RE: Simulation vs. actual data

From: "Kowalski, Ken" Ken.Kowalski@pfizer.com Subject: RE: [NMusers] Simulation vs. actual data Date: Tue, July 5, 2005 12:30 pm Hi Nick, Yes, the method you describe comes closest to what I describe as a tolerance interval in that you are trying to characterize the variability (distribution) of individual responses. Whereas confidence intervals try to characterize the variability (uncertainty) in the population mean response. In the univariate normal case the distinction boils down to whether you construct intervals of the form Xbar +/- k*SD or Xbar +/- k*SE (where SE = SD/sqrt(n)) where the former characterizes the variability in the individual observations (SD) while the latter characterizes the variability or uncertainty in the mean response (SE). If Yano et al. define a "degenerate" tolerance interval as one that does not attach any confidence statement to the interval because parameter uncertainty is not taken into account I suppose I can accept that definition. Prediction intervals are more closely related to confidence intervals in that we are often trying to characterize uncertainty in some kind of mean response (not necessarily the population mean response) rather than the distribution of individual responses. The mean response in this case is conditioned on some design or sample of a fixed number of future (predicted) observations. When this fixed number of predicted observations approaches infinity then the mean we are characterizing is the population mean and the prediction interval (for a mean of an infinite number of predicted observations) collapses to a confidence interval on the population mean. When the fixed number of observations is small, then the distribution of the mean of n predicted responses will be considerably wider than the distribution of the population mean prediction. Moreover, when n=1 then the prediction interval is on a single observation which can be considerably wider than tolerance intervals as I noted in my previous message. I hope this helps clarify the distinctions between these types of intervals. Ken