I wrote multiplying with n’ is probably in the right direction. I meant dividing with square root of n’ is probably be in the right direction.

A site mean with fewer effective samples might be expected to have a lower (temperature)signal to noise ratio. Then the signal part has lower amplitude after normalization than in a mean from a site with more effective samples. If all site means were adjusted to compensate for this before the total mean is calculated, the signal part in the total mean would be independent of which sites that are included at a specific time.

How this compensation should look I don’t know, but multiplying with n’ is probably in the right direction.

But, this compensation should be done uniformally over time but different on different site means before calculating the total mean. I can’t see that this is what they do.

Frank et al:

Multiplication of the mean timeseries with the square root of Neff at every time t theoretically results in variance that is independent of sample size.

Seems that Eq (6) is not correct, X and Y mixed (??)

Steve wrote:

Because there is little reason to believe that the annual variance in the early period was substantially greater than at present, Briffa and Osborn [1999] proposed a variance adjustment methodology (applied here) as follows.

Because Briffa and Osborn have never heard of filtering theory (specifically, the problem of estimating the state of a stochastic dynamical system from noisy observations), they decided to go the easy way and just scale the observation so that the result looks good.

For example, and perhaps simple mindedly, if a given ring width is produced by average temperatures and average precipitation or by above average temperatures and above average precipitation, how do you separate temperature and precipitation? But I assume that this is so obvious a point dendrochronologists must have addressed it, would they not? It sounds like, for example, they choose sites based on some assumptions that attempt to control for other factors like precipitation – but frankly the logic of assuming constant that which inherently fluctuates is very puzzling. This is, I assume, part of the argument for up-to-date records so that this assumption of independence can be tested.

Amazingly, I don’t think the dendrochronologists HAVE addressed this fundamental and extremely important issue. They seem to avoid the question like the plague. Tree rings are often very good proxies for moisture. I don’t think they are generally valuable as “thermometers” for many reasons.

that means that they are doing the opposite to what I said earlier, in fact amplifying the mean where you have fever effective samples.

What I don’t understand is how this can cause “…the variance Var(Y) to be independent of sample size” if

If one assumes that each sample contains a signal plus noise, doesn’t different scaling for different years distort the signal?

Yes. This adjustment leads to a biased estimate.

Isn’t the effect of the correction that you lower the signal amplitude for periods where you have less data!? Instead of increasing the error bars!!

Yes. In the past, we have sparser data. Past variations will be scaled towards zero. Increasing the error bars is not a legal move in climate science. Those bars might reach the current temperature levels, that wont do.

#7

Why is this procedure being used repeatedly if it hasn’t been shown to be a valid statistical technique?

Because it makes the results look nice.