## “It is a little depressing that after many years of being criticised for their insufficiently good understanding of statistics and lack of close engagement with the statistical community, the climate science community appears still not to have solved this issue.”

The recently published open-access paper “How accurately can the climate sensitivity to CO2 be estimated from historical climate change?” by Gregory et al.[i] makes a number of assertions, many uncontentious but others in my view unjustified, misleading or definitely incorrect. Perhaps most importantly, they say in the Abstract that “The real-world variations mean that historical EffCS [effective climate sensitivity] underestimates CO_{2} EffCS by 30% when considering the entire historical period.” But they do not indicate that this finding relates only to effective climate sensitivity in GCMs, and then only to when they are driven by one particular observational sea surface temperature dataset.

However, in this article I will focus on one particular statistical issue, where the claim made in the paper can readily be proven wrong without needing to delve into the details of GCM simulations.

Gregory et al. consider a regression in the form *R* = *α* *T*, where *T* is the change in global-mean surface temperature with respect to an unperturbed (i.e. preindustrial) equilibrium, and *R* is the radiative response of the climate system to change in *T*. *α* is thus the climate feedback parameter, and *F*_{2xCO2 }/ *α* is the EffCS estimate, *F*_{2xCO2} being the effective radiative forcing for a doubling of preindustrial atmospheric carbon dioxide concentration.

The paper states that “that estimates of historical α made by OLS [ordinary least squares] regression from real-world *R* and *T* are biased low”. OLS regression estimates *α* as the slope of a straight line fit between *R* and *T *data points (usually with an intercept term since the unperturbed equilibrium climate state is not known exactly), by minimising the sum of the squared errors in *R*. Random errors in *R* do not cause a bias in the OLS slope estimate. Thus in the below chart, with *R* taken as plotted on the y-axis and *T *on the x-axis, OLS finds the red line that minimizes the sum of the squares of the lengths of the vertical lines.

However, some of the variability in measured *T* may not produce a proportionate response in *R*. That would occur if, for example, *T* is measured with error, which happens in the real world. It is well known that in such an “error in the explanatory variable” case, the OLS slope estimate is (on average) biased towards zero. This issue has been called “regression dilution”.

Regression dilution is one reason why estimates of climate feedback and climate sensitivity derived from warming over the historical period often instead use the “difference method”.[ii] [iii] [iv] [v] The difference method involves taking the ratio of differences, Δ*T *and Δ*R*, between *T *and *R* values late and early in the period. In practice Δ*T *and ΔR are usually based on differencing averages over at least a decade, so as to reduce noise.

I will note at this point that when a slope parameter is estimated for the relationship between two variables, both of which are affected by random noise, the probability distribution for the estimate will be skewed rather than symmetric. When deriving a best estimate by taking many samples from the error distributions of each variable, or (if feasible) by measuring them each on many differing occasions, the appropriate central measure to use is the sample median not the sample mean. Physicists want measures that are invariant under reparameterization[vi], which is a property of the median of a probability distribution for a parameter but not, when the distribution is skewed, of its mean. Regression dilution affects both the mean and the median estimates of a parameter, although to a somewhat different extent.

So far I agree with what is said by Gregory et al. However, the paper goes on to state that “The bias [in *α* estimation] affects the difference method as well as OLS regression (Appendi*x *D.1).” This assertion is wrong. If true, this would imply that observationally-based estimates using the difference method would be biased slightly low for climate feedback, and hence biased slightly high for climate sensitivity. However, the claim is *not *true.

The statistical analyses in Appendi*x *D consider estimation by OLS regression of the slope *m *in the linear relationship *y*(*t*) = *m x*(*t*), where *x *and y are time series the available data values of which are affected by random noise. Appendi*x *D.1 considers using the difference between the last and first single time periods (here, it appears, of a year), not of averages over a decade or more, and it assumes for convenience that both *x *and *y* are recentered to have zero mean, but neither of these affect the point of principle at issue.