Analytical Procedure for Factorial Experiments with Logistic and Gompertz Error Distributions

Abstract

Factorial experiments are widely used in agriculture and allied sciences. In the present study, 2×3^2 factorial experiments have been considered under the assumption that the error follows non-normal distribution. Under the assumption of normality and independence of observations, the normal equations obtained from the maximum likelihood function are linear and 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 difficult to handle.

The present study focuses on the development of an analytical procedure for the factorial experiments in order to tackle the situations where error term violates normality assumptions. Here, factorial experiments have been considered where error follows non-normal distribution. Two non-normal distributions have been considered from which one is generalized logistic distribution and another is Gompertz distribution. The theory of modified maximum likelihood estimation has been applied and efficient estimators have been developed. New modified maximum likelihood estimates have been developed and the estimates of parameters are obtained for both the situations of non-normality. The developed procedure is applied in the analysis of 2×32 factorial experiments in which error follows the generalized logistic and Gompertz error distributions. Data have been generated for the simulation studies for which error follows generalized logistic distribution. Three data sets have been generated for parameter values (θ =0.5, 1 and 2) in 2×32 factorial set up where the data are positively skewed, symmetric and negatively skewed respectively. In the same way, two data sets have been generated with the parameter values (η =1 and 2) where error follows Gompertz distribution. These data sets are analyzed through developed procedure. SAS codes have been developed for analysis of the data sets generated through 2×32 factorial experiments where error follows logistic and Gompertz distributions. The output for the data sets of all mentioned five parameter values i.e. θ =0.5, 1, 2 and η =1, 2 are given in table 4.4, 4.5, 4.6, 4.10 and 4.11 of Chapter 4 where the sum of squares and F* statistics have been given. The probability P (F*>F0.05) (v1, v18) is calculated empirically for the developed F* statistics. Further, the size of the test is computed with 5000 Monte Carlo runs using re-sampling technique.

This present investigation would help scientists, research scholars and students under NARES dealing with factorial experiments where error follows generalized logistic distribution and Gompertz distribution. For easy accessibility by the users, the SAS codes have been developed which provide a readymade solution.