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.

This study compared three methods of detecting test item bias, the chi-square approach, the transformed item difficulties approach, and the Linn-Harnish three-parameter item response approach which is the only Item Response Theory (IRT) method that can be utilized with minority samples relatively small in size. The items on two tests which measured writing and reading skills were examined for evidence of sex and ethnic bias. Eight sets of samples, four from each test, were randomly selected from the population (N=7287) of sixth, seventh, and eighth grade students enrolled in a large, urban school district in the southwestern United States. Each set of samples, male/female, White/Hispanic, White/Black, and White/White, contained 800 examinees in the majority group and 200 in the minority group. In an attempt to control differences in ability that may have existed between the various population groups, examinees with scores greater or less than two standard deviations from their group's mean were eliminated. Ethnic samples contained equal numbers of each sex. The White/White sets of samples were utilized to provide baseline bias estimates because the tests could not logically be biased against these groups. Bias indices were then calculated for each set of samples with each of the three methods. Findings of this study indicate that the percent agreement between the Linn-Harnish IRT method and the chisquare and transformed difficulties methods is similar to that found in previous studies comparing the latter approaches with other IRT methods requiring large minority samples. Therefore, it appears that the Linn-Harnish IRT approach can be used in lieu of other more restrictive IRT methods. Ethnic bias appears to exist in the two tests as measured by the large mean bias indices for the White/Hispanic and White/Black samples. Little sex bias was found as evidenced by the low mean bias indices of the male/ female samples and the fact that the male/female mean bias indices were lower than those of the White/White in 33% of the samples.