Additive isotonic regression models in epidemiology

Morton‐Jones, Tony; Diggle, Peter J.; Parker, Louise; Dickinson, Heather O; Binks, K.

doi:10.1002/(sici)1097-0258(20000330)19:6<849::aid-sim410>3.0.co;2-d

Cited by 35 publications

(13 citation statements)

References 11 publications

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“…Banerjee (2007) and Banerjee (2009) studied monotone regression models in which the conditional distribution of a response given a covariate is assumed to come from a regular parametric model and an exponential family respectively. There have been several papers studying additive isotonic regression, including Bacchetti (1989), Morton-Jones et al (2000), Tutz and Leitenstorfer (2007), Cai and Dunson (2007), Mammen and Yu (2007), Brezger and Steiner (2008), Cheng (2009), Cheng et al (2012, Fang and Meinshausen (2012), Rueda (2013) and Yu (2014). The recent work of Meyer (2013a) develops similar methodology (but not theory) to ours in the Gaussian, non-index setting.…”

Section: Introductionmentioning

confidence: 78%

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Generalized Additive and Index Models with Shape Constraints

Chen

Samworth

2015

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary. We study generalized additive models, with shape restrictions (e.g. monotonicity, convexity and concavity) imposed on each component of the additive prediction function. We show that this framework facilitates a non-parametric estimator of each additive component, obtained by maximizing the likelihood. The procedure is free of tuning parameters and under mild conditions is proved to be uniformly consistent on compact intervals. More generally, our methodology can be applied to generalized additive index models. Here again, the procedure can be justified on theoretical grounds and, like the original algorithm, has highly competitive finite sample performance. Practical utility is illustrated through the use of these methods in the analysis of two real data sets. Our algorithms are publicly available in the R package scar, short for shape-constrained additive regression.

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Section: Introductionmentioning

confidence: 78%

“…There have been several papers studying additive isotonic regression, including Bacchetti (), Morton‐Jones et al . (), Tutz and Leitenstorfer (), Cai and Dunson (), Mammen and Yu (), Brezger and Steiner (), Cheng (), Cheng et al . (), Fang and Meinshausen (), Rueda () and Yu ().…”

Section: Introductionmentioning

confidence: 99%

Generalized Additive and Index Models with Shape Constraints

Chen

Samworth

2015

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…With the notation I r,k ≡ −g [r] k for the incremental changes at the k-th cycle in the r-th iteration, equations (30) and (31) form a Dykstra algorithm to solve the following optimization problem: (32) min…”

Section: A2 Proof Of Theoremmentioning

confidence: 99%

Additive isotone regression

Mammen¹,

Yu²

2007

Institute of Mathematical Statistics Lecture Notes - Monograph Series

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This paper is about optimal estimation of the additive components of a nonparametric, additive isotone regression model. It is shown that asymptotically up to first order, each additive component can be estimated as well as it could be by a least squares estimator if the other components were known. The algorithm for the calculation of the estimator uses backfitting. Convergence of the algorithm is shown. Finite sample properties are also compared through simulation experiments.Comment: Published at http://dx.doi.org/10.1214/074921707000000355 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

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“…We mention Dykstra and Robertson (1982); Matthews and Crowther (1998); Jamshidian (2004); Tan et al (2007); Taylor et al (2007), or Balabdaoui et al (2009) discussing computation of order restricted estimates in specific regression problems, and Terlaky and Vial (1998); Balabdaoui and Wellner (2004) or Rufibach (2007) for estimation of probability densities under order restrictions. Additionally, generalizations of the pool-adjacent-violaters algorithm (PAVA) to inclusion of continuous isotonic covariates are discussed in Bacchetti (1989); Morton-Jones et al (2000); Ghosh (2007); Cheng (2009) in the context of "additive isotonic regression". Estimation in this type of model is usually performed using the cyclical PAVA in connection with backfitting.…”

Section: Introductionmentioning

confidence: 99%

An active set algorithm to estimate parameters in generalized linear models with ordered predictors

Rufibach¹

2010

Computational Statistics & Data Analysis

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In biomedical studies, researchers are often interested in assessing the association between one or more ordinal explanatory variables and an outcome variable, at the same time adjusting for covariates of any type. The outcome variable may be continuous, binary, or represent censored survival times. In the absence of precise knowledge of the response function, using monotonicity constraints on the ordinal variables improves efficiency in estimating parameters, especially when sample sizes are small. An active set algorithm that can efficiently compute such estimators is proposed, and a characterization of the solution is provided. Having an efficient algorithm at hand is especially relevant when applying likelihood ratio tests in restricted generalized linear models, where one needs the value of the likelihood at the restricted maximizer. The algorithm is illustrated on a real life data set from oncology.

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Additive isotonic regression models in epidemiology

Cited by 35 publications

References 11 publications

Generalized Additive and Index Models with Shape Constraints

Generalized Additive and Index Models with Shape Constraints

Additive isotone regression

An active set algorithm to estimate parameters in generalized linear models with ordered predictors

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