Regression Models and Life-Tables
01 Jan 1972-Journal of the royal statistical society series b-methodological (JOURNAL OF ROYAL STATISTICAL SOCIETY)-Vol. 34, Iss: 2, pp 187-220
TL;DR: The analysis of censored failure times is considered in this paper, where the hazard function is taken to be a function of the explanatory variables and unknown regression coefficients multiplied by an arbitrary and unknown function of time.
Abstract: The analysis of censored failure times is considered. It is assumed that on each individual arc available values of one or more explanatory variables. The hazard function (age-specific failure rate) is taken to be a function of the explanatory variables and unknown regression coefficients multiplied by an arbitrary and unknown function of time. A conditional likelihood is obtained, leading to inferences about the unknown regression coefficients. Some generalizations are outlined.
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TL;DR: The method of classifying comorbidity provides a simple, readily applicable and valid method of estimating risk of death fromComorbid disease for use in longitudinal studies and further work in larger populations is still required to refine the approach.
39,961 citations
Book•
01 Jan 2001
TL;DR: This is the essential companion to Jeffrey Wooldridge's widely-used graduate text Econometric Analysis of Cross Section and Panel Data (MIT Press, 2001).
Abstract: The second edition of this acclaimed graduate text provides a unified treatment of two methods used in contemporary econometric research, cross section and data panel methods. By focusing on assumptions that can be given behavioral content, the book maintains an appropriate level of rigor while emphasizing intuitive thinking. The analysis covers both linear and nonlinear models, including models with dynamics and/or individual heterogeneity. In addition to general estimation frameworks (particular methods of moments and maximum likelihood), specific linear and nonlinear methods are covered in detail, including probit and logit models and their multivariate, Tobit models, models for count data, censored and missing data schemes, causal (or treatment) effects, and duration analysis. Econometric Analysis of Cross Section and Panel Data was the first graduate econometrics text to focus on microeconomic data structures, allowing assumptions to be separated into population and sampling assumptions. This second edition has been substantially updated and revised. Improvements include a broader class of models for missing data problems; more detailed treatment of cluster problems, an important topic for empirical researchers; expanded discussion of "generalized instrumental variables" (GIV) estimation; new coverage (based on the author's own recent research) of inverse probability weighting; a more complete framework for estimating treatment effects with panel data, and a firmly established link between econometric approaches to nonlinear panel data and the "generalized estimating equation" literature popular in statistics and other fields. New attention is given to explaining when particular econometric methods can be applied; the goal is not only to tell readers what does work, but why certain "obvious" procedures do not. The numerous included exercises, both theoretical and computer-based, allow the reader to extend methods covered in the text and discover new insights.
28,298 citations
Cites methods from "Regression Models and Life-Tables"
...Cox (1972) obtained a partial maximum likelihood estimator for b that does not require estimating l0ð Þ....
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TL;DR: This article proposes methods for combining estimates of the cause-specific hazard functions under the proportional hazards formulation, but these methods do not allow the analyst to directly assess the effect of a covariate on the marginal probability function.
Abstract: With explanatory covariates, the standard analysis for competing risks data involves modeling the cause-specific hazard functions via a proportional hazards assumption Unfortunately, the cause-specific hazard function does not have a direct interpretation in terms of survival probabilities for the particular failure type In recent years many clinicians have begun using the cumulative incidence function, the marginal failure probabilities for a particular cause, which is intuitively appealing and more easily explained to the nonstatistician The cumulative incidence is especially relevant in cost-effectiveness analyses in which the survival probabilities are needed to determine treatment utility Previously, authors have considered methods for combining estimates of the cause-specific hazard functions under the proportional hazards formulation However, these methods do not allow the analyst to directly assess the effect of a covariate on the marginal probability function In this article we pro
11,109 citations
TL;DR: A simple coronary disease prediction algorithm was developed using categorical variables, which allows physicians to predict multivariate CHD risk in patients without overt CHD.
