Curvatures for Parameter Subsets in Nonlinear Regression

Cook, R. Dennis; Goldberg, Miriam L.

doi:10.1214/aos/1176350166

Cited by 51 publications

(37 citation statements)

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“…Since parameter δ δ δ is the parameter subset of this model, the subset curvatures for δ δ δ can be found by the methods described in Cook and Goldberg (1986).…”

Section: Accuracy Of Linear Approximationmentioning

confidence: 99%

“…In order to obtain simple reduced forms for the curvatures, Cook and Goldberg (1986) assume that the intrinsic curvature of h h h atδ δ δ is negligible. This assumption seems to be a reasonable approximation for many nonlinear models, as in the examples given in numerous literatures such as Bates and Watts (1980).…”

Section: Accuracy Of Linear Approximationmentioning

confidence: 99%

“…Following Ratkowsky (1983, p.18) and Cook and Goldberg (1986), 1/{2 √ F α (m, n − p − m)} may be used as a rough guide for judging the size of these curvatures. This method can be used to judge the adequacy of the test procedures which are based on the linear approximation.…”

Section: Accuracy Of Linear Approximationmentioning

confidence: 99%

“…Bates and Watts (1980) propose measures of intrinsic and parameter-effects curvature for assessing the adequacy of the linear approximation. These ideas are extended and refined by Cook and Goldberg (1986). They develop measures for assessing the agreement between linear and likelihood regions for an arbitrary subset of parameters from a nonlinear model.…”

Section: Introductionmentioning

confidence: 99%

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Accuracy of Multiple Outlier Tests in Nonlinear Regression

Kahng¹

2011

Communications for Statistical Applications and Methods

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The original Bates-Watts framework applies only to the complete parameter vector. Thus, guidelines developed in that framework can be misleading when the adequacy of the linear approximation is very different for different subsets. The subset curvature measures appear to be reliable indicators of the adequacy of linear approximation for an arbitrary subset of parameters in nonlinear models. Given the specific mean shift outlier model, the standard approaches to obtaining test statistics for outliers are discussed. The accuracy of outlier tests is investigated using subset curvatures.

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“…Since parameter δ δ δ is the parameter subset of this model, the subset curvatures for δ δ δ can be found by the methods described in Cook and Goldberg (1986).…”

Section: Accuracy Of Linear Approximationmentioning

confidence: 99%

Section: Accuracy Of Linear Approximationmentioning

confidence: 99%

Section: Accuracy Of Linear Approximationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Accuracy of Multiple Outlier Tests in Nonlinear Regression

Kahng¹

2011

Communications for Statistical Applications and Methods

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“…The notion of level of nonlinearity ("mild" or "strong" as mentioned in this paper) is derived from the term "close to linear" introduced when evaluating a nonlinear model's behaviour (Fletcher and Powell, 1963). Measures of nonlinearity have been studied in several publications (Bates and Watts, 1980;Cook and Goldberg, 1986;Smyth, 2002) in order to evaluate whether the "close to linear" condition is satisfied and to indicate if the linear approximation is reasonable or questionable.…”

Section: Introductionmentioning

confidence: 99%

Evaluation of the Fisher information matrix in nonlinear mixed effect models using adaptive Gaussian quadrature

Nguyễn

2014

Computational Statistics & Data Analysis

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. Evaluation of the Fisher information matrix in nonlinear mixed effect models using adaptive Gaussian quadrature. Computational Statistics and Data Analysis, Elsevier, 2014, 80, pp.57 -69 AbstractNonlinear mixed effect models (NLMEM) are used in model-based drug development to analyse longitudinal data. To design these studies, the use of the expected Fisher information matrix (M F ) is a good alternative to clinical trial simulation. Presently, M F in NLMEM is mostly evaluated with first-order linearisation. The adequacy of this approximation is, however, influenced by model nonlinearity. Alternatives for the evaluation of M F without linearisation are proposed, based on Gaussian quadratures. The M F , expressed as the expectation of the derivatives of the log-likelihood, can be obtained by stochastic integration. The likelihood for each simulated vector of observations is approximated by Gaussian quadrature centred at 0 (standard quadrature) or at the simulated random effects (adaptive quadrature). These approaches have been implemented in R. Their relevance was compared with clinical trial simulation and linearisation, using dose-response models, with various nonlinearity levels and different number of doses per patient. When the nonlinearity was mild, three approaches based on M F gave correct predictions of standard errors, when compared with the simulation. When the nonlinearity increased, linearisation correctly predicted standard errors of fixed effects, but over-predicted, with sparse designs, standard errors of some variability terms. Meanwhile, quadrature approaches gave correct predictions of standard errors overall, but standard Gaussian quadrature was very time-consuming when there were more than two random effects. To conclude, adaptive Gaussian quadrature is a relevant alternative for the evaluation of M F for models with stronger nonlinearity, while being more computationally efficient than standard quadrature.

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References

1997

Geometrical Foundations of Asymptotic Inference

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Curvatures for Parameter Subsets in Nonlinear Regression

Cited by 51 publications

References 0 publications

Accuracy of Multiple Outlier Tests in Nonlinear Regression

Accuracy of Multiple Outlier Tests in Nonlinear Regression

Evaluation of the Fisher information matrix in nonlinear mixed effect models using adaptive Gaussian quadrature

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