Critical values of some nonparametric statistics used for tests of total stochastic independence

The problem of testing for total independence of the variates of a stochastic p(≧3) component vector using rank correlation statistics is considered. Two distribution free statistics are considered, one based on the determinant of the matrix of rank correlation statistics, the second on their sum of squares. Tables of critical values are given for p=3,4 for the cases when (a) ranks, and (b) exponential scores are used to replace the ordered observations within each variate. Some approximations to the critical values are proposed and evaluated.     

The Gaussian rank correlation estimator: robustness properties

The Gaussian rank correlation equals the usual correlation coefficient computed from the normal scores of the data. Although its influence function is unbounded, it still has attractive robustness properties. In particular, its breakdown point is above 12%. Moreover, the estimator is consistent and asymptotically efficient at the normal distribution. The correlation matrix obtained from pairwise Gaussian rank correlations is always positive semidefinite, and very easy to compute, also in high dimensions. We compare the properties of the Gaussian rank correlation with the popular Kendall and Spearman correlation measures. A simulation study confirms the good efficiency and robustness properties of the Gaussian rank correlation. In the empirical application, we show how it can be used for multivariate outlier detection based on robust principal component analysis.     

Rank tests for the k-sample problem with restricted alternatives

An asymptotically maximin most powerful rank test among somewhere asymptotically most powerful linear rank tests with scores generating function cf> is derived for each of the simple order alternative, the simple loop alternative and the simple tree alternative in the k-sample problem. The comparisons of the tests obtained with the rank analogues of the Bartholomew's xv tests are made in terms of local asymptotic relative efficiency. It is found that our tests are better than the rank analogues of the xk tests. Furthermore, the asymptotic equivalence of the ranking by the pooled sample to the ranking in pairs are discuss¬ed and the tests which are asymptotically equivalent to ours are given.     

A Variable Selection Criterion for Two Sets of Principal Component Scores in Principal Canonical Correlation Analysis

Canonical correlation analysis (CCA) is often used to analyze the correlation between two random vectors. However, sometimes interpretation of CCA results may be hard. In an attempt to address these difficulties, principal canonical correlation analysis (PCCA) was proposed. PCCA is CCA between two sets of principal component (PC) scores. We consider the problem of selecting useful PC scores in CCA. A variable selection criterion for one set of PC scores has been proposed by Ogura (2010), here, we propose a variable selection criterion for two sets of PC scores in PCCA. Furthermore, we demonstrate the effectiveness of this criterion.     

On multivariate linear regression shrinkage and reduced-rank procedures

An alternate derivation of the canonical analysis shrinkage prediction procedure of Breiman and Friedman (1997. J. Roy. Statist. Soc. B 59, 3–54) is presented for the multivariate linear model. It is based on consideration of prediction mean square error matrix, and bias of the squared sample canonical correlations. A modified procedure involving partial canonical correlation analysis is also introduced and discussed.     

Testing linear inequality constraints in the standard linear model

In this paper we are concerned with the problem of testing whether the â-parameters of the standard linear model satisfy the linear equality constraints R = r when they are known to satisfy the corresponding linear inequality constraints Râ ? r. In particular we will show that the exact finite sample null distributions of the Likelihood Ratio, Wald and Kuhn-Tucker

statistics are known when R is of full row rank but not known when R has less than full row rank. The less than full row rank problem has not been discussed previously but it is of considerable potential importance.

This paper contains several simple numerical examples which illustrate the computational details of the tests     

Rank correlation methods for missing data

The subject of rank correlation has had a rich history. It has been used in numerous applications in tests for trend and for independence. However, little has been said about how to define rank correlation when the data are incomplete. The practice has often been to ignore missing observations and to define rank correlation for the smaller complete record. We propose a new class of measures of rank correlation which are based on a notion of distance between incomplete rankings. There is the potential for a significant increase in efficiency over the approach which ignores missing observations as demonstrated by a specific case.     