Abstract: Background—The objective of this study was to examine the association of Joint National Committee (JNC-V) blood pressure and National Cholesterol Education Program (NCEP) cholesterol categories with coronary heart disease (CHD) risk, to incorporate them into coronary prediction algorithms, and to compare the discrimination properties of this approach with other noncategorical prediction functions. Methods and Results—This work was designed as a prospective, single-center study in the setting of a community-based cohort. The patients were 2489 men and 2856 women 30 to 74 years old at baseline with 12 years of follow-up. During the 12 years of follow-up, a total of 383 men and 227 women developed CHD, which was significantly associated with categories of blood pressure, total cholesterol, LDL cholesterol, and HDL cholesterol (all P,.001). Sex-specific prediction equations were formulated to predict CHD risk according to age, diabetes, smoking, JNC-V blood pressure categories, and NCEP total cholesterol and LDL cholesterol categories. The accuracy of this categorical approach was found to be comparable to CHD prediction when the continuous variables themselves were used. After adjustment for other factors, ’28% of CHD events in men and 29% in women were attributable to blood pressure levels that exceeded high normal ($130/85). The corresponding multivariable-adjusted attributable risk percent associated with elevated total cholesterol ($200 mg/dL) was 27% in men and 34% in women. Conclusions—Recommended guidelines of blood pressure, total cholesterol, and LDL cholesterol effectively predict CHD risk in a middle-aged white population sample. A simple coronary disease prediction algorithm was developed using categorical variables, which allows physicians to predict multivariate CHD risk in patients without overt CHD. (Circulation. 1998;97:1837-1847.)
9,227 citations
References
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TL;DR: In this article, the product-limit (PL) estimator was proposed to estimate the proportion of items in the population whose lifetimes would exceed t (in the absence of such losses), without making any assumption about the form of the function P(t).
Abstract: In lifetesting, medical follow-up, and other fields the observation of the time of occurrence of the event of interest (called a death) may be prevented for some of the items of the sample by the previous occurrence of some other event (called a loss). Losses may be either accidental or controlled, the latter resulting from a decision to terminate certain observations. In either case it is usually assumed in this paper that the lifetime (age at death) is independent of the potential loss time; in practice this assumption deserves careful scrutiny. Despite the resulting incompleteness of the data, it is desired to estimate the proportion P(t) of items in the population whose lifetimes would exceed t (in the absence of such losses), without making any assumption about the form of the function P(t). The observation for each item of a suitable initial event, marking the beginning of its lifetime, is presupposed. For random samples of size N the product-limit (PL) estimate can be defined as follows: L...
52,450 citations
"Regression Models and Life-Tables" refers background in this paper
...The functions (7) and (8) are maximum-likelihood estimates in the family of all possible distributions (Kaplan and Meier, 1958)....
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...…A(u)du}= 1I7I j1-i (8) u aO t(i)K (i For uncensored ata this is the usual sample survivor function; some of the asymptotic properties of (8) are given by Kaplan and Meier (1958) and by Efron (1967) and can be used to adapt to the censored case tests based on sample cumulative distribution function....
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...Kaplan and Meier (1958) gave a comprehensive review of earlier work and many new results....
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TL;DR: In this paper, the role and limitations of retrospective investigations of factors possibly associated with the occurrence of a disease are discussed and their relationship to forward-type studies emphasized, and examples of situations in which misleading associations could arise through the use of inappropriate control groups are presented.
Abstract: The role and limitations of retrospective investigations of factors possibly associated with the occurrence of a disease are discussed and their relationship to forward-type studies emphasized. Examples of situations in which misleading associations could arise through the use of inappropriate control groups are presented. The possibility of misleading associations may be minimized by controlling or matching on factors which could produce such associations; the statistical analysis will then be modified. Statistical methodology is presented for analyzing retrospective study data, including chi-square measures of statistical significance of the observed association between the disease and the factor under study, and measures for interpreting the association in terms of an increased relative risk of disease. An extension of the chi-square test to the situation where data are subclassified by factors controlled in the analysis is given. A summary relative risk formula, R, is presented and discussed in connection with the problem of weighting the individual subcategory relative risks according to their importance or their precision. Alternative relative-risk formulas, R I , R2, Ra, and R4/ which require the calculation of subcategory-adjusted proportions ot the study factor among diseased persons and controls for the computation of relative risks, are discussed. While these latter formulas may be useful in many instances, they may be biased or inconsistent and are not, in fact, overages of the relative risks observed in the separate subcategories. Only the relative-risk formula, R, of those presented, can be viewed as such an average. The relationship of the matched-sample method to the subclassification approach is indicated. The statistical methodolo~y presented is illustrated with examples from a study of women with epidermoid and undifferentiated pulmonary ccrclnomc.e-J. Nat. Cancer Inst, 22: 719748, 1959.
14,433 citations
"Regression Models and Life-Tables" refers background or methods in this paper
...The procedures proposed are, especially for the two-sample problem, closely related to procedures for combining contingency tables; see Mantel and Haenzel (1959), Mantel (1963) and, especially for the application to life tables, Mantel (1966)....
