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A general random effects model is proposed that allows for continuous as well as discrete distributions of the responses. Responses can be unrestricted continuous, bounded continuous, binary, ordered categorical or given in the form of counts. The distribution of the responses is not restricted to exponential families, which is a severe restriction in generalized mixed models. Generalized mixed models use fixed distributions for responses, for example the Poisson distribution in count data, which has the disadvantage of not accounting for overdispersion. By using a response function and a thresholds function the proposed mixed thresholds model can account for a variety of alternative distributions that often show better fits than fixed distributions used within the generalized linear model framework. A particular strength of the model is that it provides a tool for joint modeling, responses may be of different types, some can be discrete, others continuous. In addition to introducing the mixed thresholds model parameter sparsity is addressed. Random effects models can contain a large number of parameters, in particular if effects have to be assumed as measurement-specific. Methods to obtain sparser representations are proposed and illustrated. The methods are shown to work in the thresholds model but could also be adapted to other modeling approaches.

A strong tool for the selection of items that share a common trait from a set of given items is proposed. The selection method is based on marginal estimates and exploits that the estimates of the standard deviation of the mixing distribution are rather stable if items are from a Rasch model with a common trait. If, however, the item set is increased by adding items that do not share the latent trait the estimated standard deviations become distinctly smaller. A method is proposed that successively increases the set of items that are considered Rasch items by examining the estimated standard deviations of the mixing distribution. It is demonstrated that the selection procedure is on average very reliable and a criterion is proposed, which allows to identify items that should not be considered Rasch items for concrete item sets. An extension of the method allows to investigate which groups of items might share a common trait. The corresponding hierarchical clustering procedure is considered an exploratory tool but works well on average.

The Rasch model is the most prominent member of the class of latent trait models that are in common use. The main reason is that it can be considered as a measurement model that allows to separate person and item parameters, a feature that is referred to as invariance of comparisons or specific objectivity. It is shown that the property is not an exclusive trait of Rasch type models but is also found in alternative latent trait models. It is distinguished between separability in the theoretical measurement model and empirical separability with empirical separability meaning that parameters can be estimated without reference to the other group of parameters. A new type of pairwise estimator with this property is proposed that can be used also in alternative models. Separability is considered in binary models as well as in polytomous models.

A general framework of latent trait item response models for continuous responses is given. In contrast to classical test theory models, which traditionally distinguish between true scores and error scores, the responses are clearly linked to latent traits. It is shown that classical test theory models can be derived as special cases but the model class is much wider. It provides, in particular, appropriate modelling of responses that are restricted in some way, for example, if responses are positive or are restricted to an interval. Restrictions of this sort are easily incorporated in the modeling framework. Restriction to an interval is typically ignored in common models yielding inappropriate models, for example, when modeling Likert-type data. The model also extends common response time models, which can be treated as special cases. Properties of the model class are derived and the role of the total score is investigated, which leads to a modified total score. Several applications illustrate the use of the model including an example, in which covariates that may modify the response are taken into account.

Non-probability sampling, for example in the form of online panels, has become a fast and cheap method to collect data. While reliable inference tools are available for classical probability samples, non-probability samples can yield strongly biased estimates since the selection mechanism is typically unknown. We propose a general method how to improve statistical inference when in addition to a probability sample data from other sources, which have to be considered non-probability samples, are available. The method uses specifically tailored regression residuals to enlarge the original data set by including observations from other sources that can be considered as stemming from the target population. Measures of accuracy of estimates are obtained by adapted bootstrap techniques. It is demonstrated that the method can improve estimates in a wide range of scenarios. For illustrative purposes, the proposed method is applied to two data sets.

A comprehensive class of models is proposed that can be used for continuous, binary, ordered categorical and count type responses. The difficulty of items is described by difficulty functions, which replace the item difficulty parameters that are typically used in item response models. They crucially determine the response distribution and make the models very flexible with regard to the range of distributions that are covered. The model class contains several widely used models as the binary Rasch model and the graded response model as special cases, allows for simplifications, and offers a distribution free alternative to count type items. A major strength of the models is that they can be used for mixed item formats, when different types of items are combined to measure abilities or attitudes. It is an immediate consequence of the comprehensive modeling approach that allows that difficulty functions automatically adapt to the response distribution. Basic properties of the model class are shown. Several real data sets are used to illustrate the flexibility of the models.

A general class of models is proposed that is able to estimate the whole predictive distribution of a dependent variable $Y$ given a vector of explanatory variables $\xb$. The models exploit that the strength of explanatory variables to distinguish between low and high values of the dependent variable may vary across the thresholds that are used to define low and high. Simple linear versions of the models are generalizations of classical linear regression models but also of widely used ordinal regression models. They allow to visualize the effect of explanatory variables in the form of parameter functions. More general models are based on efficient nonparametric approaches like random forests, which are more flexible and are strong prediction tools. A general estimation method is given that can use all the estimation tools that have been proposed for binary regression, including selection methods like the lasso or elastic net. For linearly structured models maximum likelihood estimates are derived. The usefulness of the models is illustrated by simulations and several real data set.

