Adjusted Pearson residuals in exponential family nonlinear models

In this paper, we give matrix formulae of order 𝒪(n ^?1), where n is the sample size, for the first two moments of Pearson residuals in exponential family nonlinear regression models [G.M. Cordeiro and G.A. Paula, Improved likelihood ratio statistic for exponential family nonlinear models, Biometrika 76 (1989), pp. 93–100.]. The formulae are applicable to many regression models in common use and generalize the results by Cordeiro [G.M. Cordeiro, On Pearson's residuals in generalized linear models, Statist. Prob. Lett. 66 (2004), pp. 213–219.] and Cook and Tsai [R.D. Cook and C.L. Tsai, Residuals in nonlinear regression, Biometrika 72(1985), pp. 23–29.]. We suggest adjusted Pearson residuals for these models having, to this order, the expected value zero and variance one. We show that the adjusted Pearson residuals can be easily computed by weighted linear regressions. Some numerical results from simulations indicate that the adjusted Pearson residuals are better approximated by the standard normal distribution than the Pearson residuals.     

The extreme residuals in logistic regression models

Goodness-of-fit tests for logistic regression models using extreme residuals are considered. Approximations to the moments of the Pearson residuals are given for model fits made by maximum likelihood, minimum chi-square and weighted least squares and used to define modified residuals. Approximations to the critical values of the extreme statistics based on the ordinary and modified Pearson residuals are developed and assessed for the case of a single explanatory variable.     

Characterization of the Pearson system of distributions based on reliability measures

LetX be a random variable andX ^(w) be a weighted random variable corresponding toX. In this paper, we intend to characterize the Pearson system of distributions by a relationship between reliability measures ofX andX ^(w), for some weight functionw>0.     

Simulating dependent discrete data

This article describes a method for simulating n-dimensional multivariate non-normal data, with emphasis on count-valued data. Dependence is characterized by either Pearson correlations or Spearman correlations. The simulation is accomplished by simulating a vector of correlated standard normal variates. The elements of this vector are then transformed to achieve the target marginal distributions. We prove that the method corresponds to simulating data from a multivariate Gaussian copula. The simulation method does not restrict pairwise dependence beyond the limits imposed by the marginal distributions and can achieve any Pearson or Spearman correlation within those limits. Two examples are included. In the first example, marginal means, variances, Pearson correlations, and Spearman correlations are estimated from the epileptic seizure data set of Diggle et al. [P. Diggle, P. Heagerty, K.Y. Liang, and S. Zeger, Analysis of Longitudinal Data, Oxford University Press, Oxford, 2002]. Data with these means and variances are simulated to first achieve the estimated Pearson correlations and then achieve the estimated Spearman correlations. The second example is of a hypothetical time series of Poisson counts with seasonal mean ranging between 1 and 9 and an autoregressive(1) dependence structure.     

Measures of lack of fit from tests of chi-squared type

If X² is the Pearson chi-squared statistic for testing fit, then $\text{X}{}^2\text{n}$ has long been considered an associated measure of the degree of lack of fit. Here we consider two classes of statistics of chi-squared type, each having X² as a member. The first is a class of directed divergence statistics discussed by Cressie and Read, the second consists of nonnegative definite quadratic forms in the standardized cell frequencies. We investigate the large sample behavior of $\text{T}\text{n}$, where T is any of these statistics. A number of auxiliary results on the Cressie-Read statistics are also obtained. The measures are illustrated by application to data from classical physics compiled by Stigler.     

SINGULAR ELLIPTICAL DISTRIBUTION: DENSITY AND APPLICATIONS

ABSTRACT

The paper present an explicit expression for the density of a n-dimensional random vector with a singular Elliptical distribution. Based on this, the densities of the generalized Chi-squared and generalized t distributions are derived, examining the Pearson Type VII distribution and Kotz Type distribution (as specific Elliptical distributions). Finally, the results are applied to the study of the distribution of the residuals of an Elliptical linear model and the distribution of the t-statistic, based on a sample from an Elliptical population.     

Characterizations based on generalized cumulative residual entropy functions

The Shannon entropy and the cumulative residual entropy (CRE) of a random variable are useful tools in probability theory. Recently, a new concept called generalized cumulative residual entropy (GCRE) of order n was introduced and studied. It is related with the record values of a sequence of i.i.d. random variables and with the relevation transform. In this paper, we show that, under some assumptions, the GCRE function of a fixed order n uniquely determines the distribution function. Some characterizations of particular probability models are obtained from this general result.     

