Standard errors and confidence intervals of parameters

Standard errors of best-fit parameters

Interpreting the standard errors of parameters

The only real purpose of the standard errors is as an intermediate value used to compute the confidence intervals. If you want to compare Prism's results to those of other programs, you will want to include standard errors in the output. Otherwise, we suggest that you ask Prism to report the confidence intervals only (choose on the Diagnostics tab). The calculation of the standard errors depends on the sum-of-squares, the spacing of X values, the choice of equation, and the number of replicates.

'Standard error' or 'standard deviation' ?

Prism reports the standard error of each parameter, but some other programs report the same values as 'standard deviations'. Both terms mean the same thing in this context.

When you look at a group of numbers, the standard deviation (SD) and standard error of the mean (SEM) are very different. The SD tells you about the scatter of the data. The SEM tells you about how well you have determined the mean. The SEM can be thought of as "the standard deviation of the mean" -- if you were to repeat the experiment many times, the SEM (of your first experiment) is your best guess for the standard deviation of all the measured means that would result.

When applied to a calculated value, the terms "standard error" and "standard deviation" really mean the same thing. The standard error of a parameter is the expected value of the standard deviation of that parameter if you repeated the experiment many times. Prism (and most programs) calls that value a standard error, but some others call it a standard deviation.

Confidence intervals of parameters

Do not ignore the confidence intervals

In most cases, the entire point of nonlinear regression is to determine the best-fit values of the parameters in the model. The confidence interval tells you how tightly you have determined these values. If a confidence interval is very wide, your data don't define that parameter very well. Confidence intervals are computed from the standard errors of the parameters.

How accurate are the standard errors and confidence intervals?

The standard errors reported by Prism (and virtually all other nonlinear regression programs) are based on some mathematical simplifications. They are called "asymptotic" or "approximate" standard errors. They are calculated assuming that the equation is linear, but are applied to nonlinear equations. This simplification means that the intervals can be too optimistic. You can test the accuracy of a confidence interval using simulations.

Sometimes Prism reports "very wide" instead of reporting the confidence interval

If you see the phrase 'very wide' instead of a confidence interval, you will also see the phrase 'ambiguous' at the top of the results tables. This means that the data do not unambiguously define the parameters. Many sets of parameters generate curves that fit the data equally well. The curve may fit well, making it useful artistically or to interpolate unknowns, but you can't rely on the best-fit parameter values.

Confidence intervals of transformed parameters

In addition to reporting the confidence intervals of each parameter in the model, Prism can also report confidence intervals for transforms of those parameters. For example, when you fit an exponential model to determine the rate constant, Prism also fits the time constant tau, which is the reciprocal of the rate constant.

When you write your own equation, or clone an existing one, choose between two ways to compute the confidence interval of each transformed parameter. If you pick a built-in equation, Prism always reportd asymmetrical confidence intervals of transformed parameters.

Do not mix up confidence intervals and confidence bands

It is easy to mix up confidence intervals and confidence bands. Choose both on the Diagnostics tab.

The 95% confidence interval tells you how precisely Prism has found the best-fit value of a particular parameter. It is a range of values, centered on the best-fit value. Prism can display this range in two formats:

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. It is closely related to the 95% prediction bands , which 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.