Explaining Lord's Paradox in Introductory Statistical Theory Courses
- Steven Kim
AbstractWhen two groups are compared in a pre-post study, two different conclusions can be drawn between the two-sample t-test and the analysis of covariance (ANCOVA). It is known as Lord's Paradox, and it occurs because the parameter in the two-sample t-test and the parameter of interest in the ANCOVA model are not the same quantity. The difference between the two parameters can be explained by the covariance of linearly combined random variables which is an important topic in introductory statistical theory courses. Lord's paradox is frequently observed in practice, and it is very important for students (future researchers) to have clear understanding of the paradox. The objective of this article is to explain Lord's Paradox using the covariance of linearly combined random variables. The paradox is explained using three scenarios in the context of educational research. The first scenario is when the average baseline (pre-score) is greater in the treatment group than the control group, the second scenario is when the average baseline is lower in the treatment group than the control group, and the third scenario is when the average baseline is same between the two groups by randomization. This article is written at the level of introductory statistical theory courses for undergraduate and graduate statistics students to help understanding the difference between the parameter of interest in the two-sample t-test and the parameter of interest in the ANCOVA model.
This work is licensed under a Creative Commons Attribution 4.0 License.
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