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I ask this question referring to the post: Bootstrap prediction interval, where a step by step method for calculating the prediction interval for linear regression models is explained.

In the explanation taken from Section 6.3.3 of Davidson and Hinckley (1997), Bootstrap Methods and Their Application, the variance-adjusted residuals are used instead of the raw residuals. Why is that and are the methods that use the raw residuals wrong or underestimating the prediction interval?

Example of algorithms where the raw residuals are used are: YouTube video and lecture slides

On the same topic: after trying the method described here: Bootstrap prediction interval and the one, simpler, described in the YouTube video linked above, on 1000 bootstrap replications, I obtain the Prediction Intervals shown in the figure.

enter image description here

They are pretty similar, though the method from Davidson and Hinckley seems to return slightly broader PI. What is the benefit of using the first method over the second? Can it be that with a bigger sample size (I have only 22 observations), the difference between the two Prediction Intervals would increase?

I have then a question on the practical use of the prediction intervals. I use the data collected by a robot to predict a clinical score. I am interested in the test-retest reliability of the prediction. How are the PI related to the absolute reliability (e.g. to the Standard Error of Measurement or to the Limits of Agreement)?

Thank you for clarifying.

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