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Measuring the progress of equality of educational opportunity in absence of cardinal comparability

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Abstract

There are many situations in which measures of the full diversity of a collection of distributions is necessary and where simple comparisons of limited numbers of distributional moments are inadequate since they cast a veil of ignorance over the full extent of distributional differences. An example is the equality of opportunity imperative which demands equal chances for diverse circumstance groups. It requires comparison of distributional differences over the full range of their variation since only then can complete equality of chances be guaranteed. Here new techniques in the form of Gini-like coefficients for quantifying multilateral distributional differences in absence of cardinal comparability are introduced and employed to study changes in the German educational system in the first decade of this century.

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Notes

  1. This simple formulation nicely illustrates the problem highlighted in the Carniero et al. [13] critique in that just seeking equality in means ignores or masks potential variability in variance which would reflect important differences in high and low achievers.

  2. The choice of normal densities depend on the assumption of normality in effort. However this is not an overly strong assumption since, any continuous distribution can be approximated to some desired degree of accuracy by an appropriate finite Gaussian mixture [34].

  3. It is well known that the likelihood function of normal mixtures is unbounded and the global maximizer does not exist [25]. Therefore, the maximum likelihood estimator of \({\varvec{\Psi }}\) should be the root of the likelihood equation corresponding to the largest of the local maxima. The usual solution is to apply a range of starting values for the iterations. In this paper, randomly selected large sample, non-hierarchical clustering-based starts, are employed [21].

  4. Assuming for convenience that \(f_{krn}(x)\) has positive support over the whole real line, these tests are closely related to Integrated Squared Difference tests [14], and Pearson goodness of fit tests [20].

  5. A matrix T has a quasi-maximal diagonal when there exists positive \(\alpha _1, \ldots , \alpha _k, \ldots , \alpha _K\) such that \(\alpha _k t_{kk} \ge \alpha _j t_{kj}\), for all k, j. Restricting the analysis to the subset of quasi-maximal diagonal matrices reconciles the axioms of Perfect Mobility and Monotonicity as shown for the Shorrocks’s mobility index [38].

  6. When classes are large in number it is possible to study and assess polarization and convergence to multiple poles in a similar fashion but it is not necessary in this case.

  7. German school tracking, long viewed as in institutional device reinforcing the intergenerational persistence in educational achievements across different social classes has been the object of considerable study (for an excellent survey see [23]).

  8. It would be also interesting, and left for future research, to investigate the progress in EO separately for gender and other students’ characteristics. Moreover, in this paper we are interested in an overall evaluation of the skills mastered by students at the age of 15, but a more comprehensive understanding could emerge from the analysis of each discipline since similar achievement index may reflect different scores for each field. Actually, correlation between test scores in each field is not strong. Correlation between test scores in 2003 shows a moderate positive correlation between reading and mathematical literacy (\(\rho =0.534\)), a moderate correlation between reading and science (\(\rho =0.367\)), while correlation between science and math is negligible (\(\rho =0.067\)). Interestingly, in 2009 correlation between science and math becomes positive and significant (\(\rho =0.460\)), the correlation between reading and science becomes negative (\(\rho =-0.277\)), while correlation between reading and math remains moderately positive (\(\rho =0.405\)).

  9. An extensive discussion of context indicators is, e.g., in [18].

  10. Circumstances could be treated in a similar semiparametric fashion but the technique yielded a variable which was effectively discrete with just 20 points of support.

  11. A Gaussian kernel density estimator was employed. The bandwidth has been estimated using the plug-in procedure of [37].

  12. Widely-used parsimony-based criteria (AIC, AIC3, CAIC, BIC) confirm that in 2003 the prevalent choice is three components and that in 2009 is definitely four components.

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Acknowledgements

We acknowledge Thomas Fruehauf for his research assistance and for having raised the issue of evaluating the German education system when only ordinal comparability over time is allowed. We would like to thank two anonymous referees, and participants of the Equal Chances International Conference (Bari), the World Bank Conference on “Equity and Development: Ten Years On” (Washington) and economics seminars at Cambridge University and the University of Toronto for their useful comments and suggestions. Obviously we are the solely responsible of any further errors and omissions.

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  1. Department of Economics, University of Toronto, Toronto, Canada

    Gordon Anderson

  2. Department of Statistical Sciences, Sapienza University of Rome, Rome, Italy

    Maria Grazia Pittau & Roberto Zelli

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  1. Gordon Anderson
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Anderson, G., Pittau, M.G. & Zelli, R. Measuring the progress of equality of educational opportunity in absence of cardinal comparability. METRON 78, 155–174 (2020). https://doi.org/10.1007/s40300-020-00172-8

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