Are mass shootings really random events? A look at the US numbers
empiricalzeal.com
empiricalzeal.com
1) Yes, we can use the chi-squared test here - the small number of events is built into the p value.
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.
3) These statistics prove only a single thing: mass shootings in USA are likely random events and have a mean value of ~2.
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.
EDIT: Or not - nobody is questioning the real possible problem here: the data itself?
http://www.bradycampaign.org/xshare/pdf/major-shootings.pdf
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.
" ... found a p-value of 0.18. What does this mean? It suggests that there is no evidence of clustering beyond what you would expect from a random process ... "
Above is incorrect. What the author is actually testing is the following statement:
The mass shooting data points come from a Poisson
distribution with an average value of 2
All the author has proven is that there is an 18% chance that the data does not come from a Poisson distribution. ~1 in 5 chance of being wrong is not something you assume to be right. Author needs to learn statistics.
What is pretty clear is that the US has an obscenely high rate of gun violence, which is what we ahould be debating. Several thousand children die each year, and nearly one and a half Sandy Hook's occur each day.
This isn't right either. You've committed the inverse fallacy [1].
The correct way to read that p-value is: "If the data came from a Poisson(2) process, it would be 18% likely to have deviations at least as different from poisson(2) as the observed data."
The test gives no insight into the probability that the true distribution is Poisson or Poisson(2).
Why should americans expect an average of 2 mass shootings a year? What the heck is wrong with that country that you can expect TWO mass murderings a year?
If you compare the statistics to a poisson distribution with a mean of say, 0 (you know, like most countries, you don't expect ANY mass shootings in a year AT ALL), then you cannot claim randomness any more.
The premise is what bugged me the most - that a mean of 2 is acceptable
This analysis has nothing to do with whether or not there are too many shootings. It's about whether the shootings occur at a fixed rate, and are independent of each other.
You are correct that the analysis is about whether the shootings occur at a fixed rate. The author had set the expected value of 2 at a given time period, which is to say the author assumes that the 'natural rate' of a mass shooting murder is 2 a year.
The mean across the years has got nothing to do with this IMO. It may have been a good rule of thumb to compare against, but it's not a good one to do an analysis upon. It leads to all sorts of mistaken conclusions, like the one the author made.
>The mean across the years has got nothing to do with this IMO.
Well, that's all well and good, but your opinion is irrelevant. This is simply how this statistical tool is used. And crucially, the data fits this model, so objecting to its completely standard parameters is very silly.
Norway has had 0 mass murders since ww2. Yet 18 months ago Norway faced one of the worst mass murdrs in peacetime. Maybe the percentage of people wanting to do such stupid things are the same in Norway as in US, only the population is so small it takes years between everytime there is someone with high enough iq to plan this kind of cruelty and low enough respect of life to actually do it.
Also take a look at Japan. Few shootings but at least during the nineties a couple of attempted mass murders using poisenous gas.
Aum Shinrikyo were a remarkable terrorist cult who achieved pretty scarry things. (https://en.wikipedia.org/wiki/Aum_Shinrikyo)
Assume a lower base rate, and you make just 1 mass murder in the 67 years since WWII a fairly plausible outcome. I'd note that before Breivik, the previous record-setting mass murder was a South Korean policeman, also a small country not known for violence or crime.
The double negative here and concept of "distinguishing from" is important. In particular, this result does not rule out being underpowered by which we'd mean to say that given infinite tragic observations we may be able to make a case to distinguish the data from the random distribution, but since we only had a little that effect was buried.
Again, you cannot even say things like "are likely not" and "should expect" because those are epistemologically reversed from what we can state. What we can say, in parallel, is
- We do not have evidence here suggesting that mass shootings are copycat crimes (if we buy that copycat crimes would lead to non-Poisson distribution of mass shootings).
- We have not been able to show it flawed to predict 10 mass shootings over the next 5 years.
The whole "it is random" BS is based on the pre-assigned value of 2 per year. Why pre-suppose that 2 per year is an acceptable norm, that is not problematic in itself?
Consider a US with a mean value of 10,000 mass shootings per year, with the occasional 0, 20.000 or 50,000 etc mass shootings fitting the poisson distribution. Using the same methodology as the article, the same "conclusion" would have been reached, that mass shootings are random.
This is a major misunderstanding at what "random" implies in this context. It just means that the motives and decisions at the individual level to "go for it" are triggered independently and with the specific year bearing no influence. That is, it proves that the specific events are not co-ordinated.
