# A Linear Digression

Guest Post by Willis Eschenbach [SEE UPDATE AT END]

In my most recent post, called “Where Is The Top Of The Atmosphere“, I used what is called “Ordinary Least Squares” (OLS) linear regression. This is the standard kind of linear regression that gives you the trend of a variable. For example, here’s the OLS linear regression trend of the CERES surface temperature from March 2000 to February 2021.

Figure 1. OLS regression, temperature (vertical or “Y” axis) versus time (horizontal or “X” axis). Red circles mark the ends of the correct regression trend line.

However, there’s an important caveat about OLS linear regression that I was unaware of. Thanks to a statistics-savvy commenter on my last post, I found out that there is something that must always be considered regarding the use of OLS linear regression.

It only gives the correct answer when there is no error in the data shown on the X-axis.

Now, if you’re looking at some variable on the Y-axis versus time on the X-axis, this isn’t a problem. Although there is usually some uncertainty in the values of a variable such as the global average temperature shown in Figure 1, in general we know the time of the observations quite accurately.

But suppose, using the exact same data, we put time on the Y-axis and the temperature on the X-axis, and use OLS regression to get the trend. Here’s that result.

Figure 2. OLS regression, time (vertical or “Y” axis) versus temperature (horizontal or “X” axis). As in Figure 1, red circles mark the ends of the correct regression trend line.

YIKES! That is way, way wrong. It greatly underestimates the true trend.

Fortunately, there is a solution. It’s called “Deming regression”, and it requires that you know the errors in both the X and Y-axis variables. Here’s Figure 2, with the Deming regression trend line shown in red.

Figure 3. OLS and Deming regression, time (vertical or “Y” axis) versus temperature (horizontal or “X” axis). As in Figure 1, red circles mark the ends of the correct regression trend line.

As you can see, the Deming regression gives the correct answer.

And this can be very important. For example, in my last post, I used OLS regression in a scatterplot comparing top-of-atmosphere (TOA) upwelling longwave (Y-axis) with surface temperature (X-axis). The problem is that both the TOA upwelling LW and the temperature data contain errors. Here’s that plot:

Figure 4. Scatterplot, monthly top-of-atmosphere upwelling longwave (TOA LW) versus surface temperature. The blue line is the incorrect OLS regression trend line.

But that’s not correct, because of the error in the X-axis. Once the commenter pointed out the problem, I replaced it with the correct Deming regression trend line.

Figure 5. Scatterplot, monthly top-of-atmosphere upwelling longwave (TOA LW) versus surface temperature. The yellow line is the correct Deming regression trend line.

And this is quite important. Using the incorrect trend shown by the blue line in Figure 4, I incorrectly calculated the equilibrium climate sensitivity as being 1°C for a doubling of CO2.

But using the correct trend shown by the blue line in Figure 5, I calculate the equilibrium climate sensitivity as being 0.6 °C for a doubling of CO2 … a significant difference.

I do love writing for the web. No matter what subject I pick to write about, I can guarantee that there are people reading my posts who know much more than I do about the subject in question … and as a result, I’m constantly learning new things. It’s the world’s best peer-review.

[UPDATE] My friend Rud said in the comments below:

First, CERES is too short a data set to estimate ECS.

I replied that climate sensitivity depends on the idea that temperature must increase to offset the loss of upwelling TOA LW. What I’ve done is measure the relationship between temperature and TOA LW. I asked him to please present evidence that that relationship has changed over time … because if it has not, why would a longer dataset help us?

Of course, me being me, I then had to go take a look at a longer dataset. NOAA has records of upwelling TOA longwave since 1979, and Berkeley Earth has global gridded temperatures since 1850. So I looked at the period of overlap between the two, which is January 1979 to December 2020. Here’s that graph.

Figure 6. Scatterplot, NOAA monthly top-of-atmosphere upwelling longwave (TOA LW) versus Berkeley Earth surface temperature. The yellow line is the correct Deming regression trend line.

Would you look at that. Instead of using CERES data for the graph, I’ve used two completely different datasets—upwelling TOA longwave from NOAA and global gridded temperature data from Berkeley Earth. And despite that, I get the exact same answer to the nearest tenth of a watt per square meter— 3.0 W/m2 per °C.

My thanks to the commenter who put me on the right path, and my best regards to all,

w.

via Watts Up With That?

January 9, 2022