Confidence Intervals

Confidence and Prediction Intervals

The Review Curve-Fit Graph offers the option of displaying confidence and/or prediction bands about the fitted curve.

The Set Confidence/Prediction Intervals, % Confidence button is used to set confidence interval type and percentage.

The option opens a simple dialog in which you can individually check Confidence Intervals and Prediction Intervals, and select from 90%, 95%, 99%, 99.9%, and 99.99% Confidence.

Since confidence and prediction intervals assume the normality of residuals, you are strongly encouraged to use a Residuals Stabilized Normal Probability Plot to verify the validity of intervals. While important at all confidence levels, this confirmation of normality is especially critical for the higher confidence levels.

The Show Confidence/Prediction Intervals button toggles the currently set intervals on and off.

The selected confidence level applies not only to the confidence and prediction intervals in the graph, but also to the initial confidence limits used for the parameters in the Numeric Summary and to the initial confidence and prediction ranges for the data in the Data Summary.

Confidence Intervals

Confidence intervals are useful for replicated data in which, for each X, there are either multiple Y observations or a single Y value which represents an average from multiple Y observations (if each Y value in the data set consists of an average from multiple observations at the same X, the point should be weighted with the inverse square of the standard deviation). With many Y observations at a given X, the average Y can be said to approach the true Y value, since the errors will sum to zero.

A 95% confidence interval is the Y-range for a given X that has a 95% probability for containing the true Y value.

Prediction Intervals

Prediction intervals are useful for predicting, for a given X, the Y value of the next experiment. It is often used when a fit represents a single experiment, where each Y value is a single observation rather than an average. In this case, the weight for each Y value isn't based upon a standard deviation from multiple observations, but rather is inversely related to the experimental uncertainty for the individual measurement, if such is known. If the uncertainty of the Y measurement is unknown or thought to be equal for all X, all points can use equal 1.0 weights.

A 95% prediction interval is the Y range for a given X where there is a 95% probability that the next experiment's Y value will occur, based upon the fit of the present experiment's data.

Local Measure of Error

Confidence and prediction intervals measure the confidence only at a specific X, not for the entire X data region. These must be computed for each X value in the curve. For fits containing models that are computationally intensive, it may take TableCurve 2D some time to compute these intervals. This is especially true with the high precision rational and polynomial models, since the intervals must also be computed using high precision math.

Relation to Standard Error

There is often a strong similarity between points which lie outside 2 standard errors and the 95% prediction interval. These are not identical, however.

The standard error of fit is based upon the overall fit. It does not matter if a point is near a portion of the curve that is well characterized by the fit, or a point that is poorly characterized by the fit. A certain standard error in Y exists for the overall fit, and a point with a residual whose magnitude lies beyond a given multiple of this value is drawn in a certain color.

The prediction interval, on the other hand, is a more localized measure of error. Points near a region of the curve strongly determined by the fit will have a slightly tighter confidence interval than those points near a region of the curve that is only weakly determined in the fit. Generally, strongly determined regions of the fitted curve will have a good number of accurate data points whereas poorly determined regions of the curve will have relatively few points and these will have higher errors.

Drawing Intervals

If the standard uncertainties are negligible, these intervals will not be drawn. On very good fits, it may still appear that no intervals were drawn simply because they are graphically covered by the curve-fit line.

Errors in Intervals

If a math error occurs in constructing these intervals, a vertical error line of a different color from that of a standard math error will be drawn at the X-value where the error occurred. The calculation of the intervals is sensitive to floating point precision limits. When both the equation error and intervals errors are drawn, the equation itself is undefined. When only the intervals errors are drawn, the failure may be due to the limitations of machine precision. The latter case is especially true in regions outside the actual data area.