Ibrahim A. Salama

Age-period-cohort analysis: an illustration of the problems in assessing interaction in one observation per cell data

This paper discusses the specific problems of age-period-cohort (A-P-C) analysis within the general framework of interaction assessment for two-way cross-classified data with one observation per cell. The A-P-C multiple classification model containing the effects of age groups (rows), periods of observation (columns), and birth cohorts (diagonals of the two-way table) is characterized as one of a special class of models involving interaction terms assumed to have very specific forms. The so-called A-P-C identification problem, which results from the use of a particular interaction structure for detecting cohort effects, is shown to manifest itself in the form of an exact linear dependency among the columns of the design matrix. The precise relationship holding among these columns is derived, as is an explicit formula for the bias in the parameter estimates resulting from an incorrect specification of an assumed restriction on the parameters required to solve the normal equations. Current methods for model...

A nonparametric comparison of two multiple regressions by means of a weighted measure of correlation

Let (R1j,....,Rmj) j=1,2, be two rankings of m items; let Tk , k=l,...,m. be the number of items with rank ≤ k in both rankings; and let T = ∑Tk/k. Then T is a measure of rank correlation which gives greater weight to items of low rank than high. Such a measure is particularly useful in comparing the ordering of the regressors in two multiple regressions. We discuss the distribution of T, presenting both exact tables and practical approximations, and extend the concept to other situations

Using weighted rankings to test against ordered alternatives in complete blocks

Consider testing the hypothesis of no treatment effects against a postulated ranking of the treatments, given data from n complete blocks. A suitable test statistic is the weighted average rank correlation W = where Ci correlation between the postulated ranking and the ranking observed within the i-th block, Qi is the rank of the i-th block with respect to credibility, and the bs are weights such that 0<=b1<=...<=bn. Tests using Spearman and Kendall correlation are proposed and their distributions are obtained for both small and large experiments. These tests and others are compared in small experiments with respect to expected significance level. Finally, a simple illustrative example is presented.

The Spearman footrule and a Markov chain property

An equivalent representation of the Spearman footrule is considered and a characterization in terms of a Markov chain is established. A martingale approach is thereby incorporated in the study of the asymptotic normality of the statistics.

Spearman's footrule under progressive censoring

Spearmans footrule, a well-known measure of rank correlation, is extended here to progressively censored rankings. Under the hypothesis of randomness ( i.e. , under random ranking), a martingale characterization is exploited in the formulation of a functional central limit theorem, and its applications in incomplete rankings are illustrated.

On expected sample range from heterogeneous symmetric distributions

Abstract It is shown that for independent (but not necessarily identically distributed) random variables with distributions symmetric about the respective medians (means), the expected value of the sample range is a minimum when these means are all equal.