# _Sample selection issues_

Although we have established that the country ranking that can be derived from Table 1 is ordinally equivalent to the pre-standardization ranking, the issue of PISA sample selection remains a potential problem. As noted in Section 2, coverage rates range from a low of 0.47 in Turkey, to 1.02 in Switzerland.<sup>11</sup> Selection would not be a problem if one were interested _exclusively_ in the performance of 15 year-olds that are in school, and within a reasonable range of their expected grade of attendance. But this is likely to be an excessively narrow prism through which to assess a country’s educational system and – even more so – to make international comparisons. Consider the example of two hypothetical “educational strategies”, illustrated by countries A and B, which have identical distributions of school and family characteristics, as well as of underlying ability in the population of 15 year-olds. Country A seeks to be inclusive, and allocates resources towards retaining as many students as possible in school, and towards promoting learning by those with the lowest demonstrated achievement. Country B, on the other hand, actively discourages enrollment by those with lower ability, and seeks to retain only the top half of performers in school by age 15. Looking only at the test scores for the samples of enrolled fifteen year-olds will naturally suggest that Country B has both a higher mean and a lower variance than country A, and thus a superior educational system altogether.

This is not to suggest, of course, that Brazil, Indonesia, Mexico, Turkey, or any of the other countries with low coverage rates in Table 1 actively pursue an exclusionary strategy like that of hypothetical country B. But dropping out and lagging behind are, nevertheless, extremely likely to be selective processes, in the sense that they are correlated with family and student characteristics that also affect test scores. If one is interested in comparing the educational achievement of the _population_ of fifteen year-olds across countries, therefore, the PISA samples suffer from selection bias.

Correcting for such biases is never simple, and even less so when non-participants are not observed at all in the sample (unlike, say, when seeking to correct for labor force participation on the basis of surveys that contain information on both earners and non-participants). While we do not offer a sample selection bias correction procedure for all countries in the PISA sample in this paper, we propose a simple two-sample non-parametric mechanism for assessing the sensitivity of our inequality measures to alternative assumptions about the sample selection process.

Denote the (density of the) distribution of test scores _y_ in a particular country _j_ by . Consider a vector of covariates _X_ that is observed both in the PISA sample and in an ancillary household survey, which is representative of the full population of 15 year-olds. Note that the density of test scores in the PISA sample can be written as:

> 11 One presumes that coverage rates in excess of 1.00 must be due either to statistical discrepancies in the estimates of 15 year-olds in the total population, or to errors of inclusion in the sample of test-takers.

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