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Guaranteed Local Maximum Likelihood Detection of a Change Point in Nonparametric Logistic Regression

A. Vexler, G. Gurevich

2006
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Communications in Statistics - Theory and Methods
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The problems of identifying changes at unknown times and of estimating the location of changes in stochastic processes are referred to as "the change-point problem" or, in the Eastern literature, as "disorder". The change-point problem, first introduced in the quality control context, has since developed into a fundamental problem in the areas of statistical control theory, stationarity of a stochastic process, estimation of the current position of a time series, testing and estimation of
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... in the patterns of a regression model, and most recently in the comparison and matching of DNA sequences in microarray data analysis. Numerous methodological approaches have been implemented in examining changepoint models. Maximum-likelihood estimation, Bayesian estimation, isotonic regression, piecewise regression, quasi-likelihood and non-parametric regression are among the methods which have been applied to resolving challenges in changepoint problems. Grid-searching approaches have also been used to examine the change-point problem. Statistical analysis of change-point problems depends on the method of data collection. If the data collection is ongoing until some random time, then the appropriate statistical procedure is called sequential. If, however, a large finite set of data is collected with the purpose of determining if at least one change-point occurred, then this may be referred to as non-sequential. Not surprisingly, both the former and the latter have a rich literature with much of the earlier work focusing on sequential methods inspired by applications in quality control for industrial processes. In the regression literature, the change-point model is also referred to Regression Analysis and Change-point The general problem of regression can perhaps be best described as fitting a function to a set of random pairs {(Y t , X t )} t∈T , where T is a subset of R k , though most applications are only concerned with k = 1. Having such description in mind, one may consider the following model, where f is a link function and Y t and ε t are random vectors, while X t can be a random or deterministic vector, or it might have both random and deterministic components. The simplest possible scenario is when the link function f is linear and X t is a deterministic vector. A case of special interest in many applications is the so-called linear regression with random slope which pertains to the case where X t has a mixture of stochastic and deterministic components. It often happens that a linear link function cannot adequately explain the possible relationship between X t and Y t . In such cases, one may choose nonlinear link functions should a specific choice of link function be plausible. When such a choice is not readily available, one may resort to nonparametric regression. There is a vast literature pertaining to nonparametric regression. Choosing an appropriate link function that can encompass the most important features of the data is at the heart of regression analysis. In many applications, a smooth link function by which we mean a C 1 map, cannot describe the possible relationship between X t and Y t , and one has to fit different models in different subregions. The points at which the link function is not smooth are of special interest, since they often represent a change in the pattern of data. Such points are often called change-points. There has been a surge of research over the past several decades on locating and making inferences about the change-points as well as the pattern of the data before and after the change-points. We have collected an annotated list of articles written on change-points and related subjects. We have mostly confined ourselves to the literature on change-point problems pertaining to regression analysis. The list is, by no means, an exhaustive list. It reflects only those articles which happen to be closer to some applications we have had in mind. The standard change-point problem in regression models consists of (1) testing the null hypothesis that no change in regimes has taken place against the alternative that observations were generated by two (or possibly more) distinct regression equations, and (2) estimating the two regimes that gave rise to the data. Literature on this topic is divided between models in which continuity is assumed and those which allow a discontinuity at the point of change. The regression model without the restriction of continuity is, in fact, the generalization of the mean-shift problem in which the interest is testing and estimating the shift in mean in a sequence of random variables. Generally speaking, change-point regression is a regression problem in which the expected Claudia Kirch and Josef Steinebach(2006-1) Permutation principles for the change analysis of stochastic processes under strong invariance Journal of Computational and Applied Mathematics, 186, Abstract: Approximations of the critical values for change-point tests are obtained through permutation methods. Both, abrupt and gradual changes are studied in models of possibly dependent observations satisfying a strong invariance principle, as well as gradual changes in an i.i.d. model. The theoretical results show that the original test statistics and their corresponding permutation counterparts follow the same distributional asymptotic. Some simulation studies illustrate that the permutation tests behave better than the original tests if performance is measured by the a-and -error, respectively. Goldenshluger, A.; Tsbakov, A. and Zeevi, A.