Thursday, September 21, 2017

Great Minds in Statistics: Georg Rasch

Happy birthday, Georg Rasch, who would have been 116 today! Rasch was a Danish mathematician who contributed a great deal to statistics, and specifically a branch of statistics known as psychometrics. In fact, his most important contribution bears his name - the Rasch measurement model, which is a model that we use to analyze measures of ability (and later, other characteristics like personality).

Explaining everything about the Rasch model and using it to analyze measurement data would take more blog posts than this one. (Don't worry, reader, I'm planning a series of posts.) But my goal for today is to explain the basic premise and tie it to things I've blogged about previously.

The Rasch model was originally developed for ability tests, where the outcomes are correct (1) or incorrect (0). Remember binary outcomes? This is one of those instances where the raw scale is both binary and ordinal. But you don't use items in isolation. You use them as part of a measure: the SAT, the GRE, a cognitive ability test.

So you might argue the outcome is no longer ordinal. It's the total number of correct answers. But not all items are created equal. Some items are easier than others. And for adaptive exams, which determine the next item to administer based on whether the previous item was answered correctly, you need to take into account item difficulty to figure out a person's score.

Adding a bunch of ordinal variables together doesn't necessarily mean you have a continuous variable. That final outcome could also be ordinal.

Rasch developed a logarithmic model that converts ordinal scales to interval scales. Each individual item now has an interval scale measure, and overall score (number of items correct) also has an interval scale. It does this by converting to a scale known as logits, which are log odds ratios. Item difficulty (the interval scale for items) and person ability (the interval scale for overall score) are placed on the same metric, so that you can easily determine whether a person will respond to a certain item correctly. If your ability, on the Rasch scale, is a logit of 1.5, that means an item of difficulty 1.5 on the Rasch scale is perfectly matched to your ability.

What does that mean in practice? It means you have a 50% chance of responding correctly to that item. That's how person ability is typically determined in the Rasch model; based on the way you respond to questions, your ability becomes the difficulty level where you answer about half the questions correctly.

Even better, if I throw an item of difficulty 1.3 at you, you'll have a better than 50% chance of responding.

But I can be even more specific about that. Why? Because these values are log odds ratios and there's a great reason person ability and item difficulty are on the same scale. First, I subtract item difficulty (which we symbolize as D) from your ability level (which we symbolize as B): B-D = 1.5-1.3. The resulting different (0.2) is also a log odds ratio. It is the log transformation of the odds ratio that a person of B ability will answer an item of D difficulty correctly. I convert that back to a proportion, to get the probability that you will answer the item correctly, using this equation:

where P(X=1) refers to the probability of getting an item correct. This equation is slightly different from the one I showed you in the log odds ratio post (which was the natural number e raised to the power of the log odds ratio). Remember that equation was to convert a log odds ratio back to an odds ratio. This equation includes one additional step to convert back to a proportion.

If I plug my values into this equation, I can tell that you have a 55% chance of getting that question correct. This is one of the reasons the Rasch model does beautifully with missing data (to a point); if I know your ability level and the difficulty level of an item, I can compute how you most likely would have responded.

Stay tuned for more information on Rasch later! And be sure to hug a psychometrician today!

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