonmalignant disorders in the A-bomb survivors. Affected individuals were ascertained through the three-digit codes of the International Classification of Diseases which are encoded in the AHS database subsequent to diagnoses made on the basis of general laboratory tests, physical examinations, and history-taking conducted during biennial AHS examinations. The disease onset time was estimated using the mid-point between the AHS examination data when the disease was initially reported and the previously attended disease-free examination date. Dosimetry System 86 organ doses judged to be most appropriate were used. Tests of dose effects were performed assuming a linear relative risk model with stratified background incidence. For the entire study period, significant excess risk was detected for uterine myoma (P < 0.001), chronic liver disease and cirrhosis (P = 0.006), and thyroid disease (P < 0.0001), defined broadly as the presence of one or more of certain noncancerous thyroid conditions. The incidence of myocardial infarction was shown to be increased (P = 0.03) in later years (1968-1986) among the younger heavily exposed AHS subjects, confirming the results of the recent Life Span Study (LSS) noncancer mortality report on coronary heart disease. The findings for uterine myoma may serve as additional evidence indicating benign tumor growth as a possible consequence of radiation exposure. Our results indicating the involvement of radiation in the development of liver diseases are consistent with the report of increased mortality from liver cirrhosis with radiation dose in the LSS cohort. An effect of age at exposure was detected for nonmalignant thyroid disease (P = 0.02), with an increased risk for those exposed who were under 20 years of age, but not for older survivors. Thus the AHS data suggest that thyroid glands in the young are more radiosensitive not only to the development of malignancies, but also to the development of nonmalignant disorders as well. The findings hold independently of the dose effects observed for thyroid cancer. This study also shows that for the period 1958-1986 new occurrences of lens opacity are not increased with radiation dose (P = 0.39) in the AHS subjects.

Mortality among the Offspring(F1) of Atomic Bomb Survivors, 1946-85.

We compare the mortality in the years 1946-85 in a cohort of 31,159 children born to parents one or both of whom were exposed to the atomic bombing of Hiroshima and Nagasaki (a parental gonadal dose greater than or equal to 0.01 Sv) with that in a control group of 41,069 children. The average gonadal dose for the exposed parents was 0.435 Sv. The mean age of the cohorts was 28.8 years. In the greater than or equal to 0.01 Sv dose group 1,253 deaths were observed in the subset of children both of whose parents have been assigned DS86 doses. 3.2% were attributed to cancers, 72.9% to all diseases except neoplasms. These proportions in the 0 Sv dose group were about the same. Based on a linear relative risk model, no statistically significant increase in the mortality attributable to diseases other than neoplasms is noted following parental exposure, the excess relative risk being 0.030 (+/- 0.046) per sievert based on the DS86 doses (RBE of neutrons = 20). For fatal cancer, no statistically significant effect of parental radiation dose was also observed. An analysis based on the full sample, using not only the DS86 dose group but also ad hoc dose group, yields essentially the same results as the analysis restricted to the DS86 dose group.

Cigarette smoking and cancer mortality risk in Japanese men and women--results from reanalysis of the six-prefecture cohort study data.

In 1965 a cohort of 265,000 residents of 29 public health districts in six prefectures throughout Japan was established and followed between 1966 and 1981. By using survival analysis based on Poisson regression models adjusted for age, prefecture of residence, and occupation, a statistically significant dose-response relationship between cigarette smoking and mortality rate was found for cancers of the liver, pancreas, and lung in both sexes; cancers of the oral cavity, esophagus, stomach, larynx, and bladder in men; and cancer of the uterus in women. The magnitudes of relative risks were only slightly affected by adjustment for prefecture of residence or occupation. Analysis using linear relative risk models revealed that the inclusion of a quadratic term for the amount of daily cigarette consumption in addition to the linear term improved the fit of the model significantly for cancers of the esophagus and stomach in men and cancer of the stomach in women. The sex ratio of gastric cancer mortality was higher among smokers than among nonsmokers. When the follow-up period was divided into four 4-year intervals, it was noted that the relative mortality risk associated with lung cancer among males increased significantly during these time periods.

Lung cancer incidence among Japanese A-bomb survivors, 1950-80.

The incidence of lung cancer during 1950-80 in a cohort of Japanese A-bomb survivors and controls was investigated. A total of 1, 057 cases were identified; 608 of these diagnoses were based on histopathological examination, and 442 were confirmed by the present investigators. The distributions of histologic types varied significantly between the sexes (p < .001), with adenocarcinoma more frequent among women and epidermoid and small cell carcinoma more frequent among men. The distributions of primary sites did not differ significantly between the sexes. The relative risk (RR) of lung cancer increased significantly with A-bomb radiation dose (p < .0001): based on tentative 1965 dose estimates as revised in 1978 (T65DR) and a linear RR model, the estimated RR at 100 rad (±s.e.) is 1.41 ± 0.09. Among Hiroshima survivors the women experienced radiation-related excess RR nearly twice as great as men (p = .06). RR increased with decreasing age at the time of bombing (ATB; p = .07), and after allowing for this effect, there was no significant evidence that RR varied systematically with attained age. Small cell carcinoma displayed somewhat greater sensitivity to radiation than did adenocarcinoma or epidermoid carcinoma; however the variation between the histology-specific RR functions was not statistically significant (p = .44).

On the Application of Linear Relative Risk Regression Models

Motivations for considering linear relative risk models are described. These include covariate data reduction and the testing of additivity of covariate effects on the relative risk. A simulation study was conducted in order to study properties of asymptotic distributional approximations and iterative convergence properties. Parameter transformations and likelihood ratio approximations are considered for confidence interval calculation.