Topic 5:

Common CDMs

• 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

• 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

• 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

0.75

0.5

0.25

0

1

00 10 01 11

conjunctivenon-compensatory

0.75

0.5

0.25

0

1

00 10 01 11

not conjunctivenon-compensatory

0.75

0.5

0.25

0

1

00 10 01 11

disjunctivecompensatory

0.75

0.5

0.25

0

1

00 10 01 11

not disjunctivecompensatory

0.75

0.5

0.25

0

1

00 10 01 11

neither conjunctive nor disjunctivenot fully compensatory

• 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

• 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

• 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

0.75

0.5

0.25

0

1

00 10 01 11

DINA Model

.10 .10 .10

.90

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

• 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

0.75

0.5

0.25

0

1

00 10 01 11

NIDA Model

.06

.16

.27

.72

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

• 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

0.75

0.5

0.25

0

1

00 10 01 11

NIDA Model

.06

.16

.27

.72

Reduced RUM

• 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

• 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

0.75

0.5

0.25

0

1

00 10 01 11

DINO Model

.10

.90 .90 .90

• 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