Barry H. Margolin

An Analysis of Variance for Categorical Data

Abstract A measure of variation for categorical data is discussed. We develop an analysis of variance for a one-way table, where the response variable is categorical. The data can be viewed alternatively as falling in a two-dimensional contingency table with one margin fixed. Components of variation are derived, and their properties are investigated under a common multinomial model. Using these components, we propose a measure of the variation in the response variable explained by the grouping variable. A test statistic is constructed on the basis of these properties, and its asymptotic behavior under the null hypothesis of independence is studied. Empirical sampling results confirming the asymptotic behavior and investigating power are included.

An Analysis of Variance for Categorical Data, II: Small Sample Comparisons with Chi Square and other Competitors

Abstract Exact small sample behavior in two-way contingency tables is investigated for Pearsons chi-square statistic (X 2), Light and Margolins C statistic and its related R 2 measure of association, Kullbacks minimum discrimination information statistic (2I), and Goodman and Kruskals Lambda. R 2 is shown to be identical to Goodman and Kruskals tb , leading to a test for independence based on tb. In small samples from a product of multinomials model, the null distribution of C is better approximated by a χ2 distribution than is the null distribution of X 2; both are considerably better approximated by a χ2 distribution than is the null distribution of 2I. It is proved for tables with two columns and any number of rows that if the column totals are equal, then X 2 ≤ 2I; thus, X 2 is more conservative than 2I. Hence, use of 2I should be avoided in testing independence in tables with small samples.

Analysis of free-storage algorithms

Central to successful operation of a computer system based heavily on reentrant (or pure) procedures is reliable and efficient dynamic management of free storage. Such systems must allocate, use, and release one or more blocks of free storage for each task or system operation, e.g., each I/O task or each request for supervisor services. For any list-processing system, such as AED or LISP, efficient management of free storage is a fundamental problem. The consequence of errors in allocation and release is usually total collapse of the system; that of mismanagement is usually processor inefficiency or under-utilization of the free-storage pool. The processor inefficiency resulting from a poor or ill-chosen management algorithm is usually tolerated, even though it is high relative to that of other system functions; under-utilization of free storage is less tolerable, as “lock-up” can be encountered. This circumstance must be averted by task deferral or “garbage collection” procedures, both of which are costly to system performance.

Results on Factorial Designs of Resolution IV for the 2 n and 2 n 3 m Series

Factorial designs of resolution III are such that all main effects are estimable, ignoring two-factor interactions and all higher order interactions. Designs of resolution IV are such that all main effects are estimable with no two-factor interactions as aliases, ignoring all higher order interactions. The general technique for producing 2 n designs of resolution IV is a specific application of the Box-Wilson “fold-over” theorem. Recent work by Steve Webb on resolution IV designs for two-level factors is discussed and extended. The miniium run requirement for a 2 n resolution IV design is 2 n . It is proved that if a minimal resolution IV 2 n design has each factor occurring equally often at its high and low levels, then the design must be a fold-over design. A proof that the minimum run requirement for a resolution IV 2 n 3 m design, m > 0, is 3(n + 2m − 1) also is included. Minimal 2 n resolution IV designs are presented for various values of n. All these designs can be run in n blocks of size 2 each.

Orthogonal Main-Effect 2 n 3 m Designs and Two-Factor Interaction Aliasing

Methods are presented for the determination of the alias matrix of two-factor interactions for the orthogonal main-effect 2 n 3 m plans catalogued by Addelman and Kempthorne. This catalogue includes Placket-Burman designs and designs obtained by replacement in 2 n–p plans or collapsing in 3 m–m plans. Systematic methods are included to facilitate the data computations. For standard r n–p factorial designs, techniques are given to determine a set of live factors, a generating set of linear sum congruences and the alias matrix. Additional orthogonal main-effect 2 n 3 m designs are constructed to supplement the Addelman-Kempthorne catalogue of designs.

Orthogonal Main-Effect Plans Permitting Estimation of All Two-Factor Interactions for the 2n3m Factorial Series of Designs

If we assume no higher order interactions for the 2n3m factorial series of designs, then relaxing the restrictions concerning equal frequency for the factors and complete orthogonality for each estimate permits considerable savings in the number of runs required to estimate all the main effects and two-factor interactions. Three construction techniques are discussed which yield designs providing orthogonal estimates of all the main effects and allowing estimation of all the two-factor interactions. These techniques are: (i) collapsing of factors in symmetrical fractionated 3m–p designs, (ii) conjoining fractionated designs, and (iii) combinations of (i) and (ii). Collapsing factors in a design either maintains or increases the resolution of the original design, but does not decrease it. Plans are presented for certain values of (n, m) as examples of the construction techniques. Systematic methods of analysis are also discussed.

Design and Analysis of Factorial Experiments Via Interactive Computing in APL

This paper discusses the use of interactive computing in APL for the design and analysis of an ad hoc experiment. As an example, a project involving a 16 × 4 × 2 experiment in 8 blocks of size 8 is reviewed. The desirability of utilizing single-degreeof-freedom pseudo-factors in both the design and analysis phases of such experiments is exhibited, and the detection of and compensation for heterogeneity of variance and outliers are discussed.