This article focuses on four-parameter logistic (4PL) model as an extension of the usual three-parameter logistic (3PL) model with an upper asymptote possibly different from 1. For a given item with fixed item parameters, Lord derived the value of the latent ability level that maximizes the item information function under the 3PL model. The purpose of this article is to extend this result to the 4PL model. A generic and algebraic method is developed for that purpose. The result is practically illustrated by an example and several potential applications of this result are outlined.
BartonM. A.LordF. M. (1981). An upper asymptote for the three-parameter logistic item-response model. Princeton, NJ: Educational Testing Service.
2.
BirnbaumA. (1968). Some latent trait models and their use in inferring an examinee’s ability. In LordF. M.NovickM. R. (Eds.), Statistical theories of mental test scores (pp. 397-479). Reading, MA: Addison-Wesley.
3.
ChangH.-H.YinZ. (2008). To weight or not to weight? Balancing influence of initial items in adaptive testing. Psychometrika, 73, 441-450.
4.
GreenB. F. (2011). A comment on early student blunders on computer-based adaptive tests. Applied Psychological Measurement, 35, 165-174.
JacobsonN. (1985). Basic algebra I (2nd ed.). San Francisco, CA: W. H. Freeman.
7.
LiaoW.-W.HoR.-G.YenY.-C.ChengH.-C. (2012). The four-parameter logistic item response theory model as a robust method of estimating ability despite aberrant responses. Social Behavior and Personality, 40, 1679-1694.
8.
LinacreJ. M. (2004). Discrimination, guessing and carelessness: Estimating IRT parameters with Rasch. Rasch Measurement Transactions, 18, 959-960.
9.
LokenE.RulisonK. L. (2010). Estimation of a four-parameter item response theory model. British Journal of Mathematical and Statistical Psychology, 63, 509-525.
10.
LordF. M. (1980). Applications of item response theory to practical testing problems. Hillsdale, NJ: Lawrence Erlbaum.
11.
LunnD. J.ThomasA.BestN.SpiegelhalterD. (2000). WinBUGS—A Bayesian modeling framework: Concepts, structure, and extensibility. Statistics and Computing, 10, 325-337.
12.
MagisD. (2012). Some robust ability estimators with logistic item response models. Unpublished manuscript, Department of Education, University of Liège, Liège, Belgium.
13.
MagisD.RaîcheG. (2012). Random generation of response patterns under computerized adaptive testing with the R package catR. Journal of Statistical Software, 48, 1-19.
14.
MislevyR. J.BockR. D. (1982). Biweight estimates of latent ability. Educational and Psychological Measurement, 42, 725-737.
15.
OsgoodD.McMorrisB. J.PotenzaM. T. (2002). Analyzing multiple-item measures of crime and deviance I: Item response theory scaling. Journal of Quantitative Criminology, 18, 267-296.
16.
R Development Core Team. (2012). R: A language and environment for statistical computing [Computer software]. Vienna, Austria: R Foundation for Statistical Computing.
17.
ReiseS. P.WallerN. G. (2003). How many IRT parameters does it take to model psychopathology items?Psychological Methods, 8, 164-184.
18.
RulisonK. L.LokenE. (2009). I’ve fallen and I can’t get up: Can high ability students recover from early mistakes in computerized adaptive testing?Applied Psychological Measurement, 33, 83-101.
19.
RuppA. A. (2003). Item response modeling with BILOG-MG and MULTILOG for Windows. International Journal of Testing, 3, 365-384.
20.
SchusterC.YuanK.-H. (2011). Robust estimation of latent ability in item response models. Journal of Educational and Behavioral Statistics, 36, 720-735.
21.
TavaresH. R.de AndradeD. F.PereiraC. A. (2004). Detection of determinant genes and diagnostic via item response theory. Genetics and Molecular Biology, 27, 679-685.
22.
TendeiroJ. N.MeijerR. R. (2012). A CUSUM to detect person misfit: A discussion and some alternatives for existing procedures. Applied Psychological Measurement, 36, 420-442.
23.
WallerN. G.ReiseS. P. (2009). Measuring psychopathology with non-standard IRT models: Fitting the four parameter model to the MMPI. In EmbretsonS.RobertsJ. S. (Eds.), New directions in psychological measurement with model-based approaches (pp. 147-173). Washington, DC: American Psychological Association.
24.
YenY.-C.HoR.-G.LiaoW.-W.ChenL.-J.KuoC.-C. (2012). An empirical evaluation of the slip correction in the four parameter logistic models with computerized adaptive testing. Applied Psychological Measurement, 36, 75-87.