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...If0A, = 0 for all u, then the appropriate test statistic is the one discussed by Cochran (1954) and Mantel and Haenzel (1959)....
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...To test for the presence of a difference between the two samples the information from the separate tables can then be combined (Cochran, 1954; Mantel and Haenzel, 1959; Mantel, 1963)....
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Journal Article•
7,476 citations
"Regression Models and Life-Tables" refers background or methods in this paper
...The procedures proposed are, especially for the two-sample problem, closely related to procedures for combining contingency tables; see Mantel and Haenzel (1959), Mantel (1963) and, especially for the application to life tables, Mantel (1966)....
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...Z( ))) (38) t(i 1- 1T 7Ti)exp (- 0() For an estimate at a given non-zero z, replace exp (- lz(f)) by exp {f(zAlternative simpler procedures would be worth having (Mantel, 1966)....
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...The application of this to life tables is discussed especially by Mantel (1966)....
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TL;DR: In this article, the authors discuss two kinds of failure to make the best use of x2 tests which I have observed from time to time in reading reports of biological research, and propose a number of methods for strengthening or supplementing the most common uses of the ordinary x2 test.
Abstract: Since the x2 tests of goodness of fit and of association in contingency tables are presented in many courses on statistical methods for beginners in the subject, it is not surprising that x2 has become one of the most commonly-used techniques, even by scientists who profess only a smattering of knowledge of statistics. It is also not surprising that the technique is sometimes misused, e.g. by calculating x2 from data that are not frequencies or by errors in counting the number of degrees of freedom. A good catalogue of mistakes of this kind has been given by Lewis and Burke (1). In this paper I want to discuss two kinds of failure to make the best use of x2 tests which I have observed from time to time in reading reports of biological research. The first arises because x2 tests, as has often been pointed out, are not directed against any specific alternative to the null hypothesis. In the computation of x2, the deviations (fi mi) between observed and expected frequencies are squared, divided by mi in order to equalize the variances (approximately), and added. No attempt is made to detect any particular pattern of deviations (fi mi) that may hold if the null hypothesis is false. One consequence is that the usual x2 tests are often insensitive, and do not indicate significant results when the null hypothesis is actually false. Some forethought about the kind of alternative hypothesis that is likely to hold may lead to alternative tests that are more powerful and appropriate. Further, when the ordinary x2 test does give a significant result, it does not direct attention to the way in which the null hypothesis disagrees with the data, although the pattern of deviations may be informative and suggestive for future research. The remedy here is to supplement the ordinary test by additional tests that help to reveal the significant type of deviation. In this paper a number of methods for strengthening or supplementing the most common uses of the ordinary x2 test will be presented and illustrated by numerical examples. The principal devices are as follows:
3,351 citations
"Regression Models and Life-Tables" refers background in this paper
...If0A, = 0 for all u, then the appropriate test statistic is the one discussed by Cochran (1954) and Mantel and Haenzel (1959)....
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...To test for the presence of a difference between the two samples the information from the separate tables can then be combined (Cochran, 1954; Mantel and Haenzel, 1959; Mantel, 1963)....
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TL;DR: Some comparisons are made for five cases of varying degrees of censoring and tying between probabilities from the exact test and those from the proposed test and these suggest the test is appropriate under certain conditions when the sample size is five in each group.
Abstract: H2: F1(t) F2(t) (t F2(t) (t < T). The asymptotic efficiency of the test relative to the efficient parametric test when the distributions are exponential is at least 0 75 and increases with degree of censoring. When Ho is true, the test is not seriously affected by real differences in the percentage censored in the two groups. Some comparisons are made for five cases of varying degrees of censoring and tying between probabilities from the exact test and those from the proposed test and these suggest the test is appropriate under certain conditions when the sample size is five in each group. A worked example is presented and some discussion is given to further problems.
3,318 citations
"Regression Models and Life-Tables" refers background or methods or result in this paper
...The critical ratio of juist over 4 compares with about 3X6 for the generalized Wilcoxon test of Gehan (1965)....
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...This is different from the procedure of Gehan who adapted the Wilcoxon test to censored data (Gehan, 1965; Efron, 1967; Breslow, 1970)....
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...TABLE 1 Times of remission (weeks) of leukemia patients (Gehan, 1965, from Freireich et al.)...
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...AN EXAMPLE To illustrate some of the above results, it is convenient o take data of Freireich et al. used by Gehan (1965) and several subsequent authors....
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