Ralph A. Bradley

Bio: Ralph A. Bradley is an academic researcher from University of Georgia. The author has contributed to research in topics: Block design & Multivariate statistics. The author has an hindex of 24, co-authored 65 publications receiving 4368 citations. Previous affiliations of Ralph A. Bradley include Florida State University.

Rank-Sum Tests for Dispersions

Abstract: This paper deals with non-parametric two-sample tests on dispersions. Two samples, $X$- and $Y$-samples of $m$ and $n$ independent observations from populations with continuous cumulative distribution functions $F(u)$ and $G(u)$ respectively, are considered. It is required for the basic test that the difference in locations (medians) of the two populations be known and, when this is so, the two samples may be adjusted to have equal locations. Taking these location parameters to be zero without loss of generality, we test the hypothesis that $G(u) \equiv F(u)$ against alternatives of the form $G(u) \equiv F(\theta u), \theta eq 1$. The two samples are ordered in a single joint array and ranks are assigned from each end of the joint array towards the middle. The statistic used is $W$, the sum of ranks for the $X$-sample. The distribution of $W$ is studied and tables of significant values of $W$ are provided for $m + n \leqq 20$ and both upper- and lower-tail significance levels .005, .01, .025 and .05. The first four moments of $W$ are developed and a normal approximation to the null distribution of $W$ is devised. Large-sample properties of the $W$-test are considered. A proof of limiting normality is based on a theorem of Chernoff and Savage. Consistency of the $W$-test is indicated and its relative efficiency in comparison with the variance-ratio $F$-test is obtained as $6/\pi^2$ when $F(u)$ is the normal distribution function. Other non-parametric tests of dispersions are reviewed. The $W$-test is less efficient asymptotically than some of these other tests but is easier to apply, particularly with the tables provided. A modified test is suggested for the case where the difference in population locations is not known. This involves replacing the two original samples by two corresponding samples of deviations from sample medians. The procedure of the $W$-test is applied to the two samples of deviations. The properties of the modified test have not been investigated except for a sampling study of rather limited scope. That study indicates that the moments of $W$ for the modified test are not greatly different from those under the basic procedure.

The asymptotic properties of ML estimators when sampling from associated populations

Abstract: The method of maximum likelihood was proposed as a general method of estimation by Fisher (1912). However, rigorous proofs of the asymptotic properties of the maximum likelihood estimators (hereafter called ML estimators) were not developed until the work of Dugue (1946), Wald (1943, 1949), Cramer (1946), Huzurbazar (1948) and Chanda (1954), all of whom published within the last 25 years. Each of these authors assumed observations from a single population. Neyman & Scott (1948) and Kraft & LeCam (1956) have pointed out situations wherein the ML estimator is inefficient and not consistent. Their examples involved sampling from a number of subpopulations with different distributions but with only a finite number of observations from each of the subpopulations. Some situations arise where observations do not come from a single population but from distinct but related populations, related in the sense that some populations reasonably may be assumed to have some parameters in common. We shall call such populations associated. We consider two cases of associated populations. In the first, a finite number of populations are considered and limiting results regarding ML estimators are obtained as numbers of observations from the distinct populations become large in constant ratios. In the second, one observation is obtained from each population and the number of populations sampled becomes large. The second case may include the first as some populations sampled are identical but for applications to investigations of specific ML estimators it seems better to consider the two cases separately. In this paper we extend the basic theory on ML estimation to associated populations. Under regularity conditions developed, the ML estimators for parameters in associated populations are shown to be consistent and asymptotically normal with a variance-covariance matrix as derived. The asymptotic X2-distribution of -2 ln A, A being the likelihood ratio, holds and, for the non-null distributions, the parameter of non-centrality is set forth. Our objective is to set forth results as simply as possible and in usable forms; more generalized theorems could be obtained. We limit consideration to ML estimators that are roots of the normal equations for the maximization process. The consistency of the more generally defined estimator can be proved for associated populations following the approach of Wald (1943). Our demonstrations follow those of Chanda closely; most of the results for associated populations are easy generalizations of results for a single population and we state results without complete proofs when this is so. Two simple examples involving associated populations are included. One of these applies to the work of Dykstra (1960) which is dependent upon results given here.