Existing ordinal trees and random forests typically use scores that are assigned to the ordered categories, which implies that a higher scale level is used. Versions of ordinal trees are proposed that take the scale level seriously and avoid the assignment of artificial scores. The basic construction principle is based on an investigation of the binary models that are implicitly used in parametric ordinal regression. These building blocks can be fitted by trees and combined in a similar way as in parametric models. The obtained trees use the ordinal scale level only. Since binary trees and random forests are constituent elements of the trees one can exploit the wide range of binary trees that have already been developed. A further topic is the potentially poor performance of random forests, which seems to have ignored in the literature. Ensembles that include parametric models are proposed to obtain prediction methods that tend to perform well in a wide range of settings. The performance of the methods is evaluated empirically by using several data sets.

Different voters behave differently, different governments make different decisions, or different organizations are ruled differently. Many research questions important to political scientists concern choice behavior, which involves dealing with nominal-scale dependent variables. Drawing on the principle of maximum random utility, we propose a flexible and general heterogeneous multinomial logit model for studying differences in choice behavior. The model systematically accounts for heterogeneity that is not captured by classical models, indicates the strength of heterogeneity, and permits examining which explanatory variables cause heterogeneity. As the proposed approach allows incorporating theoretical expectations about heterogeneity into the analysis of nominal dependent variables, it can be applied to a wide range of research problems. Our empirical example uses data on multiparty elections to demonstrate the benefits of the model in the study of heterogeneity in spatial voting.

A common framework is provided that comprises classical ordinal item response models as the cumulative, sequential and adjacent categories models as well as nominal response models and item response tree models. The taxonomy is based on the ways binary models can be seen as building blocks of the various models. In particular one can distinguish between conditional and unconditional model components. Conditional models are by far the larger class of models containing the adjacent categories model and the whole class of hierarchically structured models. The latter is introduced as a class of models that comprises binary trees and hierarchically structured models that use ordinal models conditionally. The study of the binary models contained in latent trait models clarifies the relation between models and the interpretation of item parameters. It is also used to distinguish between ordinal and nominal models by giving a conceptualization of ordinal models. The taxonomy differs from previous taxonomies by focusing on the structured use of dichotomizations instead of the role of parameterizations.

The potential of location-shift models to find adequate models between the proportional odds model and the non-proportional odds model is investigated. It is demonstrated that these models are very useful in ordinal modeling. While proportional odds models are often too simple, non proportional odds models are typically unnecessary complicated and seem widely dispensable. The class of location-shift models is also extended to allow for smooth effects. The additive location-shift model contains two functions for each explanatory variable, one for the location and one for dispersion. It is much sparser than hard-to-handle additive models with category-specific covariate functions but more flexible than common vector generalized additive models.

A flexible semiparametric class of models is introduced that offers an alternative to classical regression models for count data as the Poisson and negative binomial model, as well as to more general models accounting for excess zeros that are also based on fixed distributional assumptions. The model allows that the data itself determine the distribution of the response variable, but, in its basic form, uses a parametric term that specifies the effect of explanatory variables. In addition, an extended version is considered, in which the effects of covariates are specified nonparametrically. The proposed model and traditional models are compared by utilizing several real data applications., Comment: 24 pages, 8 figures, 4 tables

In binary and ordinal regression one can distinguish between a location component and a scaling component. While the former determines the location within the range of the response categories, the scaling indicates variance heterogeneity. In particular since it has been demonstrated that misleading effects can occur if one ignores the presence of a scaling component it is important to account for potential scaling effects in the regression model, which is not possible in available recursive partitioning methods. The proposed recursive partitioning method yields two trees, one for the location and one for the scaling. They show in a simple interpretable way how variables interact to determine the binary or ordinal response. The developed algorithm controls for the global significance level and automatically selects the variables that have an impact on the response. The modelling approach is illustrated by several real-world applications.

Ordered item response models that are in common use can be divided into three groups, cumulative, sequential and adjacent categories model. The derivation and motivation of the models is typically based on the assumed presence of latent traits or underlying process models. In the construction frequently binary models play an important role. The objective of this paper is to give motivations for the models and to clarify the role of the binary models for the various types of ordinal models. It is investigated which binary models are included in an ordinal model but also how the models can be constructed from a sequence of binary models. In all the models one finds a Guttman space structure, which has previously been investigated in particular for the partial credit model. The consideration of the binary models adds to the interpretation of model parameters, which is helpful, in particular, in the case of the partial credit model, for which interpretation is less straightforward than for the other models. A specific topic that is addressed is the ordering of thresholds in the partial credit model because for some researchers reversed ordering is an anomaly, others disagree. It is argued that the ordering of thresholds is not a constitutive element of the partial credit model.

