An investigation of statistical matching

This Paper examines the file-merging technique of statistical matching. Details are given of both its practical application and its numerical accuracy. In particular, it is shown that the ability of statistical matching to recreate correlations between variables existing only in different files is dependent on some respective assumptions which are unlikely to be satisfied in practice. These theoretical properties are illustrated by results from a simulated match where the estimates of correlations, as well as other measures of association, are found to be grossly inaccurate.     

A new measure of association for bivariate survival data

Time dependent association measures between variables are of interest in bivariate survival data. Several such measures have been proposed in literature for the modelling and analysis of survival data. In this paper, we introduce a new measure of association for bivariate survival data using product moment residual life function and mean residual life function. Various properties of the proposed measure and its relationship with existing measures are discussed. We also develop a non-parametric estimator of the measure and study its asymptotic properties. The application of the result is illustrated using a real life data. Finally, a stimulation study is carried out to assess the performance of the estimator.     

Measuring association between nominal categorical variables: an alternative to the Goodman–Kruskal lambda

As a measure of association between two nominal categorical variables, the lambda coefficient or Goodman–Kruskal's lambda has become a most popular measure. Its popularity is primarily due to its simple and meaningful definition and interpretation in terms of the proportional reduction in error when predicting a random observation's category for one variable given (versus not knowing) its category for the other variable. It is an asymmetric measure, although a symmetric version is available. The lambda coefficient does, however, have a widely recognized limitation: it can equal zero even when there is no independence between the variables and when all other measures take on positive values. In order to mitigate this problem, an alternative lambda coefficient is introduced in this paper as a slight modification of the Goodman–Kruskal lambda. The properties of the new measure are discussed and a symmetric form is introduced. A statistical inference procedure is developed and a numerical example is provided.     

Aspects of two group concordance

A definition of concordance is derived from Rao's concept of a perfect diversity measure in order to identify aspects about which two populations of judges agree. In the case where each judge independently ranks a fixed set of items, the class of concordance measures based on the marginal distributions of the ranks is characterized by bi-affine similarity functions that measure how well pairs of judges tend to agree. This class contains population versions of several familiar indices of concordance, including Kendall's W. Concordance between two populations, refered to as intergroup concordance, is also scaled against its corresponding intragroup measures. Small sample properties of estimators of the ratio of inter-to-intra group concordance are investigated in a Monte Carlo study. An example is given to illustrate components of concordance attributable to subsets of items, and to contrast the proposed methods with previous analyses.     

On a multiple endpoints testing problem

In a clinical trial comparing drug with placebo, where there are multiple primary endpoints, we consider testing problems where an efficacious drug effect can be claimed only if statistical significance is demonstrated at the nominal level for all endpoints. Under the assumption that the data are multivariate normal, the multiple endpoint-testing problem is formulated. The usual testing procedure involves testing each endpoint separately at the same significance level using two-sample t-tests, and claiming drug efficacy only if each t-statistic is significant. In this paper we investigate properties of this procedure. We show that it is identical to both an intersection union test and the likelihood ratio test. A simple expression for the p-value is given. The level and power function are studied; it is shown that the test may be conservative and that it is biased. Computable bounds for the power function are established.     

Sensitivity analysis for two control groups

In many experiments where data have been collected at two points in time (pre-treatment and post-treatment), investigators wish to determine if there is a difference between two treatment groups. In recent years it has been proposed that an appropriate statistical analysis to determine if treatment differences exist is to use the post-treatment values as the primary comparison variables and the pre-treatment values as covariates. When there are several outcome variables, we propose new tests based on residuals as alternatives to existing methods and investigate how the powers of the new and existing tests are affected by various choices of covariates. The limiting distribution of the test statistic of the new test based on residuals is given. Monte Carlo simulations are employed in the power comparisons.     

On Two Forms of Fisher's Measure of Information

ABSTRACT

Fisher's information number is the second moment of the “score function” where the derivative is with respect to x rather than Θ. It is Fisher's information for a location parameter, and also called shift-invariant Fisher information. In recent years, Fisher's information number has been frequently used in several places regardless of parameters of the distribution or of their nature. Is this number a nominal, standard, and typical measure of information? The Fisher information number is examined in light of the properties of classical statistical information theory. It has some properties analogous to those of Fisher's measure, but, in general, it does not have good properties if used as a measure of information when Θ is not a location parameter. Even in the case of location parameter, the regularity conditions must be satisfied. It does not possess the two fundamental properties of the mother information, namely the monotonicity and invariance under sufficient transformations. Thus the Fisher information number should not be used as a measure of information (except when Θ a location parameter). On the other hand, Fisher's information number, as a characteristic of a distribution f(x), has other interesting properties. As a byproduct of its superadditivity property a new coefficient of association is introduced.     

