Local Independence

Basic Concepts and Importance of Local Independence

Classical true score theory assumes that the observed scores are equal to the true scores plus the error scores (O = T + E). Local independence in classical true score theory means that the error scores are uncorrelated to each other given the examinee’s true score, also referred to as local independence of item scores. IRT models formulate the probability of a response to an item as a function of an examinee’s latent trait and an item’s features (i.e., item difficulty, discrimination, and guessing).

In IRT models, local independence assumes that the probability of a response pattern of all items is a product of the probabilities of individual items, given the examinee’s ability level. Mathematical expression of local independence is presented as

$$\begin{array}{l}p(X_1,X_2,\cdots ,X_n|\theta \text{)}=p(X_1|\theta )\times p(X_2|\theta )\\ \times \cdots \times p(X_n|\theta ),\end{array}$$

where p is the probability of a response pattern or an individual response (X1, X2,…, Xn) is a vector of a response pattern for all items, X1, X2,…, Xn are individual responses to Item 1, Item 2,…, Item n, and θ is the ability level. Local independence is essential in IRT because many IRT models are formulated based on the local independence assumption using this mathematical equation. Local independence also fits the multidimensional IRT models. In sum, the assumption behind the local independence is that the latent variable(s) being measured by test items is the only factor that affects students’ performance on the test. Thus, local independence is also related to the dimensionality assumption.

Potential Sources and Statistical Detection Methods of Violations of Local Independence

Potential sources of violations of local independence have been proposed for several decades. Wendy M. Yen broadly discussed some of the sources for LID in her journal article in 1993. Some are related to examinees, such as external assistance, speededness, fatigue, and practice. Some are related to the test or test items, such as item or response format, passage dependence, item chaining, explanation of previous answer, scoring rubrics or raters, and exposure in the curriculum of testing content, knowledge, and abilities. The key idea of these additional effects causing violations of local independence is that they consistently disturb the performance of some students on some test items to a great degree. It would not lead to LID if they have equal effects to all examinees and/or to all test items. The statistics, Q3 proposed by Yen as well as Pearson’s chi-square and the likelihood rate G2 developed by Wen-Hung Chen and David Thissen, are commonly used to detect LID. The testlet IRT models are also developed to fit the [Page 988]data that inevitably occur LID, such as reading comprehension items with the same passage.

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