C. Investigation of DimensionalityAll of the models that we will be describing on this web site will make one common assumption: That the constructs that we are measuring are unidimensional, that is they have one prominent factor underlying them, and that all other factors are functionally insignificant. A number of approaches have been proposed for assessing unidimensionality (see Hattie [1985] for a review). Commonly used methods include determining the number of eigenvalues greater than one, examining scree plots, and considering the ratio of the first eigenvalue to the second. Some techniques, such as Modified Parallel Analysis (MPA; Drasgow & Lissak, 1984) and DIMTEST (Stout, 1987), were specifically developed for IRT analyses. Interested readers may consult original sources to learn about these relatively complex procedures.In our tutorial, we focus on the scree plot method that uses factor analytic techniques to assess unidimensionality. We suggest using Principle Axis Factoring (PAF) in determining the number of factors in a measure as an alternative to Principal Components Analysis (PCA). We make this suggestion due to the fact that factors in PAF only account for common variance, as opposed to common and unique variance in PCA. Furthermore, estimates of the communalities (the squared multiple correlations of a variable with the other variables in the matrix) are in the diagonal of the matrix to be analyzed instead of 1's. PAF is widely available through the common statistical analysis software packages. In examining the results of a PAF analysis in light of the IRT models presented in this tutorial, we want to compare the magnitude of the first and second eigenvalues. The first value should be significantly higher than the second. Many software packages provide a scree plot to help in the evaluation of the contribution of factors other than the first factor. We will present scree plots in our examples that follow. We will now walk through two examples of investigating dimensionality with the Agreeableness scale via PAF. For these examples -- and for the examples that follow pertaining to estimating IRT parameters -- we will be using our sample data set available in the downloads section and described in the data preparation section. There are two broad categories of IRT models, those dealing with dichotomously scored measures (or binary data, much like what we find with the SAT or ACT) and those dealing with polytomous data (where more than 2 options are recorded and considered, as with Likert scales or biodata measures). When investigating dimensionality, it is important to consider the nature of your data, as an appropriate factor analysis is dependent on this:
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