Proportional hazards tests and diagnostics based on weighted residuals
TL;DR: In this article, Chen et al. showed that a treatment effect that decreases with time can be directly visualized by smoothing an appropriate residual plot, which can be expressed as a weighted least-squares line fitted to the residual plot.
Abstract: SUMMARY Nonproportional hazards can often be expressed by extending the Cox model to include time varying coefficients; e.g., for a single covariate, the hazard function for subject i is modelled as exp { fl(t)Zi(t)}. A common example is a treatment effect that decreases with time. We show that the function /3(t) can be directly visualized by smoothing an appropriate residual plot. Also, many tests of proportional hazards, including those of Cox (1972), Gill & Schumacher (1987), Harrell (1986), Lin (1991), Moreau, O'Quigley & Mesbah (1985), Nagelkerke, Oosting & Hart (1984), O'Quigley & Pessione (1989), Schoenfeld (1980) and Wei (1984) are related to time-weighted score tests of the proportional hazards hypothesis, and can be visualized as a weighted least-squares line fitted to the residual plot.
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TL;DR: In this article, an easily interpretable index of predictive discrimination as well as methods for assessing calibration of predicted survival probabilities are discussed, which are particularly needed for binary, ordinal, and time-to-event outcomes.
Abstract: Multivariable regression models are powerful tools that are used frequently in studies of clinical outcomes. These models can use a mixture of categorical and continuous variables and can handle partially observed (censored) responses. However, uncritical application of modelling techniques can result in models that poorly fit the dataset at hand, or, even more likely, inaccurately predict outcomes on new subjects. One must know how to measure qualities of a model's fit in order to avoid poorly fitted or overfitted models. Measurement of predictive accuracy can be difficult for survival time data in the presence of censoring. We discuss an easily interpretable index of predictive discrimination as well as methods for assessing calibration of predicted survival probabilities. Both types of predictive accuracy should be unbiasedly validated using bootstrapping or cross-validation, before using predictions in a new data series. We discuss some of the hazards of poorly fitted and overfitted regression models and present one modelling strategy that avoids many of the problems discussed. The methods described are applicable to all regression models, but are particularly needed for binary, ordinal, and time-to-event outcomes. Methods are illustrated with a survival analysis in prostate cancer using Cox regression.
7,879 citations
TL;DR: The gene-expression profile studied is a more powerful predictor of the outcome of disease in young patients with breast cancer than standard systems based on clinical and histologic criteria.
Abstract: Background A more accurate means of prognostication in breast cancer will improve the selection of patients for adjuvant systemic therapy. Methods Using microarray analysis to evaluate our previously established 70-gene prognosis profile, we classified a series of 295 consecutive patients with primary breast carcinomas as having a gene-expression signature associated with either a poor prognosis or a good prognosis. All patients had stage I or II breast cancer and were younger than 53 years old; 151 had lymph-node–negative disease, and 144 had lymph-node–positive disease. We evaluated the predictive power of the prognosis profile using univariable and multivariable statistical analyses. Results Among the 295 patients, 180 had a poor-prognosis signature and 115 had a good-prognosis signature, and the mean (±SE) overall 10-year survival rates were 54.6±4.4 percent and 94.5±2.6 percent, respectively. At 10 years, the probability of remaining free of distant metastases was 50.6±4.5 percent in the group with a...
5,902 citations
24 Aug 2005
TL;DR: An easily interpretable index of predictive discrimination as well as methods for assessing calibration of predicted survival probabilities are discussed, applicable to all regression models, but are particularly needed for binary, ordinal, and time-to-event outcomes.
Abstract: Multivariable regression models are powerful tools that are used frequently in studies of clinical outcomes. These models can use a mixture of categorical and continuous variables and can handle partially observed (censored) responses. However, uncritical application of modelling techniques can result in models that poorly fit the dataset at hand, or, even more likely, inaccurately predict outcomes on new subjects. One must know how to measure qualities of a model's fit in order to avoid poorly fitted or overfitted models. Measurement of predictive accuracy can be difficult for survival time data in the presence of censoring. We discuss an easily interpretable index of predictive discrimination as well as methods for assessing calibration of predicted survival probabilities. Both types of predictive accuracy should be unbiasedly validated using bootstrapping or cross-validation, before using predictions in a new data series. We discuss some of the hazards of poorly fitted and overfitted regression models and present one modelling strategy that avoids many of the problems discussed. The methods described are applicable to all regression models, but are particularly needed for binary, ordinal, and time-to-event outcomes. Methods are illustrated with a survival analysis in prostate cancer using Cox regression.
4,905 citations
01 Jan 1996
TL;DR: An easily interpretable index of predictive discrimination as well as methods for assessing calibration of predicted survival probabilities are discussed, which are applicable to all regression models, but are particularly needed for binary, ordinal, and time-to-event outcomes.
