Rasch is a mathematical model that allows researchers to compare data that measure a unidimensional trait or ability (Bond & Fox, 2007). When data fit the Rasch model, it is mathematically proven that the item difficulty estimates are independent of the sample of respondents. The purpose of this study was to test the robustness of the Rasch model with regards to its ability to maintain invariant item difficulty estimates when real (data that does not perfectly fit the Rasch model), polytomous scored data is used. The data used in this study comes from a university alumni questionnaire that was collected over a period of five years. The analysis tests for significant variation between (a) small samples taken from a larger sample, (b) a base sample and subsequent (longitudinal) samples and (c) variation over time with confounding variables. The confounding variables studied include (a) the gender of the respondent and (b) the respondent's type of major at the time of graduation. The study used three methods to assess variation: (a) the between-fit statistic, (b) confidence intervals around the mean of the estimates and (c) a general linear model. The general linear model used the person residual statistic from the Winsteps' person output file as a dependent variable with year, gender and type of major as independent variables. Results of the study support the invariant nature of the item difficulty estimates when polytomous data from the alumni questionnaire is used. The analysis found comparable results (within sampling error) for the between-fit statistics and the general linear model. The confidence interval method was limited in its usefulness due to small confidence bands and the limitation of the plots. The linear model offered the most valuable data in that it provides methods to not only detect the existence of variation but to assess the relative magnitude of the variation from different sources. Recommendations for future research include studies regarding the impact of sample size on the between-fit statistic and confidence intervals as well as the impact of large amounts of systematic missing data on the item parameter estimates.



College and Department

David O. McKay School of Education; Instructional Psychology and Technology



Date Submitted


Document Type





Rasch, Invariance, Polytomous, Rating Scale, Item Difficulties