Latent regression models
Where does the probability come from?
• Why isn’t the model deterministic.• Each item tests something unique
– We are interested in the average of what the items assess
• Stochastic subject argument• Random sampling of subjects
Two different models
Pr ; |ni ni i nX x
Pr ; ,ni ni i nX x
Random sampling
Stochastic subject
A Random Effects (Sampling) Model -- 1
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2222
1; , exp22
g
Part 1: A Model for the population
equivalently
A Random Effects Model -- 2
Part 2: A Model for the item response mechanism
Pr ; | ; |
exp1 exp
ni ni i ni i
ni i
i
X x f x
x
Example: SLM (but can be any)
A Random Effects Model -- 3
Part 3: Putting them together
2 2; , , ; | ; ,ni i ni if x f x g d
The unknowns are: 2, iand is not an unknown, it is variable of integration
What is analysed is item response, what is estimated is itemparameters and population parameters
Why do this?
• Solves some theoretical estimation problems– For non-Rasch models
• Provides better estimates of population characteristics
Problems with point estimates
ˆvar var
var var
var
n n n
n n
n
e
e
Problem with discreteness
• For a 6-item test, there are only 7 possible ability estimates to assign to people, those getting a score of 0,1,2,3,4,5,6. (raw score is sufficient statistic for ability)
• Suppose we want to know where the 25th percentile point is. That is, 25% of the population are below this point. We need extrapolation.
The Resulting JML Ability Distribution
Score 0
Score 1
Score 2
Score 3Score 4
Score 5
Score 6
Proficiency on Logit Scale
Distribution for a six item test
Score 0
Score 1
Score 2
Score 3Score 4
Score 5
Score 6
Proficiency on Logit Scale
Traditional approach is a two-step analyses
First estimate abilities
Then compute population estimates such as mean, variance, percentiles using
leads to biased results due to measurement error.
In the case of the population variance, we can correct the bias (disattenuate) by multiplying by the reliability.
But in other cases, it is less obvious how to correct for the bias caused by measurement error.
n̂
n̂
Distribution of Estimates is Discrete
• One ability for each raw score• Ability estimates have a discrete distribution• We imagine (and the model’s premise) is a
continuous distribution• The distribution of ability estimates is
distorted by measurement error
Solutions
• Direct estimation of population parameters (directly via item responses, and not through the estimated abilities)
• Complicated analyses that take into account the error– Not always possible
MML: How it works — 1
• Item Response Model for item i:
• Population Model (discrete)
i
iixf
exp1
exp/1
g()-1.5 0.1-0.5 0.2
0 0.40.5 0.21.5 0.1
MML: How it works — 2
1 1/ 1.5 1.5
1/ 0.5 0.5
1/ 1.5 1.5
1/
/
i i
i
i
i
f x f x g
f x g
f x g
f x g
f x g d continuous case
The Implications — 1
• If , then contains parameters – Note that no ability parameters are involved, only
population parameters.• Use maximum likelihood estimation method
to estimate the item difficulty parameters and population parameters.
• Thus, we directly estimate population parameters through the item responses
2,~ Ng xf
,,,,, 21 I
Bayes Theorem
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Pr
Pr Pr | Pr
A BA B
B
A B A B B
PrPr |
Pr
Pr Pr | Pr
A BB A
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A B B A A
Pr | Pr Pr | Pr
Pr | PrPr |
Pr
A B B B A A
B A AA B
B
The Idea of Posterior Distribution
• If a student’s item response pattern is x then the posterior distribution is given by
Pr | PrPr |
PrB A A
A BB
| ||
|
f g f gh
f f g d
x x
xx x
Pr | PrPr |
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x
xx
The Idea of Posterior Distribution
• Instead of obtaining a point estimate for ability, there is now a (posterior) probability distribution
• incorporates measurement error for
the uncertainty in the estimate.
n̂
|h x
|h x
The Resulting JML Ability Distribution
Score 0
Score 1
Score 2
Score 3Score 4
Score 5
Score 6
Proficiency on Logit Scale
Resulting MML Posterior Distributions
Score 0
Score 1
Score 2
Score 3 Score 4
Score 5
Score 6
Proficiency on Logit Scale
MML EAP Estimates – an aside
Score 0
Score 1
Score 2
Score 3 Score 4
Score 5
Score 6
Proficiency on Logit Scale
MML EAP Estimates – an aside
• Biased at the individual level• Discrete scale, bias & measurement error
leads to bias in population parameter estimates
• Requires assumptions about the distribution of proficiency in the population
Distribution for a six item test
Score 0
Score 1
Score 2
Score 3Score 4
Score 5
Score 6
Proficiency on Logit Scale
Estimating proportions below a point based up posterior distributions
More General Form of the Model
expPr ; , , ,
expn
n nn n n
n
z
x b AξX x b A ξ
z b Aξ
2
2
~ ,
~ ,
N x y z
N
Y β
Item response model
Population model
The underlying population distribution is a mixture of two normal distributions, with different means ( and ).
Population Not Normal
• E.g., sample consists of grades 5 and 8.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69
1 2
• where x=0 if a student is in group 1 and x=1 if a student is in group 2. In this case, we estimate , and . Note that is the difference between the means of the two distributions. That is, group 1 has mean (as x=0), and group 2 has mean (as x=1).
Latent Regression - 1
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2
Latent Regression - 2
• We call “x” a “regressor”, or a “conditioning variable”, or a “background variable”. We can generalise to include many conditioning variables.
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