Ronald L. Iman

Rank Transformations as a Bridge between Parametric and Nonparametric Statistics

Abstract Many of the more useful and powerful nonparametric procedures may be presented in a unified manner by treating them as rank transformation procedures. Rank transformation procedures are ones in which the usual parametric procedure is applied to the ranks of the data instead of to the data themselves. This technique should be viewed as a useful tool for developing nonparametric procedures to solve new problems.

A distribution-free approach to inducing rank correlation among input variables

A method for inducing a desired rank correlation matrix on a multivariate input random variable for use in a simulation study is introduced in this paper. This method is simple to use, is distribution free, preserves the exact form of the marginal distributions on the input variables, and may be used with any type of sampling scheme for which correlation of input variables is a meaningful concept. A Monte Carlo study provides an estimate of the bias and variability associated with the method. Input variables used in a model for study of geologic disposal of radioactive waste provide an example of the usefulness of this procedure. A textbook example shows how the output may be affected by the method presented in this paper.

Small sample sensitivity analysis techniques for computer models.with an application to risk assessment

As modeling efforts expand to a broader spectrum of areas the amount of computer time required to exercise the corresponding computer codes has become quite costly. This costly process can be directly tied to the complexity of the modeling, which makes the relationships among the input variables not mathematically tractable. In this setting it is desired to perform sensitivity studies of the input-output relationships. A variety of situations require that decisions and judgments be made in the face of uncertainty, such as lack of knowledge about probability distributions associated with input variables, different hypothesized future conditions, or different strategies associated with a decision making process. In this paper a generalization of Latin hypercube sampling is given that allows these areas to be investigated without making additional computer runs.

Approximations of the critical region of the fbietkan statistic

The Friedman (1937) test for the randomized complete block design is used to test the hypothesis of no treatment effect among k treatments with b blocks. Difficulty in determination of the size of the critical region for this hypothesis is com¬pounded by the facts that (1) the most recent extension of exact tables for the distribution of the test statistic by Odeh (1977) go up only to the case with k6 and b6, and (2) the usual chi-square approximation is grossly inaccurate for most commonly used combinations of (k,b). The purpose of this paper 2 is to compare two new approximations with the usual x2 and F large sample approximations. This work represents an extension to the two-way layout of work done earlier by the authors for the one-way Kruskal-Wallis test statistic.

The Use of the Rank Transform in Regression

The rank transform is a simple procedure which involves replacing the data with their corresponding ranks. The rank transform has previously been shown by the authors to be useful in hypothesis testing with respect to experimental designs. This study shows the results of using the rank transform in regression. Two sets of data given by Daniel and Wood [8] are considered for purposes of illustrating the rank transform in simple and multiple regression. Also given are the results of a Monte Carlo study which compares regression on ranks with some published Monte Carlo results on isotonic regression. This Monte Carlo study is also modified to compare regression on ranks with robust regression. Another illustration gives the results of analyses on large computer codes by regression on ranks. The rank transform is a simple, repeatable process that compares favorably with other methods such as given by Andrews [1]. Our studies indicate the method works quite well on monotonic data.

On some alternative procedures using ranks for the analysis of experimental designs

The analysis of data from eseperisental designs is often hampered by the lack of more than one procedure available for the analysis, especially when that procedure is based on assumptions which do not apply in the situation at hand. In this paper tvo classes of alternative procedures are discussed and compared, One is the aligned ranks procedure which first standardises the data by subtracting an appropriate estimate of location, then replaces the data with ranks t and finally uses an appropriate test statistic which has asymptotically a chi-square distribution The second procedure is the rank transform which first replaces all of the data with the ranks, and then employs the usual parametric methods, but computed on the ranks instead of the data Some Monte Carlo simulations for a test of interaction in a two way layout with replication enable the robustness and pover of these tvo methods to be compared with the usual analysis of variancs.

A measure of top-down correlation

Many situations exist in which n objects are ranked by two or more independent sources, where interest centers primarily on agreement in the top rankings and disagreements on items at the bottom of the rankings are of little or no importance. A problem with Spearmans rho or Kendalls coefftcient of concordance in this setting is that they are equally influenced by disagreement on the assignment of rankings at all levels. In this article, a concordance measure is provided that is more sensitive to agreement on the top rankings. The statistics used in this setting are functions of the ordinary correlation coeRicient computed on Savage (1956) scores. The asymptotic distributions of these statistics are presented, and a summary of the quantiles of the exact distribution for the two sample case are provided for n = 3(1)14. The statistic for the two-sample case is shown to provide a locally most powerful rank test for a model given by Hajek and Sidak (1967).

Expert opinion in risk analysis; The NUREG-1150 methodology

The Reactor Risk Reference Document (US Nuclear Regulatory Commission, 1987) is the most comprehensive study and application of probabilistic risk analysis and uncertainty analysis methods for nuclear power generation safety since the Reactor Safety Study (US Nuclear Regulatory Commission, 1975). Many of the issues addressed in PRA work such as NUREG-1150 involve phenomena that have not been studied through experiment or observation to an extent that makes possible a definitive analysis. In many instances, the rarity or severity of the phenomena make resolution impossible at this time. In these instances, the best available information resides with experts who have studied the phenomena in question. This paper is about a reasoned approach to the acquisition of expert opinion for use in PRA work and other public policy areas.

The rank transformation as a method of discrimination with some examples

The procedure of statistical discrimination Is simple in theory but so simple in practice. An observation x0possibly uiultivariate, is to be classified into one of several populations π1,…,πk which have respectively, the density functions f1(x), • • • , fk(x). The decision procedure is to evaluate each density function at X0 to see which function gives the largest value fi(X0) , and then to declare that X0 belongs to the population corresponding to the largest value. If these den-sities can be assumed to be normal with equal covariance matricesthen the decision procedure is known as Fisher’s linear discrimi-nant function (LDF) method. In the case of unequal covariance matrices the procedure is called the quadratic discriminant func-tion (QDF) method. If the densities cannot be assumed to be nor-mal then the LDF and QDF might not perform well. Several different procedures have appeared in the literature which offer discriminant procedures for nonnormal data. However, these pro-cedures are generally difficu...

Comparison of Asymptotically Distribution-Free Procedures for the Analysis of Complete Blocks

Abstract The Friedman test (or sign test when k = 2) depends entirely on within-block rankings. In a recent paper, Quade (1979) attempted to provide a test with more power than the Friedman test by considering a k-sample extension of the Wilcoxon signed ranks test. This is done by taking advantage of the between-block information. A third way to approach the problem and still retain both the within- and between-block information is first to transform all the observations to ranks from 1 to bk (b blocks and k treatments) and then to apply the parametric F test to the ranks. This approach is shown to be asymptotically distribution-free under suitable conditions. Computer simulation results indicate that this procedure is both robust and powerful for small sample sizes.