Are mass shootings really random events? A look at the US numbers

All these comments about statistics don't seem to even begin to understand statistics.<p>1) Yes, we can use the chi-squared test here - the small number of events is built into the p value.<p>2) No, we do not need to include all murders, as we are not testing for murders. We are testing for mass shootings. This line of reasoning is the same as saying we cannot test for rotten apples only, we can only test if all fruit is rotten. Statistics on categories is acceptable and meaningful if apples are a specific kind of fruit, or mass shootings are a specific kind of crime. We can't draw an conclusion about crime, but we can draw a conclusion about mass shootings.<p>3) These statistics prove only a single thing: mass shootings in USA are likely random events and have a mean value of ~2.<p><pre><code> The following conclusions are applicable: - Mass shootings are likely not a 'copycat' crime, and each event is likely completely independent of any other event occurring. - You should expect about 10 mass shootings over the next 5 years. Not so nice... Following conclusions are NOT applicable: - Gun laws have any effect. Gun laws may or may not decrease the average, these statistics do not say. - No measures that have been put in place to reduce mass shootings (I assume there are?) have had any effect so far. They may have had a positive or negative effect, but these effects may be small or may be cancelled out by other, opposite effects. - The chance of a mass shooting is stable, and no increase or decrease in events seems to be happening. The distribution fit test only shows us on the aggregate data, and that it does fit that distribution. The correct way to check for this is to break the data set in half, and compare the first set against the second set. </code></pre> And now we return you to our regular statistics hate...<p>EDIT: Or not - nobody is questioning the real possible problem here: the data itself?<p><a href="http://www.bradycampaign.org/xshare/pdf/major-shootings.pdf" rel="nofollow">http://www.bradycampaign.org/xshare/pdf/major-shootings.pdf</a><p>This seems to imply there are FAR more mass shootings per year than indicated by the data. None of my conclusions above are correct if the data itself is wrong, and I don't even live in the USA so I can't vouch for the correctness of the data.

This seems like trying to use a hammer (statistics) because you have a hammer, not because it's the right tool for the job.<p>Essentially, if you look at the incidents, you see enough common factors (increasingly, using semiautomatic carbines, carrying multiple weapons, attacking schools, wearing armor or load bearing gear, etc.) to think there is some common factor at work. The population of random people on the street doesn't pick the AR-15 to do <i>anything</i>, and certainly doesn't pick a school as a target for anything. The solution space here isn't "spree shootings at schools through time", it is traits of spree shootings themselves -- location, methods, etc. They're pretty tightly clustered.<p>Either there is a common hidden factor, or these incidents are feeding on each other.<p>I personally don't think gun control is the major tool to deal with this, and don't think violent video games are the problem, but rather the non-stop multi-day press coverage by the media of each of these incidents.<p>Some insignificant douchebags from a Colorado school became about as famous as the 9/11 terrorists (and far more than fortune 50 CEOs or scientists or classical musicians) by murdering their classmates.<p>(Columbine essentially as as big a deal for the 'how to respond to shootings' world as 9/11 was to aviation security; previously, you cordoned off the area and called in SWAT to negotiate, thinking it was a hostage situation -- now, the first 1-2 responders on scene move directly to the threat with whatever weapons they have on them at the time, ignore any wounded victims, close, and engage/destroy -- similarly, hijacked airliners are now viewed as air to ground missiles vs. hostage negotiations.)<p>Every time the media talks about the shooters in one of these situations, making them famous, it reinforces the rational (if defective) choice of someone who wants to be famous at the cost of doing evil to copycat.<p>The mythological/historical example is Herostratus, who burned the temple of artemis just to be famous.

Mass shootings are rare enough that you're going to get poor stats due to granularity. It might be more interesting to look at stats like homicide rate.<p>The U.S. has a homicide rate that is second only to Russia in the G8, and is more than 3 times higher that of any G8 nation besides Russia. This sets off warning bells for me...<p>Normally I'd love to dig a little deeper, but it's time for me to play Santa. Merry Christmas!

