The Standard Error of the Mean and the Difference between Means for Finite Populations

This article discusses a representation of Pearson's chi-square for independence in two-way contingency tables in terms of conditional probabilities of two categorical random variables and proposes a functional interpretation of Pearson's chi-square. This representation is suggested for use in the teaching of statistical independence between categorical variables.     

Test of Association Between Two Ordinal Variables While Adjusting for Covariates

We propose a new set of test statistics to examine the association between two ordinal categorical variables X and Y after adjusting for continuous and/or categorical covariates Z. Our approach first fits multinomial (e.g., proportional odds) models of X and Y, separately, on Z. For each subject, we then compute the conditional distributions of X and Y given Z. If there is no relationship between X and Y after adjusting for Z, then these conditional distributions will be independent, and the observed value of (X, Y) for a subject is expected to follow the product distribution of these conditional distributions. We consider two simple ways of testing the null of conditional independence, both of which treat X and Y equally, in the sense that they do not require specifying an outcome and a predictor variable. The first approach adds these product distributions across all subjects to obtain the expected distribution of (X, Y) under the null and then contrasts it with the observed unconditional distribution of (X, Y). Our second approach computes "residuals" from the two multinomial models and then tests for correlation between these residuals; we define a new individual-level residual for models with ordinal outcomes. We present methods for computing p-values using either the empirical or asymptotic distributions of our test statistics. Through simulations, we demonstrate that our test statistics perform well in terms of power and Type I error rate when compared to proportional odds models which treat X as either a continuous or categorical predictor. We apply our methods to data from a study of visual impairment in children and to a study of cervical abnormalities in human immunodeficiency virus (HIV)-infected women. Supplemental materials for the article are available online.     

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.     

A Bayesian test of independence in a two-way contingency table using surrogate sampling

We consider a Bayesian approach to the study of independence in a two-way contingency table which has been obtained from a two-stage cluster sampling design. If a procedure based on single-stage simple random sampling (rather than the appropriate cluster sampling) is used to test for independence, the p-value may be too small, resulting in a conclusion that the null hypothesis is false when it is, in fact, true. For many large complex surveys the Rao–Scott corrections to the standard chi-squared (or likelihood ratio) statistic provide appropriate inference. For smaller surveys, though, the Rao–Scott corrections may not be accurate, partly because the chi-squared test is inaccurate. In this paper, we use a hierarchical Bayesian model to convert the observed cluster samples to simple random samples. This provides surrogate samples which can be used to derive the distribution of the Bayes factor. We demonstrate the utility of our procedure using an example and also provide a simulation study which establishes our methodology as a viable alternative to the Rao–Scott approximations for relatively small two-stage cluster samples. We also show the additional insight gained by displaying the distribution of the Bayes factor rather than simply relying on a summary of the distribution.     

Tests of independence in incomplete multi-way tables using likelihood functions

Kang (2006) and Kang and Larsen (in press) used the log likelihood function with Lagrangian multipliers for estimation of cell probabilities in two-way incomplete contingency tables. This paper extends results and simulations to three-way and multi-way tables. Numerous studies cross-classify subjects by three or more categorical factors. Constraints on cell probabilities are incorporated through Lagrangian multipliers. Variances of the MLEs are derived from the matrix of second derivatives of the log likelihood with respect to cell probabilities and the Lagrange multiplier. Wald and likelihood ratio tests of independence are derived using the estimates and estimated variances. In simulation results in Kang and Larsen (in press), for data missing at random, maximum likelihood estimation (MLE) produced more efficient estimates of population proportions than either multiple imputation (MI) based on data augmentation or complete case (CC) analysis. Neither MLE nor MI, however, lead to an improvement over CC analysis with respect to power of tests for independence in two-way tables. Results are extended to multidimensional tables with arbitrary patterns of missing data when the variables are recorded on individual subjects. In three-way and higher-way tables, however, there is information relevant for judging independence in partially classified information, as long as two or more variables are jointly observed. Simulations study three-dimensional tables with three patterns of association and two levels of missing information.     

Matched survival data in a co-twin control design

When using the co-twin control design for analysis of event times, one needs a model to address the possible within-pair association. One such model is the shared frailty model in which the random frailty variable creates the desired within-pair association. Standard inference for this model requires independence between the random effect and the covariates. We study how violations of this assumption affect inference for the regression coefficients and conclude that substantial bias may occur. We propose an alternative way of making inference for the regression parameters by using a fixed-effects models for survival in matched pairs. Fitting this model to data generated from the frailty model provides consistent and asymptotically normal estimates of regression coefficients, no matter whether the independence assumption is met.     

