Analysis of experimental designs with t-family of error distributions

Abstract

Theory of designed experiments has been developed under certain basic assumptions. These basic assumptions are that error in the model is normally distributed with mean zero and constant variance; error is independently distributed; effects are additive in nature. But many times the data collected in agricultural experiments do not follow these assumptions of analysis of variance (ANOVA). In it, one of the important assumptions is that the observations are independently and identically distributed as normal. In a study made under NARS, it is observed that more than 20% of the agricultural experiments conducted do not follow the assumptions of ANOVA and in 10 to 12% of the experiments the assumption of normality is violated. Unfortunately, the practitioners of statistics often do not test the assumption of normality. Thus blindly following statistical procedures without understanding the underlying assumptions may result in misleading or incorrect inference from the statistical analysis. Normality in data is violated due to many reasons. Some of these reasons are outliers in the data, truncated distribution, asymmetric distribution, student’s – t family of symmetric distribution, etc. Here we relax normality by other reasons and for the time being concentrate on t - family of symmetric distributions. Under the assumption of normality and independence of observations, the normal equations obtained from of maximum likelihood function are linear and are solvable. On the other hand when the data do not follow the normal distribution, the equations obtained of MLE are not linear and so these equations are not easy to handle. In any designed experiment, when the error in the model follows t - family of symmetric distribution, the normal equations obtained from the derivative of log-likelihood function with respect to parameters do not yield explicit solutions for the parameters due non-linearity of the function. Generally, these equations are not easy to solve by iterative method because of slow convergence, multiple roots, and convergence to incorrect values or even divergence. Tiku (1967) and Tiku and Stewart (1977) have developed the theory of modified maximum likelihood estimation (MMLE). The theory of MMLE has an explicit solution of these equations and is asymptotically identical with MLE. Tiku and Suresh (1992) have extended modified maximum likelihood method for Student’s – t-family of distributions. It has been shown Tiku et al. (1986) that modified maximum likelihood estimates (MMLEs) are almost as efficient as maximum likelihood estimates (MLEs). In this investigation, some of the experimental situations have been discussed where the experimental data is non-normal. These experimental situations strengthen the need to develop the procedures when the data is not normally distributed. It has been illustrated by using a set of data that the analysis performed by using MML procedure is efficient than the analysis performed on the original data and data obtained after Box-Cox transformation. These procedures have been developed for the designs of one-way elimination of heterogeneity when the error follows the t-family of symmetric distribution. These procedures are based on the solution of modified maximum likelihood estimations. When the derivative of log-likelihood function with respect to parameters do not yield explicit solutions for the parameters due non-linearity of the function, the non-linear function has been expanded using the Taylor’s expansion of the first order. By this procedure the function becomes linear and the equations are solvable. Further, the test statistics and their distributions have been worked out. Some of the experimental situations of designs of two-way elimination of heterogeneity have been discussed where such designs are generally used. Commonly used designs of two-way elimination of heterogeneity are Latin square designs. Similar to the designs of one-way elimination of heterogeneity, the theory of MML estimators have been developed for the designs of two-way elimination of heterogeneity. Procedures for the analysis of designs of two-way elimination of heterogeneity have been developed. Also the test statistics for testing the null hypothesis of the effects of treatments rows and columns for latin square have been developed. One of the most commonly used types of factorial designs is the 2^k factorial experiments. These experiments are quite useful in agricultural experiments. For factorial experiments, we started with the model of 2^2 factorial experiments when the error follows the t-family of symmetric distribution. Contrasts of main effects and sum of squares of contrasts for two factor interactions have been worked out. Variance of the error components has also been worked out. For testing hypothesis of the main effects and effects two factor interaction, test statistics have been developed. Further, this method of MMLE of 2^2 factorial experiment has been extended for factorial experiments and have been generalized for the factorial experiment with k factors each at 2 levels.