Simple correspondence analysis using adjusted residuals

Correspondence analysis is a versatile statistical technique that allows the user to graphically identify the association that may exist between variables of a contingency table. For two categorical variables, the classical approach involves applying singular value decomposition to the Pearson residuals of the table. These residuals allow for one to use a simple test to determine those cells that deviate from what is expected under independence. However, the assumptions concerning these residuals are not always satisfied and so such results can lead to questionable conclusions.One may consider instead, an adjustment of the Pearson residual, which is known to have properties associated with the standard normal distribution. This paper explores the application of these adjusted residuals to correspondence analysis and determines how they impact upon the configuration of points in the graphical display.     

Distribution of the correlation coefficient for the class of bivariate elliptical models

We consider n pairs of random variables (X₁₁,X₂₁),(X₁₂,X₂₂),… (X_1n,X_2n) having a bivariate elliptically contoured density of the form where θ₁ θ₂ are location parameters and Δ = ((λ_ik)) is a 2 × 2 symmetric positive definite matrix of scale parameters. The exact distribution of the Pearson product-moment correlation coefficient between X₁ and X₂ is obtained. The usual case when a sample of size n is drawn from a bivariate normal population is a special case of the abovementioned model.     

A generalized Q–Q plot for longitudinal data

Most biomedical research is carried out using longitudinal studies. The method of generalized estimating equations (GEEs) introduced by Liang and Zeger [Longitudinal data analysis using generalized linear models, Biometrika 73 (1986), pp. 13–22] and Zeger and Liang [Longitudinal data analysis for discrete and continuous outcomes, Biometrics 42 (1986), pp. 121–130] has become a standard method for analyzing non-normal longitudinal data. Since then, a large variety of GEEs have been proposed. However, the model diagnostic problem has not been explored intensively. Oh et al. [Modeldiagnostic plots for repeated measures data using the generalized estimating equations approach, Comput. Statist. Data Anal. 53 (2008), pp. 222–232] proposed residual plots based on the quantile–quantile (Q–Q) plots of the χ²-distribution for repeated-measures data using the GEE methodology. They considered the Pearson, Anscombe and deviance residuals. In this work, we propose to extend this graphical diagnostic using a generalized residual. A simulation study is presented as well as two examples illustrating the proposed generalized Q–Q plots.     

Bias correction and improved residuals for non-exponential family nonlinear models

In this paper we derive general formulae for the biases to order n ^?1 of the parameter estimates in a general class of nonlinear regression models, where n is the sample size. The formulae are related to those of Cordeiro and McCullagh (1991) and Paula (1992) and may be viewed as extensions of their results, Correction factors are derived for the score and deviance component residuals in these models. The practical use of such corrections is illustrated for the log-gamma model.     

Improved likelihood ratio tests and pearson chi-square tests for independence in two dimensional contingency tables

When an I×J contingency table has many cells having very small frequencies, the usual chi-square approximation to the upper tail of the likelihood ratio goodness-of-fit statistic, G² and Pearson chi-square statistic, X², for testing independence, are not satisfactory. In this paper we consider the problem of adjusting G² and X². Suitable adjustments are suggested on the basis of analytical investigation of asymptotic bias terms for G² and X². A Monte Carlo simulation is performed for several tables to assess the adjustments of G² and X² in order to obtain a closer approximation to the nominal level of significance.     

Estimating the correlation coefficient in the presence of correlated observations from a bivariate normal population

Estimation of the correlation coefficient between two variates (p) in the presence of correlated observations from a bivar iate normal population is considered The estimated maximum likelihood estimator (EMLE), an estimate based on the maximum likelihood estimator (MLE), is proposed and studied for the estimation of p For the large sample case , approximate expressions foi the variance and the bias of the Pearson estimate of the correlation coefficient are derived. These expressions suggests that the Pearson’s estimator possesses high mean square error (MSE) in estimating ρ in comparison to the MLE The MSE is particularly high when the observations within clusters aie highly correlated. The Pearson’s estimate, the MLE, and the EMLE aie evaluated in a simulation study This study shows that the proposed EMLE pefoims bettei than the Pearson’s correlation coefficient except when the number of clusters is small.     