This is mighty fine, but it doesn't at all mean that the cultural / law / etc climate that makes even considering an attempt at one of these events possible (much more for a median of 2 a year) is "random" or non changeable by human intervention.
Way to misapply statistics. As they say, there are lies, damned lies, and statistics.
As an extreme example, in a country where there are no guns (either legal or illegal), there would be no mass shootings AT ALL. No "spontaneous random activity" pertaining to mass shootings could change that.
In a similar, but more practical way, if other western countries have a median of about 0 mass shootings per year for decades (with the occasional exception), this tells us that even if the number of mass shootings per year in the US is random, the fact that it has mass shooting at all is not random but a cultural/structural result.
No one does. The rest of your comment is a riff on that point, so I won't respond to that. This (brief) analysis was trying to answer the question: are there more mass shootings now than in the past? The answer was no, and that there are about 2 a year.
This point is entirely independent from the question, what can we do to decrease the number of mass shootings?
There is no such presupposition. The way that you estimate the mean parameter in a Poisson distribution is by taking the average over your sample. "2 per year" is not a normative statement; it's an observation of history.
In statistics, you don't get to make a moral judgment when estimating parameters. If you do that, you will get the Wrong Answer.
There is many widely different definitions of what a mass-shooting is (about apples we don't); the article is using a definition from the motherjones website; their definition is strict but randomly chosen; like somebody claiming that only red apples are real apples and the other ones are not; with no data to support that this definition creates any useful subset that can or should be statistically analysed; the statistical sample is already biased by absolutist and arbitrary boolean conditionals which render null any conclusion extracted from its statistical analysis (e.g shooter killed 3 people instead of 4 and therefore is not a mass-shooter)
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 anything, 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.
Either there is a common hidden factor, or these incidents are feeding on each other.
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.
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.
(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.)
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.
The mythological/historical example is Herostratus, who burned the temple of artemis just to be famous.
Given what we know of spree shooters around the world and in the US, they strike me as predominantly desperate, mentally ill people. We had the Columbine shooters who seemed to be more vengeful than fame-seeking. We had the Aurora shooter who didn't do any grandstanding when apprehended, or in fact talk to the media at all. And of course, we had the Newtown shooter who didn't leave behind any notes, manifestos, or in fact any evidence that he wanted to be infamous (even if posthumously).
These don't strike me as people who are driven by the ensuing media coverage.
Don't get me wrong - I've been appalled by the news media surrounding the Newtown shooting, particularly CNN's journalistically abhorrent fingering of an innocent individual, but I don't think the coverage is really the crux of it, and I don't think taking it away will stop them.
However, pro for your argument, there is some reporting bias -- we hear about school shootings for days, but we rarely hear about suicides, or murder/suicides, which are far more common; even things like the Sikh temple murders went out of the news in a day or two. Schools are just particularly likely to get coverage and stay in the news.
But if ending the media deification of spree killers converts some of them back into simple suicides, that's a win, really. On the other hand, it seems to have channeled them into a local maximum of "pretend like you're playing Call of Duty, without any real skill or training" -- I'd be a lot more afraid of a halfway competent chemist with no tactical experience deciding to go the full McVeigh, vs. trying to play soldier. I'm personally pretty comfortable with a rifle, but if my goal were mass casualties, it'd be bombs or other technology based attacks for sure.
a) keep the news of violence to local community only b) do not glorify the attacker by negatively praising him (like a ninja, moved quickly...) or showing him in positions of power (brandishing guns or wearing clothes that hide his face) c) telling how many people he killed.
Each time one of those news is broadcast, he added, they see a spike in violence. And it seems consistent with how humans regularly behave.
* http://cad.sagepub.com/content/48/1/46
* http://books.google.rs/books?hl=en&lr=&id=3B4lTTZE58...
* http://en.wikipedia.org/wiki/Social_contagion
They are cited as references on this page:
*http://en.wikipedia.org/wiki/Copycat_crime
As for other evidence, you'd need to look at crimes and if there is a large number of similar attempts of violence (not necessarily school, but overall mass shootings), grouped around highly prominent cases covered by media. That kind of research and evidence is sadly outside my scope. I'm not sure where you'd find sources, nor how to process them statistically correct.
He left plenty of evidence that pretty much guaranteed he'd be infamous. Nothing will do that like several dozen dead bodies. Notes and manifestos are optional.
