Statistical methodology for regression model with measurement error

PhD Thesis

Saqr, Anwar A. Mohamad. 2013. Statistical methodology for regression model with measurement error. PhD Thesis Doctor of Philosophy. University of Southern Queensland.
Title Statistical methodology for regression model with measurement error PhD Thesis Saqr, Anwar A. Mohamad Khan, Professor Shahjahan Langlands, Dr Trevor University of Southern Queensland Doctor of Philosophy 223 2013 This thesis primarily deals with the estimation of the slope parameter of the simple linear regression model in the presence of measurement errors (ME) or error-in-variables in both the explanatory and response variables. It is a very old and difficult problem which has been considered by a host of authors since the third quarter of the nineteenth century. The ME poses a serious problem in fitting the regression line, as it directly impacts on estimators and their standard error (see eg Fuller, 2006, p. 3). The standard linear regression methods, including the least squares or maximum likelihood, work when the explanatory variable is measured without error. But in practice, there are many situations where the variables can only be measured with ME. For example, data on the medical variables such as blood pressure and blood chemistries, agricultural variables such as soil nitrogen and rainfall etc can hardly be measured accurately. The apparent observed data represents the manifest variable which measures the actual unobservable latent variable with ME.The ME model is divided into two general classifications, (i) functional model if the explanatory (ξ) is a unknown constant, and (ii) structural model if ξ is independent and identically distributed random variable (cf Kendall, 1950,1952). The most important characteristic of the normal structural model is that the parameters are not identifiable without prior information about the error variances as the ratio of error variances (λ) (see Cheng and Van Nees, 1999, p. 6). However, the non-normal structural model is identifiable without any prior information. The normal and non-normal structural modelswith ME in both response and explanatory variables are considered in this research.There are a number of commonly used methods to estimate the slope parameter of the ME model. None of these methods solves the estimation problem in varying situations. A summary of the well known methods is provided in Table 1.The first two chapters of this thesis cover an introduction to the ME problem, background, and motivation of the study. From Chapter 3 we provide a new methodology to fit the regression line using the reflection of the explanatoryvariable about the fitted regression line with the manifest variables. The asymptotic consistency and the mean absolute error (MAE) criteria are usedTable 1: A summary of commonly used methods to handle the ME model problemto compare the new estimators and the relevant existing estimators under different conditions.One of the most commonly used methods to deal with the ME model is the instrumental variable (IV) method. But it is difficult to find valid IV that is highly correlated to the explanatory but uncorrelated with the error term.Therefore, in Chapter 4 we propose a new method to find a good IV based on the reflection of explanatory variable. The new method is easy to implement, and performs much better than the existing methods. The superiority ofthis method is demonstrated both analytically and via numerical as well as graphical illustrations under certain assumptions.In Chapter 5, a commonly used method to deal with the normal structural model, namely the orthogonal regression (OR) (which is the same the maximum likelihood solution when λ = 1) method under the assumption of known λ is discussed. But the OR method does not work well (inconsistent) if λ is misspecified and/or the sample size is small. We provide an alternative method based on the reflection method (RM) of estimation for measurement error model. The RM uses a new transformed explanatory variable which is derived from the reflection formula. This method is equivalent or asymptotically equivalent to the orthogonal regression method, and nearly asymptotically unbiased and efficient under the assumption that λ is equalto one and the sample size is large. If λ is misspecified the RM method is better than the OR method under the MAE criterion even if the sample size is small.Chapter 6 considers the Wald method (two grouping method) which is still widely used, in spite of increasing criticism on the efficiency of the estimator. To address this problem, we introduce a new grouping method based on the reflection grouping (RG) approach. The proposed method provides new grouping process to modify Wald method in order to increase its efficiency. The RG method introduces a new way of dividing the data using the rank of the reflection of the explanatory variable. The method recommends different grouping criteria depending on the value of λ to be one or more/less than one. The RG method significantly increases the efficiency of Wald method,and it is more precise than the other competing methods and works well for different sample sizes and for different values of λ. Moreover, the RG method also removes the shortcomings of the maximum likelihood method when λ ismisspecified and sample size is small.The geometric mean (GM) regression is covered in Chapter 7. The GM method is widely used in many disciplines including medical, pharmacology, astrometry, oceanography, and fisheries researches etc. This method is known by many names such as reduced major axis, standardized major axis, line of organic correlation etc. We introduce a new estimator of the slope parameter when both variables are subject to ME. The weighted geometric mean (WGM) estimator is constructed based on the reflection and the mathematical relationship between the vertical and orthogonal distancesof the observed points and the regression line of the manifest model. The WGM estimator possesses better statistical properties than the geometric mean estimator, and OLS-bisector estimator. The WGM estimator is stableand work well for different values of λ and for different sample sizes.The properties of the proposed reflection estimators are investigated in Chapters 3-7. Also, these estimators are compared with the relevant existing estimators by simulation studies. The computer package Matlab is used for all computations and preparation of graphs. Based on the asymptotic consistency and MAE criteria the proposed reflection estimators perform better than the existing estimators, in some cases, even the standard assumption on λ and sample size are violated.Chapter 8 provides some concluding summaries remarks. Matlab; measurement errors; ME; estimation of the slope parameter; simple linear regression; regression methods; statistical methodolgy 490501. Applied statistics 490504. Forensic evaluation, inference and statistics School of Agricultural, Computational and Environmental Sciences

