Bulletin of the Computational Statistics of Japan

Recently some visual analyses of multivariate data have been studied in the field of statistical graphics. Several dynamic graphical methods have been proposed. Moving or rotating the graphical data points continuously, we can explore the latent structure of the data interactively. One of these methods, Linked Lines Rotation Graphics (LLRG) has been discussed by Wakimoto (1993). In this paper, we propose a visual method for finding clusters of multivariate data with LLRG. Numerical examples of the proposed method are shown.

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Missing values are almost unavoidable in many practical data analyses. Particularly in the analysis of experimental data, since the number of observations is generally not so large, the influence of missing should be carefully checked. This paper discusses several graphical methods which are useful in assessing the effect of missing values in the analysis of variance of randomized block design. Some techniques such as simple imputation and multiple imputation have been proposed in the literature to cope with the missing values. However, with the use of recent computer facilities, more effective methods which utilize graphical representation seem to be called for. Techniques considered in this paper are functional representations of F-ratios and P-values against imputed values and graphical representations of the result of Monte Carlo simulation as an approximation to the Bayesian posterior distribution of missing value. Numerical examples are shown to illustrate the techniques. It is also claimed that the present methods can be used not only to assess the effect of missing values but also to evaluate the sensitiveness of obtained data.

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An experiment was carried out to investigate the utility of graphical methods such as face graph, radar chart and letter graph for classifying multivariate data. To do this 3, 5 and 7 dimensional observations were generated based on models N (μk, I) of three groups, i.e., k=1, 2 and 3, where the distances among the groups were set as 3, 4 and 5 regardless of their dimensions. For each kind of graph 40 subjects were asked to classify 30 observations from 3 groups, which were expressed in a graph, into 3 groups, and correct/incorrect numbers of classification were counted. In face graph and radar chart, the rate of correct classification was the best in the case of 5 variables while the rate was the best in the case of 3 variables in letter graph. These facts suggest that the rate of correct classification decreased as the number of variables increase over a fixed number. It was also shown that the human's ability of classification was inferior to a numerical method such as principal component analysis.

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In this paper, we propose and discuss three kinds of approximate confidence intervals for Coefficient of Variation (CV), the classical confidence interval based on Normal approximation, the confidence intervals based on second-order and third-order Cornish-Fisher expansions. All the intervals are based on the jackknife-t quantity, a jackknife bias-adjusted random variable Studentized by its jackknife estimate of standard deviation. Within a practical range of CV, in terms of coverage probability, the approximate confidence interval based on third-order Cornish-Fisher expansion of the jackknife-t quantity improves the one based on second-order Cornish-Fisher expansion, which in turn improves the one based on basic Normal approximation. Our numerical experiment is carried out by assuming the original basic random variable being Normal. Approximations for the distribution function of the jackknife-t quantity are also discussed. The conclusion is similar to the case of confidence interval estimation: the third-order Edgeworth approximation is better than its second-order counterpart, which behaves better than Normal approximation, provided that the true coefficient of variation is reasonably stable (not too large).

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