On multiple-choice tests, even examinees with very low ability have a nonzero probability of answering correctly by guessing. The three-parameter logistic (3PL) model, introduced by Birnbaum (1968), accommodates this reality by adding a lower asymptote parameter c to the two-parameter model. The parameter c represents the probability that an examinee with extremely low ability answers the item correctly, often interpreted as the "guessing" or "pseudo-chance" parameter.
The Model
where c_i = lower asymptote (pseudo-guessing parameter)
a_i = discrimination, b_i = difficulty
As θ → −∞, P_i(θ) → c_i (not 0)
The item characteristic curve of the 3PL model is a logistic function that has been rescaled and shifted vertically. The lower asymptote is c rather than 0, meaning the curve never drops below c. For a four-option multiple-choice item, random guessing would yield c = 0.25, though the estimated c parameter often differs from 1/(number of options) because guessing is rarely purely random — examinees may eliminate implausible distractors before guessing.
Estimation Challenges
The addition of the c parameter substantially complicates estimation. With three parameters per item, larger samples are required for stable estimates — typically 1,000 or more examinees. The c parameter is poorly identified in the data because few examinees have ability low enough to provide direct information about the lower asymptote. Consequently, c estimates can be unstable, and estimation algorithms often incorporate Bayesian prior distributions on c (e.g., a beta prior centered near 0.20) to regularize the estimates.
Maximum information is reduced relative to the 2PL
and shifted to a higher ability level than b_i
The interpretation and even the existence of the c parameter have been debated. Some psychometricians argue that guessing is not a stable property of an item but depends on examinee strategies, test context, and motivation. Others have proposed alternative parameterizations — such as mixture models with a separate guessing class — that may better capture the psychology of guessing behavior. Despite these debates, the 3PL remains the standard model in large-scale educational assessment programs such as the SAT, GRE, and many state accountability tests.
Applications and Comparisons
The 3PL model is most appropriate for multiple-choice items where guessing is a genuine concern. For constructed-response items (essays, short-answer), the lower asymptote is essentially zero and the 2PL or graded response model is preferred. In operational testing, the 3PL is used for item calibration, equating, and adaptive testing. Its item information function shows that the guessing parameter reduces information at all ability levels, with the greatest reduction occurring at the lower end of the ability continuum.
Model comparison between the 2PL and 3PL can be conducted via likelihood-ratio tests, information criteria (AIC, BIC), or by examining item-level fit statistics. The 3PL generally fits multiple-choice data better than the 2PL, but the improvement must be weighed against the increased estimation demands and potential instability of the c parameters. In practice, many testing programs use a constrained 3PL where the c parameter is fixed or given a strong prior, achieving a compromise between the 2PL's parsimony and the 3PL's flexibility.