https://research.usq.edu.au/item/q3185/statistical-methodology-for-regression-model-with-measurement-error

Published Version
 Saqr_2013_whole.pdf File access level: Anyone

total views
• 2
views this month

Related outputs

Estimation of slope for measurement error model with equation error: applications on serum kanamycin data
Saqr, Anwar and Khan, Shahjahan. 2017. "Estimation of slope for measurement error model with equation error: applications on serum kanamycin data." Bakar, Shaiful Anuar Abu, Yunus, Rossita Mohamad and Mohamed, Ibrahim (ed.) 3rd ISM International Statistical Conference 2016 (ISM III): Bringing Professionalism and Prestige in Statistics. Kuala Lumpur, Malaysia 09 - 11 Aug 2016 United States. https://doi.org/10.1063/1.4982859
New weighted geometric mean method to estimate the slope of measurement error model
Saqr, Anwar and Khan, Shahjahan. 2014. "New weighted geometric mean method to estimate the slope of measurement error model." Journal of Applied Statistical Science. 22 (3-4), pp. 261- 280.
Slope estimator for the linear error-in-variables model
Saqr, Anwar and Khan, Shahjahan. 2012. "Slope estimator for the linear error-in-variables model." Ahmad, Munir (ed.) 12th Islamic Countries Conference on Statistical Sciences . Doha, Qatar 19 - 22 Dec 2012 Lahore, Pakistan.
Weighted reduced major axis method for regression model
Saqr, Anwar and Khan, Shahjahan. 2012. "Weighted reduced major axis method for regression model." Ahmad, Munir (ed.) 12th Islamic Countries Conference on Statistical Sciences . Doha, Qatar 19 - 22 Dec 2012 Lahore, Pakistan.
Instrumental variable estimator of the slope parameter when the explanatory variable is subject to measurement error
Saqr, Anwar and Khan, Shahjahan. 2012. "Instrumental variable estimator of the slope parameter when the explanatory variable is subject to measurement error." Ahmed, Munir (ed.) 11th Islamic Countries Conference on Statistical Sciences. Lahore, Pakistan 19 - 22 Dec 2011 Lahore, Pakistan.
Mathematical reflection approach to instrumental variable estimation method for simple regression model
Saqr, Anwar and Khan, Shahjahan. 2016. "Mathematical reflection approach to instrumental variable estimation method for simple regression model." Pakistan Journal of Statistics. 32 (1), pp. 37-48.
Reflection method of estimation for measurement error models
Saqr, Anwar and Khan, Shahjahan. 2012. "Reflection method of estimation for measurement error models." Journal of Applied Probability and Statistics. 7 (2), pp. 71-88.