In longitudinal clinical trials that compare treatments of chronic diseases missing data occur mainly because of dropouts, where patients stop participating in the trial before the completion due to various reasons. Such incomplete data are often analyzed by using so-called completer analysis and/or LOCF (Last Observation Carried Forward). However, such procedures require strong assumptions for their validity. Multiple imputation (MI) (Rubin, 1987) is a valid method under MAR (Missing At Random). This method consists of three steps (“imputation”, “analysis” and “combination”) and various methods for MI also have been proposed. In this paper, we evaluate the performance of four methods for MI contrasted with completer analysis and LOCF via Monte-Carlo simulations in the context of small-sample longitudinal clinical trials for comparison of two treatments. The performance of these methods with non-normal data (i.e. mixture of responders and non-responders) is also examined.

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Individuals' choices of categories observed on two occasions are described by transition frequency matrices. In this paper, a penalized optimal scaling method is presented to analyze a set of the matrices obtained from multiple sources and graphically represent a transition trend for each source as a vector. This method finds scores of individuals, those of categories, and vectors of trends, in such a way that individuals' scores become homogeneous to the scores of chosen categories and trend vectors become homogeneous to the inter-occasion changes in individuals' scores. The resulting low-dimensional configuration of trend vectors allows us easily to grasp transition trends. Further, the projection of category scores onto trend vectors gives the unidimensional scales of categories useful for scrutinizing transition trends.

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For square contingency tables with ordered categories, Tomizawa, Miyamoto and Hatanaka (2001) considered a measure that represents the degree of departure from symmetry. This paper extends the measure to multi-way tables with ordered categories. The measure proposed is expressed by using the Cressie-Read power-divergence or the Patil-Taillie diversity index. The measure could be useful for comparing the degrees of departure from symmetry in several multi-way tables with ordered categories. Examples are given.

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We consider methods for parameter estimation of the shifted power transformation. The ordinary likelihood function is unbounded and then fails to have a local maximum. This is a non-regular problem in likelihood because the range of observations depends on the unknown shift parameter. To avoid such a difficulty, we discuss the group likelihood method and the maximum product of spacings method, in a univariate case, assuming the power-normal distribution as an underlying distribution for observations. We describe the computational procedures for parameter estimation. To evaluate the performance of the estimates from the two methods, we perform a simulation study. In addition, two examples are given to illustrate some aspects of the two methods.

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The additive regression model assumes additivity among explanatory variables and other rigid requirements, which might give poor estimation of regression functions. Transforming response variables is a useful method to diagnose additivity and other requirements. From a practical point of view, parametric transformations such as the Box-Cox power transformation would give more helpful suggestions in interpreting results of analysis than nonparametric transformations.
The power additive smoothing spline (PASS) model is proposed to diagnose the validity of assuming additivity in the additive regression model. The smooth functions (and often regression parameters) are estimated with a penalized likelihood approach, and the power and the smoothing parameters, which govern global nonlinear regression structure, are estimated with the empirical Bayes method, in which a Laplace approximation of the marginal likelihood is developed. The PASS model is applied to some data sets, and also its performance is examined through a simulation experiment. It is shown that the PASS model can extract an appropriate regression structure if true structure is additive after a Box-Cox power transformation of responses.

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