Confidence intervals of the parameters

If you reason to run nonlinear regression is to interpolate unknown values, you won't really care about the values of the parameters so won't care about the confidence interval of the parameters. But if you do care about the values of the parameters, we suggest that you always ask Prism to report confidence intervals, as inspecting the confidence intervals of best-fit parameters is an essential part of evaluating any nonlinear fit.

Prism offers two methods to compute the confidence intervals.

• Asymptotic approximate symmetrical confidence intervals. These are also called Wald confidence intervals. These were the only confidence intervals reported by Prism 6 and earlier and by most programs. But since the true uncertainty in the parameter's value is often asymmetrical, these symmetrical intervals are not always accurate. We suggest choosing them only when you need to compare Prism's results with other programs, when you need to be consistent with earlier work, or when you have so much data that the profile likelihood method is too slow.

•Asymmetrical (and thus more accurate) profile likelihood confidence intervals. These do a better job of quantifying how precisely you know the parameters value, so we suggest using them routinely. The only disadvantage is that the calculations are more complex so are noticeably slower with huge data sets (especially with user-defined equations).

Don't confuse the two choices here with the two choices for reporting the CI of parameter transforms.

What happens if the fit is ambiguous? You can ask Prism to compute profile likelihood confidence intervals and it will try to do so but this may take a while and the results are unlikely to be useful. So we recommend not checking this option unless you have a strong reason to.

Prism can report the confidence intervals in two ways: as a range or as separate blocks of lower and upper confidence limits (useful if you want to paste the results into another program). The former is easier to read. The latter might be better if you are tabulating the results elsewhere.

The 95% confidence bands enclose the area that you can be 95% sure contains the true curve. It gives you a visual sense of how well your data define the best-fit curve.

The 95% prediction bands enclose the area that you expect to enclose 95% of future data points. This includes both the uncertainty in the true position of the curve (enclosed by the confidence bands), and also accounts for scatter of data around the curve. Therefore, prediction bands are always wider than confidence bands. When you have lots of data points, the discrepancy is huge.

Learn more about confidence and prediction bands.

Standard errors are intermediate values used to compute the symmetrical confidence intervals, but are not very useful by themselves.

You may want to include standard errors in the results to compare Prism's results to those of another program that doesn't report confidence intervals or to collaborate with colleagues who don't understand confidence intervals. But we recommend routinely turning off reporting of the standard errors, because they do a poor job of conveying the precision of the best-fit parameter values.