1. Latin squares
Latin Square designs will require us to think back to what we learned about Randomized Controlled Block Designs, as they'll expand on that design. Remember that the goal in RCBDs was to block on a factor to reduce the variance of the outcome due to that factor. Let's look at Latin square designs to see how they expand on that idea.
2. Latin squares
Latin square designs are, luckily, not that ominous: they expand on RCBDs by blocking on a second factor. Now, by designing an experiment in this way, we'll be able to control both of the blocking factors' variability and study our actual outcome of interest. Of note -- the treatment and both blocking factors need to have the same number of levels, which is a limitation of the Latin square design.
One key assumption of a Latin square experiment is that there is no interaction between any combination of the treatment and blocking factors. If this turns out not to be true, the error term of the model will be inflated or overestimated.
One nice thing about Latin square is you can analyze them just as you would a RCBD, by building a linear or other appropriate type of model, then examining it with anova() or using summary to examine the results if you used aov().
3. Latin square diagram
You may notice that a Latin Square looks a lot like a RCBD, and you're absolutely right about that. However, in the case of a Latin Square experiment, both the rows and columns of this grid are blocking factors. We know that this is a valid Latin square experiment because all treatments, A through D, appear four times each across the grid, and each treatment appears only once per block. There is only one A, B, C, and D per both row and column. Take a second to verify that for yourself.
4. Why is it a Latin square?
It may be easier to verify quickly that this is a Latin square if we used shapes to represent each treatment. Personally, I can see a lot faster that there's only one circle per row and column than I can with letters.
5. Intro to NYC scores
For this chapter, we'll be using the nyc_scores dataset, which is an open dataset from the city of New York giving the SAT scores for the Reading, Writing, and Math sections from all accredited high schools in the 2014 to 2015 school year. It also includes variables giving the school's borough and stats on enrollment by race and ethnicity, among other variables.
We'll use it in this chapter to conduct some educational experiments on a Tutoring Program's effect on the scores. Given that we did not collect this data ourselves, the tutoring program data has been fabricated, and will not be available in the original data should you download it yourself.
6. Let's practice!
Let's go to the exercises. We'll begin to get familiar with the nyc_scores dataset and build and analyze a Latin square experiment.