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Standard Error of Equating

Standard Error of Equating. Lisa Lendway 2/20/2008. Definition. “The standard deviation of equated scores over hypothetical replications of an equating procedure in samples from a population or populations of examinees.” -Kolen & Brennan (p. 232).

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Standard Error of Equating

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  1. Standard Error of Equating Lisa Lendway 2/20/2008

  2. Definition “The standard deviation of equated scores over hypothetical replications of an equating procedure in samples from a population or populations of examinees.” -Kolen & Brennan (p. 232)

  3. Closed-Form EquationsStandard Error of Equipercentile Equating • X & Y administered to N1 & N2 examinees • Given x0, find y’ that has same sample cumulative frequency. • Goal: find asymptotic sampling variance of y’ • Solution: Use delta method • Formula below is for continuous case, similar for discrete case. Sample values substituted Lord, Frederick M. (1982). Journal of Educational Statistics.

  4. Closed-Form EquationsStandard Error of Equipercentile Equating • Tested against numerical results • 1,000 pairs of pseudorandom standardized normal bivariate deviates drawn from a population with correlation = .9 • y’ found separately for x0=0,0.5,1.0,1.5,2.0,2.5 • Repeated 1,000 times with individually drawn random samples • Empirical standard deviation calculated and compared to theoretical values Lord, Frederick M. (1982). Journal of Educational Statistics.

  5. Use synthetic population Estimator of equipercentile equivalent of x0 (integer score on X plus .5) using linear interpolation y0 (y’ for sample) is the integer score on Y s.t. Gs(y0-1)<Fs(x0)<Gs(y0) Delta Method provides formula for SE of y’’ (lengthy!) Also performed simulation – errors are similar. Closed-Form EquationsSE of Equipercentile – common item Similar to Kolen & Brennan, formula 2.18 Jarjoura, D. & Kolen, M. (1985). Journal of Educational Statistics.

  6. Closed-Form EquationsSE of Equipercentile – common item Jarjoura, D. & Kolen, M. (1985). Journal of Educational Statistics.

  7. Small Sample Equating • Used actual test data file from Early Childhood Ed., Art Ed., Music Ed., Math, & Spanish • Samples of 15, 25, 50, & 100, 1000 times • Calculate bootstrap SE • Smaller sample size  larger SE Parshall, Du Bose Houghton, Kromrey, (1995). Journal of Educational Measurement.

  8. Small Sample Equating Parshall, Du Bose Houghton, Kromrey, (1995). Journal of Educational Measurement.

  9. Thoughts • Bootstrap is computer intensive, but can always be used • Equations are often messy and are only asymptotically true, but may be more accurate. • What about confidence intervals? We should be able to bootstrap those as well, but couldn’t find anything in the literature.

  10. Questions or comments?

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