These assholes may have many different drives but the one thing they seem to have in common is that they pick their targets to guarantee that they will become notorious. Stomping on an ant hill will kill many more but won't put you on CNN.
Apropos CNN, the one thing the media could do with cases like these is to restrain themselves in the way they cover these incidents. I'm pretty sure there is a large element of copy-cat behaviour in here that could potentially be reduced by the amount of media coverage given.
You seem to be confusing cause and effect.
He more likely chose his target because his mother worked there. Even if every shooter is choosing randomly we will only talk about the "media seeking" ones the news covers.
Still, the decision to attack the school, it's students and not simply one individual is why we are hearing about these.
That's circular reasoning. You're reusing evidence in a way that does not check out.
The media decide who gets to be (in)famous and by behaving in a certain way you can get your 15 minutes. Behaving in this particular way pretty much guarantees that. The link between gore and media coverage is firmly established by now so even if in the first few cases you could have said 'that's circular reasoning' after several such cycles you can no longer tell cause from effect.
Hence copycat crimes and other stupidities.
All it took for dumb, angry children to be set on this path was the attention given to the Columbine shooters. Quite a few of these idiots - if not all of them - think those guys were heroes and perceive the media attention given to them (movies, incessant press coverage) as positive.
They want that for themselves to compensate for some perceived inferiority or other shortcoming and so they go the same route.
I think today it is safe to assume that a hunger for fame is at least a component in these shootings.
No, we mostly know them in the US, by an order of magnitude more than other western countries. The same with serial killers: the US is up there with some third world nations for more prolific killers, and unsurpassed for total serial killer count, so much that Wikipedia devotes its own page to US serial killers which is larger than the sum page for the rest of the world.
And, desperate, mentally ill people? Could also be the products of a desperate, mentally ill society.
> So much that Wikipedia devotes its own page to US serial killers.
From Wikipedia, school shootings in Europe in this century, last two numbers are killed and wounded:
R., Georg, 18, Sep. 17 2009, Germany, 0, 10-15
Kretschmer, Tim, 17, March 11, 2009, Germany, 15, 9-13
De Gelder, Kim, 20, Jan. 16/23,, 2009, Belgium, 4, 12
Saari, Matti Juhani, 22, Sep. 23, 2008, Finland, 10, 1-3
Auvinen, Pekka-Eric, 18, Nov. 7, 2007, Jokela, Finland, 8, 1
Bosse, Bastian, 18, Nov. 20, 2006, Germany, 0, 22
Steinhäuser, Robert, 19, April 26, 2002, Germany, 16 1
Merah, Mohammed, 23, March 11–22, 2012, Toulouse & Montauban, France, 7, 8
Amrani, Nordine, 33, Dec. 13, 2011, Liège, Belgium, 6, 123 (he also used grenades)
Breivik, Anders Behring, 32, July 22, 2011, Norway, 75, 242 (a bomb killed 8)
van der Vlis, Tristan, 24, April 9, 2011, Netherlands, 6, 17
Radmacher, Sabine, 41, Sep. 19, 2010, Germany, 3, 18 (arson was also used)
Bird, Derrick, 52, June 2, 2010, United Kingdom, 12, 11
Sacco, Angelo Secondo, 54, June 28, 2005, Italy, 3, 9
Antonello, Mauro, 40, Oct. 15, 2002, Italy, 7
Durn, Richard, 33, March 27, 2002, France, 8, 19
Selamet, Ozan, 42, Jan. 18, 2002, Belgium, 6 (2 strangled)
Roux-Durrafourt, Jean-Pierre, 44, Oct. 29, 2001, France, 4, 7
Leibacher, Friedrich, 57, Sep. 27, 2001, Switzerland, 14, 18
Kaya, Hakan, 24, Dec. 22, 2000, Germany, 61) There are enough guns out there now that a ban on new production would have only a minimal effect, a ban on transfers would have only somewhat more effect, and a ban on possession, given that criminals are willing to break laws, wouldn't have a huge effect.
For spree shooters, even a pump-action shotgun or lever carbine and a 10-rd handgun or two (or even just a couple revolvers with speedloaders) would be pretty adequate to kill 50+ unarmed people, even if they weren't 6 year old children. So you'd need to address not just "assault rifles and 100 round drums" but basically most firearms (maybe leave single-shot shotguns/rifles).