For anyone who's interested, this blog post piqued a very morbid interest in me, and so I decided to have a look at the data.<p>Here's a distribution of the number of days before the previous incidents (as reported by MotherJones. I'm personally not familiar with the number of shootings in the US, which according to different sources ranges from very few to way too many): <a href="http://imgur.com/Onoxv" rel="nofollow">http://imgur.com/Onoxv</a> . The colours represent the group of how many incidents happened in the 180 days prior to the incident.<p>Here is the distribution of the number of incidents 180 days prior to an incident: <a href="http://imgur.com/9VTjQ" rel="nofollow">http://imgur.com/9VTjQ</a><p>Draw your own conclusions

There is absolutely an observed trend of copycat school shootings which flies in the face of randomness.<p>That said, I think this trend is hidden from this dataset because it relates specifically to <i>mass</i> shootings, including only those incidents in which the shooter took the lives of at least four people.<p>I would hypothesize that each major shooting is followed in the succeeding months by a number of slapdash copycat shootings, in many of which less than four people are killed by the shooter.

Since mass shootings are such a rare events, the data is overdispersed, so a negative binomial distribution is likely more appropriate than a Poisson distribution. This, along with quasi poisson, is commonly used by criminologists to account for such problems.<p>See this paper for a more detailed discussion:<p><a href="http://www.crim.upenn.edu/faculty/papers/berk/regression.pdf" rel="nofollow">http://www.crim.upenn.edu/faculty/papers/berk/regression.pdf</a>

This was a great article, but I'm not convinced mass shootings are completely independent. Given the media attention, it is very likely that event n+1 was, in some way, impacted by event n.<p>I don't have any evidence to back this up, so I could be (and I hope I am) completely wrong. But if I'm right, then I think it would imply we need to reduce the correlation between events. And it is likely the media that provides that correlation. Instead of reporting about the &#60;insert your word here&#62; that causes these events, report on the <i>impacts</i> of these events. It would, hopefully, stop deifying the perpetrator, which might reduce the likelihood of other perpetrators from doing the same.

This hits HN as another shooting covers front pages. A new dark twist for extra newsworthiness. <a href="http://www.nytimes.com/2012/12/25/nyregion/2-firefighters-killed-in-western-new-york.html?hp&#38;_r=0" rel="nofollow">http://www.nytimes.com/2012/12/25/nyregion/2-firefighters-ki...</a><p>Edit: it is unclear at this stage if this event would qualify for inclusion in the numbers according to the criteria in the article. Below are the used criteria.<p>The killings were carried out by a lone shooter. (Except in the case of the Columbine massacre and the Westside Middle School killings, both of which involved two shooters.)<p>The shootings happened during a single incident and in a public place. (Public, except in the case of a party in Crandon, Wisconsin, and another in Seattle.) Crimes primarily related to armed robbery or gang activity are not included.<p>The shooter took the lives of at least four people. An FBI crime classification report identifies an individual as a mass murderer—as opposed to a spree killer or a serial killer—if he kills four or more people in a single incident (not including himself), and typically in a single location.<p>If the shooter died or was hurt from injuries sustained during the incident, he is included in the total victim count. (But we have excluded cases in which there were three fatalities and the shooter also died, per the previous criterion.)<p>We included six so-called "spree killings"—prominent cases that fit closely with our above criteria for mass murder, but in which the killings occurred in multiple locations over a short period of time.

The auther does a horrible job. He takes it as an assumption that the reader doesn't know statistics, and when he gets to the actual meat he just states as fact, "i calculate k=32.5 m/j" " what does this mean? It means I'm right" he should as a minimum either assume completely that the reader knows everything nessesary to interpret the hypothesis test or actually make an effort to explain to layman what the p-value means.

Kudos to the author for attacking the question this way, it's always important to look at data and try to model it. (and it's a lot of work!)<p>Unfortunately, I'm pretty sure there's a fatal flaw in the analysis.<p>Suppose we had two types of years, both following a Poisson Distributino, but one with a higher incidence, the other with a lower incidence, and they alternated[1].<p>Now sample each type of year separately. Each will give you a Poisson distribution. To get the distribution of all years you'd just add bins of the two types together.<p>The sum of two poisson distributions is a poisson distribution (with a different mean), therefore, we cannot conclude anything about how the mean value is changing with time (and thus, answering the question: is the incidence rate rising) unless we actually bin the data by year, and see if the mean value is changing with time.<p>Unfortunately, by splitting time up, we reduce the already small sample sizes for each bracket. Time series analysis is a more appropriate tool to tackle this question.<p>[1] This is unchanged if the rate is continuously increasing, the setup was so that you can actually think of collecting the same data over multiple years.

