A Primer on Design Matrices

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A Primer on Design Matrices

• Cathleen Kennedy
• 4-15-03

http//bear.soe.berkeley.edu/kennedyc/primer.pdf
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Dichotomous item
3
Dichotomous item
4
Dichotomous item
q1
q coefficients become the Scoring Matrix.
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Dichotomous item
di
d coefficients become the Design Matrix.
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3-category partial credit item
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3-category partial credit item
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Focus on the Numerators
P(x0) exp(0q (0di1 0di2)))   P(x1) exp(1q
(1di1 0di2))   P(x2) exp(2q (1di1
1di2))
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Focus on the Numerators
P(x0) exp(0q (0di1 0di2)))   P(x1) exp(1q
(1di1 0di2))   P(x2) exp(2q (1di1
1di2))
q1
q coefficients become the Scoring Matrix.
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Focus on the Numerators
P(x0) exp(0q (0di1 0di2)))   P(x1) exp(1q
(1di1 0di2))   P(x2) exp(2q (1di1
1di2))
di1
di2
d coefficients become the Design Matrix.
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Another partial credit parameterization
Let di average of all dijs and tij deviation
from di at dij. Then, dij di tij.
P(x0) exp(0q (0di0ti1 0di0ti2))   P(x1)
exp(1q (1di1ti1 0di0ti2))   P(x2) exp(2q
(1di1ti1 1di1ti2))
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Rating scale is a special case of partial credit
parameterization
The deviations, t, are the same for a given step
across all items (t11 t21 t31 and t12 t22
t32 etc.).
P(x0) exp(0q (0di0t1 0di0t2))   P(x1) ex
p(1q (1di1t1 0di0t2))   P(x2) exp(2q
(1di1t1 1di1t2))
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Rating scale is a special case of partial credit
parameterization
Combine common terms
P(x0) exp(0q (0di 0t1 0t2))   P(x1) exp(1
q (1di 1t1 0t2))   P(x2) exp(2q (2di
1t1 1t2))
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Rating scale is a special case of partial credit
parameterization
Combine common terms
P(x0) exp(0q (0di 0t1 0t2))   P(x1) exp(1
q (1di 1t1 0t2))   P(x2) exp(2q (2di
1t1 1t2))
q1
q coefficients become the Scoring Matrix.
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Rating scale is a special case of partial credit
parameterization
Combine common terms
P(x0) exp(0q (0di 0t1 0t2))   P(x1) exp(1
q (1di 1t1 0t2))   P(x2) exp(2q (2di
1t1 1t2))
di
t1
t2
d coefficients become the Design Matrix.
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Multiple Items
A
B
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Multiple Dimensions- Between Item
Item 1 Item 2 Item 3
q1 q2
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Multiple Dimensions- Between Item
Assessment Scoring Matrix q1
q2 item 1, category 1 0 0 item 1, category
2 1 0 item 1, category 3 2 0 item 2, category
1 0 0 item 2, category 2 0 1 item 2, category
3 0 2 item 3, category 1 0 0 item 3, category
2 1 0 item 3, category 3 2 0 item 3, category
4 3 0  
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Multiple Dimensions- Between Item
Assessment Design Matrix d11 d12 d21
d22 d31 d32 d33 item 1, category 1 0 0
0 0 0 0 0 item 1, category 2 1 0 0 0 0
0 0 item 1, category 3 1 1 0 0 0 0
0 item 2, category 1 0 0 0 0 0 0 0 item 2,
category 2 0 0 1 0 0 0 0 item 2, category
3 0 0 1 1 0 0 0 item 3, category 1 0 0 0
0 0 0 0 item 3, category 2 0 0 0 0 1 0
0 item 3, category 3 0 0 0 0 1 1 0 item 3,
category 4 0 0 0 0 1 1 1    
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Multiple Dimensions- Within Item
q1 q2
Item 1
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Multiple Dimensions- Within Item
Item Scoring Matrix q1 q2 category 1
(1,1) 0 0 category 2 (1,2) 0 1 category 3
(1,3) 0 2 category 4 (2,1) 1 0 category 5
(2,2) 1 1 category 6 (2,3) 1 2 category
12 (4,3) 3 2
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Multiple Dimensions- Within Item
Item Design Matrix dq1,1 dq1,2 dq1,3
dq2,1 dq2,2 (ddimension,step) category 1
(1,1) 0 0 0 0 0 category 2 (1,2) 0
0 0 1 0 category 3 (1,3) 0 0 0
1 1 category 4 (2,1) 1 0 0 0 0
category 5 (2,2) 1 0 0 1
0 category 6 (2,3) 1 0 0 1
1 category 12 (4,3) 1 1 1 1 1