Relative Risk Function Research Articles

Estimation of relative risk functions

The risk associated with different levels of a quantitative factor X is often measured relative to the level corresponding to X = 0. There are situations, however, where there is no natural zero for X, for example where the risk factor is the age of an individual. In this case it is more natural to measure risk relative to an overall average for the study population. To use the whole population in this way also raises the possibility of regarding X as truly continuous, rather than as a grouped variable. This gives rise to the concept of a relative risk function. Methods for estimating such functions are discussed, concentrating for the most part on the discrete case. The extension to higher dimensions permits the investigation of joint effects of several factors, while the problem of controlling for confounding variables can be handled by fitting multiplicative risk models. Relating the latter to the log-linear model permits the estimation of adjusted relative risk functions. The method is illustrated using data on childhood cancer. The continuous case can in principle be handled in a similar way using density estimation techniques.

Radiation dose and leukaemia risk: General relative risk techniques for dose‐response models in a matched case‐control study

Generalized relative risk functions were used to model radiation dose-response information from a large matched case-control study of leukaemia occurring after treatment for cervical cancer. Models suggested by radiobiological theory were investigated and compared to standard analyses of categorical dose-response to the linear model. Local radiation doses to each of fourteen bone marrow compartments for each patient were incorporated into the models, and the corresponding risks were summed. Conditional maximum likelihood methods were used to estimate risk parameters. Unique features of the analysis include modelling both induction and reduction of risk as a function of radiation dose absorbed by different parts of the body within individuals. Detailed statistical aspects of these analyses are presented and discussed.

An application of density estimation to geographical epidemiology

A relative risk function over a geographical region is defined and it is shown that it can be estimated effectively using kernel density estimation separately for the spatial distribution of disease cases and for a sample of controls. This procedure is demonstrated using data on childhood leukaemia in the vicinity of the Sellafield nuclear reprocessing plant in Cumbria, U.K. Various modifications to the method are proposed, including the use of an adaptive kernel. The final plot demonstrates a sharp peak at Sellafield and a reasonably smooth surface over the rest of the region, despite the small number of cases in the series.

Nonparametric Survival Analysis with Time-Dependent Covariate Effects: A Penalized Partial Likelihood Approach

Techniques are developed for nonparametric analysis of data under a Cox-regression-like model permitting time-dependent covariate effects determined by a regression function $\beta_0(t)$. Estimators resulting from maximization of an appropriate penalized partial likelihood are shown to exist and a computational approach is outlined. Weak uniform consistency (with a rate of convergence) and pointwise asymptotic normality of the estimators are established under regularity conditions. A consistent estimator of a common baseline hazard function is presented and used to construct a consistent estimator of the asymptotic variance of the estimator of the regression function. Extensions to multiple covariates, general relative risk functions and time-dependent covariates are discussed.

Validity of international, time trend, and migrant studies of dietary factors and disease risk

A linear form relative risk model is used to identify circumstances in which various types of aggregate data lead to valid inferences on relative risk parameters. Upon making a random effects assumption, international or time trend data can lead to appropriate relative risk parameter estimation using iteratively reweighted least-squares procedures. Adequate confounding factor control, however, will typically require data on the distribution of confounding factors in each country or time period. For a simple interpretation of relative risk parameters one may also require data on the joint distribution of primary and confounding factors in each country or time period. Hence disease rate data need to be supplemented by dietary and risk factor survey data in order to avoid confounding bias. Measurement error in individual dietary assessment may, however, limit the ability to quantify the dependence of relative risk on dietary factors, unless the relative risk function is approximately linear in the dietary factors of interest. Most studies of migrant populations involve a comparison of migrant mortality rates with those of their countries of emigration and immigration, with little or no data collection on the dietary habits and risk factors of the migrants themselves. The potential of more comprehensive aggregate data and analytic migrant studies in the diet and disease area is briefly indicated. These issues and methods are illustrated using various types of data pertinent to the association between dietary fat and breast cancer.

Origin-invariant relative risk functions for case-control and survival studies

SUMMARY Several parametric families of relative risk functions have been proposed as models for matched case-control and survival data. Some advantages accrue to those in which rela-tive risks are invariant under arbitrary translations of the origin of the covariate space, such as in the reassignment of values to dichotomous factors. It is shown in this paper that the family proposed by Guerrero & Johnson, which includes the commonly-used exponential and linear relative risk functions, has the simplest form of this origin invariance property. A differential equation that general origin-invariant relative risk functions must satisfy is derived. The effect of covariate translations on the parameter space is also discussed.

General relative risk regression models for epidemiologic studies.

Three parametric families of relative risk functions for the analysis of case-control data are discussed. A desirable feature for any general relative risk function is that inference based on it be independent of the coding of a binary covariate. Only one of the three families considered has this property. Additionally, when the relative risk is not multiplicative, methods of inference based on the asymptotic covariance matrix are likely to be seriously misleading unless the sample size is very large, as has been noted previously in other papers. This is illustrated by means of examples. Likelihood-based procedures should routinely be employed when nonmultiplicative relative risk functions are used for analysis of case-control data.