An RKHS framework for functional data analysis

Linear combinations of random variables play a crucial role in multivariate analysis. Two extension of this concept are considered for functional data and shown to coincide using the Loève–Parzen reproducing kernel Hilbert space representation of a stochastic process. This theory is then used to provide an extension of the multivariate concept of canonical correlation. A solution to the regression problem of best linear unbiased prediction is obtained from this abstract canonical correlation formulation. The classical identities of Lawley and Rao that lead to canonical factor analysis are also generalized to the functional data setting. Finally, the relationship between Fisher's linear discriminant analysis and canonical correlation analysis for random vectors is extended to include situations with function-valued random elements. This allows for classification using the canonical Y scores and related distance measures.     

A bayesian wls approach to generalized linear models

We use a Bayesian approach to fitting a linear regression model to transformations of the natural parameter for the exponential class of distributions. The usual Bayesian approach is to assume that a linear model exactly describes the relationship among the natural parameters. We assume only that a linear model is approximately in force. We approximate the theta-links by using a linear model obtained by minimizing the posterior expectation of a loss function.While some posterior results can be obtained analytically considerable generality follows from an exact Monte Carlo method for obtaining random samples of parameter values or functions of parameter values from their respective posterior distributions. The approach that is presented is justified for small samples, requires only one-dimensional numerical integrations, and allows for the use of regression matrices with less than full column rank. Two numerical examples are provided.     

Designing cancer immunotherapy trials with delayed treatment effect using maximin efficiency robust statistics

The indirect mechanism of action of immunotherapy causes a delayed treatment effect, producing delayed separation of survival curves between the treatment groups, and violates the proportional hazards assumption. Therefore using the log‐rank test in immunotherapy trial design could result in a severe loss efficiency. Although few statistical methods are available for immunotherapy trial design that incorporates a delayed treatment effect, recently, Ye and Yu proposed the use of a maximin efficiency robust test (MERT) for the trial design. The MERT is a weighted log‐rank test that puts less weight on early events and full weight after the delayed period. However, the weight function of the MERT involves an unknown function that has to be estimated from historical data. Here, for simplicity, we propose the use of an approximated maximin test, the V₀ test, which is the sum of the log‐rank test for the full data set and the log‐rank test for the data beyond the lag time point. The V₀ test fully uses the trial data and is more efficient than the log‐rank test when lag exits with relatively little efficiency loss when no lag exists. The sample size formula for the V₀ test is derived. Simulations are conducted to compare the performance of the V₀ test to the existing tests. A real trial is used to illustrate cancer immunotherapy trial design with delayed treatment effect.     

Sample Size Calculation for Rank Tests Comparing K Survival Distributions

Rank tests, such as logrank or Wilcoxon rank sum tests, have been popularly used to compare survival distributions of two or more groups in the presence of right censoring. However, there has been little research on sample size calculation methods for rank tests to compare more than two groups. An existing method is based on a crude approximation, which tends to underestimate sample size, i.e., the calculated sample size has lower power than projected. In this paper we propose an asymptotically correct method and an approximate method for sample size calculation. The proposed methods are compared to other methods through simulation studies.     

Rank estimation of regression coefficients using iterated reweighted least squares

This paper is concerned with the rank estimator for the parameter vector β in a linear model which is obtained by the minimization of a rank dispersion function. The rank estimator has many advantages over the regular least squares estimator, but the inaccessibility of software to carry out its computation has limited its use. An iterated reweighted least squares algorithm is presented for the computation of the rank estimator. The method is simple in concept and can be carried out readily with a wide variety of statistical software. Details of the method are discussed along with some results on its asymptotic distribution and numerical stability. Some examples are presented to show advantages of the rank method.     

