Goodness of Fit in Item Factor Analysis: Effect of the Number of Response Alternatives

Abstract

The power of the chi-square test statistic used in structural equation modeling decreases as the absolute value of excess kurtosis of the observed data increases. Excess kurtosis is more likely the smaller the number of item response categories. As a result, fit is likely to improve as the number of item response categories decreases, regardless of the true underlying factor structure or χ2-based fit index used to examine model fit. Equivalently, given a target value of approximate fit (e.g., root mean square error of approximation ≤ .05) a model with more factors is needed to reach it as the number of categories increases. This is true regardless of whether the data are treated as continuous (common factor analysis) or as discrete (ordinal factor analysis). We recommend using a large number of response alternatives (≥ 5) to increase the power to detect incorrect substantive models.