Linear Regression R Squared

One of the tests of climate models predicting drought in my review of the Drought Exceptional Circumstances Report was the correlation of predicted area under drought with actual observed area under drought. Lazar criticized my inclusion of the R-Squared (r2) coefficient, an issue I didn’t follow up at the time.

… correlating model predictions for individual years of exceptional rainfall with observed years of exceptional rainfall! This ignores noise (internal variability in the climate system and GCM climate simulations) and that the CSIRO report predicted frequency. Steve MicIntrye and the auditors repeat this mistake here, with the obligatory snark from Steve

The objection is that it unreasonable to expect climate models to predict ‘year-to-year variation’ with drought using this test. To set the record straight, I have run a small test demonstrating conclusively that the r2 does detect trends in frequency of intermittent events, (as opposed to trends in actual values) and consequently the test does not only rely only on year-to-year variations.

Below is a short R script where I represent a trend of increasing drought frequency with two independent sequences of numbers (0,1). These are plotted below. The results of fitting a linear regression to the sequences follow.


o runif(100)
e runif(100)
l<- lm(e~o)
plot(o,col="blue",type="l")
lines(e,col="red")
print(summary(l))


> source("rtest.R")

Call:
lm(formula = e ~ o)

Residuals:
Min 1Q Median 3Q Max
-0.6739 -0.3148 -0.3148 0.3261 0.6852

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.31481 0.06412 4.910 3.64e-06 ***
oTRUE 0.35910 0.09454 3.798 0.000253 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.4712 on 98 degrees of freedom
Multiple R-Squared: 0.1283, Adjusted R-squared: 0.1194
F-statistic: 14.43 on 1 and 98 DF, p-value: 0.0002527

The run shown produced a non-zero R-squared of 0.128, and significant slope (***), demonstrating that a similar trend in frequency of values does produce a positive correlation r2. In comparison, the r2 between modeled and drought data was essentially zero, indicating no detectable common trend in drought frequency using this method.

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0 thoughts on “Linear Regression R Squared

  1. If it would not be too much trouble, could you reproduce the R code and summary with a font based on uniform letter spacing? Perhaps some equivalent of Courier? The above font is difficult to line up, and copy and pasting into another package introduces possible misalignment and unnecessary friction into the information extraction process.
    Thanks.

  2. If it would not be too much trouble, could you reproduce the R code and summary with a font based on uniform letter spacing? Perhaps some equivalent of Courier? The above font is difficult to line up, and copy and pasting into another package introduces possible misalignment and unnecessary friction into the information extraction process.
    Thanks.

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