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Are higher marks in mathematics or science indicative of higher marks in other subjects, such as reading and spelling? And conversely, are higher marks in reading or spelling indicative of higher marks in, say, mathematics?
These questions are among the many that have long interested educators and psychologists, and the literature on such topics is abundant. But a natural statistical question arises: what are real-life data telling us, regardless of the wishful thinking of curricula developers? With a huge raw data set in its possession, Hefei Gemei Culture and Education Technology Co. Ltd., an educational tech company based in China, wanted to explore such questions with the help of theoretical and data-crunching statisticians.
Traditionally, mathematical correlations, such as Pearson, intra-class, and Spearman, have been used in such studies to describe and analyze associations in data. But the use of these correlations presented serious issues when it came to assessing associations in data that clearly exhibited non-monotonic patterns and were suggestive of non-interchangeability of underlying variables (e.g., mathematics may affect reading differently than the other way around).
To exemplify the problems posited to us by the company, we may think of two groups of students who are taught two subjects (e.g., mathematics and reading) using two different teaching approaches; let’s call them “traditional” and “modern.” Student knowledge in the two subjects is tested, and the resulting paired data (mathematics versus reading) give rise to two scatterplots, which are then compared.
As is the case in most real-life data sets that we have worked with, the scatterplots exhibit non-linear and even non-monotonic relationships between underlying variables, such as mathematics and reading. Despite these non-monotonic relationships, we would nevertheless tend to claim, based on visual assessment, that one of the scatterplots exhibits a more increasing pattern than the other one. Of course, if the patterns were linear, then we could substantiate such claims by comparing the slopes of the underlying straight lines, but what does a “more increasing” pattern mean in the case of non-monotonic relationships?
To better understand such relationships and patterns, and to also quantify their lack of monotonicity, in a recent paper, Yuri Davydov of the Chebyshev Laboratory in Russia and I (Ricardas Zitikis, Western University) employed geometric insights and mathematical considerations that gave rise to indices of (lack of) increase, which are functional distances of the observed non-monotonic patterns — such as those exhibited by the aforementioned scatterplots — from the hypothetical set of all increasing patterns. Subsequently, my PhD student Jiang Wu pursued a collaboration on this topic with Hefei Gemei Culture and Education Technology Co. Ltd., supported by Mitacs’ Globalink Partnership Award.
During the project, Jiang analyzed massive educational data and developed software for analyzing such data. Upon his return to Canada, Jiang provided feedback to us (Dr. Zitikis and Lingzhi Chen, PhD student) that allowed us to refine, calibrate, and modify existing techniques, and we have published our results in the Education Sciences journal. Since the original Chinese data analyzed by Jiang remain proprietary, our published proof-of-concept was accomplished using publicly available small-scale data.
The Globalink Partnership Award played a pivotal role in connecting industry with the Zitikis Research Group, which works, among other consulting projects, on problems of educational measurement and assessment. The group has since expanded, and in addition to Jiang and Lingzhi, it now includes a diverse group of professors in Canada and around the world. We expect the group to expand even further, given the interest from our industry partner to pursue further research, and due to a wealth of interesting and challenging mathematical, statistical, and probabilistic problems that have arisen as a result of this project.