Item Response Theory

The Item Response Function

Item response theory relates characteristics of items and characteristics of individuals to the probability of affirming, endorsing, or correctly answering individual items. The cornerstone of IRT is the item response function (IRF), which is the graphical representation [Page 372]of a mathematical formula that relates the probability of affirming item i with the value of a latent trait, θ. Figure 1 presents a graphical representation of a hypothetical IRF.

Figure 1 Graphical Representation of a Hypothetical Item Response Function

In Figure 1, the x-axis relates to the level of the characteristic being measured by the test. This trait, θ, is typically scored like a z score, with scores at zero being average and scores above zero being above average; scores below zero are below average. Typical θ distributions range from −3 to +3. The y-axis relates to the probability of affirming an item. For ability items, the y-axis measures the probability of answering an item correctly. For items without correct answers (e.g., attitude or personality items), the y-axis refers to the probability of choosing the keyed option (i.e., the option that taps high conscientiousness). The IRF relates the level of θ with probability of affirming the item. As can be seen in Figure 1, as θ increases, the probability of affirming the item also increases. This property of monotonicity is common with IRT models. As can be seen in Figure 1, an individual with a θ = 0 would have an expected probability of affirming the item of roughly 50%. The corresponding probabilities for a person of θ= −3 and θ= +3 are roughly 0% and 100%, respectively.

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