In the linking theories above, the anchoring process is done by adjusting results from the same population between two tests, or with the same items used in different tests. In our case, we use the former approach. We examine the same population between two tests to determine the relationship between _Reference Test X_ and _Anchored Test Y_ . To this end, we compare the same countries at the same point in time which took an ISAT and an RSAT. Since ISATs and RSATs are psychometrically-robust, sample-based test designed to be nationally representative, they represent the same underlying population at the country-level. Thus, by comparing _doubloon_ countries which participate in both tests being linked, we can index difficulty and scales across tests. Table 2 provides the list of countries that overlap in assessments<sup>12</sup> . This enables inclusion of Regional Standardized Achievement Tests (RSATs) from Latin America and Sub-Saharan Africa and thus international comparison. This is a significant addition, since many developing countries have participated in RSATs (LLECE, SERCE, PASEC, and SACMEQ) but rarely or never in ISATs (PISA, TIMMS, PIRLS). Transformation of regional scores into an internationally comparable value is most accurate the more _doubloon_ countries are available. If our index relies on just one _doubloon_ country (if it is the only country participating in both surveys), it is ambitious to convert all other regional scores using this quotient. We provide robustness tests on the sensitivity to the number of doubloon countries.

In addition to indexing learning outcomes across assessments using _doubloon_ countries, we anchor assessments across time using the _National Assessment of Educational Progress_ (NAEP) in the United States. This is possible since the U.S. participated in NAEP and various international achievement test at every interval. For example, if the performance of the United States changed in NAEP in a given year but did not in the same subject and year in which in the IEA assessment the U.S. took part in, it would mean that the IEA study is upward or downward biased. To correct for this under- or over-estimation, we adjust old IEA studies by trends on NAEP results.<sup>13</sup> We only include the NAEP adjustment for scores before the 1990s since standardized ISATs began to be conducted consistently from the 1990s onwards and are therefore comparable over time.

> 12 For example, when linking PISA 2003 and TIMSS 2013 the 15 overlapping countries are: countries are Australia, Hong-Kong, Hungary, Indonesia, Italy Japan, Republic of Korea, Latvia, Netherlands, New Zealand, Norway, Russian Federation, Slovakia, Sweden, Tunisia and the United States).

> 13 A similar methodology was used for linking PASEC assessments over time. We used results for a national assessment in Burkina Faso, which provides over-time comparable scores, and also took part at PASEC in 2006 and 2014. After linking PASEC assessments onto a single scale, we used the participation of Mauritius in both PASEC and SACMEQ for linking PASEC to our internationally anchored scale. However, PASEC is an assessment for Francophone countries, while SACMEQ focuses on Anglophone countries. This might bias the anchoring process for adjusted reading scores. Since Mauritius has been tested in both languages in PASEC (English and French) we can use this to correct for language differences.

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