topic 5: common cdms. in addition to general models for cognitive diagnosis, there exists several...
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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
( 1| ) ( )P X P
{00},{10},{01},{11}
( )P
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0.75
0.5
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conjunctivenon-compensatory
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not conjunctivenon-compensatory
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disjunctivecompensatory
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not disjunctivecompensatory
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neither conjunctive nor disjunctivenot 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
( 1| )j cP X
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• 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,
( 1)jc
1
, jk
Kq
c j jc ckk
q
( 0)jc
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
(1 )( 1| ) ( 1| ) (1 )jc jc
j c jc jc j jP X P X g s
jsjg
.1jg .1js
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DINA Model
.10 .10 .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 kgks
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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
1
1
( 1| ) (1 )jk
ck ck
qK
j c k kk
P X s g
1 1 2 2.3, .2, .2, .1g s g s
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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* , 1,jkr k K *
j
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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
k
*j
*jkr
* (1 )*
1
( | ) jk ck
Kq
j c j jkk
P X r
* * *1 1.72, .22, .38j j jr r
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NIDA Model
.06
.16
.27
.72
Reduced RUM
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• 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,
( 1)jc
1
, 1 (1 ) jk
Kq
c j jc ckk
q
( 0)jc
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
*(1 ) *( 1| ) ( 1| ) (1 )jc jc
j c jc jc j jP X P X g s *js*
jg
* .1jg * .1js
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DINO Model
.10
.90 .90 .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