# Dangers of Forcing Regressions Through the Origin

Posted on the 17 October 2017 by Bradshaw @conservbytes

I 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:

__

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:

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).

Sounds reasonable, and if you take it on face value, it justifies the purported positive relationship shown in the original article. Suddenly, the slope is now positive (1.0308 ± 0.1236) and supported statistically.

But slow-down there, matey! There’s a problem here, and it means everything for the interpretation of the relationship. Just because it’s intuitive and you can force the relationship to go through the origin, it doesn’t mean that the data support this approach.

If you look closely, you’ll see that the origin is far away from the nearest real data — in other words, we’re taking a huge leap by assuming the relationship continues from the lowest values in the dataset back to the origin. You’ll also notice that the outlier (the USA: point farthest to the right of the plot) seems to be driving much of this relationship.

What we need to do is examine the relative support for the 2-parameter (intercept + slope) model versus the origin model. We can do this easily by comparing the sample size-corrected Akaike’s information criterion (AICc) for each model, and then dividing the AICc for the full model by the AICc of the origin model; this is called the information-theoretic evidence ratio.

As you can see from the code, the evidence ratio is 56.8, which means that the 2-parameter model is 56 times more likely than the origin model after accounting for different the number of parameters in each.

As such, there is no statistical justification whatsoever to support the forced-origin model, even though it seems to make intuitive sense (see another useful discussion here). This is because we have overstepped the bounds of our inference capacity based on the small sample of data available.

Now, in this particular case, I admit that even I wanted this relationship to be real, for I am a firm supporter of extensive gun control (the many other data supporting why this makes sense are overwhelming). I am, in fact, a gun owner myself (for scientific and farming purposes), and I am in absolute favour of the tightest gun-control legislation possible.

But just because I want it to be true, doesn’t make it so. We have to stick to the evidence.

You Might Also Like :

• ## Why You Should Create Your Own Niche - If You Can't Find One

One day while I was filling a blogger contact form for a bloggers meet, they asked me - Select a category that fits your blog theme.and I didn't know what to... Read more

16 hours, 46 minutes ago by   Sakshi Raina
LIFE COACH, LIFESTYLE
• ## Irish Whiskey and Ginger Cake

I had my DNA done several summers ago and when I got the results, I was surprised and very pleased to discover that I was 27% Irish. Read more

17 hours, 57 minutes ago by   Mariealicerayner
FOOD & DRINK, RECIPES
• ## Finish Your Day Sparkling: Half-Seas Sparkling Canned Cocktails

There are days when a well-crafted cocktail is worth the wait and anticipation. Then there are days when you want to enjoy a well-crafted cocktail as quickly an... Read more

19 hours, 10 minutes ago by   Glasgow Skinner/ My Life On And Off The Guest List
FASHION, HAIR & BEAUTY, LIFESTYLE
• ## Fluffy Steamed Buns

Steamed Buns(recipe adapted from here)Ingredients:150g Blue Jacket Plain Flour80g milk1/4 tsp instant yeast15g fine sugar3g oilMethod:1. In a medium mixing bowl... Read more

21 hours, 52 minutes ago by   Cathysjoy
FOOD & DRINK, RECIPES
• ## National Student Walkout Against Gun Violence: Arkansas Students Receive Corpora...

My kid and two other students walked out of their rural, very conservative, public school for 17 minutes today. They were given two punishment options. They... Read more

The 16 March 2018 by   William Lindsey
LGBTQ, RELIGION, SOCIETY
• ## Defining Her by Samantha March Only 99 Cents on Amazon Kindle. Being You...

I remember watching an old movie about an actress who became a star in 1930s Hollywood and had completely changed over the years. Somebody called her Daisy... Read more

The 16 March 2018 by   Lizmckeown1955
BOOKS, CREATIVITY
• ## Cambridgeshire 1,000 BC - Must Farm - Our Very Own Pompeii?

by that, I don't mean, "Bronze age Cambridgeshire suffered a volcano," but the level of preservation in this fenland homestead is comparable to Italy's famous... Read more

The 16 March 2018 by   Freeplanet
DEBATE, SOCIETY