Confidence Regions in Curved Exponential Families: Application to Matched Case-Control and Survival Studies with General Relative Risk Function

Differential geometric methods are used to construct approximate confidence regions for curved exponential families. The $\alpha$-connection geometries discussed by Amari (1982), and another geometry introduced here, the $c$ geometry, are exploited to construct confidence regions. Survival and case-control studies with general relative risk functions are interpreted in the context of curved exponential families, and an example illustrates the construction of confidence regions for matched case-control studies. Simulations indicate that the geometric procedures have good coverage and power properties.

Lung Cancer Mortality Among U.S. Uranium Miners: A Reappraisal2

This report examines lung cancer mortality among a cohort of white underground uranium miners in the Colorado plateau and is based on mortality follow-up through December 31, 1977. The analytic methods represent a miner's annual age-specific lung cancer mortality rate as the (unspecified) rate among nonsmoking men born at the same time and with no mining history, multiplied by the relative risk factor R. This factor depends on the miner's total exposures to radon daughters [in working level months (WLM) and to cigarettes (in packs), accumulated from start of exposure until 10 years before his current age. Among those examined, the relative risk function giving the highest likelihood of the data was R = (1 + 0.31 X 10(-2) WLM)(1 + 0.51 X 10(-3) packs). This multiplicative function specifies that ratios of mortality rates for miners versus nonminers with similar age and smoking characteristics do not depend on smoking status. By contrast, differences between miners' and nonminers' mortality rates are substantially higher for smokers than for nonsmokers. The data rejected (P = .01) several additive functions for R that specify relative risk as a sum of components due to radiation and to cigarette smoking. Cumulative exposures to both radiation and cigarettes gave better fits to the data than did average annual exposure rates. Age at start of underground mining had no effect on risk, after controlling for age at lung cancer death, year of birth, and cumulative radiation and smoking exposures.

Hospital or population controls: A discussion

In conducting a case-control study, it is a good idea to consider simultaneous hospital and neighborhood controls. If one runs two independent studies, one with neighborhood controls and one with hospital controls, and if the target case populations come from different target populations (geographic or otherwise), then the comparison of control groups is meaningless. A comparison of the estimated relative risk function, in terms of a set ofcovariates, between the studies would be meaningful. Analysis along these lines can be found in [5]. When running a case-control study, it is ethically important to fully inform the subject about the study. The authors' practice of asking about the use of other irrelevant products to “attempt to make the purpose of the study less obvious to both interviewers and respondents” is questionable. There is no doubt that there is a great deal of skill and judgement required in deciding upon the appropriate control group or groups for a case-control study. Clearly, availability of resources will have impact on this decision. There is a tradeoff in choosing one type of control over another. If two separate control groups are used, there is sometimes a loss in economy of scale and as a result the total sample size may have to be decreased. In the Stravraky and Clarke studies, it might have paid off to aim at one control group only and to spend the resources thus freed on efforts to secure a higher response rate. For instance, two recalls instead of one would probably have improved the neighborhood control group response rate and representativeness.

Serial blood pressure measurements and cardiovascular disease in a Japanese cohort.

A cohort of 16,711 residents of Hiroshima and Nagasaki have participated in a program of biennial clinical examination and history taking that began in 1958. During 1958-1974, a total of 621 confirmed cases of cerebrovascular disease and 218 confirmed cases of coronary heart disease were incident. This study makes a detailed examination of the relationship between a series of biennial examination blood pressure (BP) measurements and cardiovascular disease risk. Two aspects are emphasized: the quantitative relationship between risk and both systolic blood pressure (SBP) and diastolic blood pressure (DBP), and the predictive value of BP levels some years in the past, given more recent BP determinations. Cerebral hemorrhage incidence is shown to depend markedly on recent DBP level, while earlier DBP levels make an additional important contribution to risk prediction. Corresponding SBP levels have little additional predictive value. With cerebral infarction, SBP is the more important predictor, though elevated DBP conveys some additional risk at high SBP levels. Recent BP levels are more strongly predictive for cerebral infarction than are BP levels some years earlier. Elevated SBP is also an important predictor of coronary heart disease risk in this population, while elevated DBP does not appear to be predictive at specified SBP. SBP levels several years in the past are more closely associated with coronary heart disease risk than are recent SBP readings. The dependence of BP relative risk functions on sex and age is examined and some departures from the results just listed are noted at younger ages. Implications for disease mechanism and hypertensive therapy are discussed.

Multiple relative risk functions in case-control studies.

Siegel, D. G., and S. W. Greenhouse (Nat. Inst Child Health and Human Development, NIH, Bethesda, Md. 20014.) Multiple relative risk functions in case-control studies. Am J Epidemiol 97: 324–331, 1973.—In case-control studies in which the risk of disease in the population is unknown, multiple relative risk functions similar to the multiple risk functions used in cohort studies can be obtained. This was demonstrated for two population modeLs, the linear discrimination model and the linear multiple regression model. In the first model ratios of odds are yielded; in the second model the ratios of risks can be obtained if the risks themselves are small.