A. S. Hedayat

Statistical methods in assessing agreement: Models, issues, and tools

Measurements of agreement are needed to assess the acceptability of a new or generic process, methodology, and formulation in areas of laboratory performance, instrument or assay validation, method comparisons, statistical process control, goodness of fit, and individual bioequivalence. In all of these areas, one needs measurements that capture a large proportion of data that are within a meaningful boundary from target values. Target values can be considered random (measured with error) or fixed (known), depending on the situation. Various meaningful measures to cope with such diverse and complex situations have become available only in the last decade. These measures often assume that the target values are random. This article reviews the literature and presents methodologies in terms of “coverage probability.” In addition, analytical expressions are introduced for all of the aforementioned measurements when the target values are fixed and when the error structure is homogenous or heterogeneous (proportional to target values). This article compares the asymptotic power of accepting the agreement across all competing methods and discusses the pros and cons of each. Data when the target values are random or fixed are used for illustration. A SAS macro program to compute all of the proposed methods is available for download at http://www.uic.edu/~hedayat/.

A Unified Approach for Assessing Agreement for Continuous and Categorical Data

This paper proposes several Concordance Correlation Coefficient (CCC) indices to measure the agreement among k raters, with each rater having multiple (m) readings from each of the n subjects for continuous and categorical data. In addition, for normal data, this paper also proposes the coverage probability (CP) and total deviation index (TDI). Those indices are used to measure intra, inter and total agreement among all raters. Intra-rater indices are used to measure the agreement among the multiple readings from the same rater. Inter-rater indices are used to measure the agreement among different raters based on the average of multiple readings. Total-rater indices are used to measure the agreement among different raters based on individual readings. In addition to the agreement, the paper also assess intra, inter, and total precision and accuracy. Through a two-way mixed model, all CCC, precision and accuracy, TDI, and CP indices are expressed as functions of variance components, and GEE method is used to obtain the estimates and perform inferences for all the functions of variance components. Each of previous proposed approaches for assessing agreement becomes one of the special case of the proposed approach. For continuous data, when m approaches ∞, the proposed estimates reduce to the agreement indices proposed by Barnhart et al. (2005). When m = 1, the proposed estimate reduces to the ICC proposed by Carrasco and Jover (2003). When m = 1, the proposed estimate also reduces to the OCCC proposed by Lin (1989), King and Chinchilli (2001a) and Barnhart et al. (2002). When m = 1 and k = 2, the proposed estimate reduces to the original CCC proposed by Lin (1989). For categorical data, when k = 2 and m = 1, the proposed estimate and its associated inference reduce to the kappa for binary data and weighted kappa with squared weight for ordinal data.

Sampling plans excluding contiguous units

We consider fixed size sampling plans for which the second order inclusion probabilities are zero for pairs of contiguous units and constant for pairs of non-contiguous units. A practical motivation for the use of such plans is pointed out and a statistical condition is identified under which these plans are more efficient than the corresponding simple random sampling plans. Results on the existence and construction of these plans are obtained.

2n-m designs with resolution III or IV containing clear two-factor interactions

For a fixed number of runs, not all 2n−m designs with resolution III or IV have clear two-factor interactions. Therefore, it is highly desirable to know when resolution III or IV designs can have clear two-factor interactions. In this paper, we provide a unified geometrical study of this problem and give a complete classification of the existence of clear two-factor interactions in regular 2n−m designs with resolution III or IV and reveal the structures of these designs.

A complete solution to the existence and nonexistence of Knut Vik designs and orthogonal Knut Vik designs

Abstract Hedayat and Federer ( Ann. of Statist. 3 (1975), 445–447) proved that Knut Vik designs do not exist for all even orders. They gave a simple algorithm for the construction of such designs for all other orders, except when the order of the design is divisible by 3. The existence of Knut Vik designs of orders divisible by 3 was left unsolved by these authors. It is shown here that Knut Vik designs do not also exist for all orders divisible by 3. An easy algorithm based on a result of Euler is provided for the construction of orthogonal Knut Vik designs for all orders not divisible by 2 or 3. Therefore, we can say that Knut Vik designs and orthogonal Knut Vik designs of order n exist if and only if n is not divisible by 2 or 3. The results are based on the concepts of a super diagonal and parallel super diagonals in an n × n array, which have been introduced and studied for the first time here. Other relevant results are also given.

