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- Karl Bang Christensen (Author of Rasch Models in Health)
- Introduction
- Rasch Models in Health: Christensen/Rasch Models in Health

In the RSM, the response categories are defined identically for all items, whereas they are allowed to differ in the PCM e. These measures, called thresholds, represent the point on the latent variable where adjacent response categories are equally probable.

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The thresholds express the amount of the latent variable covered by each response category and, therefore, the probability of the response category itself. Each cell of X contains the response x vi of patient v to item i. In repeated measures designs, two matrices X 1 and X 2 contain the responses observed at Time 1 and Time 2, respectively.

A seemingly straightforward approach to the measurement of change would consist of running two separate Rasch analyses on X 1 and X 2. The intra-patient differences between the patient measures could then be used as measures of individual change. Such an approach might not be feasible in practice.

Between the two time points, not only the patients might have changed but also the functioning of the instrument. The intervention does not affect the responses to all items equally, but it more strongly influences the items it is directly related to. The use of the response categories might differ across the two time points as an effect of the different health statuses of the patients before and after the intervention.

These changes would make the meaning of change uncertain. For the measurement of change to have an unambiguous numerical representation and a substantive meaning, the patient measures should be estimated and compared within a common frame of reference encompassing both time points [ 6 ]. In such a frame of reference, instrument changes are controlled by fixing the item and threshold measures to be equal in the two time points. Two approaches are available [ 25 , 26 ]. In the first one, the data from a time point are analyzed to obtain the patient measures for that time point.

Then, the data from the other time point are analyzed by anchoring the item and threshold measures to the values estimated in the previous analysis. This would provide a set of patient measures for the new time point, which are comparable with the previous ones. This approach requires the explicit identification of a time point as more decisive. If the emphasis is on making decisions about administering the intervention, Time 1 is more decisive, and then, it is measured at Time 1 and anchored at Time 2. If the emphasis is on making decisions about the outcome of the intervention success, failure , Time 2 is more decisive, and then, it is measured at Time 2 and anchored at Time 1.

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The second approach takes the more overall position that both time points are equally important. The data from the two time points are stacked on each other so that each item corresponds to one column and each time point for each patient is a row of the combined data set. In the stacked analysis, the patient measures at Time 1 and Time 2 might be influenced by local dependency across the two time points, if any exists. A simple approach for avoiding such an influence consists of the following steps [ 25 - 27 ]:. For each patient, the data for one of the two time points are selected at random so that each patient is in the selection only once but both time points are equally represented.

The Rasch analysis is run on the selected data. Given that, for each patient, only the data for one time point are considered, there will be no intra-patient dependencies across time points. The Rasch analysis is run on the complete stacked data, with the item and threshold measures anchored at the values that were estimated on the selected data. The anchor values will prevent eventual dependency from distorting the patient measures at the two time points. If the patient measures estimated in the stacked analysis with anchors do not differ from those estimated in the stacked analysis, then the effect of local dependency is negligible, and either one or the other measures can be used indifferently.

Otherwise, the former measures should be used. The analyses were run using the computer program Facets 3. Item 3 the reverse item was rescored prior to the analyses. To investigate whether the functioning of the instrument differed across the two time points, two separate analyses were run on the data collected before and after the interventions. Then, a stacked analysis was run on the data from both time points. This approach was used because it provides a frame of reference for measuring change without having to consider one time point as more important than the other.

## Karl Bang Christensen (Author of Rasch Models in Health)

The influence of local dependency was investigated by comparing the patient measures estimated in the stacked analysis with those estimated in a stacked analysis with anchors. In all analyses, Rasch-based statistics were computed, that provide useful information about the fit of data to the Rasch model, the reliability, and validity of the instrument. Outfit is more sensitive to unexpected responses on items which are far from person measure, whereas Infit is more sensitive to unexpected responses on items which are close to person measure. These statistics were computed for each patient and each item.

Infit and Outfit of the items provide evidence about the construct validity described by Messick [ 32 ]. Values greater than 2 for a particular item suggest that it is badly-formulated and confusing, or that it may measure a construct other than that measured by the other items multidimensionality [ 28 , 31 ].

The item Strata is also computed [ 33 ], which represents the number of statistically distinct groups of item measures that the patients have distinguished. If at least two groups are unable to be identified, then the variable defined by the items is hardly interpretable low construct validity [ 31 ]. Finally, the patient separation reliability R [ 33 ] was computed, which informs about reliability of the instrument. R is the Rasch equivalent of Cronbach alpha.

