Mann reported a “significant” correlation of -0.5481829 between Dongge dO18 and gridcell temperature. Today I will report on exactly how Mann calculated this “significant” correlation. In keeping with recent requests, I will refrain from making any comments on this procedure, in the confident expectation that my critics will provide some commentary on what they think of this procedure.
Let me start by describing the data sets used in the calculation. The temperature data used in the calculation is Mann’s infilled version of CRU data for gridcell 27.5N 107.5E, with the Mann version shown below against current CRU annual data (red dots). The CRU series starts in 1921, so the first half of the temperature data is not “observed” directly but has been “infilled” by Mann in a calculation that I have not yet had an opportunity to examine.
The Dongge O18 data is shown below (inverted orientation); the original data is not available annually but only in irregular years (shown in red dots), with the Mann data interpolated linearly. There are 11 values after 1921 (the start of the actual CRU instrumental record) and 33 values since 1850, the start of the infllled instrumental record. The age model used by Mann is the “tuned” age model, with the age apparently “tuned” using the method criticized by Gavin Schmidt in his critique of Loehle.
Mann’s low-frequency correlation proved to be the correlation between highly smoothed versions of both series: each series was Mannian smoothed using a Butterworth filter with f=0.05. For reference, the smoothed gridcell series is shown below (this is shown in SD Units below, together with the version extracted from clidatal, which matches very closely.)
In their SI, Mann et al say:
Owing to reduced degrees of freedom arising from modest temporal autocorrelation, the effective P value for annual screening is slightly higher …For the decadally resolved proxies, the effect is negligible because the decadal time scale of the smoothing is long compared with the intrinsic autocorrelation time scales of the data.
Obviously, the radical smoothing of these two series will reduce the number of degrees of freedom. Santer et al 2008 recently commented on the effect of autocorrelation on degrees of freedom and, presumably, one of the first observations that a co-author of Santer et al 2008 (such as Gavin Schmidt) reviewing this article would make is: ummm, Mike, can you flesh out your argument that autocorrelation in the “decadally resolved” series doesn’t matter?
Y’see, the number of years is 146 (1850-1995). The autocorrelation of the residuals in a linear regression is 0.9945544 and the resulting degrees of freedom using the Quenouille formula used in Santer et al 2008 (N(1-r)/(1+r) is only 0.399, something that must have worried Gavin Schmidt.
As an experiment, I tried the following procedure to calculate the relationship between O18 and Dongge gridcell temperature. I made the assumption that Dongge O18 could not teleconnect with future temperatures. Based on this assumption, for each year in which there was a Dongge speleo O18 reading, I calculated the average gridcell temperature for the prior years (up to the previous reading.) The results are shown below (with the “binned” temperature as red dots), compared with the original “infilled” series and the Mannian smooth.
Now I realize that this procedure does not exploit all possble covariance information between Dongge O18 and ring widths of Argentine cypress, but this is just a blog and not a “peer reviewed” publication in an esteemed journal such as PNAS. If critics will grant me the permission to proceed with the analysis on this basis, below is a scatter plot between the “binned” gridcell temperature and Dongge O18.
The r2 of the relationship is 0.0006547 (adjusted r2: -0.03158) with a t-statistic of -0.143, a value which does not meet any significance test.
Spurious correlations between smoothed series (the Slutzky-Yule effect) has been known to economists and statisticians since the 1930s. It has been mentioned in recent climate literature e.g. Gershunov et al (J Clim 2001) who state:
spurious relationships abound, especially when one deals with low-frequency phenomena diagnosed in short time series (Wunsch 1999). In general, the apparent presence of trends and periodicities in short filtered random time series is known as the ‘‘Slutsky–Yule effect’ (Stephenson et al. 2000).