Some Modelling Points with the Keeling Data

Some of you are still doing a linear regression where you fixed the coefficients of the sine and cosine terms: e.g. \[ CO_2(t) = a + bx + cx^2 + \sin(2 \pi t) + \cos(2 \pi t) \] You're dictating the amplitude -- ($\sqrt{2}$ -- and the phase shift -- (-0.125) -- rather than letting the data determine them. You're tying the regression's hands. \[ CO_2(t) = a + bx + cx^2 + d \sin(2 \pi t) + e \cos(2 \pi t) \]
In one report I read that "Some side effects of this temperature increase include ice caps melting, species dying off and ocean acidification will begin." I want to point out that these things are already happening: we're just risking making matters worse!
Some people were surprised by how high an $R^2$ can be, when the model is clearly wrong.
This is a good thing to realize early: $R^2$ is too one-dimensional a measure. We need to consider other things (e.g. residual plots, for sure). The first thing to do is look at the fit with our very clever eyes.
Some of you again failed to show confidence intervals. Parameter estimates that we get are not correct -- they're determined by data that contains errors, and so they themselves will be in error. Confidence intervals (e.g. 95% confidence intervals) tell us a range in which we can expect to find the actual value, with given confidence. A 95% confidence interval means that, 19 out of 20 times using this procedure you'll be correct (and 1/20 times you'll be wrong....).
It's frequently a good idea to generate some simulations, using a range of parameter values from within the confidence interval, to get an idea of how much "wiggle" there is in your model based on your intervals.

Website maintained by Andy Long. Comments appreciated.