I had posted this under a different topic (error correlations 3 Rb-Sr isotopes) a while ago, but would like to get a new discussion started 🙂
Iolite calculates error correlations (for isochron work) from observation rather than from differentiation. What do others think? is there some general consensus on which one is better? A few things that I have observed and would like to discuss more:
1) in synthetic datasets I can see absolutely no difference between the two methods. The calculated rho is always identical in the third or fourth decimal, so no preference there.
2) Pearson observed in 1897 that r is around 0.5 in a ratio plot of uncorrelated variables that share a common denominator, and TRUE correlation would have to be tested against the null hypothesis not being r=0, but against the spurious part (0.5 at minimum). This gets extreme when your denominator is 10x smaller than the numerator: the r calculated from sets of random (!) variables is on the order of 0.99. You would see very long slim error ellipses, and they are almost completely spurious.
This problem gets even more complex when you think about what you are actually measuring during laser ablation. Schmitz' 2007 assessment on ID TIMS UPb solution work assumes that you are measuring a solution, i.e., the same sample over and over. You also assume that the error on those multiple measurements would be normally distributed (or log-normally at least). This is not the case when we're ablating - the sample volume is strictly different every pulse, and we're not necessarily repeatedly sampling a homogeneous reservoir. This means that we should not expect normally distributed errors. I guess the term 'error' is somewhat misleading anyways, because the only error that is truely random comes from beam noise and poisson statistics at the detector. In the best case when we're ablating down we could actually get a true single mineral isochron; in the worst case we're sampling random zonation. This is then mixed with a bit of random beam noise and counting errors, leading to a little bit of 'true' error correlation.
I think given all that non-randomness we should abandon the whole concept of error correlations in laser ablation and calculate conventional isochrons not from the means of the error ellipses but using all of the raw data points.
Or, and that is probably my preferred solution, switch entirely to inverse isochron techniques where we do not have a common denominator. But this has also been suggested for more than 40 years, and didn't stick. Let me know what you all think about this.