Linear combinations come alive in crossover designs

Before learning anything about statistical inference in beginning service courses in biostatistics, students learn how to calculate the mean and variance of linear combinations of random variables. Practical precalculus examples of the importance of these exercises can be helpful for instructors, the target audience of this paper. We shall present applications to the "1-sample" and "2-sample" methods for randomized short-term 2-treatment crossover studies, where patients experience both treatments in random order with a "washout" between the active treatment periods. First, we show that the 2-sample method is preferred as it eliminates "conditional bias" when sample sizes by order differ and produces a smaller variance. We also demonstrate that it is usually advisable to use the differences in posttests (ignoring baseline and post washout values) rather than the differences between the changes in treatment from the start of the period to the end of the period ("delta of delta"). Although the intent is not to provide a definitive discussion of crossover designs, we provide a section and references to excellent alternative methods, where instructors can provide motivation to students to explore the topic in greater detail in future readings or courses.