the efficient exam shlomo yitzhaki hebrew university
Post on 27-Dec-2015
217 Views
Preview:
TRANSCRIPT
The Efficient Exam
Shlomo Yitzhaki
Hebrew University
Talk’s Structure
• Characterization of Grades in Exams
• Monotonic correlation
• Properties of Gini’s Mean Difference
• Properties of the Efficient Exam
Characterization of grades
• Grades are an Ordinal Variable• It is as if we are measuring height of
people standing behind a screen• We ask who has the X centimeter and
those that are taller than X respond positively.
• Height is the number of positive answers.• It is impossible to plot a cumulative
distribution of grades
Characterization of Grades
• If cumulative distributions of two groups intersect, then there are two alternative legitimate exams that will result in contradicting ranking of average grades.
• Hence, one can improve her country performance in international exams like PISA, by pointing out the alternative exam.
Monotonic Correlation
• It is assumed that we are examining a uni-dimensional ability
• Otherwise we have to examine whether the correlation is monotonic.
• The method to do that is based on plotting Concentration curves (A variant of Lorenz curve for two variables).
• The Method is already published (Economics Letters, 2012).
Properties of GMD
• Gini’s Mean Difference can be decomposed in a way that makes the decomposition of the variance a special case.
• This way one can find the implicit assumptions behind the variance.
• Properties described in a 540 pages book• Entitled “The Gini Methodology” by
Springer Statistics N. Y. 2013.
GMD vs. Variance
• Variance = cov(X, X)
• GMD = 4 cov(X, F(X))
• Note that F is uniform [0, 1].
• Gini covarince cov(X, F(Y)), cov(Y,F(X))
• They don’t have to have the same sign.
• Known in economics as “Index number problem.
Properties of GMD
• ANOVA Translates into ANOGI• Two correlation coefficients between two
variables two Gini Covariance, two regression coefficients, mixed GMD-OLS regression, etc..
• If the two correlations between two variables are equal, then we get an identical decomposition of the variance of a sum of random variabes
Properties of the Efficient exam
• Because of the limited number of questions, There is “Binning”
• Main proposition: To maximize between-group variability, the distribution of grades in the “efficient exam” should be Uniform.
• No proof is presented in this talk.
A sketch of the proof
• The proof is based on the proposition that the distribution of the cumultive distribution is uniform [0, 1].
• Using Lorenz curve then the question is what is the optimal size of a “bin”
• Two stages: Every “bin” should be positive. Mid-point is optimal
Transvariation
• Two possible ways to rank groups:• According to average grade• According to transvariation: The probability
of a randomly selected member of the high (low) average group to be better than the randomly selected member of the lower (higher) average group.
• Under efficient exam both criteria are equivalent.
Applications
• The arguments are relevant to any test based on ordinal variable.
• I owe this point to Emil
• This is the reason why I was invited
Thank you
• For your Patience
top related