# Statistical methodology for regression model with measurement error

PhD Thesis

*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 |
---|---|

Type | PhD Thesis |

Authors | |

Author | Saqr, Anwar A. Mohamad |

Supervisor | Khan, Professor Shahjahan |

Langlands, Dr Trevor | |

Institution of Origin | University of Southern Queensland |

Qualification Name | Doctor of Philosophy |

Number of Pages | 223 |

Year | 2013 |

Abstract | 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, 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 explanatory Table 1: A summary of commonly used methods to handle the ME model problem to 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 of 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 equal 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 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 distances 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. |

Keywords | Matlab; measurement errors; ME; estimation of the slope parameter; simple linear regression; regression methods; statistical methodolgy |

ANZSRC Field of Research 2020 | 490501. Applied statistics |

490504. Forensic evaluation, inference and statistics | |

Byline Affiliations | School of Agricultural, Computational and Environmental Sciences |

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

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##### 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.