Interval Estimation of Item Response Probabilities Along Studied Latent Dimensions

ABSTRACT

An interval estimation procedure is discussed that can be used to evaluate the probability of a particular response for a binary or binary scored item at a pre-specified point along an underlying latent continuum. The item is assumed to: (a) be part of a unidimensional multi-component measuring instrument that may contain also polytomous items, and (b) to follow a one-parameter or two-parameter logistic model. The approach is developed within the framework of latent variable modeling, and can be used in answering theoretically and empirically relevant questions concerning specific response probabilities at certain distances from the latent population mean. The method is readily and widely applicable with popular statistical software, and is illustrated using an empirical example.

KEYWORDS:

• Binary or binary scored item
• confidence interval
• logit
• one-parameter logistic model
• two-parameter logistic model
• response probability

Notes

1. It could be argued that any of the conditional probabilities appearing on the left-hand sides of Equations (2)(2) $P^{\ast }(Y_1=1|{\theta }_0)=exp[{a_1}^{\ast }({\theta }_0-{b_1}^{\ast })\left]/\{1+exp\right[{a_1}^{\ast }({\theta }_0-{b_1}^{\ast })]\}$(2) through (5(5) $P^{\ast }(Y_1=1|{\theta }_0=-1.5)=exp\left[a^{\ast }\left(-1.5-b^{\ast }\right)\right]/\left\{1+exp\left[a^{\ast }\left(-1.5-b^{\ast }\right)\right]\right\}$(5) ) is less than well defined. The basis of such an argument could be the fact that the probability of the condition stated in any of them is 0 because the underlying latent variable, θ, is continuous to begin with, while the concept of conditional probability is formally defined only for conditioning events with positive probability (e.g., Raykov & Marcoulides, 2012). We do not agree with possible implications of the latter argument as yielding meaningless probabilities in Equations (2)(2) $P^{\ast }(Y_1=1|{\theta }_0)=exp[{a_1}^{\ast }({\theta }_0-{b_1}^{\ast })\left]/\{1+exp\right[{a_1}^{\ast }({\theta }_0-{b_1}^{\ast })]\}$(2) through (5(5) $P^{\ast }(Y_1=1|{\theta }_0=-1.5)=exp\left[a^{\ast }\left(-1.5-b^{\ast }\right)\right]/\left\{1+exp\left[a^{\ast }\left(-1.5-b^{\ast }\right)\right]\right\}$(5) ). In actual fact, we find the probabilities of concern in this paper (which are those in the right-hand sides of Equations (2)(2) $P^{\ast }(Y_1=1|{\theta }_0)=exp[{a_1}^{\ast }({\theta }_0-{b_1}^{\ast })\left]/\{1+exp\right[{a_1}^{\ast }({\theta }_0-{b_1}^{\ast })]\}$(2) through (5(5) $P^{\ast }(Y_1=1|{\theta }_0=-1.5)=exp\left[a^{\ast }\left(-1.5-b^{\ast }\right)\right]/\left\{1+exp\left[a^{\ast }\left(-1.5-b^{\ast }\right)\right]\right\}$(5) )) to be entirely meaningful in the setting of relevance in the article. The reason is that these are probabilities of a certain event (“correct” response) at a given value of the argument θ that they are themselves continuous functions of, being the pertinent item characteristic curves (e.g., Raykov & Marcoulides, 2018). The particular empirical interpretation of any of the probabilities in Equations (2)(2) $P^{\ast }(Y_1=1|{\theta }_0)=exp[{a_1}^{\ast }({\theta }_0-{b_1}^{\ast })\left]/\{1+exp\right[{a_1}^{\ast }({\theta }_0-{b_1}^{\ast })]\}$(2) through (5(5) $P^{\ast }(Y_1=1|{\theta }_0=-1.5)=exp\left[a^{\ast }\left(-1.5-b^{\ast }\right)\right]/\left\{1+exp\left[a^{\ast }\left(-1.5-b^{\ast }\right)\right]\right\}$(5) ) requires expert knowledge in the subject matter area of application and is deferred to the experts in it. At the same time, the formal interpretation of the probabilities of relevance and only interest in this note, is as the value of the probability of “correct” response for a given value of the underlying latent trait that the former is a function of, i.e., as the value of a continuous function for a particular value of its argument (e.g., Stewart, 2006). (As an alternative notation, in lieu of the probabilities in the left-hand sides of Equations (2)(2) $P^{\ast }(Y_1=1|{\theta }_0)=exp[{a_1}^{\ast }({\theta }_0-{b_1}^{\ast })\left]/\{1+exp\right[{a_1}^{\ast }({\theta }_0-{b_1}^{\ast })]\}$(2) through (5(5) $P^{\ast }(Y_1=1|{\theta }_0=-1.5)=exp\left[a^{\ast }\left(-1.5-b^{\ast }\right)\right]/\left\{1+exp\left[a^{\ast }\left(-1.5-b^{\ast }\right)\right]\right\}$(5) ) one could use the joint probabilities of the event on the left of the formal conditioning bar there and that on its right.)

2. An alternative method for obtaining a standard error for the logit of the probability of concern, is based on an application of the bootstrap approach. (The confidence interval of this probability is obtainable then with the same R-function in Appendix 2.) A comparison of the delta method and bootstrap approach specifically with respect to this logit standard error (and resulting confidence interval if need be) goes beyond the confines of this article.

3. For the sake of specific substantive interpretation and illustration of the outlined estimation procedure, in this section one may think of these 5 items evaluating general mental ability say, with item 2 assumed to be representing a figural relations task (and thus tapping into our subability of mentally rotating figures and finding relationships between figure parts; e.g., Horn, 1982; cf. Raykov et al., 2019a).