On interval estimation in linear relationships with heteroscedastic measurement errors in both axes

Abstract This article investigates confidence sets for linear model with heteroscedastic measurement errors on both axes. We propose several approaches to find confidence intervals for the slope and joint confidence regions for both the intercept and the slope. The performance of the confidence set is compared in terms of both coverage probability and its length/area via theory and simulation studies. Application of these methods is also illustrated with real data sets. Guidelines for choosing suitable confidence set are provided.

[1]  S. Neumark.  Solution of Cubic and Quartic Equations , 1965 .

[2]  H Kragt,et al.  Importance of trends in the interpretation of an overall odds ratio in the meta-analysis of clinical trials. , 1992, Statistics in medicine.

[3]  R. L. Brown.  BIVARIATE STRUCTURAL RELATION , 1957 .

[4]  R. Carroll,et al.  A Note on the Efficiency of Sandwich Covariance Matrix Estimation , 2001 .

[5]  Sudhir Gupta,et al.  Statistical Regression With Measurement Error , 1999, Technometrics.

[6]  Chi-Lun Cheng,et al.  On Estimating Linear Relationships When Both Variables Are Subject to Heteroscedastic Measurement Errors , 2006, Technometrics.

[7]  N. Draper,et al.  Applied Regression Analysis , 1966 .

[8]  M. A. Creasy.  CONFIDENCE LIMITS FOR THE GRADIENT IN THE LINEAR FUNCTIONAL RELATIONSHIP , 1956 .

[9]  V. Lopez-Avila,et al.  Microwave-Assisted Extraction of Organic Compounds from Standard Reference Soils and Sediments , 1994 .

[10]  K. Kuulasmaa,et al.  Estimation of an errors‐in‐variables regression model when the variances of the measurement errors vary between the observations , 2002, Statistics in medicine.

[11]  H. Bolfarine,et al.  Consistent estimation and testing in comparing analytical bias models , 2006 .

[12]  Heleno Bolfarine,et al.  A heteroscedastic structural errors-in-variables model with equation error , 2009 .

[13]  Raymond J. Carroll,et al.  Measurement error in nonlinear models: a modern perspective , 2006 .

[14]  P. Sprent,et al.  A Generalized Least‐Squares Approach to Linear Functional Relationships , 1966 .

[15]  Alexander Kukush,et al.  Measurement Error Models , 2011, International Encyclopedia of Statistical Science.

[16]  Jordi Riu,et al.  Prediction intervals in linear regression taking into account errors on both axes , 2001 .

[17]  S D Walter,et al.  Variation in baseline risk as an explanation of heterogeneity in meta-analysis. , 1997, Statistics in medicine.

[18]  Heleno Bolfarine,et al.  Hypothesis testing in an errors‐in‐variables model with heteroscedastic measurement errors , 2008, Statistics in medicine.

[19]  Brian D. Ripley,et al.  Regression techniques for the detection of analytical bias , 1987 .

[21]  J. Riu,et al.  Detecting proportional and constant bias in method comparison studies by using linear regression with errors in both axes , 1999 .

[22]  S. Summersbee,et al.  Solution of Cubic and Quadratic Equations , 1967, The Mathematical Gazette.