NON-SYMMETRICAL CORRESPONDENCE ANALYSIS WITH CONCATENATION AND LINEAR CONSTRAINTS

Correspondence analysis is a popular statistical technique used to identify graphically the presence, and structure, of association between two or more cross-classified categorical variables. Such a procedure is very useful when it is known that there is a symmetric (two-way) relationship between the variables. When such a relationship is known not to exist, non-symmetrical correspondence analysis is more appropriate as a method of establishing the source of association. This paper highlights some tools that can be used to explore the behaviour of asymmetric categorical variables. These tools consist of confidence regions, the link between non-symmetrical correspondence analysis and the analysis of variance of categorical variables, and the effect of imposing linear constraints. We also explore the application of non-symmetrical correspondence analysis to three-way contingency tables.     

A generalization of Fisher's exact test in p×q contingency tables using more concordant relations

A randomized test for pxq contingency tables to test independence against non-independence of two categorical random variables, a generalization of Fisher's exact test, is constructed. Based on the power of the test, subsets of the alternative region are identified.     

Joint models for mixed categorical outcomes: a study of HIV risk perception and disease status in Mozambique

Two types of bivariate models for categorical response variables are introduced to deal with special categories such as ‘unsure’ or ‘unknown’ in combination with other ordinal categories, while taking additional hierarchical data structures into account. The latter is achieved by the use of different covariance structures for a trivariate random effect. The models are applied to data from the INSIDA survey, where interest goes to the effect of covariates on the association between HIV risk perception (quadrinomial with an ‘unknown risk’ category) and HIV infection status (binary). The final model combines continuation-ratio with cumulative link logits for the risk perception, together with partly correlated and partly shared trivariate random effects for the household level. The results indicate that only age has a significant effect on the association between HIV risk perception and infection status. The proposed models may be useful in various fields of application such as social and biomedical sciences, epidemiology and public health.     

Feature Screening for Ultrahigh Dimensional Categorical Data With Applications

Ultrahigh dimensional data with both categorical responses and categorical covariates are frequently encountered in the analysis of big data, for which feature screening has become an indispensable statistical tool. We propose a Pearson chi-square based feature screening procedure for categorical response with ultrahigh dimensional categorical covariates. The proposed procedure can be directly applied for detection of important interaction effects. We further show that the proposed procedure possesses screening consistency property in the terminology of Fan and Lv (2008). We investigate the finite sample performance of the proposed procedure by Monte Carlo simulation studies and illustrate the proposed method by two empirical datasets.     

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.     

Simple and multiple correspondence analysis for ordinal-scale variables using orthogonal polynomials

Correspondence analysis (CA) has gained a reputation for being a very useful statistical technique for determining the nature of association between two or more categorical variables. For simple and multiple CA, the singular value decomposition (SVD) is the primary tool used and allows the user to construct a low-dimensional space to visualize this association. As an alternative to SVD, one may consider the bivariate moment decomposition (BMD), a method of decomposition that involves using orthogonal polynomials to reflect the structure of ordered categorical responses. When the features of BMD are combined with SVD, a hybrid decomposition (HD) is formed. The aim of this paper is to show the applicability of HD when performing simple and multiple CA.     

Semiparametric efficient estimation for the auxiliary outcome problem with the conditional mean model

The authors consider semiparametric efficient estimation of parameters in the conditional mean model for a simple incomplete data structure in which the outcome of interest is observed only for a random subset of subjects but covariates and surrogate (auxiliary) outcomes are observed for all. They use optimal estimating function theory to derive the semiparametric efficient score in closed form. They show that when covariates and auxiliary outcomes are discrete, a Horvitz‐Thompson type estimator with empirically estimated weights is semiparametric efficient. The authors give simulation studies validating the finite‐sample behaviour of the semiparametric efficient estimator and its asymptotic variance; they demonstrate the efficiency of the estimator in realistic settings.     

Incomplete covariates data in generalized linear models

We consider regression analysis when part of covariates are incomplete in generalized linear models. The incomplete covariates could be due to measurement error or missing for some study subjects. We assume there exists a validation sample in which the data is complete and is a simple random subsample from the whole sample. Based on the idea of projection-solution method in Heyde (1997, Quasi-Likelihood and its Applications: A General Approach to Optimal Parameter Estimation. Springer, New York), a class of estimating functions is proposed to estimate the regression coefficients through the whole data. This method does not need to specify a correct parametric model for the incomplete covariates to yield a consistent estimate, and avoids the ‘curse of dimensionality’ encountered in the existing semiparametric method. Simulation results shows that the finite sample performance and efficiency property of the proposed estimates are satisfactory. Also this approach is computationally convenient hence can be applied to daily data analysis.     

Mutual information and redundancy for categorical data

Most methods for describing the relationship among random variables require specific probability distributions and some assumptions concerning random variables. Mutual information, based on entropy to measure the dependency among random variables, does not need any specific distribution and assumptions. Redundancy, which is an analogous version of mutual information, is also proposed as a method. In this paper, the concepts of redundancy and mutual information are explored as applied to multi-dimensional categorical data. We found that mutual information and redundancy for categorical data can be expressed as a function of the generalized likelihood ratio statistic under several kinds of independent log-linear models. As a consequence, mutual information and redundancy can also be used to analyze contingency tables stochastically. Whereas the generalized likelihood ratio statistic to test the goodness-of-fit of the log-linear models is sensitive to the sample size, the redundancy for categorical data does not depend on sample size but depends on its cell probabilities.     

