We present a forward sufficient dimension reduction method for categorical or ordinal responses by extending the outer product of gradients and minimum average variance estimator to multinomial generalized linear model. Previous work in this direction extend forward regression to binary responses, and are applied in a pairwise manner to multinomial data, which is less efficient than our approach. Like other forward regression-based sufficient dimension reduction methods, our approach avoids the relatively stringent distributional requirements necessary for inverse regression alternatives. We show consistency of our proposed estimator and derive its convergence rate. We develop an algorithm for our methods based on repeated applications of available algorithms for forward regression. We also propose a clustering-based tuning procedure to estimate the tuning parameters. The effectiveness of our estimator and related algorithms is demonstrated via simulations and applications.
A prescription is presented for a new and practical correlation coefficient, $ϕ_K$, based on several refinements to Pearson's hypothesis test of independence of two variables. The combined features of $ϕ_K$ form an advantage over existing coefficients. First, it works consistently between categorical, ordinal and interval variables. Second, it captures non-linear dependency. Third, it reverts to the Pearson correlation coefficient in case of a bi-variate normal input distribution. These are useful features when studying the correlation between variables with mixed types. Particular emphasis is…
Causal discovery for purely observational, categorical data is a long-standing challenging problem. Unlike continuous data, the vast majority of existing methods for categorical data focus on inferring the Markov equivalence class only, which leaves the direction of some causal relationships undetermined. This paper proposes an identifiable ordinal causal discovery method that exploits the ordinal information contained in many real-world applications to uniquely identify the causal structure. The proposed method is applicable beyond ordinal data via data discretization. Through real-world and …
Regression models for categorical data are specified in heterogeneous ways. We propose to unify the specification of such models. This allows us to define the family of reference models for nominal data. We introduce the notion of reversible models for ordinal data that distinguishes adjacent and cumulative models from sequential ones. The combination of the proposed specification with the definition of reference and reversible models and various invariance properties leads to a new view of regression models for categorical data.
This article proposes a powerful scheme to monitor a large number of categorical data streams with heterogeneous parameters or nature. The data streams considered may be either nominal with a number of attribute levels or ordinal with some natural order among their attribute levels, such as good, marginal, and bad. For an ordinal data stream, it is assumed that there is a corresponding latent continuous data stream determining it. Furthermore, different data streams may have different number of attribute levels and different values of level probabilities. Due to high dimensionality, traditiona…
In categorical data analysis, several regression models have been proposed for hierarchically-structured response variables, e.g. the nested logit model. But they have been formally defined for only two or three levels in the hierarchy. Here, we introduce the class of partitioned conditional generalized linear models (PCGLMs) defined for any numbers of levels. The hierarchical structure of these models is fully specified by a partition tree of categories. Using the genericity of the (r,F,Z) specification, the PCGLM can handle nominal, ordinal but also partially-ordered response variables.
The literature on clustering for continuous data is rich and wide; differently, that one developed for categorical data is still limited. In some cases, the problem is made more difficult by the presence of noise variables/dimensions that do not contain information about the clustering structure and could mask it. The aim of this paper is to propose a model for simultaneous clustering and dimensionality reduction of ordered categorical data able to detect the discriminative dimensions discarding the noise ones. Following the underlying response variable approach, the observed variables are con…
Generalized Estimation Equations (GEE) are a well-known method for the analysis of categorical longitudinal responses. GEE method has computational simplicity and population parameter interpretation. In the presence of missing data it is only valid under the strong assumption of missing completely at random. A doubly robust estimator (DRGEE) for correlated ordinal longitudinal data is a nice approach for handling intermittently missing response and covariate under the MAR mechanism. Independent working correlation is the standard way in DRGEE. However, when covariate is not time stationary, ef…
A lot of studies on the summary measures of predictive strength of categorical response models consider the likelihood ratio index (LRI), also known as the McFadden-$R^2$, a better option than many other measures. We propose a simple modification of the LRI that adjusts for the effect of the number of response categories on the measure and that also rescales its values, mimicking an underlying latent measure. The modified measure is applicable to both binary and ordinal response models fitted by maximum likelihood. Results from simulation studies and a real data example on the olfactory percep…
This study deals with the problem of outliers in ordinal response model, which is a regression on ordered categorical data as the response variable. ``Outlier" means that the combination of ordered categorical data and its covariates is heterogeneous compared to other pairs. Although the ordinal response model is important for data analysis in various fields such as medicine and social sciences, it is known that the maximum likelihood method with probit, logit, log-log and complementary log-log link functions, which are often used, is strongly affected by outliers, and statistical analysts are…
In this work we handle with categorical (ordinal) variables and we focus on the (in)dependence relationship under the marginal, conditional and context-specific perspective. If the first two are well known, the last one concerns independencies holding only in a subspace of the outcome space. We take advantage from the Hierarchical Multinomial Marginal models and provide several original results about the representation of context-specific independencies through these models. By considering the graphical aspect, we take advantage from the chain graphical models. The resultant graphical model is…
In applications involving ordinal predictors, common approaches to reduce dimensionality are either extensions of unsupervised techniques such as principal component analysis, or variable selection procedures that rely on modeling the regression function. In this paper, a supervised dimension reduction method tailored to ordered categorical predictors is introduced. It uses a model-based dimension reduction approach, inspired by extending sufficient dimension reductions to the context of latent Gaussian variables. The reduction is chosen without modeling the response as a function of the predi…
Biplot Methods allow for the simultaneous representation of individuals and variables of a data matrix. For Binary or Nominal data, Logistic biplots have been recently developed to extend the classical linear representations for continuous data. When data are ordinal, linear, binary or nominal logistic biplots are not adequate and techniques as Categorical Principal Component Analysis (CATPCA) or Item Response Theory (IRT) for ordinal items should be used instead. In this paper we extend the Biplot to ordinal data. The resulting method is termed Ordinal Logistic Biplot (OLB). Row scores are co…
A first step when fitting multilevel models to continuous responses is to explore the degree of clustering in the data. Researchers fit variance-component models and then report the proportion of variation in the response that is due to systematic differences between clusters. Equally they report the response correlation between units within a cluster. These statistics are popularly referred to as variance partition coefficients (VPCs) and intraclass correlation coefficients (ICCs). When fitting multilevel models to categorical (binary, ordinal, or nominal) and count responses, these statistic…
