Browsing by Author "Kanjo, Anis Ismail"

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    Recovery of interblock information
    Kanjo, Anis Ismail (Virginia Polytechnic Institute, 1965)
    We know that the best linear combination of the intra- and inter-block estimates is Intra-estimatexInter-variance+Inter-estimatexIntra-variance / Intra-variance + Inter-variance ; however, this combined estimate is merely theoretical, since we do not know in practice the exact inter- and intra-variances. A reasonable solution is to use a random weight which can be computed from the data of our experiment, but so far there has been no practical solution without severe restrictions on the size of the experiment, and no solution at all for a clear answer to the question of how much we recovered. In fact, the experimenter applying the methodology available to him now, cannot be sure that he is really improving the accuracy of his estimation. This research has achieved the following: 1. A new method of combining two independent estimates has been developed. This method has its use in incomplete block designs, in similar experiments, and in randomized block designs with heterogeneous variances. The improvement introduced by this method is very satisfactory, compared with the utmost possible theoretical improvement. 2. A procedure for recovering the inter-block information in B.I.B. designs was given, which is applicable in experiments as small as t = 4. 3. It has been proven that the practical utilization of inter-block information is possible in any P.B.I.B. with seven treatments or more. 4. A general procedure for recovering the inter-block information in P.B.I.B.'s with two associate classes was given. 5. An inter-block analysis of singular and semi-regular group divisible designs was discussed, which makes a partial utilization of the inter-information possible. In general, this work has two merits: 1. It makes possible the utilization of the interblock information in small and moderate size experiments. 2. As a ratio of the utmost possible theoretical recovery (by combining linearly), either exactly or a lower bound of the ratio of recovery is always computable. Tables which enable the experimenter to use the procedures described in this dissertation were given. The ratios of recovery listed in these tables show that the new method gives good results where the old method is not applicable, and when the old method starts, hopefully, to be valid, the ratio of recovery achieved by the new method starts to approach the theoretical value that can be achieved, assuming the intra- and inter-variance are known.
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    The theory and application of transformation in statistics
    Kanjo, Anis Ismail (Virginia Polytechnic Institute, 1962)
    This paper is a review of the major literature dealing with transformations of random variates which achieve variance stabilization and approximate normalization. The subject can be said to have been initiated by a genetical paper of R. A. Fisher (1922) which uses the angular transformation Φ = 2 arcsin√p to deal with the analysis of proportions p with E(p) = P. Here it turns out that Var Φ is almost independent of P and so stabilizes the variance. Some fourteen years later Bartlett introduced the so-called square-root transformation which achieves variance stabilization for variates following a Poisson distribution. These two transformations and their ramifications in theory and application are fully discussed. along with refinements introduced by later writers, notably Curtiss (1943) and Anscombe (1948). Another important transformation discussed is one which refers to an analysis of observations on to a logarithmic scale, and here there are uses in analysis of variance situations and theoretical problems in the field of estimation: in the case of the latter, the work of D. J. Finney (1941) is considered in some detail. The asymptotic normality of the transformation is also considered. Transformations primarily designed to bring about ultimate normality in distribution are also included. In particular, there is reference to work on the chi-square probability integral (Fisher), (Wilson and Hilferty (1931)) and the logarithmic transformation of a correlation coefficient (Fisher (1921)). Other miscellaneous topics referred include i. the probability integral transformation (Probits), with applications in bioassay: ii. applications of transformation theory to set up approximate confidence intervals for distribution parameters (BIom (1954)): iii. transformations in connection with the interpretation of so-called 'ranked' data.