2. Most general divide-by-total item response
theory model
NRM has received the least attention
Can be used to address important
psychometric questions
Useful in exploratory item response data
Currently unclear how researchers should
approach hypothesis testing of specific
parameters.
3. EXP ix cix
Pix m
EXP c
ix ix
x 0
ix = cix = 0
4. 1
Pix | x x or x '
1 exp( c )
* *
where, * = x - x’
and c* = (cx’ – cx)/ *
5. is a category slope
There are four for a 4-Point Item
* is a category discrimination
There are three for a 4-Point Item
They represent the discrimination of
three dichotomies
1* = 2 - 1 1 vs. 2
2* = 3 - 2 2 vs. 3
3* = 4 - 3 3 vs. 4
6. Rating scale model
◦ constrain all c* parameters to be
equal across items
Partial Credit Model
Generalized Partial Credit Model
7. Rating Scale model
Partial Credit model
◦ * is constrained to be equal
within and between items
Generalized Partial Credit model
8. Rating Scale model
Partial Credit model
Generalized Partial Credit model
(G-PCM):
◦ a* parameters are constrained
within an item, but not between
items
9. The NRM will be evaluated as a
method of hypothesis testing
◦ Evaluate the assumption of the G-PCM of
equal category discriminations within
items
◦ Using PROMIS data as an example of
testing the assumption
◦ Power to detect different category
discrimination parameters within an item
10. Part 1: Evaluation the assumption of the G-
PCM of equal category discriminations within
items
◦ Manipulated variables
Category discrimination parameter
Intersection parameters
Number of items
Sample size
Distribution of θ
11. Part 2: Using PROMIS data to test assumption
◦ PROMIS Depression Inventory
768 individuals
28 items
G-PCM was fit to data using PARSCALE
Data simulated using produced parameter estimates
◦ Manipulated variables
Distribution of θ
Sample size
12. Part 3: Power to detect different category
discrimination parameters within an item
◦ Manipulated variables
Average category discrimination
Category discrimination variance
Different forms of too many response options
One discrimination too many
Multi-point item should be a dichotomy
13. Estimate the G-PCM for all simulated data
and identify the log-likelihood
Free up the category discriminations one item
at a time and identify the log-likelihood
Evaluate the change in log-likelihood
Difference in log-likelihood should be chi-
square distributed (M=df, σ2 = 2df)
14. For all conditions with normal θ distribution
������ = .05
������ = 2.00
������ 2 = 4.01
For all conditions with skewed θ distribution
������ = .31
������ = 5.18
������ 2 = 16.56
16. PROMIS data parameters
������ = 2.25 ������1 = 0.36 ������2 = 0.81 ������3 = 1.67
L-R test results
Sample Size
500 1,000 2,000
M(σ2) 2.28 (167.43) 2.28 (12.91) 2.32 (9.57)
Normal θ
.07 .07 .09
Type I error
M(σ2) 1.90 (247.09) 3.5 (14.21) 5.58 (21.83)
Skewed θ
.14 .19 .37
Type I error
17. Average category discrimination
◦ α* = 1.75 ������ = .63
◦ α* = 1.25 ������ = .67
◦ α* = 0.75 ������ = .76
Category discrimination variance
◦ α* variance = 0.5 ������ = .26
◦ α* variance = 2.0 ������ = .63
Different forms of too many response options
◦ One discrimination too many
������ = 1.00
For all conditions
◦ Multi-point item should be a dichotomy
������ = .77
For all conditions
18. For all conditions under a normal θ
distribution, the LR-difference test appears to
be valid
The LR-difference test appears to have
adequate power to detect unequal
discrimination parameters
The LR-difference test has excellent power to
detect when an item has one too many
discrimination parameters (α4 = 0)
High category discriminations and skewed θ
distribution appears to present some
problems
