Erling B. Andersen

A goodness of fit test for the rasch model

The Rasch model is an item analysis model with logistic item characteristic curves of equal slope,i.e. with constant item discriminating powers. The proposed goodness of fit test is based on a comparison between difficulties estimated from different scoregroups and over-all estimates.Based on the within scoregroup estimates and the over-all estimates of item difficulties a conditional likelihood ratio is formed. It is shown that—2 times the logarithm of this ratio isx2-distributed when the Rasch model is true.The power of the proposed goodness of fit test is discussed for alternative models with logistic item characteristic curves, but unequal discriminating items from a scholastic aptitude test.

Sufficient statistics and latent trait models

For questionnaires with two answer categories, it has been proven in complete generality that if a minimal sufficient statistic exists for the individual parameter and if it is the same statistic for all values of the item parameters, then the raw score (or the number of correct answers) is the minimal sufficient statistic. It follows that the model must by of the Rasch type with logistic item characteristic curves and equal item-discriminating powers.This paper extends these results to multiple choice questionnaires. It is shown that the minimal sufficient statistic for the individual parameter is a function of the so-called score vector. It is also shown that the so-called equidistant scoring is the only scoring of a questionnaire that allows for a real valued sufficient statistic that is independent of the item parameters, if a certain ordering property for the sufficient statistic holds.

Environmental effects of agri-environmental schemes in Western Europe

Agri-environmental schemes (AES) have been introduced as part of European Unions (EU) Common Agricultural Policy and are now an important part of this. A methodological approach to analyse the policy effects of AES is outlined, in which we distinguish between performance effects (on agricultural practices) and outcome effects (environmental impact). The performance effects are further approached including measurement of improvement and protection effects based on 12 indicators on changes/maintenance of land use and agricultural management. Data from personal interviews of participating and non-participating farmers in AES measures in nine EU Member States and Switzerland were used to analyse policy effects, including single indicator effects on agricultural practices as well as combined effects at the agreement level. Significant effects were found for mineral N-fertiliser use, stocking density reduction, maintenance of a minimum livestock density and pesticides. For AES agreements regulating grassland management, fertiliser use and pesticides, clear indications of combined improvement and protection effects were found. In addition clear improvement effects of agreements regulating fertiliser and pesticides use on mainly arable lands were revealed. It is concluded that the approach presented including the 12 selected indicators has proven to be operational.

The statistical analysis of categorical data

This book is about the analysis of categorical data with special emphasis on applications in economics, political science and the social sciences. The book gives a brief theoretical introduction to log-linear modeling of categorical data, then gives an up-to-date account of models and methods for the statistical analysis of categorical data, including recent developments in logistic regression models, correspondence analysis and latent structure analysis. Also treated are the RC association models brought to prominence in recent years by Leo Goodman. New statistical features like the use of association graphs, residuals and regression diagnostics are carefully explained, and the theory and methods are extensively illustrated by real-life data. The book introduces readers to the latest developments in categorical data analysis, and are shown how real life data can be analysed, how conclusions are drawn and how models are modified.

Estimating the parameters of the latent population distribution

Under consideration is a test battery of binary items. The responses ofn individuals are assumed to follow a Rasch model. It is further assumed that the latent individual parameters are distributed within a given population in accordance with a normal distribution. Methods are then considered for estimating the mean and variance of this latent population distribution. Also considered are methods for checking whether a normal population distribution fits the data. The developed methods are applied to data from an achievement test and from an attitude test.

Estimating latent correlations between repeated testings

A model for longitudinal latent structure analysis is proposed. We assume that test scores for a given mental or attitudinal test are observed for the same individuals at two different points in time. The purpose of the analysis is to fit a model that combines the values of the latent variable at the two time points in a two-dimensional latent density. The correlation coefficient between the two values of the latent variable can then be estimated. The theory and methods are illustrated by a Danish dataset concerning psychic vulnerability.

Introduction to the statistical analysis of categorical data

This book deals with the analysis of categorical data. Statistical models, especially log-linear models for contingency tables and logistic regression, are described and applied to real life data. Special emphasis is given to the use of graphical methods. The book is intended as a text for both undergraduate and graduate courses for statisticians, applied statisticians, social scientists, economists and epidemiologists. Many examples and exercises with solutions should help the reader to understand the material.

The Logit Model

In chapters 4, 5 and 6 the categorical variables appeared in the model in a symmetrical way. In many situations, for example in examples 6.1 and 6.2 in chapter 6, one of the variable is of special interest. For the survival data in example 6.1, survival is the variable of special interest, and the problem is to study if the other three variables have influenced the chance of survival. Variable B in example 6.1 may, therefore, be called a response variable and variables A, C and D explanatory variables. This terminology is the same as the one used in regression analysis, and when survival is regarded as a response variable the data in example 6.1 can in fact be analysed by a regression model. In example 6.2 the position on the truck of the collision can be regarded as a response variable. We are here primarily interested in the effect of explanatory variable A, i.e. the introduction of the safety measure in November 1971, but have to take into account that the other explanatory variables, i.e. whether the truck was parked or not and what the light conditions were, may be of importance for the location of the collision. When the response variable is binary and the explanatory variables are categorical, the appropriate regression model is known as the logit model. More precisely the assumptions for a logit model are: (a) The response variable is binary. (b) The contingency table formed by the reponse variable and the explanatory variables can be described by a log-linear model.

A Generic Bio-Economic Farm Model for Environmental and Economic Assessment of Agricultural Systems

Bio-economic farm models are tools to evaluate ex-post or to assess ex-ante the impact of policy and technology change on agriculture, economics and environment. Recently, various BEFMs have been developed, often for one purpose or location, but hardly any of these models are re-used later for other purposes or locations. The Farm System Simulator (FSSIM) provides a generic framework enabling the application of BEFMs under various situations and for different purposes (generating supply response functions and detailed regional or farm type assessments). FSSIM is set up as a component-based framework with components representing farmer objectives, risk, calibration, policies, current activities, alternative activities and different types of activities (e.g., annual and perennial cropping and livestock). The generic nature of FSSIM is evaluated using five criteria by examining its applications. FSSIM has been applied for different climate zones and soil types (criterion 1) and to a range of different farm types (criterion 2) with different specializations, intensities and sizes. In most applications FSSIM has been used to assess the effects of policy changes and in two applications to assess the impact of technological innovations (criterion 3). In the various applications, different data sources, level of detail (e.g., criterion 4) and model configurations have been used. FSSIM has been linked to an economic and several biophysical models (criterion 5). The model is available for applications to other conditions and research issues, and it is open to be further tested and to be extended with new components, indicators or linkages to other models.

Polytomous Rasch Models and their Estimation

In this chapter, the polytomous Rasch model is introduced, based on the original formulation by Georg Rasch in the 1960 Berkeley Symposium on Mathematical Statistics and Probability. The various versions of the basic model, suggested in the literature, are briefly mentioned and compared. The main part of the chapter deals with estimation problems, theoretically as well as numerically. The connection to the RC-association models is discussed, and finally the theory is illustrated by means of data from a Danish psychiatric rating scale.