Although various polytomous item response models are considered to be ordinal models there seems no general definition of an ordinal model available. Alternative concepts of ordinal models are discussed and it is shown that they coincide for classical unidimensional models. For multidimensional models the definition of an ordinal model refers to specific traits in the multidimensional space of traits. The objective is to provide a theoretical framework for ordinal models. Practical considerations concerning the strength of the link between the latent trait and the order of categories are considered briefly.

Missing values are a common phenomenon in all areas of applied research. While various imputation methods are available for metrically scaled variables, methods for categorical data are scarce. An imputation method that has been shown to work well for high dimensional metrically scaled variables is the imputation by nearest neighbor methods. In this paper, we extend the weighted nearest neighbors approach to impute missing values in categorical variables. The proposed method, called $\mathtt{wNNSel_{cat}}$, explicitly uses the information on association among attributes. The performance of different imputation methods is compared in terms of the proportion of falsely imputed values. Simulation results show that the weighting of attributes yields smaller imputation errors than existing approaches. A variety of real data sets is used to support the results obtained by simulations.

The varying-coefficient model is a strong tool for the modelling of interactions in generalized regression. It is easy to apply if both the variables that are modified as well as the effect modifiers are known. However, in general one has a set of explanatory variables and it is unknown which variables are modified by which covariates. A recursive partitioning strategy is proposed that is able to deal with the complex selection problem. The tree-structured modelling yields for each covariate, which is modified by other variables, a tree that visualizes the modified effects. The performance of the method is investigated in simulations and two applications illustrate its usefulness., Comment: 20 pages, 4 figures, 10 tables

During the last decades, many methods for the analysis of functional data including classification methods have been developed. Nonetheless, there are issues that have not been adressed satisfactorily by currently available methods, as, for example, feature selection combined with variable selection when using multiple functional covariates. In this paper, a functional ensemble is combined with a penalized and constrained multinomial logit model. It is shown that this synthesis yields a powerful classification tool for functional data (possibly mixed with non-functional predictors), which also provides automatic variable selection. The choice of an appropriate, sparsity-inducing penalty allows to estimate most model coefficients to exactly zero, and permits class-specific coefficients in multiclass problems, such that feature selection is obtained. An additional constraint within the multinomial logit model ensures that the model coefficients can be considered as weights. Thus, the estimation results become interpretable with respect to the discriminative importance of the selected features, which is rated by a feature importance measure. In two application examples, data of a cell chip used for water quality monitoring experiments and phoneme data used for speech recognition, the interpretability as well as the selection results are examined. The classification performance is compared to various other classification approaches which are in common use., Comment: The first replacement is due to an update of the data links. No other changes took place with respect to the original submission. To reproduce results of the cell chip or phoneme data application, files can now be downloaded from http://agfda.userweb.mwn.de/files/Fuchs_Poessnecker_Tutz_pcMLM_code_cellchip.zip or http://agfda.userweb.mwn.de/files/Fuchs_Poessnecker_Tutz_pcMLM_code_phonemes.zip

Various methods to detect differential item functioning (DIF) in item response models are available. However, most of the methods assume that the responses are binary, for ordered response categories available methods are scarce. In the present paper DIF in the widely used partial credit model is investigated. An item-focussed tree is proposed that allows to detect DIF-items, which might affect the performance of the partial credit model. The method uses tree methodology yielding a tree for each item that is detected as DIF-item. The resulting trees show which variables induce DIF and in which way. The visualization as trees makes the results easily accessible. The method is compared to an alternative approach, simulations demonstrate the performance of the method and an application illustrates how it works for real data., Comment: 28 pages, 10 figures, 6 tables

Fixed effects models are very flexible because they do not make assumptions on the distribution of effects and can also be used if the heterogeneity component is correlated with explanatory variables. A disadvantage is the large number of effects that have to be estimated. A recursive partitioning (or tree based) method is proposed that identifies clusters of units that share the same effect. The approach reduces the number of parameters to be estimated and is useful in particular if one is interested in identifying clusters with the same effect on a response variable. It is shown that the method performs well and outperforms competitors like the finite mixture model in particular if the heterogeneity component is correlated with explanatory variables. In two applications the usefulness of the approach to identify clusters that share the same effect is illustrated., Comment: 31 pages, 9 figures, 4 tables

Detection of differential item functioning by use of the logistic modelling approach has a long tradition. One big advantage of the approach is that it can be used to investigate non-uniform DIF as well as uniform DIF. The classical approach allows to detect DIF by distinguishing between multiple groups. We propose an alternative method that is a combination of recursive partitioning methods (or trees) and logistic regression methodology to detect uniform and non-uniform DIF in a nonparametric way. The output of the method are trees that visualize in a simple way the structure of DIF in an item showing which variables are interacting in which way when generating DIF. In addition we consider a logistic regression method in which DIF can by induced by a vector of covariates, which may include categorical but also continuous covariates. The methods are investigated in simulation studies and illustrated by two applications., Comment: 32 pages, 13 figures, 7 tables