Inferences Based on a Skipped Correlation Coefficient

The most popular method for trying to detect an association between two random variables is to test H ₀ ?:?ρ=0, the hypothesis that Pearson's correlation is equal to zero. It is well known, however, that Pearson's correlation is not robust, roughly meaning that small changes in any distribution, including any bivariate normal distribution as a special case, can alter its value. Moreover, the usual estimate of ρ, r, is sensitive to only a few outliers which can mask a true association. A simple alternative to testing H ₀ ?:?ρ =0 is to switch to a measure of association that guards against outliers among the marginal distributions such as Kendall's tau, Spearman's rho, a Winsorized correlation, or a so-called percentage bend correlation. But it is known that these methods fail to take into account the overall structure of the data. Many measures of association that do take into account the overall structure of the data have been proposed, but it seems that nothing is known about how they might be used to detect dependence. One such measure of association is selected, which is designed so that under bivariate normality, its estimator gives a reasonably accurate estimate of ρ. Then methods for testing the hypothesis of a zero correlation are studied.     

Cumulative ratio information based on general cumulative entropy

In this paper, we suggest an extension of the cumulative residual entropy (CRE) and call it generalized cumulative entropy. The proposed entropy not only retains attributes of the existing uncertainty measures but also possesses the absolute homogeneous property with unbounded support, which the CRE does not have. We demonstrate its mathematical properties including the entropy of order statistics and the principle of maximum general cumulative entropy. We also introduce the cumulative ratio information as a measure of discrepancy between two distributions and examine its application to a goodness-of-fit test of the logistic distribution. Simulation study shows that the test statistics based on the cumulative ratio information have comparable statistical power with competing test statistics.     

Nonparametric tests for multivariate multi-sample locations based on data depth

Several nonparametric tests for multivariate multi-sample location problem are proposed in this paper. These tests are based on the notion of data depth, which is used to measure the centrality/outlyingness of a given point with respect to a given distribution or a data cloud. Proposed tests are completely nonparametric and implemented through the idea of permutation tests. Performance of the proposed tests is compared with existing parametric test and nonparametric test based on data depth. An extensive simulation study reveals that proposed tests are superior to the existing tests based on data depth with regard to power. Illustrations with real data are provided.     

A method for estimating lorenz curves

This paper proposes a new method for estimating the parameters of Lorenz Curves (LC’s) and fitting LC’s to observed data. The method is very general. It is applicable to any family of LC’s as long as it is given in closed form which is often the case in practice. The method can also be applied to either the LC or to its associated distribution. The estimators are easy to compute as they are obtained one at a time by solving only one equation in one unknown and in many cases the solutions are given in closed-forms. An additional advantage, that is not shared with the currently used method of estimation, is that the method is invariant as to the specification of which variable is written as a function of the other in the LC form. The method is applied to the most commonly suggested LC’s families. An example of real-life data is used to illustrate the methodology. A simulation study is performed to study the properties of the proposed estimators and to compare them with existing ones. The results seem to indicate that the proposed estimators have good properties and they often perform much better than the existing ones.     

A nonparametric measure of interclass correlation

A nonparametric measure of interclass correlation is considered and its unbiased estimator and a test based on the estimator are studied. Hie measure is an analogue of the Kendall's measure of dependence. It is shown that the variance of the estimator is small and the information loss of the test based on the estimator is not serious relative to a standard parametric test in the sense of the Pitman asymptotic relative efficiency. Furthermore, the approximate variance of the estimator is given in the normal model.     

Probability Generating Function Based Jeffrey's Divergence for Statistical Inference

Statistical inference procedures based on transforms such as characteristic function and probability generating function have been examined by many researchers because they are much simpler than probability density functions. Here, a probability generating function based Jeffrey's divergence measure is proposed for parameter estimation and goodness-of-fit test. Being a member of the M-estimators, the proposed estimator is consistent. Also, the proposed goodness-of-fit test has good statistical power. The proposed divergence measure shows improved performance over existing probability generating function based measures. Real data examples are given to illustrate the proposed parameter estimation method and goodness-of-fit test.     

