University:
California State University, Northridge
Course:
PSY 427 | Introduction to Psychological Testing
Academic year:

2024
Views:

353

Pages:

19
Author:

Nevaeh Bauer

IRT - Item Response Function Item Response Function (IRF) - characterizes the relation between a latent variable (i.e., individual differences on a construct) and the probability of endorsing an item.  The IRF models the relationship between examinee trait level, item properties and the probability of endorsing the item.  Examinee trait level is signified by the greek letter theta (θ) and typically has mean = 0 and a standard deviation = 1  IRT - Item Characteristic Curves  IRFs can then be converted into Item Characteristic Curves (ICC) which are graphical functions that represents the respondents ability as a function of the probability of endorsing the item IRF – Item Parameters Location (b) An item’s location is defined as the amount of the latent trait needed to have a .5 probability of endorsing the item.  The higher the “b” parameter the higher on the trait level a respondent needs to be in order to endorse the item  Analogous to difficulty in CTT  Like Z scores, the values of b typically range from -3 to +3  IRF – Item Parameters Discrimination (a) Indicates the steepness of the IRF at the items location  An items discrimination indicates how strongly related the item is to the latent trait like loadings in a factor analysis  Items with high discriminations are better at differentiating respondents around the location point; small changes in the latent trait lead to large changes in probability  Vice versa for items with low discriminations  1 Probability of Endorsing 0.9 0.8 Discrimination 0.7 0.6 0.5 0.4 0.3 Location 0.2 0.1 0 -3 -2 -1 0 1 Latent Trait 2 3 1 Probability of Endorsing 0.9 Same Discrimination 0.8 0.7 1 2 0.6 0.5 0.4 0.3 Different Locations 0.2 0.1 0 -3 -2 -1 0 1 Latent Trait 2 3 1 Probability of Endorsing 0.9 Different Discriminations 0.8 0.7 0.6 0.5 0.4 Different Locations 0.3 0.2 1 0.1 2 0 -3 -2 -1 0 1 Latent Trait 2 3 IRF – Item Parameters Guessing (c) The inclusion of a “c” parameter suggests that respondents very low on the trait may still choose the correct answer.  In other words respondents with low trait levels may still have a small probability of endorsing an item  This is mostly used with multiple choice testing…and the value should not vary excessively from the reciprocal of the number of choices.  1 Different Discriminations Probability of Endorsing 0.9 0.8 0.7 Lower Asymptote 0.6 0.5 0.4 Different Locations 1 0.3 0.2 2 0.1 0 -3 -2 -1 0 1 Latent Trait 2 3 IRF – Item Parameters Upper asymptote (d) The inclusion of a “d” parameter suggests that respondents very high on the latent trait are not guaranteed (i.e. have less than 1 probability) to endorse the item  Often an item that is difficult to endorse (e.g. suicide ideation as an indicator of depression)  1 0.9 0.8 Probability 0.7 0.6 0.5 0.4 0.3 0.2 2plm 3plm 4plm 0.1 0 -3 -2 -1 0 Trait Level 1 2 3 IRT - Item Response Function  The 4-parameter logistic model  Where a (θ −b ) e P( X =1 θ , a, b, c, d ) =c + (d − c) a (θ −b ) 1+ e  θ represents examinee trait level  b is the item difficulty that determines the location of the IRF  a is the item’s discrimination that determines the steepness of the IRF  c is a lower asymptote parameter for the IRF  d is an upper asymptote parameter for the IRF IRT - Item Response Function  The 3-parameter logistic model a (θ −b ) e P( X =1 θ , a, b, c) = c + (1 − c) a (θ −b ) 1+ e  If the upper asymptote parameter is set to 1.0, then the model is termed a 3PL.  In this model, individuals at low trait levels have a non-zero probability of endorsing the item. IRT - Item Response Function  The 2-parameter logistic model a (θ −b ) e = θ , a, b) P( X 1= a (θ −b ) 1+ e  If in addition the lower asymptote parameter is constrained to zero, then the model is termed a 2PL.  In the 2PLM, IRFs vary both in their discrimination and difficulty (i.e., location) parameters. IRT - Item Response Function  The 1-parameter logistic model (θ −b ) e = θ , b) P( X 1= (θ −b ) 1+ e  If the item discrimination is set to 1.0 (or any constant) the result is a 1PL  A 1PL assumes that all scale items relate to the latent trait equally and items vary only in difficulty (equivalent to having equal factor loadings across items). Quick Detour: Rasch Models vs. Item Response Theory Models  Mathematically, Rasch models are identical to the most basic IRT model (1PL), however there are some (important) differences  In Rasch the model is superior. Data which does not fit the model is discarded  Rasch does not permit abilities to be estimated for extreme items and persons  And other differences IRT - Test Response Curve  Test Response Curves (TRC) - Item response functions are additive so that items can be combined to create a TRC.  A TRC is the latent trait relative to the number of items IRT - Test Response Curve 20 18 16 Number of Items 14 12 10 8 6 4 2 0 -4 -3 -2 -1 0 Trait 1 2 3 4