Parameter Confidence Intervals in Reports
Confidence Intervals for Regression Parameters
Background – In SigmaPlot, we currently provide the (asymptotic) standard errors for the best-fit parameters in the nonlinear regression report. These errors measure the variability in the value of our parameter estimates due to the uncertainties in the observation measurements. However, we do not provide the user with a measure of the accuracy of our estimates to the true parameter values. That is the purpose of confidence intervals.
For example, a 95% confidence interval tells us, regardless of the true parameter value, that if we continuously resample our observations from their underlying distributions and compute the estimated parameter value each time, then 95% of the time the interval defined by our estimate will contain the true parameter value. In other words, the probability of obtaining an interval containing the true value is at least .95.
Computational Formula – The calculation of the confidence interval for each parameter uses our computed estimates for the parameter´s standard error. We provide two methods for computing the standard error. Both methods give the same result for unweighted regressions. For weighted regressions, the first method assumes that each weight equals the reciprocal of the variance of the underlying distribution from which the corresponding observation is sampled, up to some scale factor that is determined after the regression is performed.
This is our default method, called the reduced chi-square method. The second method assumes each weight exactly equals the reciprocal of the population variance (the scale factor is 1). The choice of method is determined by the selection of the check box in the Equation Options dialog that is launched from the Regression Wizard. It is important to verify which method has been selected since it affects the values in the confidence intervals.
Regardless of the method used to compute the standard errors, the confidence intervals (sometimes referred to as Wald confidence intervals) are computed using the following formula:
One can actually compute these confidence intervals in SigmaPlot manually. For example, suppose you want a 95% confidence interval for a parameter whose value is 2.3 with a standard error of 1.07. Also, suppose there are 20 data points and the fit model has 3 parameters. Looking at the description of the formula above, α is equal to .05 (set 100(1-α) = 95 and solve for α).
The percentile of the t-distribution can be found from the transform language by using the function tinv. In this case, we compute tinv(1-α/2,n-p) = tinv(.975,17) ≈ 2.1098. Using the above formula, we obtain the interval 0.0425 to 4.5575.