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I am trying to understand the difference between logistic regression probabilities and linear regression prediction intervals.

For example, let's say we have a database of student test scores in the range of 1 to 100 and some predictors. The goal of this study is to build a model to predict if other students will reach at least a score of 60 with an 80% confidence. To simplify, we are assuming that all the linear modeling requirements in the data are verified.

The first method would be to run a linear regression on the observed data, then calculate the 80% prediction intervals and finally determine whether a student will reach a score of 60 or higher based on the lower end of the prediction interval. The other approach is to categorize our data and run a logistic regression on each student score < 60 (0) or >= 60 (1) observation.

Is there any benefit in using the logistic regression approach in this case? Or does linear regression will result in the same level of accuracy when using prediction intervals?

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Neat question. It seems to me prediction would have to be more accurate using the linear method. The logistic would necessitate throwing away information by categorizing each person in a binary way. Such an approach would mean that the regression (if I could anthropomorphize) "wouldn't know the difference" between a 60 and a 100--only that they are each in the upper category. This would have to weaken its ability to come up with precise coefficients. No doubt others will offer more rigorous replies.

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  • $\begingroup$ If a large majority of observations is around 20 and the mean is, say, 25, then the linear regression coefficients will be optimized for this center value. This is where the prediction interval will have the smallest range. By "discretizing" the observations at the 60 score threshold the logistic regression would not suffer from the same problem? $\endgroup$ – Robert Kubrick Feb 1 '12 at 3:38

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