ROBUST 2k FACTORIAL DESIGNS WITH LOGISTIC ERROR DISTRIBUTION
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Date
2013
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IARI, IASRI, NEW DELHI
Abstract
In designed experiment it is not always true that the error of the generated data follows
the normal distribution which is one of the basic assumptions of analysis of variance.
Under such situations, the maximum likelihood equations may not be linear and so are
not solvable. The equations obtained from the first derivative of log likelihood function
with respect to parameters do not yield the explicit solutions for the estimates due to nonlinearity
of the function. Solving them by iterations is indeed problematic for reasons of
(i) multiple roots, (ii) non-convergence of iterations, and (iii) convergence to wrong
values. Therefore, methods have been developed using the modified maximum likelihood
estimates in which the maximum likelihood equations are linearized by using the Taylor’s
expansion and estimates of the parameters are obtained. These estimates are called
modified maximum likelihood estimates. These estimates are efficient under non-normal
error distribution and asymptotic to maximum likelihood estimates.
In this dissertation, the error is assumed to follow logistic distribution and the model of
factorial experiment is assumed to be fixed effect model and design considered is
completely randomized design for equal number of observations per cell. Logistic
distribution is negatively skewed, positively skewed or symmetric for the values of b<1,
b>1, b=1, respectively where b indicates the shape of generalized logistic distribution. We
started with 23 factorial experiments and obtained the modified maximum likelihood
estimates for all the effects (main effects and interaction effects) and the estimate of the
error. F-statistics have been developed for all the treatment effects for testing the
significance of parameters. These results have been generalized for the factorial
experiments with k factors each at two levels.
SAS code has been developed for the generation of data for 23 factorial experiments in
which error follows logistic distribution for different values of parameter of logistic
distribution. By using the SAS code, the generated data have been analysed and the
compared the modified maximum likelihood procedure with the usual ANOVA
procedure. Finally the size of the test is computed by using Monte Carlo simulation
technique for different values of the parameter of the distribution.
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