*Naive statistics underlie many causal claims in climate "science"*

You know who Charles Darwin is of course but you may not have heard of his mad cousin Francis Galton who did the math for Darwin’s theory of evolution. Two of the many procedures Sir Galton came up with to help him make sense of the data are still used today and are possibly the two most widely used tools in all of statistics. They are ordinary least squares (OLS) linear regression and OLS correlation. [Soon after these amazing mathematical innovations, Sir Galton retired from the evolution business and devoted the rest of his life to making the perfect cup of tea.]

Both of these statistics are measures of a linear relationship between two variables X and Y. Linear regression coefficient B of Y against X is a measure of how much Y changes on average for a unit change in X and the linear correlation R is a measure of how close the observed changes are to the average. The regression and correlation metrics are demonstrated below with data generated by Monte Carlo simulation used to control the degree of correlation.

In the HIGH (R=0.94) and VERY HIGH (R=0.98) correlation charts, linear regression tells us that on average, a unit change in X causes Y to change by about B=5 and this assessment is very consistent. The consistency in this case derives from a low variance of the regression coefficient implied by high correlation. The strong correlation also implies that the observed changes in Y for a unit increases in X is close the the average value of B=5 over the full span of the data and for any selected sub-span of the time series.

In the LOW (R=0.36) and MID (R=0.7) correlation charts, the regression coefficients are correspondingly less precise varying from B=1.8 to B=7.1 for LOW-R and B=3.5 to B=5.6 for MID-R in the five random estimates presented. The point here is that without a sufficient degree of correlation between the time series at the time scale of interest, though regression coefficients can be computed, the computed coefficients may have no interpretation. The weak correlations in these cases also imply that the observed changes in Y for a unit increases in X would be different in sub-spans of the time series. The so called “split-half” test, which compares the first half of the time series to the second half, may be used to examine the instability of the regression coefficient imposed by low correlation.

Correlation is a necessary but not sufficient evidence of causation. Although correlation may imply causation in controlled experiments, field data do not offer that interpretation. If Y is correlated with X in field data, it may mean that X causes Y, or that Y causes X, or that a third variable Z causes both X and Y, or that the correlation is a fluke of the data without a causation interpretation. However, because correlation is a necessary condition for causation, the absence of correlation serves as evidence to refute a theory of causation.

An issue specific to the analysis of time series data is that the observed correlation in the source data must be separated into the portion that derives from shared long term trends (that has no interpretation at the time scale of interest) from the responsiveness of Y to changes in X at the time scale of interest. If this separation is not made, the correlation used in the evaluation may be, and often is spurious. An example of such a spurious correlation is shown in the graphic below. It was provided by the TylerVigen collection of spurious correlations.

As is evident, the spurious correlation derives from a shared trend. The fluctuations around the trend at an appropriate time scale (whether annual or decadal) are clearly not correlated. The separation of these effects may be carried out using detrended correlation analysis. Briefly, the trend component is removed from both time series and the residuals are tested for the responsiveness of Y for changes in X at the appropriate time scale. The procedure and its motivation are described quite well in Alex Tolley’s Lecture available on Youtube.

The motivation and procedure for detecting and removing such spurious correlations in time series data are described in a short paper available for download at this link: Spurious Correlations in Time Series Data . The abstract of this paper follows: Unrelated time series data can show spurious correlations by virtue of a shared drift in the long term trend. The spuriousness of such correlations is demonstrated with examples. The SP500 stock market index, GDP at current prices for the USA, and the number of homicides in England and Wales in the sample period 1968 to 2002 are used for this demonstration. Detrended analysis shows the expected result that at an annual time scale the GDP and SP500 series are related and that neither of these time series is related to the homicide series. Correlations between the source data and those between cumulative values show spurious correlations of the two financial time series with the homicide series.

It is for these reasons the argument that “the theory that X causes Y is supported by the data because X shows a rising trend and at the same time we see that Y has also been going up” is specious because for the data to be declared consistent with causation theory it must be shown that Y is responsive to X at the appropriate time scale when the spurious effect of the shared trend is removed. Some examples from climate science are presented in the papers below along with the URL to their download sites.

Are fossil fuel emissions since the Industrial Revolution causing atmospheric CO2 levels to rise? Responsiveness of Atmospheric CO2 to Fossil Fuel Emissions

Can sea level rise be attenuated by reducing or eliminating fossil fuel emissions? A Test of the Anthropogenic Sea Level Rise Hypothesis

Can ocean acidification be attenuated by reducing or eliminating fossil fuel emissions? An Empirical Study of Fossil Fuel Emissions and Ocean Acidification

Is surface temperature responsive to atmospheric CO2 levels? #1 Validity and Reliability of the Charney Climate Sensitivity Function

Is surface temperature responsive to atmospheric CO2 levels? #2 Uncertainty in Empirical Climate Sensitivity Estimates 1850-2017

Is surface temperature responsive to atmospheric CO2 levels? #3 The Charney Sensitivity of Homicides to Atmospheric CO2: A Parody

A further caution needed in regression and correlation analysis of time series data arises when the source data are preprocessed prior to analysis. In most cases, the effective sample size of the preprocessed data is less than that of the source data because preprocessing involves using data values more than once. For example taking moving averages involves multiplicity in the use of the data that reduces the effective sample size (EFFN) and the effect of that on the degrees of freedom (DF) must be taken into account when carrying out hypothesis tests. The procedures and their rationale are described in this freely downloadable paper Illusory Statistical Power in Time Series Analysis.

Failure to correct for this effect on DF may result in a false sense of statistical power and faux rejection of the null in hypothesis tests as shown in this analysis of Kerry Emmanuel’s famous paper on what he called “increasing destructiveness” of North Atlantic hurricanes: Circular Reasoning in Climate Change Research.

An extreme case of the effect of preprocessing on degrees of freedom occurs when a time series of cumulative values is derived from the source data as in the famous Matthews paper on the proportionality of warming to cumulative emissions [Matthews, H. Damon, et al. “The proportionality of global warming to cumulative carbon emissions.” Nature 459.7248 (2009): 829].

It has been shown in these downloadable papers that the time series of cumulative values has an effective sample size of EFFN=2 and therefore there are no degrees of freedom and no statistical power.

Degrees of freedom lost in moving window preprocessing Effective Sample Size of the Cumulative Values of a Time Series

Degrees of freedom lost in a time series of the cumulative values of another time series #1 Limitations of the TCRE: Transient Climate Response to Cumulative Emissions

Degrees of freedom lost in a time series of the cumulative values of another time series #2 From Equilibrium Climate Sensitivity to Carbon Climate Response

Degrees of freedom lost in a time series of the cumulative values of another time series #3 The Spuriousness of Correlations between Cumulative Values

SOURCE