How do I transform IRT discrimination parameters of a 2PL model into factor loadings?

Question

I estimated a 2PL model for ordinal data and got discrimination as well as difficulty parameters for each item. To be able to estimate a cut-off value for the discrimination parameter, I wanted to transform it to a factor loading metric. In another thread on here I found the formula: $\frac{\alpha}{\sqrt{(1+\alpha^2)}}$, with alpha being the discrimination parameter. But I´m not quite sure if this is the right formula for a 2PL model? The factor loadings seem to be pretty high. Unfortunately I cannot open the literature given in the other thread and cant seem to find the formula anywhere.

Answer 1

The formula you provided, $\lambda$ = $\frac{\alpha}{\sqrt{1+\alpha^2}}$, is the formula to transform Item Response Theory (IRT) slope/discrimination parameters of a normal ogive model (e.g., the 2-parameter normal ogive model for dichotomous items) to the Factor Analysis (FA) "factor loading" metric. You appear to be looking for the formula to transform the IRT slope/discrimination parameter estimates obtained from the 2PL model onto the FA metric. This is done by using a scaling constant D, usually set to 1.7 or 1.702$^1$, in the formula, or $\lambda$ = $\frac{\alpha/D}{\sqrt{1+(\alpha/D)^2}}$.

A possible reason why you had trouble finding the formula is because D is largely a historic artifact$^2$ (Wirth & Edwards, 2007). Early IRT models were parameterized using the normal ogive (e.g., Bock & Lieberman, 1970; Bock & Aitkin, 1981), though more recently, likelihood-based estimation methods have used the 2PL model for computational reasons. Nowadays, most applications of normal ogive-based IRT models use Bayesian estimation methods, as the computational advantages of the 2PL models no longer apply$^3$. For more information on the similarities and differences between FA and IRT model parameterizations, see Wirth & Edwards (2007) and Kamata & Bauer (2008).

Finally, it should be noted that the proof of equivalence between FA and IRT models (Takane & De Leeuw, 1987) depends on the normality of IRT ability and FA factors. For more information on this, see Cho (2023).

$^1$ The constants 1.7 and 1.702 are values of the minimax constant rounded to 1 and 3 decimals respectively. The value of the minimax constant rounded to 5 decimal places is 1.70174. Further, other constants exist. For example Savalei (2006) proposed 1.749, which is based on minimizing the Kullback-Leibler (KL) information. Though in practice, the minimax constant is used almost exclusively.

$^2$ For more information on the history of the scaling constant see Camilli (1994) and Camilli (2017).

$^3$ This is covered in depth in Baker & Kim (2004).

References

Baker, F. B., & Kim, S. H. (Eds.). (2004). Item response theory: Parameter estimation techniques. CRC press.

Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46(4), 443-459.

Bock, R., & Lieberman, M. (1970). Fitting a response model for n dichotomously scored items. Psychometrika, 35(2), 179-197.

Camilli, G. (1994). Teacher’s corner: origin of the scaling constant d= 1.7 in item response theory. Journal of Educational Statistics, 19(3), 293-295.

Camilli, G. (2017). The scaling constant D in item response theory. Open Journal of Statistics, 7(05), 780.

Cho, E. (2023). Interchangeability between factor analysis, logistic IRT, and normal ogive IRT. Frontiers in Psychology, 14, 1267219.

Kamata, A., & Bauer, D. J. (2008). A note on the relation between factor analytic and item response theory models. Structural Equation Modeling: A Multidisciplinary Journal, 15(1), 136-153.

Savalei, V. (2006). Logistic approximation to the normal: The KL rationale. Psychometrika, 71, 763-767.

Takane, Y., & De Leeuw, J. (1987). On the relationship between item response theory and factor analysis of discretized variables. Psychometrika, 52(3), 393-408.

Wirth, R. J., & Edwards, M. C. (2007). Item factor analysis: current approaches and future directions. Psychological methods, 12(1), 58.