Log-transformation in regression with measurement error
by Tsang Siu Yin
vii, 62 leaves : ill. ; 30 cm
The aim of this thesis study is to analyse the effect of meansurement errors on the regression model. This thesis is divided into two parts: transformation of the respsonse Y and transformation of the explanatory variable....[ Read more ]
The aim of this thesis study is to analyse the effect of meansurement errors on the regression model. This thesis is divided into two parts: transformation of the respsonse Y and transformation of the explanatory variable.
In the first part, we consider a regression model with two explanatory variables and the explanatory variables are measured with errors. We find out the estimator of the slope parameters and their ratio. After fitting the model, we draw a residual plot to see whether the residual is random and constant variance. If the residual is not random, we transform the response Y to In(Y) and regress In(Y) on the explanatory variables. If measurement error does not exist in the explanatory variable, the ratio of the slope parameters in the true model asymtotically equals the slope parameters in the transformed model (Brillinger, 1983). When the explanatory variables are measured with errors, we find out the relationship of the expected ratio of the slope parameters in the true model and the ratio of slope parameters in the transformed model.
In the second part, we consider the effect on the estimation of the intercept and slope in a simple linear model when explanatory variable does not follow normal distribution. If we ignore the non-normality of explanatory variable, we simply regress the response Y on the explanatory variable ξ and the estimates of intercept and slope are found here.
It is supposed that the explanatory variable ξ follows lognormal distribution and so logarithm of ξ follows normal distribution. We regress Y on In(ξ) and the estimates of intercept and slope are also calculated in this transformed model. By inverting the estimates in the transformed model, we can introduce other estimates for the intercept and slope parameters in the true model. F'urther-more, we compare those estimates in the transformed model with that in the untransformed model. It is interesting to see that transformation can make the estimation worse.
Post a Comment