Dangers of forcing regressions through the origin

17 10 2017

correlationsI had an interesting ‘discussion’ on Twitter yesterday that convinced me the topic would make a useful post. The specific example has nothing whatsoever to do with conservation, but it serves as a valuable statistical lesson for all concerned about demonstrating adequate evidence before jumping to conclusions.

The data in question were used in a correlation between national gun ownership (guns per capita) and gun-related deaths and injuries (total deaths and injuries from guns per 100,000 people) (the third figure in the article). As you might intuitively expect, the author concluded that there was a positive correlation between gun-related deaths and injuries, and gun ownership:

image-20160307-30436-2rzo6k

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Now, if you’re an empirical skeptic like me, there was something fishy about that fitted trend line. So, I replotted the data (available here) using Plot Digitizer (if you haven’t yet discovered this wonderful tool for lifting data out of figures, you would be wise to get it now), and ran a little analysis of my own in R:

Rplot01

Just doing a little 2-parameter linear model (y ~ α + βx) in R on these log-log data (which means, it’s assumed to be a power relationship), shows that there’s no relationship at all — the intercept is 1.3565 (± 0.3814) in log space (i.e., 101.3565 = 22.72), and there’s no evidence for a non-zero slope (in fact, the estimated slope is negative at -0.1411, but it has no support). See R code here.

Now, the author pointed out what appears to be a rather intuitive requirement for this analysis — you should not have a positive number of gun-related deaths/injuries if there are no guns in the population; in other words, the relationship should be forced to go through the origin (xy = 0, 0). You can easily do this in R by using the lm function and setting the relationship to y ~ 0 + x; see code here). Read the rest of this entry »