Cure rate survival models with missing covariates: a simulation study

In this paper we study the cure rate survival model involving a competitive risk structure with missing categorical covariates. A parametric distribution that can be written as a sequence of one-dimensional conditional distributions is specified for the missing covariates. We consider the missing data at random situation so that the missing covariates may depend only on the observed ones. Parameter estimates are obtained by using the EM algorithm via the method of weights. Extensive simulation studies are conducted and reported to compare estimates efficiency with and without missing data. As expected, the estimation approach taking into consideration the missing covariates presents much better efficiency in terms of mean square errors than the complete case situation. Effects of increasing cured fraction and censored observations are also reported. We demonstrate the proposed methodology with two real data sets. One involved the length of time to obtain a BS degree in Statistics, and another about the time to breast cancer recurrence.     

Nonlinear measures of association with kernel canonical correlation analysis and applications

Measures of association between two sets of random variables have long been of interest to statisticians. The classical canonical correlation analysis (LCCA) can characterize, but also is limited to, linear association. This article introduces a nonlinear and nonparametric kernel method for association study and proposes a new independence test for two sets of variables. This nonlinear kernel canonical correlation analysis (KCCA) can also be applied to the nonlinear discriminant analysis. Implementation issues are discussed. We place the implementation of KCCA in the framework of classical LCCA via a sequence of independent systems in the kernel associated Hilbert spaces. Such a placement provides an easy way to carry out the KCCA. Numerical experiments and comparison with other nonparametric methods are presented.     

When exposure is subject to nondifferential misclassification,are validation data helpful in testing for an exposure–disease association?

Consider assessing the evidence for an exposure variable and a disease variable being associated, when the true exposure variable is more costly to obtain than an error‐prone but nondifferential surrogate exposure variable. From a study design perspective, there are choices regarding the best use of limited resources. Should one acquire the true exposure status for fewer subjects or the surrogate exposure status for more subjects? The issue of validation is also central, i.e., should we simultaneously measure the true and surrogate exposure variables on a subset of study subjects? Using large‐sample theory, we provide a framework for quantifying the power of testing for an exposure–disease association as a function of study cost. This enables us to present comparisons of different study designs under different suppositions about both the relative cost and the performance (sensitivity and specificity) of the surrogate variable. We present simulations to show the applicability of our theoretical framework, and we provide a case‐study comparing results from an actual study to what could have been seen had true exposure status been ascertained for a different proportion of study subjects. We also describe an extension of our ideas to a more complex situation involving covariates. The Canadian Journal of Statistics 47: 222–237; 2019 © 2019 Statistical Society of Canada     

Extending dual multiple factor analysis to categorical tables

This paper describes a proposal for the extension of the dual multiple factor analysis (DMFA) method developed by Lê and Pagès 15 to the analysis of categorical tables in which the same set of variables is measured on different sets of individuals. The extension of DMFA is based on the transformation of categorical variables into properly weighted indicator variables, in a way analogous to that used in the multiple factor analysis of categorical variables. The DMFA of categorical variables enables visual comparison of the association structures between categories over the sample as a whole and in the various subsamples (sets of individuals). For each category, DMFA allows us to obtain its global (considering all the individuals) and partial (considering each set of individuals) coordinates in a factor space. This visual analysis allows us to compare the set of individuals to identify their similarities and differences. The suitability of the technique is illustrated through two applications: one using simulated data for two groups of individuals with very different association structures and the other using real data from a voting intention survey in which some respondents were interviewed by telephone and others face to face. The results indicate that the two data collection methods, while similar, are not entirely equivalent.     

Grouped penalization estimation of the osteoporosis data in the traditional Chinese medicine

Both continuous and categorical covariates are common in traditional Chinese medicine (TCM) research, especially in the clinical syndrome identification and in the risk prediction research. For groups of dummy variables which are generated by the same categorical covariate, it is important to penalize them group-wise rather than individually. In this paper, we discuss the group lasso method for a risk prediction analysis in TCM osteoporosis research. It is the first time to apply such a group-wise variable selection method in this field. It may lead to new insights of using the grouped penalization method to select appropriate covariates in the TCM research. The introduced methodology can select categorical and continuous variables, and estimate their parameters simultaneously. In our application of the osteoporosis data, four covariates (including both categorical and continuous covariates) are selected out of 52 covariates. The accuracy of the prediction model is excellent. Compared with the prediction model with different covariates, the group lasso risk prediction model can significantly decrease the error rate and help TCM doctors to identify patients with a high risk of osteoporosis in clinical practice. Simulation results show that the application of the group lasso method is reasonable for the categorical covariates selection model in this TCM osteoporosis research.