Coefficient of determination for multiple measurement error models

The coefficient of determination (R2) is used for judging the goodness of fit in a linear regression model. It is the square of the multiple correlation coefficient between the study and explanatory variables based on the sample values. It gives valid results only when the observations are correctly observed without any measurement error. The conventional R2 provides invalid results in the presence of measurement errors in the data because the sample R2 becomes an inconsistent estimator of its population counterpart which is the square of the population multiple correlation coefficient between the study and explanatory variables. The goodness of fit statistics based on the variants of R2 for multiple measurement error models have been proposed in this paper. These variants are based on the utilization of the two forms of additional information from outside the sample. The two forms are the known covariance matrix of measurement errors associated with the explanatory variables and the known reliability matrix associated with the explanatory variables. The asymptotic properties of the conventional R2 and the proposed variants of R2 like goodness of fit statistics have been studied analytically and numerically.