The distribution of internally studentized statistics via Laplace transform inversion Get access BARRY H. MARGOLIN BARRY H. MARGOLIN Department of Statsitics, Yale UniversityNew Haven, Connecticut Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 64, Issue 3, December 1977, Pages 573–582, https://doi.org/10.1093/biomet/64.3.573 Published: 01 December 1977 Article history Received: 01 October 1976 Revision received: 01 March 1977 Published: 01 December 1977

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.

Summary Two techniques are presented for constructing efficient non-orthogonal main-effect designs. The first technique is an extended method of collapsing that permits non-orthogonality in the derived design. The second technique is a modification of the fold-over method of design construction to produce new main-effect plans. The designs in this paper supplement the major work on orthogonal asymmetrical main-effect plans by Addelman and Kempthorne. The new designs require fewer observations than the corresponding Addelman–Kempthorne plans; this advantage is gained at a sacrifice of orthogonality. The new plans are potentially useful in exploratory studies involving a few multi-level factors plus many two-level factors. Being more nearly saturated than the Addelman–Kempthorne plans, the new designs usually require graphical analysis or testing via a valid external measure of variability. The relationship of some of the new designs to Latin squares and Graeco-Latin squares is also discussed briefly.

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.

Summary Fractional factorial designs of resolution IV permit estimation of all the main effects with no aliasing by two-factor interactions. This paper produces a lower bound for the number of observations required for a general fractional factorial design to be of resolution IV. This lower bound agrees with a lower bound obtained by Rao for orthogonal arrays of strength 3. In addition, it is proved that this lower bound is attainable for the t. 2n factorial design series for t even and n ≡ 3 (mod 4) in plans which permit orthogonal estimation of the main effects. Finally, Webb's conjecture that there exist no resolution IV 2n factorial designs with 2n runs except for those constructed by the fold-over technique is proved to be valid.