topic Re: How to obtain prediction interval for a new observation of x with only parameter standard errors in Statistical Procedures

How to obtain prediction interval for a new observation of x with only parameter standard errors

Jean-Jacques — Tue, 25 Apr 2023 13:58:28 GMT

From a publication, I have parameter estimates for A, B, and C in the following model : Yi = A * exp(-(Xi/B)**C)* ei
The publication also provides standard errors for those three estimates, the range of x values in the data from which the parameters were estimated, and nothing else.

I am generating predicted y values for some values of x within the range of the original data.
Can I also generate standard errors of those point predictions, and predicted percentiles?

Re: How to obtain prediction interval for a new observation of x with only parameter standard errors

StatDave — Tue, 25 Apr 2023 15:30:07 GMT

Assuming the model was fit using regression assuming a normal response, you would need an estimate of the residual error. See the formula in "Statistical Background in Linear Regression / Predicted and Residual Values" in the "Statistical Background in Linear Regression" chapter at the beginning of the SAS/STAT User's Guide. Of course, your model is nonlinear, so see the formulas for LCL and UCL options in the "Details / Confidence intervals" section of the NLIN documentation. Even for a quantile-based interval, you would need an estimate of the normal variance.

Re: How to obtain prediction interval for a new observation of x with only parameter standard errors

Jean-Jacques — Tue, 25 Apr 2023 16:07:30 GMT

Thank you for the response and for the references.
I read both, with only partial understanding.

Am I understanding your response and the references correctly that obtaining a standard error or percentiles for the point predictions is not possible given what is available to me?

Re: How to obtain prediction interval for a new observation of x with only parameter standard errors

StatDave — Tue, 25 Apr 2023 16:11:06 GMT

Correct.

Re: How to obtain prediction interval for a new observation of x with only parameter standard errors

Jean-Jacques — Tue, 25 Apr 2023 16:27:19 GMT

Thank you !

Re: How to obtain prediction interval for a new observation of x with only parameter standard errors

SteveDenham — Thu, 27 Apr 2023 13:36:41 GMT

Hi (and particularly @StatDave 😞

Would it be possible to generate a bootstrap estimate of the residual variance? Assume the range of X values is uniform. Assume the parameter estimates are normally distributed, with mean = the reported estimate and standard deviation = the reported standard error. Draw at random with replacement 100 values of X, and 1000 3-ples of the parameters (this is where a big assumption comes in - that the parameter estimates are uncorrelated). From these, calculate 100 x 1000 Y values. Last step, refit the equation at a range of (X,Y) values covering the interval, making sure to include X values of interest with Y set to missing. That should give predicted Y values with standard errors and confidence intervals, and I am sure quantiles could be calculated given the residual error.

Thoughts?

SteveDenham