Bootstrapping regression models with BLUS residuals

To bootstrap a regression problem, pairs of response and explanatory variables or residuals can be resam‐pled, according to whether we believe that the explanatory variables are random or fixed. In the latter case, different residuals have been proposed in the literature, including the ordinary residuals (Efron 1979), standardized residuals (Bickel & Freedman 1983) and Studentized residuals (Weber 1984). Freedman (1981) has shown that the bootstrap from ordinary residuals is asymptotically valid when the number of cases increases and the number of variables is fixed. Bickel & Freedman (1983) have shown the asymptotic validity for ordinary residuals when the number of variables and the number of cases both increase, provided that the ratio of the two converges to zero at an appropriate rate. In this paper, the authors introduce the use of BLUS (Best Linear Unbiased with Scalar covariance matrix) residuals in bootstrapping regression models. The main advantage of the BLUS residuals, introduced in Theil (1965), is that they are uncorrelated. The main disadvantage is that only n —p residuals can be computed for a regression problem with n cases and p variables. The asymptotic results of Freedman (1981) and Bickel & Freedman (1983) for the ordinary (and standardized) residuals are generalized to the BLUS residuals. A small simulation study shows that even though only n — p residuals are available, in small samples bootstrapping BLUS residuals can be as good as, and sometimes better than, bootstrapping from standardized or Studentized residuals.     

The Pearson Score Statistic for Multinomial-Poisson Models

The score statistic S² is commonly used for general likelihood-based inference. Pearson’s Chi-squared statistic X² = ∑(O ? E)²/E is ubiquitous in contingency table inference. Because tests and confidence intervals based on S² have been shown to work well in practice and theory and because X² has such a simple and intuitively appealing form, it is of interest to know when S² is identical to X² and when X² has an approximate Chi-squared distribution. Toward these ends, this paper gives a simple proof that S² = X² for the broad class of multinomial-Poisson distributions when the alternative hypothesis is unrestricted in a certain sense. This paper also gives a sufficient condition under which the null distribution of the Pearson score statistic is approximately Chi-squared. Several examples illustrate the utility of the results and counter-examples highlight the importance of the sufficient conditions of the results.     

Identification of Multiple Outliers in Logistic Regression

The use of logistic regression modeling has seen a great deal of attention in the literature in recent years. This includes all aspects of the logistic regression model including the identification of outliers. A variety of methods for the identification of outliers, such as the standardized Pearson residuals, are now available in the literature. These methods, however, are successful only if the data contain a single outlier. In the presence of multiple outliers in the data, which is often the case in practice, these methods fail to detect the outliers. This is due to the well-known problems of masking (false negative) and swamping (false positive) effects. In this article, we propose a new method for the identification of multiple outliers in logistic regression. We develop a generalized version of standardized Pearson residuals based on group deletion and then propose a technique for identifying multiple outliers. The performance of the proposed method is then investigated through several examples.     

Additional bowman-shenton approximate percentage points for pearson distributions based on pearson type vi

The Bowman and Shenton approximate percentage points for Pearson distributions are extended to include some cases for which the skewness β¹ exceeds four.Use is made of the reciprocity property existing between the Pearson Type VI distribution and the Type I distribution     

Editor's Report

There are two common methods for statistical inference on 2 × 2 contingency tables. One is the widely taught Pearson chi-square test, which uses the well-known χ²statistic. The chi-square test is appropriate for large sample inference, and it is equivalent to the Z-test that uses the difference between the two sample proportions for the 2 × 2 case. Another method is Fisher’s exact test, which evaluates the likelihood of each table with the same marginal totals. This article mathematically justifies that these two methods for determining extreme do not completely agree with each other. Our analysis obtains one-sided and two-sided conditions under which a disagreement in determining extreme between the two tests could occur. We also address the question whether or not their discrepancy in determining extreme would make them draw different conclusions when testing homogeneity or independence. Our examination of the two tests casts light on which test should be trusted when the two tests draw different conclusions.     

A Single Form for t-Distributions and Symmetric Beta Distributions

A single parametric form is given for the symmetric distributions in the Pearson system with finite variance. In effect, these are Student's t-distributions with ν > 2 and all centered symmetric beta distributions. A different parametrization allows the inclusion of the t-distributions with ν ≤2 at the expense of symmetric beta distributions with a low shape parameter.