(2006-2) Optimal change-point estimation from indirect observations Abstract: We study nonparametric change-point estimation from indirect noisy observations. Focusing on the white noise convolution model, we consider two classes of functions that smooth apart from the change-point. We establish lower bounds on the minimax risk in estimating the change-point and develop rate optimal estimation procedures. The results demonstrate that the best achievable rates of convergence are determined both by smoothness of the function away from the change-point and by the degree of ill-posedness of the convolution operator. Optimality is obtained by introducing a new technique that involves, as a key element, detection of zero crossing of an estimate of the properly smoothed second derivative of the underlying function. Gregory Gurevich A(2006-3) Nonparametric AMOC Changepoint Tests for Stochastically Ordered Alternatives Communications in Statistics: Theory and Methods, 35, 887-903 Abstract: The problem considered is that of testing on the basis of a finite sequence of independent observations if all the observations have the same distribution versus the alternative 4 http://biostats.bepress.com/cobra/art44 that there is a unique change in the distribution and i.i.d. observations after the change are stochastically larger. The distributions before and after the possible change are continuous but not fully specified. We suggest a family of nonparametric tests based on ranks. Asymptotic approximations for the significance level of the test are obtained analytically. Monte Carlo experiments show that the rate of convergence of our asymptotics is fast. Osorio, Felipe and Galea, Manuel (2006-4) Detection of a change-point in student-t linear regression models Statistical Papers, 47, 31-48 Abstract: The Schwarz Information Criterion (SIC) is used in order to locate a change-point in linear regression models with independent errors distributed according to the Student-t distribution. The methodology is applied to data sets from the financial area. Singer, Julio da Motta and Cúri, Mariana (2006-5) Modelling regression and dispersion parameters in a complex repeated measures experiment Environmental and Ecological Statistics, 13, 53-68 Abstract :We analyze data from a complex repeated measures experiment directed at the evaluation of the response to an electric stimulus applied to mussel nerves under 5 different salinity levels. We discuss the form of the relation between the response and the different salinity levels, as well as the choice of an adequate within subjects covariance structure that includes random effects and autoregressive models. Parka, Cheolwoo and Kimb, Woo-Chul(2006-6) Wavelet estimation of a regression function with a sharp change point in a random design Abstract : In a random design nonparametric regression model, this paper deals with the detection of a sharp change point and the estimation of a regression function with a single jump point. A method based on design transformation and binning is used in order to convert a random design into an equispaced design whose number of points is a power of 2. Using the obtain its rate of convergence. Wavelet methods are well known for their good adaptivity around sudden local changes; however, in practice, the Gibbs phenomenon still exists. This difficulty is overcome by suitably adjusting the data with preliminary estimators for the location and the size of discontinuity. Global and local asymptotic results of the proposed method are obtained. The method is also tested on simulated examples and the results show that the proposed method alleviates the Gibbs phenomenon. Vexler, A.(2006-7) Guaranteed testing for epidemic changes of a linear regression model Journal of Statistical Planning and Inference, 136, 3101-3120 Abstract: The objective of this paper is to propose and examine a class of generalized maximum likelihood asymptotic power one tests for detection of various types of changes in a linear regression model. The proposed retrospective tests are based on martingales structures ShiryayevRoberts statistics. This approach is widely known in a sequential analysis of change point problems as an optimal method of detecting a change in distribution. Guaranteed non-asymptotic upper bounds for the significance levels of the considered tests are presented. Simulated data sets are used to demonstrate that the proposed tests can give good results in practice. Abstract: We consider nonparametric logistic regression and propose a generalized likelihood test for detecting a threshold effect that indicates a relationship between some risk factor and a defined outcome above the threshold but none below it. One important field of application is occupational medicine and in particular, epidemiological studies. In epidemiological studies, segmented fully parametric logistic regression models are often threshold models, where it is assumed that the exposure has no influence on a response up to a possible unknown threshold, and has an effect beyond that threshold. Finding efficient methods for detection and estimation of a threshold is a very important task in these studies. This article proposes such methods in a context of nonparametric logistic regression. We use a local version of 6 http://biostats.bepress.com/cobra/art44 unknown likelihood functions and show that under rather common assumptions the asymptotic