Can the One-Parameter Logistic Model Be a Spurious Finding for a Heterogeneous Population?

ABSTRACT

Utilizing the perspective of finite mixture modeling, this note considers whether a finding of a plausible one-parameter logistic model could be spurious for a population with substantial unobserved heterogeneity. A theoretically and empirically important setting is discussed involving the mixture of two latent classes, with the less restrictive two-parameter logistic model holding within each of the classes while the more restrictive one-parameter logistic model is plausible overall. A numerical example is presented then of obtaining the one-parameter logistic model as a spurious result in a finite mixture study. Implications for behavioral and social research are discussed in light of these findings.

KEYWORDS:

• Finite mixture
• one-parameter logistic model
• spurious finding
• restriction
• two-parameter logistic model
• unobserved heterogeneity

Acknowledgment

Thanks are due to T. Asparouhov and S. Embretson for valuable discussions on simulation studies and item response model mixtures.

Notes

1. Using the R-package ‘ltm’, the p-value for the overall goodness of fit test of the Rasch model is found to be p = .86 while the p-value associated with the test of the nested 1PL-model against the 2PL model is p = .439, which is in agreement with the pertinent results presented in the current section in the main text (e.g., Rizopoulos, 2006, 2018; the tests are carried out with the ltm-commands “GoF.rasch” and “anova”, respectively, and the last stated p-value is that associated with the ∆χ²-test reported above in this note). The largest z-value (test statistic value) across the five tests of possible guessing on any of the items within the 1PL-model is thereby 1.113, suggesting no guessing on an individual item; similarly, the overall test statistic for guessing on at least one item is p = .956 and thus non-significant as well. These findings show that formally a guessing (or related chance processes leading to a correct response) explanation cannot act against the reported plausibility of the 1PL-model. Last but not least, the highest two-way marginal residual (ratio of squared difference between observed and expected frequencies under the 1PL-model, to expected frequency) is 1.49 and thus not significant (relative to the pertinent df = 1; we note that these residuals are not independent and so it would be appropriate to de-emphasize this finding if interpreted only as lack of significance). These results provide further evidence in favor of the earlier claim of plausible fit of the 1PL-model (see also main text).