<p><pre><code> "If mass shootings are really occurring at random, then this suggests that they are extreme, unpredictable events, and are not the most relevant measure of the overall harm caused by gun violence." </code></pre> In my Undergrad years I would get crucified for arguing that a computational model works and concluding that it is therefore the way reality works.<p>Seriously though, can anyone reading this fathom a way to formalize the event "Guns are available"? What's the probability of getting a gun if it's not available in shops? You need to model a different country for that.<p>The same reasoning can be used to show that the probability of a mass shooting given the availability of guns is lower given a lack thereof. In other words, this: <a href="http://www.youtube.com/watch?v=KsN0FCXw914" rel="nofollow">http://www.youtube.com/watch?v=KsN0FCXw914</a> .<p>And if they are indeed rare events, what's the metric for that? If we have a chance of stopping a rare, unexpected event that would add quite a lot to the expected value of deaths per year, would it not matter significantly in our efforts of stopping it?

Beware: the whole point of this is article (and train of thought) is to cast a shadow of doubt over gun control. It's to remind us how these are random and extreme events. Yes, mass-shootings are random and extreme. A layperson can conclude this without having to look at any statistics.<p>But unfortunately this distracts from the real problem. Machine guns. And how they're legal, and easy, for anyone to buy. And ammo. And modifications for guns.<p>When the constitution was written to allow citizens the right to bare arms, the only arms in the legislators' wildest dreams took a minute to load one bullet, likely would miss, and probably wouldn't kill with one shot. Since the world is very different today, shouldn't the laws change too? Look at the success of gun control in the United Kingdom.<p>Enough of this nonsense. It reminds me of climate change deniers pointing out "Yeah but it's snowing now."

I think a better methodology would be using linear time clusters and measuring the possible impact of the Werther Effect.<p>Look at the regional areas where the news may not have spread out.<p>Look at the attempts that were stopped and see if they clustered. The stopped attempts are probably a better indicator as people, after hearing the news, are naturally more alert to the indicators that something may happen.<p>The other problem with the data is that there is no way, from looking at the graph, if an event happening in December affected an event happening in January, thus delimiting by years is arbitrary.

The question probably requires a more detailed treatment.<p>For example, even the distribution of killings by frequency looked poisson, if the number were uniformly increasing year by year, it would look like a trend.

Statistician here. Saying nothing about the substance of the argument or whether their data source appropriately classifies what counts as this kind of crime, the methodology is on the right track but not optimal.<p>Of course binning the events by year throws away data about specific timing. Having actual event times would allow fitting a hazard rate model.<p>There are two simple alternative hypotheses - explanations for a deviation from the Poisson assumption. What they're calling "random" is really "occuring at a homogeneous rate;" so we have to ask - as opposed to what?<p>1) clustering/overdispersion, because the events happen more often alongside each other (copycat effect or whatever; risk of a new event is a function of time since last event)<p>2) secular trend (the rate of events is changing over time)<p>We can't really distinguish between these two without explicitly modeling, and it doesn't look like we have enough data to do that. The tool to do it would be a generalized linear model with an overdispersed Poisson dependent variable.<p>It's kind of bad form to estimate the Poisson mean from the data, and then use that to fit the distribution you're testing against. You're using the data twice, so the p-value isn't what you think it is. You should be conditioning on the total number of events in the sample.<p>Also, chi-square distribution comparison is for large samples. This is a meh-kinda-borderline-midsize sample, to use a technical term.<p>The test they want that has neither of these problems is Fisher's exact test for equal proportions. If the events were generated by a Poisson process, the number in each year would be conditionally binomial, and they'd add up to the total observed event count. The test is: is the data explained by all the years getting events randomly at the same rate (null hypothesis) vs. each year having its own rate (alt hypothesis).<p>And finally, yes, you get a p-value. But you also need to think about the power of the test detect an anomaly if there was one (type II error). For something like this, you could do power analysis by a simple simulation, but you'd need to specify what kind of anomaly you'd want power to detect (e.g. all events cluster together in one year, to pick an extreme).<p>Otherwise, all you have is a design that has a correct p-value. If you reject the null hypothesis at 5%, but you have low power, you might as well toss a coin and declare the data "nonrandom" 5% of of the time. This might be as good a test as we can get with the data available, but p-value isn't the only thing you can look at.

I have flagged this article; is not only link bait using the subject everyone is talking about; is being extremely disingenuous using statistics about a so ambiguous concept as "mass shootings" to draw conclusions about a real world situation.