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I recently directed to a very good explanation on the difference between an error band and a confidence intervals, here.

My question arose from the context of using error bars/bands or confidence intervals as a means of weighting a fit of some data.

Given that I more clearly understand the difference between a an error bar and a confidence interval my question is now:

If one understands the error distribution of some data well enough, can this knowledge be used to generate a confidence interval and in turn can this confidence interval be used to weight a fit for that data set?

Does it even make sense to think/use a confidence interval in this way, or is a confidence interval better used for describing the result of a fit?

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  • $\begingroup$ My "animated confidence intervals" should be of interest, as it has some bearing on this question: zunzun.com/CommonProblems $\endgroup$ – James Phillips Dec 28 '19 at 19:15
  • $\begingroup$ @JamesPhillips Very nice visualisation, but I think these are the confidence intervals which one plots after a fit, not used do perform the fit itself. $\endgroup$ – Q.P. Dec 29 '19 at 0:51
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It depends on what the error estimates represent: errors in the measurements going into the model, or expected errors in predictions from a fitted model. For terminology, it's simplest to discuss in terms of the error variance estimates (which for a given study size bear a one-to-one relationship with the confidence-interval widths).

Standard linear regression, for example, assumes that the error variance is constant. If you have examined the distribution of measured outcome values and find that error variance actually depends on the values being measured, say with error increasing as the values increase, then you could consider a weighted linear regression to take that into account. In that case, you could be properly using information about observation variances to weight a model.

If you fit a standard linear regression, however, the estimated variance of predictions from the model are not constant across the range of predicted values even if the variances of observations going into the model and estimated variances of observed minus predicted values are constant. See this Wikipedia page for the formula for a one-predictor model. As the values of the predictor variable get farther from the mean of the original predictor-variable values, the variance of the predictions from the model necessarily increase. In that case the associated confidence intervals are best used to describe the result of the model; it would be inappropriate to use the variances of such predictions from the model to weight the results for a revised regression.

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