Transcript Moving Beyond Summative Scores: Tapping the Diagnostic

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
• 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
• 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 P( X  1|  )  P( ) be the conditional
probability of a correct response given the
attribute pattern 
• Consider P( ) for the attribute patterns
{00},{10},{01},{11}
conjunctive
non-compensatory
1
0.75
0.5
0.25
0
00
10
01
11
not conjunctive
non-compensatory
1
0.75
0.5
0.25
0
00
10
01
11
disjunctive
compensatory
1
0.75
0.5
0.25
0
00
10
01
11
not disjunctive
compensatory
1
0.75
0.5
0.25
0
00
10
01
11
1
neither conjunctive nor disjunctive
not fully compensatory
0.75
0.5
0.25
0
00
10
01
11
• All the CDMs we will consider model the
conditional probability of success on item j
given the attribute pattern of latent class c:
P( X j  1|  c )
• These models will have varying degrees of
conjunctiveness and compensation
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 ( jc  1) and those who lack
at least one of the required attributes ( jc  0)
• Specifically,
K
 c , q j   jc   
k 1
q jk
ck
• The item response function of the DINA model
is given by
(1 jc )
 jc
P( X j  1|  c )  P( X jc  1|  jc )  g j
(1  s) j
where g j and s j 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
g j  .1 and s j  .1
DINA Model
1
.90
0.75
0.5
0.25
0
.10
.10
.10
00
10
01
11
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 sk and g k
• The item response function of the NIDA model
is given by
q jk
K
P( X j  1|  c )   (1  sk )ck g k1ck 
k 1
• 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
g1  .3, s1  .2, g2  .2, s2  .1
NIDA Model
1
.72
0.75
0.5
.27
0.25
.16
.06
0
00
10
01
11
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
*
*
 j and rjk , k  1, K
• The probability of a correct response to item j
for examinees who have mastered all the
*
required attributes for the item is given by  j
• The penalty for not mastering  k is rjk*
• The item response function of the Reduced
RUM is given by
K
P( X j |  c )  
*
j
r
k 1
*q jk (1 ck )
jk
• For an item requiring, say, the first two
attributes where
  .72, r  .22, r  .38
*
j
*
j1
*
j1
Reduced
NIDA Model
RUM
1
.72
0.75
0.5
.27
0.25
.16
.06
0
00
10
01
11
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 ( jc  1) and those who
have none of the required attributes ( jc  0)
• Specifically,
K
 c , q j   jc  1   (1   ck )
k 1
q jk
• The item response function of the DINO model
is given by
*(1 jc )
*  jc
P( X j  1|  c )  P( X jc  1|  jc )  g j
(1  s j )
where g *j and s*j 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
*
*
g j  .1 and s j  .1
DINO Model
1
.90
.90
.90
10
01
11
0.75
0.5
0.25
.10
0
00
• 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