Allocation to groups: Examples of Lord's paradox

Background Educational and developmental psychologists often examine how groups change over time. Two analytic procedures – analysis of covariance (ANCOVA) and the gain score model – each seem well suited for the simplest situation, with just two groups and two time points. They can produce different results, what is known as Lord's paradox. Aims Several factors should influence a researcher's analytic choice. This includes whether the score from the initial time influences how people are assigned to groups. Examples are shown, which will help to explain this to researchers and students, and are of educational relevance. It is shown that a common method used to measure school effectiveness is biased against schools that serve students from groups that are historically poor performing. Methods and results The examples come from sports and measuring educational effectiveness (e.g., for teachers or schools). A simulation study shows that if the covariate influences group allocation, the ANCOVA is preferred, but otherwise, the gain score model may be appropriate. Regression towards the mean is used to account for these findings. Conclusions Analysts should consider the relationship between the covariate and group allocation when deciding upon their analytic method. Because the influence of the covariate on group allocation may be complex, the appropriate method may be complex. Because the influence of the covariate on group allocation may be unknown, the choice of method may require several assumptions.