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Topic 5: Common CDMs

Topic 5: Common CDMs. Introduction. In addition to general models for cognitive diagnosis, there exists several specific CDMs in the literature These CDMs have been classified as either conjuctive or disjunctive

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Topic 5: Common CDMs

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  1. Topic 5:Common CDMs

  2. Introduction • In addition to general models for cognitive diagnosis, there exists several specific CDMs in the literature • These CDMs have been classified as either conjuctive or disjunctive • Models are conjunctive if all the required attributes are necessary for successful completion of the item • CDMs have also been classified as either compensatory or non-compensatory

  3. Models are compensatory if the absence of one attribute can be made up for by the presence of other attributes • For most part, these two schemes of classifying CDMs have been used interchangeably • Specifically, conjunctive = non-compensatory disjunctive = compensatory • Depending on how the terms are defined, the two classification schemes may not be identical

  4. Let be the conditional probability of a correct response given the attribute pattern • Consider for the attribute patterns

  5. 1 0.75 0.5 0.25 0 conjunctive non-compensatory

  6. 1 0.75 0.5 0.25 0 not conjunctive non-compensatory

  7. 1 0.75 0.5 0.25 0 disjunctive compensatory

  8. 1 0.75 0.5 0.25 0 not disjunctive compensatory

  9. 1 0.75 0.5 0.25 0 neither conjunctive nor disjunctive not fully compensatory

  10. All the CDMs we will consider model the conditional probability of success on item j given the attribute pattern of latent class c: • These models will have varying degrees of conjunctiveness and compensation

  11. The DINA Model • DINA stands for the deterministic input, noisy “and” gate • Item j splits the examinees in the different latent classes into those who have all the required attributes and those who lack at least one of the required attributes • Specifically,

  12. The item response function of the DINA model is given by where and are the guessing and slip parameters of item j • The DINA model has only two parameters per item regardless of the number of attributes K • For an item requiring two attributes with and

  13. 1 0.75 0.5 0.25 0 DINA Model .90 .10 .10 .10

  14. The NIDA Model • NIDA stands for the noisy input, deterministic, “and” gate • Like the DINA model, the NIDA model is also defined by slip and guessing parameters • Unlike the DINA model, the slips and guesses in the NIDA model occur at the attribute, not the item level • The slip and guessing parameters of attribute k are given by and

  15. The item response function of the NIDA model is given by • Note that the slip and guessing parameters have no subscript for items • The NIDA model assumes that the probability of correct application of an attribute is the same for all items • For an item requiring, say, the first two attributes where

  16. 1 0.75 0.5 0.25 0 NIDA Model .72 .27 .16 .06

  17. The Reduced RUM • The Reduced RUM is a reduction of theReparameterized Unified Model • Like the NIDA model, the Reduced RUM allows each required attribute to contribute differentially to the probability of success • Unlike the NIDA model, the contribution of an attribute can vary from one item to another • The parameters of the Reduced RUM are and

  18. The probability of a correct response to item j for examinees who have mastered all the required attributes for the item is given by • The penalty for not mastering is • The item response function of the Reduced RUM is given by • For an item requiring, say, the first two attributes where

  19. 1 0.75 0.5 0.25 0 NIDA Model Reduced RUM .72 .27 .16 .06

  20. The DINO Model • DINO stands for the deterministic input, noisy “or” gate • Item j splits the examinees in the different latent classes into those who have at least one the required attributes and those who have none of the required attributes • Specifically,

  21. The item response function of the DINO model is given by where and are the guessing and slip parameters of item j • Like the DINA model, the DINO has only two parameters per item regardless of the number of attributes K • For an item requiring two attributes with and

  22. 1 0.75 0.5 0.25 0 DINO Model .90 .90 .90 .10

  23. Other models that have been presented include • NIDO Model • Compensatory RUM • Additive version of the GDM • Of these models, only the DINA model is truly conjunctive and non-compensatory • Only the DINO model is truly disjunctive and compensatory • These models can all be derived from (i.e., special cases of) general models for cognitive diagnosis

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