Measuring serial dependence in categorical time series

The analysis of time-indexed categorical data is important in many fields, e.g., in telecommunication network monitoring, manufacturing process control, ecology, etc. Primary interest is in detecting and measuring serial associations and dependencies in such data. For cardinal time series analysis, autocorrelation is a convenient and informative measure of serial association. Yet, for categorical time series analysis an analogous convenient measure and corresponding concepts of weak stationarity have not been provided. For two categorical variables, several ways of measuring association have been suggested. This paper reviews such measures and investigates their properties in a serial context. We discuss concepts of weak stationarity of a categorical time series, in particular of stationarity in association measures. Serial association and weak stationarity are studied in the class of discrete ARMA processes introduced by Jacobs and Lewis (J. Time Ser. Anal. 4(1):19–36, 1983). An intrinsic feature of a time series is that, typically, adjacent observations are dependent. The nature of this dependence among observations of a time series is of considerable practical interest. Time series analysis is concerned with techniques for the analysis of this dependence. (Box et al. 1994p. 1)     

A NEW MEASURE OF ROTATABILITY OF RESPONSE SURFACE DESIGNS

This paper proposes a new measure of rotatability of a response surface design. It is based on a comparison between the design moments of a given design and the design moments of a rotatable design closest to the given design in the least squares sense. This measure is easier to calculate than Khur?s measure. An alternative easier non-geometrical way of deriving Khuri's measure is also presented. Both measures are calculated to illustrate some designs considered by Khuri.     

A Novel Powerful Joint Analysis with Data Fusion in Two-stage Case–Control Genome-wide Association Studies

In genome-wide association studies (GWASs) to detect the disease-associated genetic variants, two-stage design has received much attention because of its cost effectiveness and high efficiency. Under the framework of a two-stage design, it has been shown that joint analysis is more powerful than replication-based analysis. Several robust tests have been proposed for joint analysis to handle the problem of unknown genetic mode of inheritance. However, existing joint analysis of combining test statistics from both stages might suffer from a loss of efficiency if the combined test statistics are not sufficient or the weight of the statistic for each stage is not appropriate. In this article, we propose a new strategy for joint analysis by combining the raw data rather than the test statistics across stages and construct a robust MAX3-based test for two-staged GWASs, which can make full use of the information of the data from both stages. Our numerical results show that the proposed procedure is more powerful and computationally much faster than the existing joint analysis procedures. An application to a type 2 diabetes dataset is used to illustrate the proposed approach.     

Bayesian Measures of the Minimum Detectable Concentration of an Immunoassay

The minimum detectable concentration (MDC) is one of the most important properties of an assay. It is a statement about the smallest physical quantity an assay can reliably measure, and is used in assay design and quality control assessments. A plethora of measures of the MDC have been reported in a widely scattered literature. Many of these were developed at a time when accuracy and relevance had to be sacrificed for computational feasibility. This paper identifies limitations of existing measures and demonstrates how Bayesian inference may be used to overcome these limitations. Several new measures of the MDC are developed. These are conceptually simpler than existing measures, and are free of analytical approximations. The recent advances in Bayesian computation make them efficient to evaluate. A procedure developed in this paper measures the difference in the quality of two assays and shows that the new Bayesian measures perform better than existing measures.     

The stability of several measures of association in small contingency tables

Measures of association are often used to describe the relationship between row and column variables in two—dimensional contingency tables. It is not uncommon in biomedical research to categorize continuous variables to obtain a two—dimensional table. In these situations it is desirable that the measure of association not be too sensitive to changes in the number of categories or to the choice of cut points. To accomplish this objective we attempt to find a measure of association that closely approximates the corresponding measure of association for the underlying distribution.Measures that are close to the underlying measure for various table sizes andcutpoints are called stable measures.     

Erlang–Lindley distribution

In this paper, Erlang–Lindley distribution (ErLD) is proposed which offers a more flexible model for waiting time data. It has the property that it can accommodate increasing, bathtub, and inverted bathtub shapes. Several statistical and reliability properties are derived and studied. The moments, its associated measures, and the limiting distributions of order statistics are derived. The model parameters are estimated by maximum likelihood and method of moments. An application of the proposed distribution to some waiting time data shows that it can give a better fit than other important lifetime models.     

Matrix trace Wielandt inequalities with statistical applications

The vector correlation coefficient and other measures of association play a very important role in statistics and especially in multivariate analysis. In this paper a new measure of association is proposed and its upper bound is presented by using a matrix trace Wielandt inequality. Also given are relevant results involving Wishart matrices widely used in multivariate analysis, and especially a new alternative for the relative gain of the covariance adjusted estimator of a vector of parameters.