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Discussion > How to calculate std. dev. or confidence intervals for (slope)^(-1) of a regression line?

I'm trying to do some calculations on the "adjustment time" of CO2 injected into the atmosphere. The IPCC says it's very long, based on what the "Bern model" says. Other people say it's very short. SkS gives arguments why this is wrong. I want to pin it down with calculations on actual data.

I know a good bit about probability computations but very little (almost nothing) about statistics. Here's my question...

I have some data on the rate of CO2 uptake as a function of the concentration of CO2 in the atmosphere. It is quite scattered but there is a clear linear trend. If I fit a straight line by regression, its slope gives me an estimate of the constant k in the differential equation for atmospheric CO2 in response to an injected CO2 dollop: dC/dt = -k.C. [Yes, I know this is a simplified model.]

I can find formulas for the std. dev. of the slope of the regression line, eg:

However, it is the reciprocal of the constant k that gives the time constant (or "adjustment time") for the solution of the differential equation and it's the latter that's the thing of interest.

My question:
Where can I find (or what is) the formula for the standard deviation (or better still, confidence intervals) of the reciprocal of the slope of a regression line?

In this case, the standard deviation of the estimate for k is comparable to the estimated value of k, so I can't use approximations that would apply if the coefficient of variation of the estimate were very small.

Any help will be gratefully received.

Jun 2, 2012 at 12:11 PM | Registered CommenterMartin A

Someone kindly pointed out to me that confidence interval for the slope of a regression line has a standard formula given in texts and available online eg

If the confidence interval for an estimate ke, of k, is (k1, k2) then I think that if 1/ke is used as an estrimate of 1/k, the confidence interval will then be (1/k1, 1/k2).

But if this is a misconception on my part, I'd be glad for it to be pointed out.

Jun 4, 2012 at 5:45 PM | Registered CommenterMartin A