Science, statistics, and paired comparisons.

Abstract: A prologue on science and statistics focuses attention on the role of statistics in science. Statisticians must be trained as scientists and to meet the needs of science. Those needs surely involve the formulation, modification and verification of stochastic models designed to represent natural phenomena. The method of paired comparisons provides a simple experimental technique but one with a literature rich in model development.

Regression Models for Ordinal Data

Abstract: SUMMARY A general class of regression models for ordinal data is developed and discussed. These models utilize the ordinal nature of the data by describing various modes of stochastic ordering and this eliminates the need for assigning scores or otherwise assuming cardinality instead of ordinality. Two models in particular, the proportional odds and the proportional hazards models are likely to be most useful in practice because of the simplicity of their interpretation. These linear models are shown to be multivariate extensions of generalized linear models. Extensions to non-linear models are discussed and it is shown that even here the method of iteratively reweighted least squares converges to the maximum likelihood estimate, a property which greatly simplifies the necessary computation. Applications are discussed with the aid of examples.

Learning to rank using gradient descent

Abstract: We investigate using gradient descent methods for learning ranking functions; we propose a simple probabilistic cost function, and we introduce RankNet, an implementation of these ideas using a neural network to model the underlying ranking function. We present test results on toy data and on data from a commercial internet search engine.

Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems

Abstract: Recent developments promise to increase greatly the popularity of maximum likelihood (ml) as a technique for estimating variance components. Patterson and Thompson (1971) proposed a restricted maximum likelihood (reml) approach which takes into account the loss in degrees of freedom resulting from estimating fixed effects. Miller (1973) developed a satisfactory asymptotic theory for ml estimators of variance components. There are many iterative algorithms that can be considered for computing the ml or reml estimates. The computations on each iteration of these algorithms are those associated with computing estimates of fixed and random effects for given values of the variance components.

Consequences of Failure to Meet Assumptions Underlying the Fixed Effects Analyses of Variance and Covariance

Abstract: The effects of violating the assumptions underlying the fixed-effects analyses of variance (ANOVA) and covariance (ANCOVA) on Type-I and Type-II error rates have been of great concern to researchers and statisticians. The major effects of violation of assumptions are now well known, after nearly four decades of research. Early summaries and reviews by Hey (1938), Garret and Zubin (1943), Grant (1944), and Gourlay (1955) and more recent reviews by Bradley (1963), Atiqullah (1967), Elashoff (1969) and Scheffe (1959, Ch. 10) can be extended and updated with recent research which provides closure to an area of active inquiry. (For a review of the effects of violation of the assumptions of the randomeffects ANOVA-a subject not reviewed here-the reader is directed to Scheffe, 1959, pp. 334-337 and Box & Anderson, 1962.) Asking whether ANOVA and ANCOVA assumptions are satisfied is not idle curiosity. The assumptions of most mathematical models are always false to a greater or lesser extent. The relevant question is not whether ANOVA assumptions are met exactly, but rather whether the plausible violations of the assumptions have serious consequences on the validity of probability statements based on the standard assumptions. Applied statistics in education and the social sciences experienced a largely unnecessary hegira to non-parametric statistics during the 1950s. Increasingly during the 1950s and early 1960s the fixed-effects, normal theory ANOVA was replaced by such comparable nonparametric techniques as the Wilcoxon test, Mann-Whitney U-test, Kruskal-Wallis one-way ANOVA, and the Friedman two-way ANOVA for ranks (Siegel, 1956). The flight to non-parametrics was unnecessary principally because researchers asked "Are normal theory ANOVA assumptions met?" instead of "How important are the inevitable violations of normal theory ANOVA assumptions?"