Franklin A. Graybill

An Analysis of a Two-Way Model with Interaction and No Replication

Abstract In this article the two-way array interaction model with one observation per cell is discussed. The model is given by Likelihood ratio tests are presented for two hypotheses: (1) no interaction (λ = 0) and (2) equality of treatments (τ1 = τ2 = … = τt). Also maximum likelihood estimators are given for all parameters including σ2 when λ ≠ 0.

Confidence intervals on linear combinations of variance components that are unrestricted in sign

A new method is proposed for constructing confidence intervals on where the sign of γ is unrestricted and unknown, and are independently distributed chi-squared random variables with ni degrees of freedom for Computer simulation is used to illustrate that the method provides a confidence coefficient that is generally close to the stated level. The method is illustrated by using it to test a main effect in a random three-factor crossed design.

Extensions of the General Linear Hypothesis Model

Abstract In the conventional general linear model it is often possible to examine a more complete model that includes nonlinear terms. This is an extension of a test for non-additivity by Tukey and a generalization of that test by Scheffe. This article shows explicitly how the theory of the well-known general linear model can be used for these problems and illustrates some extensions and computing procedures.

Confidence Bands of Uniform and Proportional Width for Linear Models

Abstract In this paper confidence bands are given for simple linear models. The confidence bands that are given are straight lines rather than curves and are either (1) parallel or (2) trapezoidal. The confidence bands are over a finite length interval.

Linear Segment Confidence Bands for Simple Linear Models

Abstract In this paper confidence bands are given for the entire line for simple linear regressions. The conventional bands that have been given in the past are curvilinear. In this paper we consider confidence bands that are straight lines. It is shown that under certain conditions the “width” of these straight line bands is less than the width of the conventional curvilinear bands.

A Note on Uniformly Best Unbiased Estimators for Variance Components

where Ynln2...nk is the observation, /i is a fixed unknown constant generally called the over-all mean, the Anl(1), A2 (2) y , enjn2 nk are independent normal variables whose means are zero and whose variances are r12, 0.22, . . . 2 respectively (a-,2 distinct). This model is commonly called a component of variance model and sometimes referred to as the Eisenhart Model II [4]. For example, if the model is a two-way balanced classification it can be written as

Confidence Intervals for Proportions of Variability in Two-Factor Nested Variance Component Models

Abstract In the two-factor nested variance component model with equal numbers in the cells given by y ijk = μ + A i + B ij + C ijk , three important functions of the variance components σA 2, σB 2, and σC 2 are the proportions of the total variation due to factors A, B, and C, which are given by σA 2/(σA 2 + σB 2 + σC 2), σB 2/(σA 2 + σB 2 + σC 2) and σC 2/(σA 2 + σB 2 + σC 2), respectively. There are no exact confidence intervals available for these proportions. In this article approximate lower and upper confidence intervals are presented.

Confidence intervals on ratios of linear combinations for non-disjoint sets of expected mean squares

Abstract A method for constructing confidence intervals on the ratio of expected mean squares p ∑ i=1 P c i θ i − ∑ j=P+1 Q d j θ j ∑ k=1 Q e k θ k , c i , d j , e k ⩾ 0 is proposed. This form of ϱ is different from other ratios for which confidence intervals have been developed because it contains the same expected mean squares in the numerator and denominator. Additionally, it allows negative coefficients in the numerator. Computer simulation is used to examine the confidence coefficients associated with the proposed intervals.

Confidence Intervals on Linear Combinations of Variance Components in the Unbalanced One-Way Classification

A problem that can occur in a variance component analysis is the estimation and construction of confidence intervals for linear combinations of the variance components. This article considers the unbalanced one-way classification model and develops a procedure that can be used to construct a confidence interval on a linear combination of the among- and within-group variances. A simulation study suggests that the procedure provides an interval that in most cases has an achieved confidence coefficient at least as great as the stated level.

Confidence intervals on ratios of positive linear combinations of variance components

In variance component models, the ratio [rho] is often of interest, where [rho] = ([Sigma]pq=1>kq[theta]q)/([Sigma]Qr=P+1kr[theta]r), kq0, kr\s>;0, and [theta]q, [theta]r are expected mean squares from an analysis of variance table, Q=1,..., P; r = P+1,..., Q. This paper pre methods for constructing approximate confidence intervals on [rho].