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Topic 5: Common CDMs

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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 Introduction

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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

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

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0.75 0.5 0.25 0 1 conjunctive non-compensatory

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0.75 0.5 0.25 0 1 not conjunctive non-compensatory

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0.75 0.5 0.25 0 1 disjunctive compensatory

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0.75 0.5 0.25 0 1 not disjunctive compensatory

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

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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

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DINA stands for the deterministic input, noisyand 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, The DINA Model

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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

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0.75 0.5 0.25 0 1 DINA Model.10.90

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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

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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

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0.75 0.5 0.25 0 1 NIDA Model.06.16.27.72

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The Reduced RUM The Reduced RUM is a reduction of the Reparameterized 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

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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

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0.75 0.5 0.25 0 1 NIDA Model.06.16.27.72 Reduced RUM

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DINO stands for the deterministic input, noisyor 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, The DINO Model

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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

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0.75 0.5 0.25 0 1 DINO Model.10.90

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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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