power of our test is one. We present a guaranteed non asymptotic upper bound for the significance level of the proposed test. If applying the test yields the acceptance of the conclusion that there was a change point (and hence a threshold limit value), we suggest using the local maximum likelihood estimator of the change point and consider the asymptotic properties of this estimator. Abstract: This paper develops a Bayes regression model having change points for the analysis of array-CGH data by utilizing not only the underlying spatial structure of the genomic alterations but also the observation that the noise associated with the ratio of the fluorescence intensities is bigger when the intensities get smaller. We show that this Bayes regression approach is particularly suitable for the analysis of cDNA microarray-CGH data, which are generally noisier than those using genomic clones. A simulation study and a real data analysis are included to illustrate this approach. Bernard Garel(2005-1) Asymptotic theory of the likelihood ratio test for the identification of a mixture Abstract: The problems that arise when using the likelihood ratio test for the identification of a mixture distribution are well known: non-identifiability of the parameters and null hypothesis corresponding to a boundary point of the parameter space. In their approach to the problem of testing homogeneity against a mixture with two components, Ghosh and Sen took into account these specific problems. Under general assumptions, they obtained the asymptotic distribution of the likelihood ratio test statistic. However, their result requires a separation condition which is not completely satisfactory. We show that it is possible to remove this condition with assumptions which involve the second derivatives of the density only. Brodskya, Boris and Darkhovsky, Boris(2005-2) Asymptotically optimal methods of change-point detection for composite hypotheses 7 Abstract:In this paper the problem of change-point detection for the case of composite hypotheses is considered. We assume that the distribution functions of observations before and after an unknown change-point belong to some parametric family. The true value of the parameter of this family is unknown but belongs to two disjoint sets for observations before and after the change-point, respectively. A new criterion for the quality of change-point detection is introduced. Modifications of generalized CUSUM and GRSh (GirshickRubin-Shiryaev) methods are considered and their characteristics are analyzed. Comparing these characteristics with an a priori boundary for the quality of change-point detection we establish asymptotic optimality of these methods when the family of distributions before the change-point consists of one element. Fernando A. Quintana; Pilar L. and Heleno Bolfarine(2005-3) Bayesian identification of outliers and change-points in measurement error models Advances in Complex Systems, 8, 433-449 Abstract: The problem of outlier and change-point identification has received considerable attention in traditional linear regression models from both, classical and Bayesian standpoints. In contrast, for the case of regression models with measurement errors, also known as error-in-variables models, the corresponding literature is scarce and largely focused on classical solutions for the normal case. The main object of this paper is to propose clustering algorithms for outlier detection and change-point identification in scale mixture of error-invariables models. We propose an approach based on product partition models (PPMs) which allows one to study clustering for the models under consideration. This includes the changepoint problem and outlier detection as special cases. The outlier identification problem is approached by adapting the algorithms developed by Quintana and Iglesias [Journal of the Royal Statistical Society: Series B (Statistical Methodology) Volume 65 Page 557 -May 2003 ] for simple linear regression models. A special algorithm is developed for the change-point problem which can be applied in a more general setup. The methods are illustrated with two applications: (i) outlier identification in a problem involving the relationship between two methods for measuring serum kanamycin in blood samples from babies, and (ii) change-point identification in the relationship between the monthly dollar volume of sales on the Boston Stock Exchange and the combined monthly dollar volumes for the New York and American Stock Exchanges. Abstract: We investigate the application of a new estimator for the tail index proposed in [Yu. Davydov, V. Paulauskas, and A. Rackauskas, More on p-stable convex sets in Banach spaces, J. Theoret. Probab., 13, 39-64 (2001). ] and [V. Paulauskas, A New Estimator for Tail Index, Acta Appl. Math., 79, 55-67 (2003). ]. Testing hypothesis of change at unknown place and detecting change in mean allow us to provide theoretical results on estimation of the changepoint in the tail index. We demonstrate the applicability of these results in practice. Grace Chiu, Richard Lockhart, and Richard Routledge (2005-6) Asymptotic theory for bent-cable regression the basic case Abstract: We use what we call the bent-cable model to describe potential change-point phenomena. The class of bent cables includes the commonly used broken stick (a bent cable without a bend segment). Theory for least-squares (LS) estimation is developed for the basic bent cable, whose incoming and outgoing linear phases have slopes 0 and 1, respectively, and are joined smoothly by a quadratic bend. Conditions on the design are given to ensure regularity of the estimation problem, despite