Covariate-adjusted nonlinear regression

Abstract

In this paper, we propose a covariate-adjusted nonlinear regression model. In this model, both the response and predictors can only be observed after being distorted by some multiplicative factors. Because of nonlinearity, existing methods for the linear setting cannot be directly employed. To attack this problem, we propose estimating the distorting functions by nonparametrically regressing the predictors and response on the distorting covariate; then, nonlinear least squares estimators for the parameters are obtained using the estimated response and predictors. Root $n$-consistency and asymptotic normality are established. However, the limiting variance has a very complex structure with several unknown components, and confidence regions based on normal approximation are not efficient. Empirical likelihood-based confidence regions are proposed, and their accuracy is also verified due to its self-scale invariance. Furthermore, unlike the common results derived from the profile methods, even when plug-in estimates are used for the infinite-dimensional nuisance parameters (distorting functions), the limit of empirical likelihood ratio is still chi-squared distributed. This property eases the construction of the empirical likelihood-based confidence regions. A simulation study is carried out to assess the finite sample performance of the proposed estimators and confidence regions. We apply our method to study the relationship between glomerular filtration rate and serum creatinine.Comment: Published in at http://dx.doi.org/10.1214/08-AOS627 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonlinear models with measurement errors subject to single-indexed distortion

“…Condition (A4) is required for asymptotic normality of the estimatorsθ andβ. Condition (A5) is necessary in the study of covariate-adjusted models, see [20,4,27]. Condition (A6) is generally true.…”

“…We omit the proof of statement (i), as it is similar to the proof of Theorem 1 in [4] and Lemma 1 in [25].…”

“…However, they did not provide theoretical justification for their approach. More recently, Cui et al [4] studied nonlinear models with a one-dimensional confounding variable. They used the traditional nonparametric regression to obtain estimators of the distortion functions, sayφ(·) and ψ(·).…”

“…We estimate the single-index θ using the recently developed estimating function method (EFM) by Cui et al [5] because this method is more efficient than its competitors in the literature, is easy to implement, and is not sensitive to initial values. We then derive profile nonlinear least squares estimators of β, establish asymptotic normality for the proposed estimators and correct a technical error in Lemma A.1 of [4], which plays a critical role in the proofs of their main theoretical results. As the asymptotic covariance matrix of the estimators of β contains several unknown components in a very complex structure, it may not be convenient for statistical inference-based on the normal approximation in practice.…”

Statistical inference on partial linear additive models with distortion measurement errors

“…For example, in a study of the relationship between fibrinogen level and serum transferrin level among hemodialysis patients, Kaysen et al [15] analyzed the fibrinogen level and serum transferrin level by firstly normalizing them by BMI, which indicates that there may exist a multiplicative fashion relationship between the unobserved primary variables and the confounding variables. For the statistical modeling of the distortion measurement errors data, readers can refer to the parametric models [3,[27][28][29][32][33][34][35][36]46,47,51,53], the partial linear models [21], the partial linear single index models [52], the dimension reduction models [54]. However, these existing literature do not consider estimation proposals for the PLAMs under the distortion measurement errors setting.…”