Introduction to plausible values National Research Coordinators Meeting Madrid, February 2010

NRC Meeting Madrid 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

NRC Meeting Madrid 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

NRC Meeting Madrid 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

NRC Meeting Madrid February 2010 IRT curves

NRC Meeting Madrid 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)

NRC Meeting Madrid February 2010 Maximum Likelihood vs. Raw Score

NRC Meeting Madrid February 2010 The Resulting Proficiency Distribution Score 0 Score 1 Score 2 Score 3 Score 4 Score 5 Score 6 Proficiency on Logit Scale

NRC Meeting Madrid 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

NRC Meeting Madrid 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

NRC Meeting Madrid February 2010 Plausible Values What are plausible values? Why do we use them? How to analyse plausible values?

NRC Meeting Madrid February 2010 Purpose of educational tests Measure particular students (minimise measurement error of individual estimates) Assess populations (minimise error when generalising to the population)

NRC Meeting Madrid February 2010 Posterior distributions for test scores on 6 dichotomous items

NRC Meeting Madrid February 2010 Empirical Bayes – Expected A-Priori estimates (EAP)

NRC Meeting Madrid 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

NRC Meeting Madrid February 2010 Plausible Values Score 0 Score 1 Score 2 Score 3 Score 4 Score 5 Score 6 Proficiency on Logit Scale

NRC Meeting Madrid 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

NRC Meeting Madrid 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

NRC Meeting Madrid 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

NRC Meeting Madrid February 2010 What is conditioning? Assuming normal posterior distribution: Model sub-populations: X=0 for boy X=1 for girl

NRC Meeting Madrid 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)

NRC Meeting Madrid 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

NRC Meeting Madrid 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

NRC Meeting Madrid 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

NRC Meeting Madrid February 2010 How to analyse PVs - 2 Measurement variance computed as: Total standard error computed from measurement and sampling variance as:

NRC Meeting Madrid February 2010 How to analyse PVs - 3  can be replaced by any statistic for instance: - SD - Percentile - Correlation coefficient - Regression coefficient - R-square - etc.

NRC Meeting Madrid 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

NRC Meeting Madrid February 2010 Questions or comments?