What is the equivalent of a standard deviation when considering a least squares fit line?

Question

I am calculating a tolerance interval following http://www.itl.nist.gov/div898/handbook/prc/section2/prc253.htm but this says to multiply the k value by the standard deviation of the sample. I have a model with a fit line, so I would think I do not want to use the standard deviation, but rather some value that reflects the residuals, and instead of using the sample mean, I will use the predicted value from my linear regression model. Is that right? What value do I use instead of the standard deviation?

I could (maybe should have) asked the question this way: Given a linear model, how do I compute a one-sided tolerance interval. I think a tolerance interval is the right thing for my problem based on this: http://www.kmjn.org/notes/tolerance_intervals.html

Edit again: I found this formula for "Assuming linear function and no replicates, the standard deviation about the regression" (from here)

Is this the right fomula to get a value to multiply by the k values?

Answer 1

As the question is now posed you are looking for the standard deviation to multiple by the appropriate tabled k for prediction one-sided tolerance interval for y given x. The appropriate standard deviation is the standard deviation of the prediction estimate of y given x not the standard deviation of the residuals. The right standard deviation is obtained by taking the variance for the fitted y given x and adding one estimate of the residual variance and taking the square root. This is because the prediction is the same as the fitted value but the actual value of a new y at the given x differs from the "true" model by an independent error term. So to take account of that the residual variance must be added to the variance of the difference between the "true" y given x and the model fit for it. The sample estimate just replaces the true variance terms with the estimates used in the regression for fit and the error term.

The resulting formulae taken from Chernick 2011 "The Essentials of Biostatistics for Physicians, Nurses, and Clinicians" pp. 102-103 is as follows:

SSx = ∑(X$_i$ - X$_b$)$^2$ where X$_b$ = ∑X$_i$/n

SSE = ∑(Y$_i$ - Y$_b$)$^2$ where Y$_b$ = ∑Y$_i$/n

Then the standard error of the estimate is S$_y$$_.$$_x$=√[SSE/(n-2)].

Next we have the standard error for the fitted Y given X=x is as follows:

SE(Y^) = S$_y$$_.$$_x$ √[(x-X$_b$)$^2$/SSx+1/n] But for prediction we need to add one more S$_y$$_.$$_x$$^2$ term to the get the variance of the prediction. Hence the standard error for prediction of Y given X=x is:

SE(Y$_p$$_r$$_e$$_d$)= S$_y$$_.$$_x$ √[1+(x-X$_b$)$^2$/SSx+1/n].

The constant you need with it will be the one for one-sided Gaussian confidence intervals for the confidence level and coverage that you specify. The tables can be found in the statistical intervals book by Hahn and Meeker.