## Friday, March 14, 2014

### Orthogonality and the SAT

[Note: 'SAT' refers to the SAT Reasoning Test]

If you spend any time following the SAT debate, you will frequently encounter some variation on the phrase:
All in all, the changes are intended to make SAT scores more accurately mirror the grades a student gets in school.

The thing is, though, there already is something that accurately mirrors the grades a student gets in school. Namely: the grades a student gets in school. A better way of revising the SAT, from what I can see, would be to do away with it once and for all.
Putting aside the questionable assumption that the purpose of a colleges selection process is to find students who will get good grades at that college, there is a major statistical fallacy here, and it reflects a common but very dangerous type of oversimplification.

When people talk about something being the "best predictor" they generally are talking about linear correlation. The linearity itself is problematic here – we are generally not that concerned with distinguishing potential A students from B students while we are very concerned with distinguishing potential C students from potential D and F students – but there's a bigger concern: The very idea of a "best" predictor is inappropriate in this context.

In our intensely and increasingly multivariate world, this idea ("if you have one perfectly good predictor, why do you need another?") is rather bizarre and yet surprisingly common. It has been the basis of arguments that I and countless other corporate statisticians have had with executives over the years. The importance of looking at variables in context is surprisingly difficult to convey.

The explanation goes something like this. If we have a one-variable model, we want to find the predictor variable that gives us the most relevant information about the target variable. Normally this means finding the highest correlation between some transformation of the variable in question and some transformation of the target where the transformation of the target is chosen to highlight the behavior of interest while the transformation of the predictor is chosen to optimize correlation. In our grading example, we might want to change the grading scale from A through F to three bins of A/B, C, and D/F. If we are limited to one predictor in our model picking, the one that optimizes correlation under these conditions makes perfect sense.

Once we decide to add another variable, however, the situation becomes completely different. Now we are concerned with how much information our new variable adds to our existing model. If our new variable is highly correlated with the variable already in the model, it probably won't improve the model significantly. What we would like to see is a new variable that has some relationship with the target but which is, as much as possible, uncorrelated with the variable already in the model.

That's basically what we are talking about when we refer to orthogonality. There's a bit more to it – – we are actually interested in new variables that are uncorrelated with functions of the existing predictor variables – but the bottom line is that when we add a variable to a model, we want it to add information that the variables currently in the model haven't already provided.

Let's talk about this in the context of the SAT. Let's say I wanted to build a model predicting college GPA and, in that model, I have already decided to include high school courses taken and their corresponding grades. Assume that there's an academic achievement test that asks questions about trigonometric identities or who killed whom in Macbeth. The results of this test may have a high correlation with future GPA but they will almost certainly have a high correlation with variables already in the model, thus making this test a questionable candidate for the model. When statisticians talk about orthogonality this is the sort of thing they have in mind.

The SAT works around this problem by asking questions that are more focused on aptitude and reasoning and which rely on basic knowledge not associated with any courses beyond junior high level. Taking calculus and AP English might help students' SAT scores indirectly by providing practice reading and solving problems so we won't get perfect orthogonality but it will certain do better in this regard than a traditional subject matter exam.

This is another of those posts that sits in the intersection of a couple of major threads. The first concerns the SAT and how we use it. The second concerns orthogonality, both in the specific sense described here and in the general sense of adding information to the system, whether through new data, journalism, analysis or arguments. If, as we are constantly told, we're living in an information-based economy, concepts like orthogonality should be a standard feature of the conversation, not just part of statistical esoterica.