Correlated log-normal composite error models for different scientific domains

Abstract

The Gaussian (normal) distribution is most often assumed to describe the random variation that occurs in the data from many scientific disciplines. However, it is rarely observed that a real data set exactly follows normal distribution. Generally, positive observations with constant variance from a continuous random variable are analyzed either by the log-normal or the gamma models. In practice, the variance is not always constant. For handling non-constant variance in the log-normally distributed continuous response random variable, some concomitant variables are included as explanatory variables in the regression models. In the present article, the response distribution is assumed to have the log-normal with compound symmetry errors. The log-normal model with composite compound symmetry errors is developed. The best linear unbiased estimators of all the regression coefficients have been derived except the intercept which is often unimportant in practice. Both the correlation coefficients (within and between groups) have been estimated. A robust (free of correlation coefficients) testing procedure for any set of linear hypotheses regarding the unknown regression coefficients has been introduced. Confidence intervals of an estimable function and confidence ellipsoids of a set of estimable functions of regression coefficients have been derived. An index of fit for the fitted regression model has also been developed. An example (with simulated data) illustrates the results derived in this report. A real example with replicated observations has been modelled based on the present developed theories. Some new data analysis theories are developed in the present report to analyze positive, skewed, correlated and also replicated data.