Mervyn G. Marasinghe

Using graphics and simulation to teach statistical concepts

Abstract The value to students of active learning has been recognized. This has led to the wide use of assignments in statistical methods courses where students use statistical software and computing equipment to analyze data. These assignments enable most students to master the mechanics of data analysis. The amount of experience that a student can get with such assignments is, however, limited. A sizable proportion of students have difficulty grasping some of the many concepts that are introduced in these courses. Nevertheless, these concepts are important for effective modeling and data analysis, and instructors should focus on them. By using current computing technology, it is possible to supplement standard data analysis assignments and algebraic derivations and have students become actively involved in the learning of important statistical concepts. The learning experience can be enhanced by giving students additional statistical “experiences” by using combinations of carefully designed and implemen...

A Multistage Procedure for Detecting Several Outliers in Linear Regression

A new statistic, Fk , is proposed for detecting multiple outliers in linear regression. This statistic is incorporated into the following multistage procedure: Initially, a subset of k observations is selected to be tested. If Fk is found to be significant, the most extreme observation in the subset as determined by the largest studentized residual is deleted and the test repeated for the (k – 1) observations in the subset using the remaining sample. The procedure is stopped when a test fails to reject the no-outlier hypothesis. A Monte Carlo study is used to evaluate the performance of this procedure.

Asymptotic tests and monte carlo studies associated with the multiplicative interaction model

Several authors have proposed approximations to percentage points required for testing certain hypotheses associated with the multiplicative interaction model. Alternative approximations based on the asymptotic joint distribution of the characteristic roots of a noncentral Wishart matrix are proposed in this paper. The type I error rates of the resulting tests and the existing procedures are then compared using Monte Carlo methods.

A Test of Incomplete Additivity in the Multiplicative Interaction Model

Abstract Consider the multiplicative interaction model defined by yij = μ + τ i + β j + λα i γ j + ∈ ij , i = 1, 2, …, t, j = 1, 2, …, b, where it is assumed that Σ i τ i = Σ j β j = Σ i α i = σ j γ j = 0 and γ i α i 2 = γ j γ j 2 = 1. It is also assumed that the ∈ ij are distributed NID (0, σ2). This article derives the likelihood ratio test of H 0: Hα = 0 and Gγ = 0 vs. Ha :Hα ≈ 0 or Gγ ≈ 0, where H is a q × t matrix of row contrasts of rank q and G is a r × b matrix of row constrasts of rank r. An approximation to the critical points of the test statistic is given and tables are given for a few selected values of b, t, q, and r. An improved estimator of σ2 is derived, and all results are illustrated with an example.

Assessment of haptics-based interaction for assembly tasks in virtual reality

This paper investigates the benefits of haptics-based interaction for performing assembly-related tasks in a virtual environment. The research examines the context in which haptic feedback affects user performance and identifies assembly operations that are influenced. Forty participants completed three experiments relevant to virtual assembly: weight discrimination, part positioning, and manual assembly. Each experiment featured a series of trials generated from factorial combinations of performance variables. Subjects were assessed based on task completion time and different measures of accuracy. When compared to visual-only methods, quantitative results show that haptics-based interaction is beneficial in improving performance by reducing completion times for weight discrimination, permits higher placement accuracy when positioning virtual objects, and enables steadier hand motions along three-dimensional trajectories. The results also indicate that user accuracy in weight discrimination is dependent on hand dominance when manipulating the virtual object combined with the provided sensory information.

Testing Subhypotheses in the Multiplicative Interaction Model

The problem of analyzing a two-way cross-classified treatment structure with only one observation per treatment combination is considered. A test procedure is given that will enable the data analyst to determine subareas of the data in which the data are additive. The procedure is developed by assuming that a multiplicative interaction model adequately fits the data. Such a model is given by y ij , = μ + τ i + β j + λα i γ j + ∊ ij ; where i = 1, 2, …, t and j = 1, 2, …, b. It is assumed that the ∊ ij , are distributed independently and normally with mean zero and variance σ2. The other parameters are assumed to be unknown constants. In general, the problem may be stated as one of testing H 0,: H α = 0 versus H a ,: H α ≠ 0, where α = (α1, …, α t )′ and H is a q × t contrast matrix. A likelihood ratio statistic for this testing problem is derived and approximate critical points are given for the cases q = 1 and q = 2. The procedures are illustrated with an example.

Analysis of Nonadditive Multiway Classifications

Abstract This article considers the problems of testing additivity and estimating σ2 in unreplicated multiway classifications. To model nonadditivity and jointly estimate σ2, the interaction parameter space must be restricted; otherwise the model is saturated. The parameterization we use is a multiway extension of the two-way multiplicative interaction model of Mandel (1971) and Johnson and Graybill (1972a). For example, in a three-way classification, we model interaction as θijk = λδ 1i δ2j δ3k . This structure is a special case of the k-mode principal components model, which has received considerable attention in the psychometric literature (Kapteyn, Neudecker, and Wansbeek 1986). We construct an exact test of λ = 0 and propose an estimator of σ2 that can be used when interaction has been detected. Our test is an approximation to the likelihood ratio test (LRT) of Ho : λ = 0. The proposed test has essentially the same power as the LRT but is easier to compute, and the exact null distribution of the test...

A three-degree of freedom test of additivity in three-way classifications

Abstract This article proposes a new interaction model for nonreplicated three-way classifications. A simulation study is used to show that a three-degree of freedom score test based on the new model compares favorably with existing one-degree of freedom score and likelihood ratio tests of additivity. The tests are illustrated through an analysis of a data set where it is shown how the new model may reveal a specific structure of three-factor interaction. This structure may be exploited to suggest possible explanations for the nonadditivity.

Estimation of σ2 in the multiplicative interaction model

Suppose that the model is found to adequately fit two-way cross-classification data with one observation per cell. Johnson and Graybill (1972) proposed an estimator of the error variance σ2. A test procedure was given by Marasinghe and Johnson (1981) which enables one to find rows (or columns) in the data which may be additive. In this paper an improved estimator of σ2 is proposed by making use of his additional information.

Dynamic Graphical Tools for Teaching Experimental Design and Analysis Concepts

In 1996, computer-based modules were presented that use interactive graphics and simulation to teach basic statistical concepts for undergraduate courses. This article extends that work to develop new modules that can be used to demonstrate important statistical concepts underlying the design and analysis of experiments. Details of their implementation, content and usage are discussed.