Regression Slopes

Author

Archi Howlader

1. Why Are Both Slopes Less Than 1?

In the analysis of PRISM vs. Station extreme event counts, we observe:

Slope of the regression line for Station ~ PRISM: approximately 0.86
Slope of the regression line for PRISM ~ Station: approximately 0.87

Fig 1:

Comparison of extreme event counts between PRISM estimates and station annual observations (1895-2023) across different extreme transition types.

At first glance, one might expect that flipping the axes would invert the slope (i.e., slope > 1 if the other is < 1). However, this expectation only holds under perfect correlation and no measurement error.

Explanation

Linear regression theory tells us:

The slope of the regression of ( Y ) on ( X ) is:

\[ b = r \cdot \left( \frac{\sigma_Y}{\sigma_X} \right) \]

For the reverse regression of ( X ) on ( Y ):

\[ b' = r \cdot \left( \frac{\sigma_X}{\sigma_Y} \right) \]

Unless the correlation ( r = 1 ), the two slopes are not reciprocals — and in fact, both can be less than 1.

In this case:

\[ \text{Correlation: } r = 0.864 \\ \] \[ \text{Standard deviation of PRISM: } \sigma_X = 3.643 \\ \] \[ \text{Standard deviation of Station: } \sigma_Y = 3.625 \]

Expected slopes:

For Station ~ PRISM:

\[ b = 0.864 \cdot \left( \frac{3.625}{3.643} \right) \approx 0.86 \]

For PRISM ~ Station:

\[ b' = 0.864 \cdot \left( \frac{3.643}{3.625} \right) \approx 0.868 \]

This confirms that both slopes are logically and mathematically valid as being less than 1 — a known phenomenon called regression dilution due to measurement error and non-perfect correlation.

Conclusion

Both regression slopes are validly less than 1 due to regression theory under non-perfect correlation. .