Bounded Confidence Intervals from nlparci

I've calculated parameter values to a model using lsqcurvefit(). These parameters have lower bounds of 0, which I've specified to the function. Now I'd like to know the confidence interval for each.

nlparci() generates confidence intervals from the resulting residuals and jacobian from lsqcurvefit(), but the lower bounds are negative. Since the possible values cannot be negative, what is the proper way to calculate confidence intervals for these parameters?

Answer 1

[Star Strider]

The lower bounds of parameter confidence intervals indeed can be negative.

Parameter confidence intervals that include zero, such that one limit is positive and the other is negative, indicate that the particular parameter is not necessary in the model. The confidence limits would then indicate that the parameter is not significantly different from zero at the level of significance specified.

[Darren Wethington]

This is curious. I agree with what you're saying, but I'm having trouble also agreeing that this confidence interval claims that a negative number is also plausible. For illustration, if x=1 with confidence interval -1<x<3, then I agree that x is not sufficiently different than 0, but the idea that x=-0.5 is also plausible seems impossible. We shouldn't even be testing alternative hypotheses like x=-0.5 if the range is [0,Inf).

I'll continue to think about this, but if you have additional thoughts that may clarify feel free to share. Thank you for your response!

[Star Strider]

I assume that the ‘x’ you’re referring to is a parameter, since you didn’t specifically describe it.

The parameter values the functions return represent the ‘best fit’, usually in the least-squares sense. Every estimated parameter (and the regression estimates themselves) involve some uncertainty with respect to the actual value.

The confidence intervals are not the range of values a parameter may take, but the uncertainty involved in their estimation. So ‘... if x=1 with confidence interval -1<x<3 .,..’ indicates that at the given probability, the ‘true’ value of the parameter is somewhere in those limits. For any given model, the difference between the limits will decrease as the number of observations increases.

I can’t tell how implausible ‘x=-0.5’ may be in your model. The confidence intervals of (-1<x<3) simply indicate that ‘x’ is not required in your model, and simplifying your model to exclude that parameter would not impair your model’s ability to describe your data. In that context, the actual values it may take aren’t important.