Extending the Box–Cox transformation to the linear mixed model

Summary.  For a univariate linear model, the Box–Cox method helps to choose a response transformation to ensure the validity of a Gaussian distribution and related assumptions. The desire to extend the method to a linear mixed model raises many vexing questions. Most importantly, how do the distributions of the two sources of randomness (pure error and random effects) interact in determining the validity of assumptions? For an otherwise valid model, we prove that the success of a transformation may be judged solely in terms of how closely the total error follows a Gaussian distribution. Hence the approach avoids the complexity of separately evaluating pure errors and random effects. The extension of the transformation to the mixed model requires an exploration of its potential effect on estimation and inference of the model parameters. Analysis of longitudinal pulmonary function data and Monte Carlo simulations illustrate the methodology discussed.     

I Want to Pretend I'm Eleven Years Younger: Subjective Age and Seniors' Motives for Vacation Travel

Subjective age–as distinct from chronologicalAge–is a relatively recent area of scientificinquiry within quality-of-life studies. Thepurpose of this empirical study was to seewhether the size of the gap between actual andself-perceived age could be predicted from aknowledge of a senior's reasons for wanting totravel on vacation, and his or her valuepriorities and sense of physical well-being. Asurvey of 356 Australian seniors ranging in agebetween 56 and 93 yielded data on theirsubjective age, measured in terms of (a) howold they felt; (b) the activities they wouldchoose to experience or enjoy while on avacation; and (c) the age they wished to betreated and recognized as–as well as theirpsychological motives for vacation travel,personal values, and self-rated health. Theresults of multivariate analyses indicate thatthe gap between actual age and subjective ageis a function of seniors' particular motivesfor travel and what they value in life, and,for certain aspects of subjective age, howhealthy they feel and their gender.     

Changes in executive function after acute bouts of passive cycling in Parkinson's disease

Individuals with Parkinson's disease (PD) often experience cognitive declines. Although pharmacologic therapies are helpful in treating motor deficits in PD, they do not appear to be effective for cognitive complications. Acute bouts of moderate aerobic exercise have been shown to improve cognitive function in healthy adults. However, individuals with PD often have difficulty with exercise. This study examined the effects of passive leg cycling on executive function in PD. Executive function was assessed with Trail-Making Test (TMT) A and B before and after passive leg cycling. Significant improvements on the TMT-B test occurred after passive leg cycling. Furthermore, the difference between times to complete the TMT-B and TMT-A significantly decreased from precycling to postcycling. Improved executive function after passive cycling may be a result of increases in cerebral blood flow. These findings suggest that passive exercise could be a concurrent therapy for cognitive decline in PD.     

Product Positioning Using a Self‐Organizing Map and the Rings of Influence

In this article, we propose a new product positioning method based on the neural network methodology of a self‐organizing map. The method incorporates the concept of rings of influence, where a firm evaluates individual consumers and decides on the intensity to pursue a consumer, based on the probability that this consumer will purchase a competing product. The method has several advantages over earlier work. First, no limitations are imposed on the number of competing products and second, the method can position multiple products in multiple market segments. Using simulations, we compare the new product positioning method with a quasi‐Newton method and find that the new method always approaches the best solution obtained by the quasi‐Newton method. The quasi‐Newton method, however, is dependent on the initial positions of the new products, with the majority of cases ending in a local optimum. Furthermore, the computational time required by the quasi‐Newton method increases exponentially, while the time required by the new method is small and remains almost unchanged, when the number of new products positioned increases. We also compute the expected utility that a firm will provide consumers by offering its products. We show that as the intensity with which a firm pursues consumers increases, the new method results in near‐optimal solutions in terms of market share, but with higher expected utility provided to consumers when compared to that obtained by a quasi‐Newton method. Thus, the new method can serve as a managerial decision‐making tool to compare the short‐term market share objective with the long‐term expected utility that a firm will provide to consumers, when it positions its products and intensifies its effort to attract consumers away from competition.     

Existence of unbiased covariance components estimators

The condition of PINCUS (1974) for the estimability of covariance components in normal models is extended to the case of singular covariance matrices     

Combining an Internal Pilot with an Interim Analysis for Single Degree of Freedom Tests

An internal pilot with interim analysis (IPIA) design combines interim power analysis (an internal pilot) with interim data analysis (two stage group sequential). We provide IPIA methods for single df hypotheses within the Gaussian general linear model, including one and two group t tests. The design allows early stopping for efficacy and futility while also re-estimating sample size based on an interim variance estimate. Study planning in small samples requires the exact and computable forms reported here. The formulation gives fast and accurate calculations of power, type I error rate, and expected sample size.     

Fast Computation by Block Permanents of Cumulative Distribution Functions of Order Statistics from Several Populations

The joint cumulative distribution function for order statistics arising from several different populations is given in terms of the distribution functions of the populations. The computational cost of our formula in the case of two populations is still exponential in the worst case, but it is a dramatic improvement compared to the general formula by Bapat and Beg. In the case when only the joint distribution function of a subset of the order statistics of fixed size is needed, the complexity is polynomial, for the case of two populations.     

Expected Power for the False Discovery Rate with Independence

The Benjamini–Hochberg procedure is widely used in multiple comparisons. Previous power results for this procedure have been based on simulations. This article produces theoretical expressions for expected power. To derive them, we make assumptions about the number of hypotheses being tested, which null hypotheses are true, which are false, and the distributions of the test statistics under each null and alternative. We use these assumptions to derive bounds for multiple dimensional rejection regions. With these bounds and a permanent based representation of the joint density function of the largest p-values, we use the law of total probability to derive the distribution of the total number of rejections. We derive the joint distribution of the total number of rejections and the number of rejections when the null hypothesis is true. We give an analytic expression for the expected power for a false discovery rate procedure that assumes the hypotheses are independent.     

On the distribution of summary statistics for missing data

Under an assumption that missing values occur randomly in a matrix, formulae are developed for the expected value and variance of six statistics that summarize the number and location of the missing values. For a seventh statistic, a regression model based on simulated data yields an estimate of the expected value. The results can be used in the development of methods to control the Type I error and approximate power and sample size for multilevel and longitudinal studies with missing data.