non-differentiability of the model's first partial derivatives (with respect to the covariate and model parameters). Under such conditions, we show that the LS estimators (i) are consistent, regardless of a zero or positive true bend width; and (ii) asymptotically follow a bivariate normal distribution, if the underlying cable has all three segments. In the latter case, we show that the deviance statistic has an asymptotic chi-squared distribution with two degrees of freedom. Gregory Gurevicha and Albert Vexler(2005-7) Change point problems in the model of logistic regression Abstract: The paper considers generalized maximum likelihood asymptotic power one tests which aim to detect a change point in logistic regression when the alternative specifies that a change occurred in parameters of the model. A guaranteed non-asymptotic upper bound for the significance level of each of the tests is presented. For cases in which the test supports the conclusion that there was a change point, we propose a maximum likelihood estimator 9 Hosted by The Berkeley Electronic Press of that point and present results regarding the asymptotic properties of the estimator. An important field of application of this approach is occupational medicine, where for a lot chemical compounds and other agents, so-called threshold limit values (or TLVs) are specified. We demonstrate applications of the test and the maximum likelihood estimation of the change point using an actual problem that was encountered with real data. Abstract: Applications of bootstrap with and without replacement in change point analysis in linear regression models are discussed. Particularly, bootstrap based approximations for critical values for two classes of M -type test procedures are treated. As a particular case, we obtain L 1 procedures and regression quantile procedures. Their asymptotic performance is investigated and finite sample properties are checked in a simulation study. Jing-rung Yu, Gwo-hshiung Tzeng and Han-Lin Li(2005-9) Interval piecewise regression model with automatic change-point detection by quadratic programming International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 13, 347-361 Abstract: To handle large variation data, an interval piecewise regression method with automatic change-point detection by quadratic programming is proposed as an alternative to Tanaka and Lee's method. Their unified quadratic programming approach can alleviate the phenomenon where some coefficients tend to become crisp in possibilistic regression by linear programming and also obtain the possibility and necessity models at one time. However, that method can not guarantee the existence of a necessity model if a proper regression model is not assumed especially with large variations in data. Using automatic change-point detection, the proposed method guarantees obtaining the necessity model with better measure of fitness by considering variability in data. Without piecewise terms in estimated model, the proposed method is the same as Tanaka and Lee's model. Therefore, the proposed method is an alternative method to handle data with the large variations, which not only reduces the number of crisp coefficients of the possibility model in linear programming, but also simultaneously obtains the fuzzy regression models, including possibility and necessity models with better fitness. Two examples are presented to demonstrate the proposed method. Lai, Tze Leung, Liu, Haiyan and Xing,Haipeng (2005-10) Autoregressive models with piecewise constant volatility and regression parameters Statistica Sinica, 15, 279-301 Abstract : We introduce herein a new class of autoregressive models in which the regression parameters and error variances may undergo changes at unknown time points while staying constant between adjacent change-points. Assuming conjugate priors, we derive closed-form recursive Bayes estimates of the regression parameters and error variances. Approximations to the Bayes estimates are developed that have much lower computational complexity and yet are comparable to the Bayes estimates in statistical efficiency. We also address the problem of unknown hyperparameters and propose two practical methods for simultaneous estimation of the hyperparameters, regression parameters and error variances. Lee, Ji-Hyun; Qaqish, Bahjat F(2005-11) A Latent Changepoint Model Using A Generalized Estimating Equations Approach Communications in Statistics: Theory and Methods, 2005-, 34 1233-1242. Abstract: We propose a latent changepoint model for the analysis of longitudinal biomarker data in relation to progression or recurrence of disease. A parametric model that contains a random changepoint in the expected biomarker values is considered. In this article, estimation through generalized estimating equations is proposed. The procedure allows estimation of the biomarker trend over time and the changepoint distribution. We provide the details of the estimation procedure. Through a Monte Carlo simulation study, several aspects of the small sample performance of the estimates are investigated. Loschi, R. H. and Cruz, F. R. B.(2005-12) Bayesian identification of multiple change points in poisson data. Advances in Complex Systems, 8, 465-482 Abstract: The identification of multiple change points is a problem shared by many subject areas, including disease and criminality mapping, medical diagnosis, industrial control, and finance. An algorithm based on the Product Partition Model (PPM) is developed to solve the multiple change point identification problem in Poisson data sequences. In order to address the PPM, a simple and easy way to implement Gibbs sampling scheme is derived. 