Statistical tools for measuring agreement

Introduction. - Basic approach for paired continuous data when target values are random or fixed. - Sample size and power. - Unified approach for continuous and categorical data.

Pairwise and variance balanced incomplete block designs

SummaryThe purpose of this paper is three-fold. The first purpose is to compile and to systematize published and dispersed results on two aspects of balancing in incomplete block designs, i.e., pairwise balance and variance balance. This was done in order to establish the status of these two concepts of balance in published literature and to put them in a form which is useful for further work in this area. Also, the results in this form are necessary for the development of the remainder of the paper.The second purpose of this paper is to present a method of constructing unequal replicate and/or unequal block size experiment designs for which the variance balance property is achieved. The method of construction involves the union of blocks from two or more block designs and the augmentation of some of the blocks with additional treatments; the method is denoted asunionizing block designs. A straight-forward extension of the method would produce a partially balanced block design with unequal replicate and/or unequal block designs. The enlargement of the concept and availability of variance blanced block designs to accommodate unequal replication and/or unequal block sizes is important to the researcher, the teacher, and the experimenter needing such designs. For example, an animal nutritionist or a psychologist is no longer required to have constant litter or family sizes for the blocks and may have unequal replication on the treatments for those treatments with insufficient material and still attain the goal of equal variances on all normalized treatment contracts.The third purpose of the paper is to apply the unionizing block designs method to construct a family of unequal replicate and unequal block size variance balanced designs. Some comments are given on the extension of the unionizing block designs method to construct other families of variance balanced or partially balanced block designs.

A-Optimal Incomplete Block Designs for Control-Test Treatment Comparisons

A-optimal designs for comparing v test treatments with a control in b blocks of size k each are considered. Several series of A-optimal designs are given when the parameters are in the range 2 ≤ k ≤ 8, k ≤ v ≤ 30, v ≤ b ≤ 50. A-optimal designs in blocks of size 2 are extensively studied through a combination of theoretical results and numerical investigations. Tables of approximately A-optimal designs are given when A-optimal designs are not easily available for the case k = 2.

Repeated measurements designs for a model with self and simple mixed carryover effects

Abstract We study the problem of design and analysis of two-period repeated measurements (crossover or changeover) designs based on two or more treatments. In our model, we allow the appearance of two types of carryovers (residuals) in the observations collected in the second period. It is assumed that each treatment has three types of effects: direct effect, self-carryover effect, and simple mixed carryover effect. The direct effect will manifest itself no matter where and when the treatment is applied. Carryover effects will only appear in the second period. The nature and the magnitude of this carryover effect from a treatment in the first period which will appear in the second period depends on the treatment into which it carries this carryover effect. It is called a self-carryover effect if it carries to itself. Otherwise, it is called a simple mixed carryover effect. It is proved that if the study is properly designed, we can efficiently estimate any contrast in direct treatment effects even if there are self and simple mixed carryover effects in the data. Further, it is shown that the most efficient way of constructing a two-period repeated measurements design is equivalent to constructing an efficient block design in a special class of block designs based on t (⩾2) treatments utilizing in total a fixed number of experimental units. In addition, if we are interested in contrasts involving self-carryover effects only, then it is easy to design the study so that we can estimate these contrasts. The problem of identifying small designs that allow the unbiased estimation of independent contrasts in simple mixed carryover effects is addressed.

Modeling and Identifying Optimum Designs for Fitting Dose-Response Curves Based on Raw Optical Density Data

Abstract For raw optical density (ROD) data, such as those generated in biological assays using a 96-well plate reader, D-optimal designs are identified for a family of homogeneous nonlinear models with two parameters developed for fitting and studying dose-response curves. Further, within the class of k-point equally spaced and uniform designs, D-optimal designs are specified. Also, the D-efficiency of the D-optimal equally spaced and uniform k-point designs are compared to D-optimal design for 3 ≤ k ≤ 7. The efficiency robustness of these optimal designs with respect to initial nominal values of the parameters are investigated. In addition, D-optimal designs are derived for a family of heterogeneous nonlinear models.