Ordinal test fidelity estimated by an item sampling model

A test theory using only ordinal assumptions is presented. It is based on the idea that the test items are a sample from a universe of items. The sum across items of the ordinal relations for a pair of persons on the universe items is analogous to a true score. Using concepts from ordinal multiple regression, it is possible to estimate the tau correlations of test items with the universe order from the taus among the test items. These in turn permit the estimation of the tau of total score with the universe. It is also possible to estimate the odds that the direction of a given observed score difference is the same as that of the true score difference. The estimates of the correlations between items and universe and between total score and universe are found to agree well with the actual values in both real and artificial data.Part of this paper was presented at the June, 1989, Meeting of the Psychometric Society. The authors wish to thank several reviewers for their suggestions. This research was mainly done while the second author was a University Fellow at the University of Southern California.     

A note on the unbiased estimation of the intraclass correlation

The intraclass correlation,, is a parameter featured in much psychological research. Two commonly used estimators of, the maximum likelihood and least squares estimators, are known to be negatively biased. Olkin and Pratt (1958) derived the minimum variance unbiased estimator of the intraclass correlation, but use of this estimator has apparently been impeded by the lack of a closed form solution. This note briefly reviews the unbiased estimator and gives a FORTRAN 77 subroutine to calculate it.The first author was supported by an All-University Fellowship from the University of Southern California.     

Common functional principal components analysis: a new approach to analyzing human movement data

In many human movement studies angle-time series data on several groups of individuals are measured. Current methods to compare groups include comparisons of the mean value in each group or use multivariate techniques such as principal components analysis and perform tests on the principal component scores. Such methods have been useful, though discard a large amount of information. Functional data analysis (FDA) is an emerging statistical analysis technique in human movement research which treats the angle-time series data as a function rather than a series of discrete measurements. This approach retains all of the information in the data. Functional principal components analysis (FPCA) is an extension of multivariate principal components analysis which examines the variability of a sample of curves and has been used to examine differences in movement patterns of several groups of individuals. Currently the functional principal components (FPCs) for each group are either determined separately (yielding components that are group-specific), or by combining the data for all groups and determining the FPCs of the combined data (yielding components that summarize the entire data set). The group-specific FPCs contain both within and between group variation and issues arise when comparing FPCs across groups when the order of the FPCs alter in each group. The FPCs of the combined data may not adequately describe all groups of individuals and comparisons between groups typically use t-tests of the mean FPC scores in each group. When these differences are statistically non-significant it can be difficult to determine how a particular intervention is affecting movement patterns or how injured subjects differ from controls. In this paper we aim to perform FPCA in a manner allowing sensible comparisons between groups of curves. A statistical technique called common functional principal components analysis (CFPCA) is implemented. CFPCA identifies the common sources of variation evident across groups but allows the order of each component to change for a particular group. This allows for the direct comparison of components across groups. We use our method to analyze a biomechanical data set examining the mechanisms of chronic Achilles tendon injury and the functional effects of orthoses.