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"author": "David N. Mashburn",
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"description": "# IRT Parameter Estimation routines\n\nThis package implements parameter estimation for logistic Item Characteristic Curves (ICC) from Item Response Theory (IRT).\n\nMaximum likelihood estimation (MLE) fitting routines for the following logistic models are implemented:\n \n * 1PL - 1 parameter logistic (Rausch) model\n * b (difficulty)\n * 2PL - 2 parameter logistic model\n * a (discrimination) and b (difficulty)\n * 3PL - 3 parameter logistic (Birnbaum) model\n * a (discrimination), b (difficulty) and c (pseudo-guessing)\n\nIn addition, fitting using the zeta/lamdba/c formulation is also implemented.\nThe difference here boils down to the logistic exponent.\nThe conversion is:\n\na (\u03b8j - b) = \u03b6 + \u03bb \u03b8j\n\n```a * (theta_j - b) = zeta + lambda * theta_j```\n\nThis seemingly insignificant change has drastic effects on the convergence\nproperties (especially in the 2PL case, but also the 3PL case).\n\nMany of the methods in this package are derived from work by\nFrank B. Baker and Seock-Ho Kim:\n\n_Item Response Theory: Parameter Estimation Techniques_\n\nhttp://www.crcpress.com/product/isbn/9780824758257\n\nThe exception is the 3 parameter zeta/lambda/c implementation which to our\nknowledge has not been derived or documented before.\nFor this reason, we include the mathematical derivations here:\n\n[irt-zlc-formuation.pdf](doc/zlc-irt-formulation.pdf)\n\nThe original BASIC code that work was derived from can be downloaded here:\n\nhttp://www.crcpress.com/downloads/DK2939/IRTPET.zip\n\nThe original python port of these hand-coded iterative schemes can be\nfound in the file baker_mle.py (imported as \"baker\").\nThese are mainly useful for comparative purposes (for instance, the 2PL\nversion matches 1-1 with the original routine's published values).\n\nThe main routines in this package (zlc_mle.py and abc_mle.py)\nuse scipy.optimize.root instead for greater efficiency and accuracy.\nzlc uses a hybrid (zeta, lambda, c) formulation which makes the 2PL\nand 3PL systems converge much more much stably.\nThis version is the one imported at the top level.\nThis version also includes an \"abc emulation\" mode where zlc is still\nused internally, but automatic conversion is used so that the function\ntakes a/b/c as arguments.\n\nabc (in abc_mle.py) uses the (a,b,c) formulation, which may also be\nuseful (try both for 3PL!).\nIf you want to really dig into this code, it is very informative to use\na side-by-side difference tool\n(like Vimdiff, KDiff3, WinMerge, FileMerge, or Meld)\nto compare abc_mle.py with zlc.mle.py.\n\nAll common utilities are found in util.py.",
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