11 Hosted by The Berkeley Electronic Press A sensitivity analysis is performed, for different prior specifications. The algorithm is then applied to the analysis of a real data sequence. The results show that the method is quite effective and provides useful inferences Myung Hwan Naa, Jongwoo Jeonb and Dong Ho Park(2005-13) Testing whether failure rate changes its trend with unknown change points Abstract: The problem of testing the trend change of failure rate is of great interest in the reliability and survival analysis. In this paper, we develop a new test procedure for testing whether or not the failure rate changes its trend. One big advantage of this test is that neither the change points nor the proportions at which the trend changes occur need to be known. We establish the asymptotic null distribution of the proposed test statistic to obtain the asymptotic null critical values for the test to be applied. To study the performance of the new test procedure, we conduct Monte Carlo simulations to compute the powers of the test against the lognormal alternatives and the Hjorth alternatives and to compare these powers with those of other existing tests. An example is presented to illustrate the application of the test. Preminger, Arie and Wettstein, David (2005-14) Using the penalized likelihood method for model selection with nuisance parameters present only under the alternative: An application to switching regression models Journal of Time Series Analysis, 26, 715-741 Abstract: We study the problem of model selection with nuisance parameters present only under the alternative. The common approach for testing in this case is to determine the true model through the use of some functionals over the nuisance parameters space. Since in such cases the distribution of these statistics is not known, critical values had to be approximated usually through computationally intensive simulations. Furthermore, the computed critical values are data and model dependent and hence cannot be tabulated. We address this problem by using the penalized likelihood method to choose the correct model. We start by viewing the likelihood ratio as a function of the unidentified parameters. By using the empirical process theory and the uniform law of the iterated logarithm (LIL) together with sufficient conditions on the penalty term, we derive the consistency properties of this method. Our approach generates a simple and consistent procedure for model selection. This methodology is presented in the context of switching regression models. We also provide some Monte Carlo simulations to analyze the finite sample performance of our procedure. Salazar, Diego, Venkatesan, G. and Moen, David (2005-15) Switching linear models: A general approach Communications in Statistics: Simulation and Computation, 34, 309-320 Abstract: With reference to switching linear models, using the notion of a switching interval, the posterior distributions of all the parameters in the model are obtained. This includes the beginning and end points of the switching interval and the parameters determining the nature of the switch. This is done by studying three cases of the problem: a permanent switch in a finite interval, a permanent switch in an infinite interval, and a temporary switch in a finite interval. The analysis is general in the sense that it can be applied to any problem that can be formulated as a linear model. A numerical study illustrates the methodology. Shurenkov, G.(2005-16) Asymptotic behavior of median estimators of multiple change points Theory Probability and Mathematical Statistics, 70 , 167-176. Abstract: We consider the problem of posterior estimation of multiple change points in the case of only two distributions. We find the asymptotic distribution of the difference between the median estimator of a single change point and the true change point and show that the distribution does not change if the unknown parameter is estimated by a median of the sample. We generalize the results to the case of multiple change points. Sofronov, G. Yu.(2005-17) An asymptotically d-optimal test of a posteriori change-point detection Theory of probability and its application, 49, 367-371 Abstract: We consider the problem of a posteriori change-point detection for a sequence of independent identically distributed random variables. We propose to use d-risks instead of error of the first type and error of the second type. We construct an asymptotically optimal test minimizing one d-risk and guaranteeing another Tartakovsky, A. G. and Veeravalli, V. V.(2005-18) General asymptotic bayesian theory of quickest change detection. Abstract: The optimal detection procedure for detecting changes in independent and identically distributed (i.i.d.) sequences in a Bayesian setting was derived by Shiryaev in the 1960s. However, the analysis of the performance of this procedure in terms of the average detection delay and false alarm probability has been an open problem. In this paper, we develop a general asymptotic change-point detection theory that is not limited to a restrictive i.i.d. assumption. In particular, we investigate the performance of the Shiryaev procedure for general discrete-time stochastic models in the asymptotic setting, where the false alarm probability approaches zero. We show that the Shiryaev procedure is asymptotically optimal in the general non-i.i.d. case under mild conditions. We also show that the two popular non-Bayesian detection procedures, namely the Page and the Shiryaev-Roberts-Pollak procedures, are generally not optimal (even asymptotically) under the Bayesian criterion. The results of this study are shown to