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Abstract
Odds ratio (OR) and the educational Gini coefficient (EGINI) defined on a specific social grouping scheme can both be used to gauge educational inequality. In this study, we systematically review the scholarship that underpins the utilities of OR and EGINI and evaluate their properties across four research scenarios of allocating newly created opportunities under educational expansion. Formal analyses suggest that (1) both OR and EGINI monotonically increase (decrease) if the newly created educational positions are entirely delivered to the upper (lower) class. However, the rate of change differs between OR and EGINI. (2) When educational opportunities within expansion are allocated according to the relative populations of social classes, the probabilities of upper and lower classes to receive better education rise by the same extent, suggesting that all social classes benefit from the proliferation of educational credentials. This would drive down the EGINI, but change in OR is not monotonic. (3) EGINI remains unchanged if the newly created educational opportunities are allocated based on the initial class composition of better educated individuals before educational expansion. However, OR is driven to increase by the “Matthew Effect” where the increment in educational probability for the upper class is larger than that for the lower class. The discrepancy of empirical patterns between OR and EGINI is explained, and the practical implications are discussed using a case study of the educational policy evaluation in Britain.
Notes
1By educational inequality, we in this article mean the unequal access to educational opportunities across groups defined in terms of a certain social grouping scheme, e.g., educational inequality between class groups or races. This is the most common type of educational inequality that is frequently examined by sociologists and policy makers. Besides the social-category-based definition, educational inequality, as often defined in the economic research, can also describe an unequal status without referring to a specific social grouping, such as the distribution of educational attainment across population quintiles. This type of educational inequality, however, is not discussed in this study.
2The Mare's educational transition model confronts critiques, but this is beyond the scope of this article. For methodological discussions on the Mare's model, please see Lucas, Fucella, and Berends (Citation2011).
3Nevertheless, this school of studies does not necessarily analyze educational inequality across discernable social groups.
4There are some other measures used by economists to gauge educational equality, such as the Theil index.
5In the case of two classes, if their probabilities of attending college p1 and p2 are always the same, OR, as a function of solely p1 and p2, would not change.
6One reviewer suggests a research scenario where the ratio of p1 and p2 is set to be constant. This scenario, according to the reviewer, can be used to attest the marginal insensitivity property of OR, as well as to investigate whether or not EGINI are insensitive to the variation in dependent variable (college entrance). However, a constant ratio of p1 and p2 cannot be used to reveal the marginal-insensitivity of OR. A simple example can illustrate this point of view. Suppose we consider two situations where situation one is featured by p1 = 0.3 and p2 = 0.1, and for situation two, p1 = 0.6 and p2 = 0.2. Evidently, p1/p2 is the same for these two situations, but their ORs differ (3.86 for situation one and six for situation two). As for EGINI, it is kind of straightforward to show its sensitivity to the distribution of the dependent variable. The distribution of college attendees, using the notations in Table , is A*p1 + (1 − A)*p2. This is exactly a component of EGINI, as EGINI's formula is . Hence, EGINI is mathematically not independent from the variation in dependent variable.
7Scenario D is practically possible as long as (1) college education is always desirable, and (2) college entrance is featured by full competition of individuals from different classes. In this case, people from the upper class are always advantageous over the lower class, irrespective of change in the specific college entrance criterions.
8One challenge in the following simulations as pointed out by one reviewer is the scaling effect of OR, that is, odds ratios are identified only up to scale (e.g., Allison, Citation1999; Mood, Citation2010). In order to evaluate this concern, we conducted a series of supplementary analyses by standardizing the variance of Y (that is, the access to higher education) across simulation scenarios. Although the scaled ORs differ from the original ORs, they are strongly correlated. More importantly, the basic patterns of the four research scenarios do not materially change.
9In Figure and , we normalize the x-axis and the y-axis by the division by their respective maximum value. After this operation, the maximum value for both axes is fixed at one. This operation is necessary to avoid “false” graph patterns (Watts & Strogatz, Citation1999). This normalization operation, however, is not needed for Figure and because we do not compare subfigures. In Figure , we only normalize the y-axis. In our supplementary analysis not shown here, we examined the non-normalized patterns, which suggested that the normalization operation actually did not introduce appreciable changes to the empirical results.
10ALC totally eliminates the advantages of the upper class, which is not consistent with the theoretical argument of MMI. So we do not take it into account when defining Phase I.
11This is for illustrative purpose and the values are decided arbitrarily. However, different starting values do not affect our simulation results.
12Remember that both p1 and p2 will be added Δ under the scenario of PP. However, p1 is larger than p2 by definition, so the expansion of higher education will promote p1 to be close to 100% first.
13As discussed earlier, the class composition of college students before educational expansion proxies the capabilities of attending college between classes.
14Gini = 2Φ([0.5*Variance]0.5) − 1, where Φ is the cumulative distribution function of the standard normal distribution.
15The variance function model is featured by fitting two models, one is the generic regression model where the independent variable of interest x is used to predictor dependent variable y, as in y = βx + ε. Then, the variance of predicted values is computed to stand for the between-group variance. Subsequently, the variance of the error termε, denoted by σ2, in the generic regression model (log transformed) is regressed on the predictors in a gamma model. The predicted error variance is then the within-group variance.