Accounting for uncertainty in heteroscedasticity in nonlinear regression

Abstract

Toxicologists and pharmacologists often describe toxicity of a chemical using parameters of a nonlinear regression model. Thus estimation of parameters of a nonlinear regression model is an important problem. The estimates of the parameters and their uncertainty estimates depend upon the underlying error variance structure in the model. Typically, a priori the researcher would know if the error variances are homoscedastic (i.e., constant across dose) or if they are heteroscedastic (i.e., the variance is a function of dose). Motivated by this concern, in this article we introduce an estimation procedure based on preliminary test which selects an appropriate estimation procedure accounting for the underlying error variance structure. Since outliers and influential observations are common in toxicological data, the proposed methodology uses M-estimators. The asymptotic properties of the preliminary test estimator are investigated; in particular its asymptotic covariance matrix is derived. The performance of the proposed estimator is compared with several standard estimators using simulation studies. The proposed methodology is also illustrated using a data set obtained from the National Toxicology Program.

Using Weighted Entropy to Rank Chemicals in Quantitative High-Throughput Screening Experiments

“…However, ranking substances based on AC 50 estimates derived from fits to the Hill Equation was always resulted in poor performance. Other studies have demonstrated that the Hill Equation model parameters can be unreliable estimates due to large variations in observed responses at higher doses, irregular dose spacing or data collected over an incomplete dose range 5,8,31 . In contrast to AC 50 , WES is based strictly on the observed responses and never relies on estimates of the behavior of a nonlinear function outside of the tested concentration range.…”

Robust Analysis of High Throughput Screening (HTS) Assay Data

“…The above test statistic can be approximated as L OLSE = { η̂ t ( Ĥ−H 0 ) η̂ /3}/{ η̂ t ( I n − Ĥ ) η̂ /( n −4)}+ o P (1) = L OLSE + o P (1), where η̂ = Y − β̂ 1 , Y = ( y 11 ,…, y k,n k ) t , β̂ = ȳ , 1 = (1,…, 1) t , Ĥ = F̂ ( F̂ t F̂ ) −1 F̂ t , F̂ = f θ ( θ̂ ) = { ∂f ( x i , θ )/ ∂θ j ∣ θ=θ̂ }, and H 0 = 1 ( 1 t 1 ) −1 1 t . Since L OLSE is not robust to outliers and influential observations, in this paper we make it robust by replacing OLSE by OME θ̃ n , in the above calculations, where OME is defined as (Lim et al 2012): θ̃ n = Argmin {Σ i,j h 2 ( y ij − f ( x i , θ )) : θ ∈ ℜ p } where h is taken to be the Huber-score function. For a pre-specified positive constant k 0 , $h\left(u\right)=u/\sqrt{2}$, if | u | < k 0 , otherwise h ( u ) = { k 0 (| u |− k 0 /2)} 1/2 .…”

“…We shall denote the resulting statistic by L OME . Since OME and OLSE are asymptotically equivalent and OME is consistent and asymptotically normally distributed (Lim et al 2012), the null distribution of L OME can also be approximated by central F -distribution with (3, n − 4) degrees of freedom. Throughout this paper we shall refer to this modified LRT as the OME based methodology.…”

“…Recently, in Lim et al (2012) we proposed the preliminary test estimation (PTE) procedure for possibly heteroscedastic nonlinear models. The basic idea is to select either OME or WME on the basis of a simple preliminary test for heteroscedasticity.…”

Incremental Localization Algorithm Based on Regularized Iteratively Reweighted Least Square

“…However, as with a number of key technologies, in practical applications the performance of incremental localization techniques still faces many technical problems, of which the most deadly one is the error accumulation problem, that a prior localization error affects the next locating performance, and the influence is cumulative. The accumulation of the errors will inevitably lead to the difference between the variance of the previous locating error and the variance of the posterior locating error, which is known as heteroscedasticity [5,13]. If heteroskedasticity appears in the process of locating and estimating, the location estimation of the unknown nodes using the OLS may not be efficient estimators, nor even asymptotically efficient estimators.…”