joris mulder
Tilburg University, MTO, Faculty Member
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
In Chapter 2, several examples of research questions in the analysis of variance (ANOVA) context were presented. The model parameters of an ANOVA are two or more population means and the common and unknown residual variance. In the... more
In Chapter 2, several examples of research questions in the analysis of variance (ANOVA) context were presented. The model parameters of an ANOVA are two or more population means and the common and unknown residual variance. In the examples, the hypotheses or research questions of interest impose inequality constraints on the means. For instance, for the four-group ANOVA in the
Research Interests:
Research Interests:
The expectations that researchers have about the structure in the data can often be formulated in terms of equality constraints and/or inequality constraints on the parameters in the model that is used. In a (M)AN(C)OVA model, researchers... more
The expectations that researchers have about the structure in the data can often be formulated in terms of equality constraints and/or inequality constraints on the parameters in the model that is used. In a (M)AN(C)OVA model, researchers have expectations about the differences between the (adjusted) group means; in a repeated measures model, expectations can be stated between the measuremenet means
Research Interests:
Research Interests:
... Item Selection Joris Mulder and Wim J. van der Linden ... This normal approximation is not unreasonable; Chang and Stout 1993) have shown that, under mild nonparamet-ric assumptions, the posterior in (4.2) is asymptotically normal... more
... Item Selection Joris Mulder and Wim J. van der Linden ... This normal approximation is not unreasonable; Chang and Stout 1993) have shown that, under mild nonparamet-ric assumptions, the posterior in (4.2) is asymptotically normal with a mean equal to ย. ...
Research Interests:
Researchers in the behavioural and social sciences often have expectations that can be expressed in the form of inequality constraints among the parameters of a structural equation model resulting in an informative hypothesis. The... more
Researchers in the behavioural and social sciences often have expectations that can be expressed in the form of inequality constraints among the parameters of a structural equation model resulting in an informative hypothesis. The question they would like an answer to is "Is the Hypothesis Correct" or "Is the hypothesis incorrect?". We demonstrate a Bayesian approach to compare an inequality-constrained hypothesis with its complement in an SEM framework. The method is introduced and its utility is illustrated by means of an example. Furthermore, the influence of the specification of the prior distribution is examined. Finally, it is shown how the approach proposed can be implemented using Mplus.
Research Interests:
Research Interests:
Research Interests:
... β a vector of length D=P(J+K), which is the vectorization of B, that is, vec(B)=β, View the MathML source is the maximum likelihood estimator of β, and (4) View the MathML source Expression (3) shows that the likelihood is... more
... β a vector of length D=P(J+K), which is the vectorization of B, that is, vec(B)=β, View the MathML source is the maximum likelihood estimator of β, and (4) View the MathML source Expression (3) shows that the likelihood is proportional to a normal-inverse-Wishart distribution. ...
Research Interests:
Research Interests:
Bayesian network models offer a large degree of flexibility for modeling dependence among observables (item outcome variables) from the same task, which may be dependent. This article explores four design patterns for modeling locally... more
Bayesian network models offer a large degree of flexibility for modeling dependence among observables (item outcome variables) from the same task, which may be dependent. This article explores four design patterns for modeling locally dependent observations: (a) no context—ignores dependence among observables; (b) compensatory context—introduces a latent variable, context, to model task-specific knowledge and use a compensatory model to combine this with the relevant proficiencies; (c) inhibitor context—introduces a latent variable, context, to model task-specific knowledge and use an inhibitor (threshold) model to combine this with the relevant proficiencies; (d) compensatory cascading—models each observable as dependent on the previous one in sequence. This article explores the four design patterns through experiments with simulated and real data. When the proficiency variable is categorical, a simple Mantel-Haenszel procedure can test for local dependence. Although local dependence can cause problems in the calibration, if the models based on these design patterns are successfully calibrated to data, all the design patterns appear to provide very similar inferences about the students. Based on these experiments, the simpler no context design pattern appears more stable than the compensatory context model, while not significantly affecting the classification accuracy of the assessment. The cascading design pattern seems to pick up on dependencies missed by other models and should be explored with further research.
Research Interests:
Research Interests:
ABSTRACT A new method is proposed for testing multiple hypotheses with equality and inequality constraints on the parameters of interest. The method is based on the fractional Bayes factor with a modification that the updated prior is... more
ABSTRACT A new method is proposed for testing multiple hypotheses with equality and inequality constraints on the parameters of interest. The method is based on the fractional Bayes factor with a modification that the updated prior is centered on the boundary of the constrained parameter space under investigation. The resulting prior adjusted default Bayes factors work as an “Ockham’s razor” when testing inequality constrained hypotheses, which is not the case for the fractional Bayes factor. Two different types of prior adjusted default Bayes factors are considered. In the first type, the updated prior is based on imaginary training data. Analytical and numerical examples show that this criterion converges fastest to a true inequality constrained hypothesis. In the second type, the updated prior is based on empirical training data. This second criterion only outperforms the fractional Bayes factor in the case of small samples.
