Browsing by Author "Glenn, William Alexander"

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    Analysis of variance of a randomized block design with missing observations
    Glenn, William Alexander (Virginia Tech, 1957-09-15)
    The estimation of several missing values in a randomized block design ls considered. The method used ls that of minimizing the error sum of squares, proposed originally by Yates (1933). Explicit equation for each absent value are derived for all cases in which not more than three values are missing. A general formula valid for any permissible number of missing observations ls given for the case in which no two values are missing in the same block or treatment, and also for the case in which all of the values missing are in a single block or treatment. A procedure for the completely general case is proposed. This, although requiring the inversion of s symmetric matrix of order equal to the number of missing observations, may prove to be less tedious in application than the iterative method proposed by Yates.
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    Some aspects of paired-comparison experiments
    Glenn, William Alexander (Virginia Polytechnic Institute, 1959)
    I. A Comparison of the Effectiveness of Tournaments. A paired-comparison experiment involving t treatments is analogous to a tournament with t players. A balanced experiment, in which every possible pair occurs once per replication, is the counterpart of a round robin tournament. When the objective is to pick the best treatment, the balanced design may prove to be more expensive than necessary. The knock-out tournament has been suggested as an alternative requiring fewer units of each treatment per replication. In this paper round robin, replicated knock-out, and double elimination tournaments are investigated for their effectiveness in selecting the best one or tour players. Effectiveness is gauged in terms of the two criteria (a) the probability that the best player wins and (b) the expected number of games. For general values of the parameters involved, expressions are derived for the evaluation of the criteria. Comparisons are made on the basis of series of assigned parameter values. Possibilities for the extension of the study are briefly discussed. II. Ties in Paired-Comparison Experiments. In making paired comparisons a judge frequently is unable to express a real preference in a number of the pairs he judges. In spite of this, some or the methods in current use do not permit the judge to declare a tie. In other methods tied observations are either ignored or divided equally or randomly between the tied members. It appears that there is a need, at least in the estimation of response-scale values, for a method which takes tied observations into account. In the Thurstone-Mosteller method the standardized distribution of the difference of two stimulus responses is normal with unit variance and mean equal to the difference or the two mean stimulus responses. In prohibiting ties the assumption is in effect made that all differences, however small, are perceptible to the judge. In this paper the assumption is made that a tie will occur whenever the difference between the judge's responses to the two stimuli lie below a certain threshold, i.e. if the difference lies between -t and t the judge will declare a tie. The parameter t and the mean stimulus responses are estimated by least squares. To overcome a difficulty presented by correlated data, an angular response law is postulated for the response-scale differences. In the resulting transformed data non-homogeneity of variances is encountered. In effecting a weighted solution, weights are first determined by using a preliminary unweighted analysis, and an iterative procedure is proposed. Large-sample variances and covariances of the estimates are obtained. A test of the validity of the model is described. A computational procedure is set up, and exemplified through application to experimental data.