Marc Aerts

Testing the Fit of a Parametric Function

Abstract General methods for testing the fit of a parametric function are proposed. The idea underlying each method is to “accept” the prescribed parametric model if and only if it is chosen by a model selection criterion. Several different selection criteria are considered, including one based on a modified version of the Akaike information criterion and others based on various score statistics. The tests have a connection with nonparametric smoothing because they use orthogonal series estimators to detect departures from a parametric model. An important aspect of the tests is that they can be applied in a wide variety of settings, including generalized linear models, spectral analysis, the goodness-of-fit problem, and longitudinal data analysis. Implementation using standard statistical software is straightforward. Asymptotic distribution theory for several test statistics is described, and the tests are shown to be consistent against essentially any alternative hypothesis. Simulations and a data exampl...

Bayesian-motivated tests of function fit and their asymptotic frequentist properties

We propose and analyze nonparametric tests of the null hypothesis that a function belongs to a specified parametric family. The tests are based on BIC approximations, π BIC , to the posterior probability of the null model, and may be carried out in either Bayesian or frequentist fashion. We obtain results on the asymptotic distribution of π BIC under both the null hypothesis and local alternatives. One version of π BIC , call it π* BIC , uses a class of models that are orthogonal to each other and growing in number without bound as sample size, n, tends to infinity. We show that √n(1 - π* BIC ) converges in distribution to a stable law under the null hypothesis. We also show that π* BIC can detect local alternatives converging to the null at the rate √log n/n. A particularly interesting finding is that the power of the π* BIC -based test is asymptotically equal to that of a test based on the maximum of alternative log-likelihoods. Simulation results and an example involving variable star data illustrate desirable features of the proposed tests.

Classification of Trends in Dose-Response Microarray Experiments Using Information Theory Selection Methods~!2009-02-24~!2009-07-09~!2009-12-23~!

Dose-response microarray experiments consist of monitoring expression levels of thousands of genes with respect to increasing dose of the treatment under investigation. The primary goal of such an experiment is to establish a dose-response relationship, while the secondary goals are to determine the minimum effective dose level and to identify the shape of the dose-response curve. Recently, Lin et al. (1) discussed several testing procedures to test for monotone trend based on isotonic regression of the observed means (2,3). Once a monotone relationship between the gene expression and dose is established, there is a set of R possible monotone models that can be fitted to the data. A selection of the best model from this set allows us to identify both the shape of dose-response curve and the minimum effective dose level. In this paper we focus on classification of dose-response curve shapes using the information theory model selection. In particular, the Order Restricted Information Criterion (ORIC) is discussed for the inference under order restriction. The posterior probability of the model is calculated using information criteria that take into account both the goodness-of-fit and the complexity of the models. The method is applied to a dose-response microarray experiment with 12 arrays (for three samples at each of the four dose levels) with 16,998 genes.

Investigating the Performance of Rule-based Models with Increasing Complexity on the Prediction of Trip Generation and Distribution

Modelling travel behaviour has always been a major research area in transportation analysis. After the second World War, due to the rapid increase in car ownership and car use in Western Europe and the United States, several models have been developed by transportation planners. In the fifties and sixties, travel was assumed to be the result of four subsequent decisions that were modelled: trip generation, trip distribution, mode choice and the assignment of trips to the road network (Ruiter & Ben-Akiva, 1978). These original tripbased models have been extended to ensuing tour-based models (Daly et al., 1983) and activity-based models (Pendyala et al., 1995; Ben-Akiva & Bowman, 1998; Kitamura & Fujii, 1998; Arentze & Timmermans, 2000; Bhat et al., 2004). In tour-based models, trips are explicitly connected in tours, i.e. chains that start and end at the same home or work base. This is carried out by introducing spatial constraints, hereby dealing with the lack of spatial interrelationship which was so apparent in the traditional four-step trip-based model. In activity-based models, travel demand is derived from the activities that individuals and households need or wish to perform. Decisions with respect to travel are driven by a collection of activities that form an activity diary. Travel should therefore be modelled within the context of the entire agenda, or as a component of the activity scheduling decision. In this way, the relationship between travel and non-travel aspects is taken into account. The reason why people undertake trips is one of the key aspects to be modelled in an activity-based model. However, every working transportation model still exists of at least these original four components of trip generation, distribution, mode choice and assignment. In order to fully understand the structure of a traditional transportation model, we need to elaborate on it some more. As shown in Figure 1, trip generation encompasses both the modelling of production (P) and attraction (A) of trips for a certain region (zone). Production is mainly being modelled at the level of the household, incorporating household characteristics (income, car ownership, household composition, ...), features of the zone (land price, degree of urbanization) and accessibility of the zone, whereas attraction is modelled at zone level,