﻿ ASYMPTOTE= item upper and lower asymptotes

# ASYMPTOTE= item upper and lower asymptotes = No

Persons responding to multiple-choice questions (MCQ) can exhibit guessing and carelessness. In the three-parameter IRT model (3-PL), guessing is parameterized as a lower asymptote to the item's logistic ogive of the probability of a correct answer. In the four-parameter IRT model (4-PL), carelessness is parameterized as an upper asymptote. Winsteps reports a first approximation to these parameter values, but does not use the estimates to alter the Rasch measures. The literature suggests that when the lower asymptote is .10 or greater, it is "substantial" (How Many IRT Parameters Does It Take to Model Psychopathology Items? Steven P. Reise, Niels G. Waller, Psychological Methods, 2003, 8, 2, 164-184).

 ASYMPTOTE=Y report the values of the Upper and Lower asymptotes in the Item Tables and IFILE= ASYMPTOTE=N do not report values for the Upper and Lower asymptotes.

Example 1: Estimate the 4-PL IRT parameters for the Knox Cube Test data:

Run Exam1.txt

After the analysis completes, use the "Specification" pull-down menu:

Enter: DISCRIM = Yes to report the Item Discrimination

Enter: ASYMP = Yes to report the asymptotes

On the "Output Tables" menu, select an item table, e.g., Table 14.

+-------------------------------------------------------------------------------------------------+

|ENTRY    RAW                        |   INFIT  |  OUTFIT  |PTMEA|ESTIM| ASYMPTOTE |              |

|NUMBER  SCORE  COUNT  MEASURE  ERROR|MNSQ  ZSTD|MNSQ  ZSTD|CORR.|DISCR|LOWER UPPER| TAP          |

|------------------------------------+----------+----------+-----+-----+-----------+--------------|

|     4     32     34   -4.40     .81| .90    .0| .35    .8|  .55| 1.09|  .00  1.00| 1-3-4        |

|     6     30     34   -3.38     .64|1.17    .6| .96    .6|  .53|  .87|  .10  1.00| 3-4-1        |

|     7     31     34   -3.83     .70|1.33    .9|2.21   1.2|  .40|  .54|  .09   .98| 1-4-3-2      |

Example 2: Polytomous data: Liking for Science

Run Example0.txt

After the analysis completes, use the "Specification" pull-down menu:

Enter: DISCRIM = Yes to report the Item Discrimination

Enter: ASYMP = Yes to report the asymptotes

On the "Output Tables" menu, select an item table, e.g., Table 14.

------------------------------------------------------------------------------------------------------------

|ENTRY   TOTAL  TOTAL          |   INFIT  |  OUTFIT  |PTMEASURE-A|ESTIM| ASYMPTOTE |                       |

|NUMBER  SCORE  COUNT  MEASURE |MNSQ  ZSTD|MNSQ  ZSTD|CORR.  EXP.|DISCR|LOWER UPPER| ACT                   |

|------------------------------+----------+----------+-----------+-----+-----------+-----------------------|

|     1    109     75    -.40  | .55  -3.5| .49  -2.5|  .64   .49| 1.52|  .00  2.00| WATCH BIRDS           |

|     5     37     75    2.42  |2.30   5.6|3.62   7.3|  .05   .61| -.54|  .49  2.00| FIND BOTTLES AND CANS |

|    23     42     75    2.18  |2.41   6.3|4.11   9.0|  .00   .61| -.90|  .56  1.64| WATCH A RAT           |

# Estimation

Item Response Theory (IRT) three-parameter and four-parameter (3-PL, 4-PL) models estimate lower-asymptote parameters ("guessability", "pseudo-guessing") and upper-asymptote parameters ("mistake-ability") and use these estimates to modify the item difficulty and person ability estimates. Rasch measurement models guessability and mistake-ability as misfit, and does not attempt to make adjustments for item difficulties and person abilities. But initial approximations for the values of the asymptotes can be made, and output by Winsteps with ASYMPTOTE=Yes.

The algebraic representation of the discrimination and lower asymptote estimate by Winsteps are similar to 3-PL IRT, but the estimation method is different, because Winsteps does not change the difficulties and abilities from their 1-PL values. Consequently, in Winsteps, discrimination and asymptotes are indexes, not parameters as they are in 3-PL.

A lower-asymptote model for dichotomies or polytomies is:

Tni = ci + (mi - ci) (Eni/mi)

where Tni is the expected observation for person n on item i, ci is the lower asymptote for item i, mi is the highest category for item i (counting up from 0), and Eni is the Rasch expected value (without asymptotes). Rewriting:

ci = mi (Tni - Eni) / (mi - Eni)

This provides the basis for a model for estimating ci. Since we are concerned about the lower asymptote, let us construct a weight, Wni ,

Bi = B(Eni=0.5) as the ability of a person who scores 0.5 on the item,

then Bni = Bn - Di and Wni = Bni - Bi for all Bn < Bi otherwise Wni = 0, for each observation Xni with expectation Eni,

ci Σ(Wni mi (Xni - Eni)) / Σ(Wni (mi - Eni))

Similarly, for di, the upper asymptote,

di Σ(Wni mi Xni) / Σ(Wni Eni)) for Bni>B(Eni=mi-0.5)

The lower asymptote is the lower of ci or the item p-value. The upper asymptote is the higher of di or the item p-value. If the data are sparse in the asymptotic region, the estimates may not be good. This is a known problem in 3-PL estimation, leading many analysts to impute, rather than estimate, asymptotic values.

Birnbaum A. (1968) Some latent trait models and their uses in inferring an examinee's ability. In F.M. Lord & M.R. Novick, Statistical theories of mental test scores (pp. 395-479). Reading, MA: Addison-Wesley.

Barton M.A. & Lord F.M. (1981) An upper asymptote for the three-parameter logistic item-response model. Princeton, N.J.: Educational Testing Service.

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