Abstract: SUMMARY Multivariable regression models are powerful tools that are used frequently in studies of clinical outcomes. These models can use a mixture of categorical and continuous variables and can handle partially observed (censored) responses. However, uncritical application of modelling techniques can result in models that poorly fit the dataset at hand, or, even more likely, inaccurately predict outcomes on new subjects. One must know how to measure qualities of a model's fit in order to avoid poorly fitted or overfitted models. Measurement of predictive accuracy can be difficult for survival time data in the presence of censoring. We discuss an easily interpretable index of predictive discrimination as well as methods for assessing calibration of predicted survival probabilities. Both types of predictive accuracy should be unbiasedly validated using bootstrapping or cross-validation, before using predictions in a new data series. We discuss some of the hazards of poorly fitted and overfitted regression models and present one modelling strategy that avoids many of the problems discussed. The methods described are applicable to all regression models, but are particularly needed for binary, ordinal, and time-to-event outcomes. Methods are illustrated with a survival analysis in prostate cancer using Cox regression. Accurate estimation of patient prognosis is important for many reasons. First, prognostic estimates can be used to inform the patient about likely outcomes of her disease. Second, the physician can use estimates of prognosis as a guide for ordering additional tests and selecting appropriate therapies. Third, prognostic assessments are useful in the evaluation of technologies; prognostic estimates derived both with and without using the results of a given test can be compared to measure the incremental prognostic information provided by that test over what is provided by prior information.' Fourth, a researcher may want to estimate the effect of a single factor (for example, treatment given) on prognosis in an observational study in which many uncontrolled confounding factors are also measured. Here the simultaneous effects of the uncontrolled variables must be controlled (held constant mathematically if using a regression model) so that the effect of the factor of interest can be more purely estimated. An analysis of how variables (especially continuous ones) affect the patient outcomes of interest is necessary to
4,782 citations
TL;DR: Bariatric surgery for severe obesity is associated with long-term weight loss and decreased overall mortality.
Abstract: Background Obesity is associated with increased mortality. Weight loss improves cardiovascular risk factors, but no prospective interventional studies have reported whether weight loss decreases overall mortality. In fact, many observational studies suggest that weight reduction is associated with increased mortality. Methods The prospective, controlled Swedish Obese Subjects study involved 4047 obese subjects. Of these subjects, 2010 underwent bariatric surgery (surgery group) and 2037 received conventional treatment (matched control group). We report on overall mortality during an average of 10.9 years of follow-up. At the time of the analysis (November 1, 2005), vital status was known for all but three subjects (follow-up rate, 99.9%). Results The average weight change in control subjects was less than ±2% during the period of up to 15 years during which weights were recorded. Maximum weight losses in the surgical subgroups were observed after 1 to 2 years: gastric bypass, 32%; vertical-banded gastropl...
4,297 citations
References
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01 Jan 1972
TL;DR: The drum mallets disclosed in this article are adjustable, by the percussion player, as to balance, overall weight, head characteristics and tone production of the mallet, whereby the adjustment can be readily obtained.
Abstract: The drum mallets disclosed are adjustable, by the percussion player, as to weight and/or balance and/or head characteristics, so as to vary the "feel" of the mallet, and thus also the tonal effect obtainable when playing upon kettle-drums, snare-drums, and other percussion instruments; and, typically, the mallet has frictionally slidable, removable and replaceable, external balancing mass means, positionable to serve as the striking head of the mallet, whereby the adjustment as to balance, overall weight, head characteristics and tone production may be readily obtained. In some forms, the said mass means regularly serves as a removable and replaceable striking head; while in other forms, the mass means comprises one or more thin elongated tubes having a frictionally-gripping fit on an elongated mallet body, so as to be manually slidable thereon but tight enough to avoid dislodgment under normal playing action; and such a tubular member may be slidable to the head-end of the mallet to serve as a striking head or it may be slidable to a position to serve as a hand grip; and one or more such tubular members may be placed in various positions along the length of the mallet. The mallet body may also have a tapered element at the head-end to assure retention of mass members especially of enlarged-head types; and the disclosure further includes such heads embodying a relatively hard inner portion and a relatively soft outer covering.
10,148 citations
9,803 citations
Book•
16 Jul 1993
TL;DR: Statistical Models Based on Counting Processes (SBP) as discussed by the authors is a monograph for mathematical statisticians and biostatisticians, although almost all methods are given in sufficient detail to be used in practice by other mathematically oriented researchers studying event histories.
Abstract: Modern survival analysis and more general event history analysis may be effectively handled in the mathematical framework of counting processes, stochastic integration, martingale central limit theory and product integration. This book presents this theory, which has been the subject of an intense research activity during the past one-and-a-half decades. The exposition of the theory is integrated with the careful presentation of many practical examples, based almost exlusively on the authors' experience, with detailed numerical and graphical illustrations. "Statistical Models Based on Counting Processes" may be viewed as a research monograph for mathematical statisticians and biostatisticians, although almost all methods are given in sufficient detail to be used in practice by other mathematically oriented researchers studying event histories (demographers, econometricians, epidemiologists, actuariala mathematicians, reliability engineers, biologists). Much of the material has so far only been available in the journal literature (if at all), and a wide variety of researchers will find this an invlauable survey of the subject.
3,012 citations
TL;DR: "Statistical Models Based on Counting Processes" may be viewed as a research monograph for mathematical statisticians and biostatisticians, although almost all methods are given in sufficient detail to be used in practice by other mathematically oriented researchers studying event histories.
Abstract: Modern survival analysis and more general event history analysis may be effectively handled in the mathematical framework of counting processes, stochastic integration, martingale central limit theory and product integration. This book presents this theory, which has been the subject of an intense research activity during the past one-and-a-half decades. The exposition of the theory is integrated with the careful presentation of many practical examples, based almost exlusively on the authors' experience, with detailed numerical and graphical illustrations. \"Statistical Models Based on Counting Processes\" may be viewed as a research monograph for mathematical statisticians and biostatisticians, although almost all methods are given in sufficient detail to be used in practice by other mathematically oriented researchers studying event histories (demographers, econometricians, epidemiologists, actuariala mathematicians, reliability engineers, biologists). Much of the material has so far only been available in the journal literature (if at all), and a wide variety of researchers will find this an invlauable survey of the subject.
2,852 citations