I. Van Mechelen

Diagnostic checks for discrete data regression models using posterior predictive simulations

Model checking with discrete data regressions can be difficult because the usual methods such as residual plots have complicated reference distributions that depend on the parameters in the model. Posterior predictive checks have been proposed as a Bayesian way to average the results of goodness‐of‐fit tests in the presence of uncertainty in estimation of the parameters. We try this approach using a variety of discrepancy variables for generalized linear models fitted to a historical data set on behavioural learning. We then discuss the general applicability of our findings in the context of a recent applied example on which we have worked. We find that the following discrepancy variables work well, in the sense of being easy to interpret and sensitive to important model failures: structured displays of the entire data set, general discrepancy variables based on plots of binned or smoothed residuals versus predictors and specific discrepancy variables created on the basis of the particular concerns arising in an application. Plots of binned residuals are especially easy to use because their predictive distributions under the model are sufficiently simple that model checks can often be made implicitly. The following discrepancy variables did not work well: scatterplots of latent residuals defined from an underlying continuous model and quantile–quantile plots of these residuals.

Testing the hypothesis of tissue selectivity

Motivation: Finding genes that are preferentially expressed in a particular tissue or condition is a problem that cannot be solved by standard statistical testing procedures. A relatively unknown procedure that can be used is the intersection–union test (IUT). However, two disadvantages of the IUT are that it is conservative and it conveys only the information of the least differing target tissue–other tissue pair. Results: We propose a Bayesian procedure that quantifies how much evidence there is in the overall expression profile for selective over-expression. In a small simulation study, it is shown that the proposed method outperforms the IUT when it comes to finding selectively expressed genes. An application to publicly available data consisting of 22 tissues shows that the Bayesian method indeed selects genes with functions that reflect the specific tissue functions. The proposed method can also be used to find genes that are underexpressed in a particular tissue. Availability: Both MATLAB and R code that implement the IUT and the Bayesian procedure in an efficient way, can be downloaded at http://ppw.kuleuven.be/okp/software/BayesianIUT/. Contact: katrijn.vandeun@psy.kuleuven.be

Approximate Galois Lattices of Formal Concepts

The Galois lattice approach has been shown to provide a rich framework for the study of monothetic formal concepts within the context of an object by attribute correspondence. Yet, the approach is hampered by two problems: First, even for data sets of moderate size the Galois lattice is usually very complex. Second, small modifications of the correspondence under study (e.g., due to error) may lead to considerable modifications of the lattice. This paper presents a procedure to construct an approximate Galois lattice with limited order and length for a given binary correspondence. The procedure, which is based on Boolean matrix theory, makes use of an algorithm for Boolean factor analysis. Goodness-of-recovery results from a simulation study suggest that it allows one to retrieve a true correspondence from error-perturbed data.

Rater Classification on the Basis of Latent Features in Responding to Situations

The present paper investigates to what extent several expansions of probability matrix decomposition models may be used to capture qualitative differences between raters. The models are applied to a specific data set about raters who indicate whether they would display hostile responses in frustrating situations.

On Algorithms for a Binary-Real (max, X) Matrix Approximation Problem

We consider algorithms to solve the problem of approximating a given matrix D with the (max, times) product of a binary (i.e., a 0-1) matrix S and a real matrix P: minSPparS odot P - Dpar, The norm to be used is the l1, l2, or linfin norm, and the (max, times) matrix product is constructed in the same way as the conventional matrix product, but with addition replaced by maximization. This approximation problem arises among others in data clustering applications where the maximal component instead of the sum of the components determines the final result. We propose several algorithms to address this problem. The binary-real (max, times) matrix approximation problem can be solved exactly using mixed-integer programming, but since this approach suffers from combinatorial explosion we also propose some alternative approaches based on alternating nonlinear optimization, and a method to obtain good initial solutions. We conclude with a simulation study in which the performance and optimality of the different algorithms are compared

Conjunctive prediction of an ordinal criterion variable on the basis of binary predictors

In this paper we propose an empirical prediction method to retrieve, for a given ordinal criterion and a set of binary predictors, a series of nested sets of predictors, each set containing all singly necessary (and, if feasible, jointly sufficient) predictors for a particular criterion value. The method extends a previously developed approach to construct approximate Galois lattice models of binary data. After sketching an outline of the new model and associated algorithm we illustrate our method with an application to real psychological data on the experience of anger.