Research Interests:
Several issues are discussed when testing inequality constrained hypotheses using a Bayesian approach. First, the complexity (or size) of the inequality constrained parameter spaces can be ignored. This is the case when using the... more
Several issues are discussed when testing inequality constrained hypotheses using a Bayesian approach. First, the complexity (or size) of the inequality constrained parameter spaces can be ignored. This is the case when using the posterior probability that the inequality constraints of a hypothesis hold, Bayes factors based on non-informative improper priors, and partial Bayes factors based on posterior priors. Second, the Bayes factor may not be invariant for linear one-to-one transformations of the data. This can be observed when using balanced priors which are centred on the boundary of the constrained parameter space with a diagonal covariance structure. Third, the information paradox can be observed. When testing inequality constrained hypotheses, the information paradox occurs when the Bayes factor of an inequality constrained hypothesis against its complement converges to a constant as the evidence for the first hypothesis accumulates while keeping the sample size fixed. This paradox occurs when using Zellner's g prior as a result of too much prior shrinkage. Therefore, two new methods are proposed that avoid these issues. First, partial Bayes factors are proposed based on transformed minimal training samples. These training samples result in posterior priors that are centred on the boundary of the constrained parameter space with the same covariance structure as in the sample. Second, a g prior approach is proposed by letting g go to infinity. This is possible because the Jeffreys-Lindley paradox is not an issue when testing inequality constrained hypotheses. A simulation study indicated that the Bayes factor based on this g prior approach converges fastest to the true inequality constrained hypothesis.
Research Interests: Psychology and Statistics
Default Bayes Factors for Type B Testing Problems Joris Mulder∗and Herbert Hoijtink∗ Abstract ... Other default Bayes factors that have been proposed are the Bayes factor based on the expected-posterior prior (Pérez & Berger,... more
Default Bayes Factors for Type B Testing Problems Joris Mulder∗and Herbert Hoijtink∗ Abstract ... Other default Bayes factors that have been proposed are the Bayes factor based on the expected-posterior prior (Pérez & Berger, 2002), the posterior Bayes factor (Aitkin, 1991), ...
Most researchers have specific expectations concerning their research questions. These may be derived from theory, empirical evidence, or both. Yet despite these expectations, most investigators still use null hypothesis testing to... more
Most researchers have specific expectations concerning their research questions. These may be derived from theory, empirical evidence, or both. Yet despite these expectations, most investigators still use null hypothesis testing to evaluate their data, that is, when analysing their data they ignore the expectations they have. In the present article, Bayesian model selection is presented as a means to evaluate the expectations researchers have, that is, to evaluate so called informative hypotheses. Although the methodology to do this has been described in previous articles, these are rather technical and have mainly been published in statistical journals. The main objective of the present article is to provide a basic introduction to the evaluation of informative hypotheses using Bayesian model selection. Moreover, what is new in comparison to previous publications on this topic is that we provide guidelines on how to interpret the results. Bayesian evaluation of informative hypotheses is illustrated using an example concerning psychosocial functioning and the interplay between personality and support from family.
Research Interests:
Most researchers have specific expectations concerning their research questions. These may be derived from theory, empirical evidence, or both. Yet despite these expectations, most investigators still use null hypothesis testing to... more
Most researchers have specific expectations concerning their research questions. These may be derived from theory, empirical evidence, or both. Yet despite these expectations, most investigators still use null hypothesis testing to evaluate their data, that is, when analysing their data they ignore the expectations they have. In the present article, Bayesian model selection is presented as a means to evaluate the expectations researchers have, that is, to evaluate so called informative hypotheses. Although the methodology to do this ...
Research Interests:
In the present article we illustrate a Bayesian method of evaluating informative hypotheses for regression models. Our main aim is to make this method accessible to psychological researchers without a mathematical or Bayesian background.... more
In the present article we illustrate a Bayesian method of evaluating informative hypotheses for regression models. Our main aim is to make this method accessible to psychological researchers without a mathematical or Bayesian background. The use of informative hypotheses is illustrated using two datasets from psychological research. In addition, we analyze generated datasets with manipulated differences in effect size to investigate how Bayesian hypothesis evaluation performs when the magnitude of an effect changes. After reading this article the reader is able to evaluate his or her own informative hypotheses.
Research Interests:
When analyzing repeated measurements data, researchers often have expectations about the relations between the measurement means. The expectations can often be formalized using equality and inequality constraints between (i) the... more
When analyzing repeated measurements data, researchers often have expectations about the relations between the measurement means. The expectations can often be formalized using equality and inequality constraints between (i) the measurement means over time,(ii) the measurement means between groups,(iii) the means adjusted for time-invariant covariates, and (iv) the means adjusted for time-varying covariates. The result is a set of informative hypotheses. In this paper, the Bayes factor is used to determine which ...