A Monte Carlo comparison of alternative methods of maximum likelihood ranking in racing sports

Applications of maximum likelihood techniques to rank competitors in sports are commonly based on the assumption that each competitor's performance is a function of a deterministic component that represents inherent ability and a stochastic component that the competitor has limited control over. Perhaps based on an appeal to the central limit theorem, the stochastic component of performance has often been assumed to be a normal random variable. However, in the context of a racing sport, this assumption is problematic because the resulting model is the computationally difficult rank-ordered probit. Although a rank-ordered logit is a viable alternative, a Thurstonian paired-comparison model could also be applied. The purpose of this analysis was to compare the performance of the rank-ordered logit and Thurstonian paired-comparison models given the objective of ranking competitors based on ability. Monte Carlo simulations were used to generate race results based on a known ranking of competitors, assign rankings from the results of the two models, and judge performance based on Spearman's rank correlation coefficient. Results suggest that in many applications, a Thurstonian model can outperform a rank-ordered logit if each competitor's performance is normally distributed.     

Likelihood Function and Canonical Correlation Analysis of the Peña–Box Model

The Peña–Box model is a type of dynamic factor model whose factors try to capture the time-effect movements of a multiple time series. The Peña–Box model can be expressed as a vector autoregressive (VAR) model with constraints. This article derives the maximum likelihood estimates and the likelihood ratio test of the VAR model for Gaussian processes. Then a test statistic constructed by canonical correlation coefficients is presented and adjusted for conditional heteroscedasticity. Simulations confirm the validity of adjustments for conditional heteroscedasticity, and show that the proposed statistics perform better than the statistics used in the existing literature.     

The generalised inverse form of canonical correlation

The generalised inverse equations for the canonical correlation problem are derived using a minimum distance principle which-is shown to be equivalent to the usual maximum correlation formulation.     

A test for independence in a multivariate exponential distribution with equal correlation coefficients

A rank statistic is considered which may be used for testing for total independence in a p-variate exponential distribution with equal correlation coefficients. Critical values for the statistic are provided for p = 3.4 and sample sizes less than or equal to 20. Finally, the small sample power performance of the rank test relative to that of the locally most powerful similar lest under the exponential alternative is evaluated.     

Test procedures for the analysis of experimental designs with correlated nonnormal data

Using Monte Carlo methods, an examination is made of two statistical methods used for hypothesis testing in a general factorial model with a known correlation structure General correlation structures are given in Smith and Lewis (1980) and Pavur and Lewis (1982) which allow the usual F statistic to be corrected by a constant. The corrected F statistic would be the usual F statistic multiplied by a correction constant. A comparison is made between this corrected f statistic and the rank transform F statistic presented by Conover and Iman (1976). When the usual F statistic and the rank transform statistic are corrected for correlationt this simulation study shows that these statistical tests behave well under a variety of situations when not all f the usual assumptions of an ANOVA are satisfied.     

Enriched biplots for canonical correlation analysis

This paper discusses biplots of the between-set correlation matrix obtained by canonical correlation analysis. It is shown that these biplots can be enriched with the representation of the cases of the original data matrices. A representation of the cases that is optimal in the generalized least squares sense is obtained by the superposition of a scatterplot of the canonical variates on the biplot of the between-set correlation matrix. Goodness of fit statistics for all correlation and data matrices involved in canonical correlation analysis are discussed. It is shown that adequacy and redundancy coefficients are in fact statistics that express the goodness of fit of the original data matrices in the biplot. The within-set correlation matrix that is represented in standard coordinates always has a better goodness of fit than the within-set correlation matrix that is represented in principal coordinates. Given certain scalings, the scalar products between variable vectors approximate correlations better than the cosines of angles between variable vectors. Several data sets are used to illustrate the results.     

CANONICAL VARIATE ANALYSIS WITH SPATIALLY-CORRELATED DATA

A general canonical variate model is derived when the observations are spatially correlated. For spatial covariance structures resulting from dependence of a pixel on its nearest neighbours, the solution reduces to an analysis of neighbour-corrected values. The usual analysis, in which spatial correlation is ignored, gives similar canonical vectors but over-estimates the canonical roots. A formula for approximating the reduction in the canonical roots to adjust for the spatial correlation is given.