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It ranges from 0 to 1. The closer the value of R is to 1, the greater the probability that differences among the patient measures express actual differences among the patient health statuses. The Rasch analyses were run on the data of all 98 patients. Infit and Outfit were smaller than 2 for all items. Item Strata ranged from 3.

On the whole, these results suggest that the instrument was reliable, valid, and substantively unidimensional. This section presents the results of the two separate analyses that were run on the data from the two time points. Greater measures indicate more severe items.

In Rasch measurement, extreme response categories always approach a probability of 1 asymptotically because it is assumed that respondents with infinitely high resp. These changes in item severities and response category probabilities make the interpretation of change ambiguous. Rasch analyses run separately on the data from the two time points.

The upper diagram depicts the item measures at Time 2 y axis plotted against those at Time 1 x axis. The lower diagram depicts the thresholds and the probability curves of response categories at Time 1 unbroken line and Time 2 broken line. This section presents the results of the stacked analysis and the stacked analysis with anchors.

Thus, in the present data, local dependency has had a negligible effect on patient measures. The patient measures obtained in the stacked analysis are considered in the following.

### Introduction

Greater measures indicate more healthy patients. Thirteen patients reported a significant improvement from Time 1 to Time 2 circled dots above the identity line , whereas 3 patients reported a significant worsening circled dots below the identity line. Thus, the Rasch analysis provided information at the individual level, allowing the distinction between patients who have improved, worsened, or who have not changed. Rasch analysis run on stacked data from the two time points. The patient measures at Time 2 y axis are plotted against those at Time 1 x axis.

Circled dots indicate statistically significant change. In the present study, the change has been measured at the individual level as well as in groups of patients who received different interventions. Patients have not been investigated with the aim of identifying the specific features of those who improved, worsened, or did not change. Future investigation will be devoted to this purpose. In the present study, precision and meaning of the measurement were derived from the interval level of the measures and the invariance of the instrument across time points.

However, some patients did not fit the Rasch model, so that the validity of their measure is questionable. Further investigation is needed to understand the causes of misfit Do these patients belong to a different population?

## Rasch Models in Health: Christensen/Rasch Models in Health

Do they have filled out the scale inaccurately. Rasch models are especially demanding of data that satisfy the requirements for constructing measures. Two alternative pathways can be pursued when the data do not fit a Rasch model [ 35 ]. The first one consists of modifying the instrument, the definition of the construct under investigation, or both, in order to generate new data that better conform to the model.

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The second one consists of identifying an alternative model, usually within the framework of item response theory, that accounts better for the given data. Research on responsiveness generally presents the patients with a battery of instruments before and after a well-known efficacious intervention and then compares their responsiveness through some indexes which are based on the measurement of patient change. Highly responsive instruments are chosen for applications in clinical trials.

Different indexes may provide different rank orderings of instrument responsiveness [ 37 ]. By taking into account aspects concerning the patients, the items, and the response scale, the Rasch models might provide a relevant contribution to the investigation of responsiveness. There are other Rasch methods to the measurement of change [ 38 - 42 ], that have not been taken into account in the present study. Future studies should compare them in health fields experiencing different degrees and direction of change.

Competing interests. PA participated in the design of the study, performed and interpreted the statistical analyses and drafted the manuscript. GV participated in the design of the study, helped to interpret the statistical analyses and drafted the manuscript. OB participated in the design of the study, contributed to the data collection and revised the manuscript critically.

GB participated in the design of the study, contributed to the data collection and revised the manuscript critically. All authors read and approved the final manuscript. Pasquale Anselmi, Email: ti. He majored in mathematical statistics in , and worked on and off at the Danish Institute of Educational Research. I Probabilistic models 1 1 The Rasch model for dichotomous items 3 1.

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Analyzing and Reporting Rasch Models Software for Rasch Analysis, Mounir Mesbah. Reporting a Rasch Analysis, Thomas Salzberger. With a background in mathematical statistics he has worked mainly within Biostatistics and Epidemiology. Inspired by the issue of measurement in social and health sciences he has published methodological work about Rasch models in journals such as Applied Psychological Measurement, the British Journal of Mathematical and Statistical Psychology and Psychometrika.

He has for some years tried to combine his interest in Rasch models with his interest in graphical models for categorical data and has developed a family of Rasch-related models that he refers to as graphical loglinear Rasch models in which several of the problems with Rasch models for social and health science data have been resolved. Rasch Models in Health. Description The family of statistical models known as Rasch models started with a simple model for responses to questions in educational tests presented together with a number of related models that the Danish mathematician Georg Rasch referred to as models for measurement.