be especially important in studying the asymptotic of decentralized change detection procedures. Tsai-Hung Fan and Wei-chen Chen(2005-19) Bayesian change points analysis on the seismic activity in northeastern Taiwan Journal of Statistical Computation and Simulation, 75, 857-868 Abstract: Bayesian change points analysis on the seismic activity in northeastern Taiwan is studied via the reversible jump Markov chain Monte Carlo simulation. An epidemic model is considered with Gamma prior distributions for the parameters. The prior distributions are essentially determined based on an earlier period of the seismic data in the same region. It is investigated that there exist two change points during the time period considered. This result is also confirmed by the BIC criteria. Wang, Zhiguo and Wang, Jinde (2005-20) Parameter estimation of some NHPP software reliability models with change-point Communications in Statistics: Simulation and Computation, 34, 121-134 Abstract: The nonhomogeneous Poisson process (NHPP) model is an important class of software reliability models and is widely used in software reliability engineering. The failure intensity function is usually assumed to be continuous and smooth. However, in many realistic situations, the failure intensity may be not continuous for many possible causes, such by the Shiryayev-Roberts procedure. (Reviewed by Lian Fen Qian) Wu, Yanhong(2005-22) Inference for Change-Point and Post-Change Mean with Possible Change in Variance Sequential Analysis; Aug 2005-, 24, 279-302 Abstract: For a sequence of independent normal random variables, we consider the estimation of the change-point and the post-change mean after a change in the mean is detected by a CUSUM procedure, subject to a possible change in variance. Conditional on the event that a change is detected and it occurred far away from the starting point and the threshold is large, the (absolute) bias of the maximum likelihood estimator for the change-point (obtained at the reference value) is found. The first-order biases for the post-change mean and variance estimators are also obtained by using Wald's Likelihood Ratio Identity and the renewal theorem. In the local case when the reference value and the post-change mean are both small, accurate approximations are derived. A confidence interval for post-change mean based on a corrected normal pivot is then discussed. Yoshiyuki Ninomiya(2005-23) Information criterion for Gaussian change-point model Statistics and Probability Letters, 72,[237][238][239][240][241][242][243][244][245][246][247] Abstract: AIC-type information criterion is generally estimated by the bias-corrected maximum log-likelihood. In regular models, the bias can be estimated by p, where p is the number of parameters. The present paper considers the AIC-type information criterion for change-point models which are not regular, the bias of which will not be the same as for regular models. The bias is shown to depend on the expected maximum of a random walk with negative drift. Furthermore, it is shown that by using an approximation to a Brownian motion, the evaluated bias is given by 3m + p m (not m + p m ), where m is the number of change-points and p m is the number of regular parameters, which differs from regular models. Young Sook Son Kim, Seong W.(2005-24) Bayesian single change point detection in a sequence of multivariate normal observations 16 Abstract: A Bayesian method is used to see whether there are changes of mean, covariance, or both at an unknown time point in a sequence of independent multivariate normal observations. Noninformative priors are used for all competing models: no-change model, mean change model, covariance change model, and mean and covariance change model. We use several versions of the intrinsic Bayes factor of Berger and Pericchi (Berger, J.O. and Pericchi, L.R., 1996, The intrinsic Bayes factor for model selection and prediction. Journal of the American Statistical Association, 91, 109-122; Berger, J.O. and Pericchi, L.R., 1998, Accurate and stable Bayesian model selection: the median intrinsic Bayes factor. Sankhya Series B , 60, 1-18.) to detect a change point. We demonstrate our results with some simulated data sets and a real data set. Andersen, Lars Bo (2004-1) Relative risk of mortality in the physically inactive is underestimated because of real changes in exposure level during follow-up Abstract: Relative risk among exposure groups in prospective cohort studies is based on the assumption that all subjects are exposed at the level recorded at baseline throughout the study. Changes in risk behavior during follow-up will dilute the relative risk. This prospective cohort study in Copenhagen, Denmark, between 1964 and 1994 included 30,640 men and women; 19,149 were examined twice, with an interval of 6.7 (standard deviation, 3.4) years. Relative risks calculated from baseline measurements for moderately active and sedentary groups compared with the highly active group were 1.11 (95% confidence interval: 1.05, 1.18) and 1.64 (95%confidence interval: 1.53, 1.75), respectively. The relative risk between the highly active group and the sedentary group decreased with increasing follow-up time. When intraindividual changes in physical activity level during follow-up were taken into account, the relative risk of physical inactivity was 24-59% higher compared with the relative risk estimated from baseline measurements. The risk of a sedentary lifestyle is underestimated when it is calculated from one baseline measurement in prospective studies, because subjects change behavior during follow-up.

doi:10.1080/03610920500498923
fatcat:jxva2knaxvbpbfmovc7yfwwlhu