# SinglePredictionBands versus MeanPredictionBands from NonlinearModelFit

From my Statistics lectures I know the term confidence and prediction interval, whereas the confidence interval is smaller than the prediction interval. If I plot the mean and single prediction bands with the same confidence level in Mathematica, than the mean prediction band is smaller than the single prediction band interval.

Anyone who can explain me what are the differences between the “SinglePredictionBands” and “MeanPredictionBands” which are standard properties from NonlinearModelFit?
Or is it just a different terminology for confidence and prediction interval used by Mathematica? If no it would be useful if someone can tell me which properties I have to apply for the confidence and the prediction interval.

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## 2 Answers

• SinglePredictionBands - confidence bands based on single observations
• MeanPredictionBands - confidence bands for mean predictions

Directly from the help centre under the Details and Options submenu Link

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Citing this Demonstration by Darren Glosemeyer (Wolfram Research):

Single prediction bands incorporate both the variation in parameter estimates and the overall variation in response values, while the mean confidence bands incorporate only the variation in parameter estimates. As a result, single prediction bands are wider than mean prediction bands for the same confidence level. Mean prediction bands also exhibit more variation in width.

Also from this MathGroups post:

The method used is very briefly described near the end of

http://reference.wolfram.com/mathematica/tutorial/StatisticalModelAnalysis.html

"Tabular results for confidence intervals are given by "MeanPredictionConfidenceIntervalTable" and "SinglePredictionConfidenceIntervalTable". These results are analogous to those for linear models obtained via LinearModelFit, again with first-order approximations used for the design matrix.

"MeanPredictionBands" and "SinglePredictionBands" give functions of the predictor variables."

The bands use the same formulas as the confidence intervals. The method is the same for 2, 3, or more parameters.

Darren Glosemeyer Wolfram Research

One can also find useful this blog post: "Two kinds of Prediction bands."

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