Introduction to plausible values National Research Coordinators Meeting Madrid, February 2010.
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Transcript of Introduction to plausible values National Research Coordinators Meeting Madrid, February 2010.
Introduction to plausible values
National Research Coordinators Meeting Madrid, February 2010
NRCMeetingMadrid
February 2010
Content of presentation
• Rationale for scaling• Rasch model and possible ability
estimates• Shortcomings of point estimates• Drawing plausible values• Computation of measurement error
NRCMeetingMadrid
February 2010
Rationale for IRT scaling of data
• Summarising data instead of dealing with many single items
• Raw scores or percent correct sample-dependent
• Makes equating possible and can deal with rotated test forms
NRCMeetingMadrid
February 2010
The ‘Rasch model’
• Models the probability to respond correctly to an item as
• Likewise, the probability of NOT responding correctly is modelled as
in
innii XP
exp1
exp)1(
)exp(1
1)0(
inniXP
NRCMeetingMadrid
February 2010
IRT curves
0
0.5
1
-4 -3 -2 -1 0 1 2 3 4
NRCMeetingMadrid
February 2010
How might we impute a reasonable proficiency value?
• Choose the proficiency that makes the score most likely– Maximum Likelihood Estimate– Weighted Likelihood Estimate
• Choose the most likely proficiency for the score– empirical Bayes
• Choose a selection of likely proficiencies for the score– Multiple imputations (plausible values)
NRCMeetingMadrid
February 2010
Maximum Likelihood vs. Raw Score
0
1
2
3
4
5
Proficiency
Scor
e
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February 2010
The Resulting Proficiency Distribution
Score 0
Score 1
Score 2
Score 3Score 4
Score 5
Score 6
Proficiency on Logit Scale
NRCMeetingMadrid
February 2010
Characteristics of Maximum Likelihood Estimates (MLE)
• Unbiased at individual level with sufficient information BUT biased towards ends of ability scale.
• Arbitrary treatment of perfects and zeroes required
• Discrete scale & measurement error leads to bias in population parameter estimates
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February 2010
Characteristics of Weighted Likelihood Estimates
• Less biased than MLE
• Provides estimates for perfect and zero scores
• BUT discrete scale & measurement error leads to bias in population parameter estimates
NRCMeetingMadrid
February 2010
Plausible Values
• What are plausible values?
• Why do we use them?
• How to analyse plausible values?
NRCMeetingMadrid
February 2010
Purpose of educational tests
• Measure particular students
(minimise measurement error of
individual estimates)
• Assess populations
(minimise error when generalising
to the population)
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February 2010
Posterior distributionsfor test scores on 6 dichotomous items
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February 2010
Empirical Bayes – Expected A-Priori estimates (EAP)
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February 2010
Characteristics of EAPs
• Biased at the individual level but unbiased population means (NOT variances)
• Discrete scale, bias & measurement error leads to bias in population parameter estimates
• Requires assumptions about the distribution of proficiency in the population
NRCMeetingMadrid
February 2010
Plausible Values
Score 0
Score 1
Score 2
Score 3 Score 4
Score 5
Score 6
Proficiency on Logit Scale
NRCMeetingMadrid
February 2010
Characteristics of Plausible Values
• Not fair at the student level
• Produces unbiased population parameter estimates– if assumptions of scaling are reasonable
• Requires assumptions about the distribution of proficiency
NRCMeetingMadrid
February 2010
Estimating percentages below benchmark with Plausible Values
Level One Cutpoint
The proportion of plausible values less than the cut-point will be a superior estimator to the EAP, MLE or WLE based values
NRCMeetingMadrid
February 2010
Methodology of PVs
• Mathematically computing posterior distributions around test scores
• Drawing 5 random values for each assessed individual from the posterior distribution for that individual
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February 2010
What is conditioning?
• Assuming normal posterior distribution:
• Model sub-populations:
X=0 for boyX=1 for girl
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
2,N
2,N X
2...,N X Y Z
NRCMeetingMadrid
February 2010
Conditioning Variables
• Plausible values should only be analysed with data that were included in the conditioning (otherwise, results may be biased)
• Aim: Maximise information included in the conditioning, that is use as many variables as possible
• To reduce number of conditioning variables, factor scores from principal component analysis were used in ICCS
• Use of classroom dummies takes between-school variation into account (no inclusion of school or teacher questionnaire data needed)
NRCMeetingMadrid
February 2010
Plausible values
• Model with conditioning variables will improve precision of prediction of ability (population estimates ONLY)
• Conditioning provides unbiased estimates for modelled parameters.
• Simulation studies comparing PVs, EAPs and WLEs show that– Population means similar results– WLEs (or MLEs) tend to overestimate variances– EAPs tend to underestimate variance
NRCMeetingMadrid
February 2010
Calculating of measurement error
• As in TIMSS or PIRLS data files, there are five plausible values for cognitive test scales in ICCS
• Using five plausible values enable researchers to obtain estimates of the measurement error
NRCMeetingMadrid
February 2010
How to analyse PVs - 1
• Estimated mean is the AVERAGE of the mean for each PV
• Sampling variance is the AVERAGE of the sampling variance for each PV
M
iiM 1
ˆ1
ˆ
M
iiM 1
2)(
2)( ˆ
1ˆ
NRCMeetingMadrid
February 2010
How to analyse PVs - 2
• Measurement variance computed as:
• Total standard error computed from measurement and sampling variance as:
25
1
2)( ˆˆ
1
1ˆ
i
iPV M
2 2ˆ ˆ ( )( ) ( )
1ˆ ˆ ˆ(1 )PV PVM
NRCMeetingMadrid
February 2010
How to analyse PVs - 3
can be replaced by any statistic for instance:- SD- Percentile- Correlation coefficient- Regression coefficient- R-square- etc.
NRCMeetingMadrid
February 2010
Steps for estimating both sampling and measurement error
• Compute statistic for each PV for fully weighted sample
• Compute statistics for each PV for 75 replicate samples
• Compute sampling error (based on previous steps)
• Compute measurement error• Combine error variances to calculate
standard error
NRCMeetingMadrid
February 2010
Questions or comments?