2) The cost of ending the media deification of spree killers is almost zero. The cost (in terms of implementation cost as well as loss of rights) of any gun control is far higher. The cost of a comprehensive gun ban and confiscation could be up to (probably localized) civil war. Pick the tool which has the least costs first, especially if it's probably more effective.
3) Even if there were zero guns, you'd have other instruments of death, assuming people had the desire. I'll accept that guns are easier for some of the more amateurish and unmotivated killers (including maybe the CT guy), but Columbine involved bombs, McVeigh was a bomb, and now that the "spree kill and become famous" thing is out there, all it would take would be another McVeigh or two to show how its done and then people would be doing bombings, or running over a bunch of kids at bus stops, or whatever.
In places where drugs and alcohol are banned, they make alcohol out of prunes, get high off glue, etc. It's a lot better to focus on demand than supply.
I think Europe generally doesn't have incidents like this, not because it's disarmed, but because even "losers" in Europe can live a somewhat comfortable life compared to losers in the US. Better healthcare, better social support, etc. Europe has certainly had some serious violence in the past, generally by organized groups that feel they are "losers" vs. individuals, more so than the US. European society seems to have less inequality and thus fewer positive and negative outliers than the US. (Similarly, there is a gender difference in distributions of most traits; men tend toward the extremes; which may be why most criminals and other horrible people seem to be male)
A recurring element in a lot of these shootings is that the shooter gets/steals his weaponry from an uncle, dad's rifle cabinet, a friend or so on. These are the kind of people that would dutifully get rid of their guns if they were banned. Thus closing up a potential source of destructive firepower for nutcases.
> I think Europe generally doesn't have incidents like this, not because it's disarmed, but because even "losers" in Europe can live a somewhat comfortable life compared to losers in the US.
Is there any indication that any of the recent shootings happened because the shooter didn't have any food to eat, no shelter or bad healthcare? Because that's the sort of comfort that a welfare state provides and I don't see the link.
Especially true for minors. True for a possession ban. Less so for sale prohibition or transfer prohibition. More so for sale/transfer prohibition with generous buybacks. (if new-sale is prohibited, prices will rise, so they'll be secured better; if transfer-ban, you'd probably see prices fall (since there's no market), so less of a reason to secure them, ironically).
> [food, shelter]
Clearly they have insufficient mental health services (in the CT case, some of the problem was that he wasn't a minor, and his mother was trying to have him committed). Not so much "how to pay for mental health" vs. general delivery issues.
They do seem to generally be middle class and not starving or anything, but I meant "failure" in the sense of social standing/competitiveness/etc., not just absolute poverty. After all, even a fairly poor person today lives (in absolute terms throughout history) fairly comfortably, but feels the relative social standing and status.
http://online.wsj.com/article/SB1000142412788732467720457818...
Having a half decent welfare program is another element in there. But with the current crisis and the upcoming pension plan troubles it remains to be seen if we can sustain that and if we can not I predict an increase in crime and associated violence.
== Interesting to note that Prision murder rates are higher than many states in the US, and many countries in europe. None of this is guns, though. It seems in the abscence of weapon x a killer will use y or z. lots of people killed this way with pens and spoons etc.
So if you're sincerely asking what differences other than gun ownership might be preventing violence, that'd be a good place to look. Time Magazine just had a good piece on the cultural differences that lead to a safe gun culture there: http://world.time.com/2012/12/20/the-swiss-difference-a-gun-...
For the patriotic side, check out Project Appleseed (http://www.appleseedinfo.org/), which conducts weekend courses on rifle marksmanship and Revolutionary War history. (It's run by volunteers, and costs $70 for a weekend.)
This culture used to be more widespread in the U.S., back when rifle shooting was a high school sport. The more we crack down on legal gun ownership, the more we stamp it out.
http://www.cdc.gov/nchs/fastats/homicide.htm
https://en.wikipedia.org/wiki/List_of_countries_by_intention...
One possible cause is demographics - more than half of our murders are committed by a demographic group making up only 10% of our population, and this subgroup is very uncommon in Europe.
http://en.wikipedia.org/wiki/Race_and_crime_in_the_United_St...
The whole thing is a vicious circle, this kind of news has to be sensationalised because we have been steadily desensitised by the media.
As a teenager in the 90s, I remember getting mildly claustrophobic at the thought of watching another Movie or TV Show about a hijacked plane or hostage situation. There were just so many building up to 9/11. Essentially, the messages were things like 'Enemies of America hijack planes', 'Radical citizens plant bombs', and now 'Angry employees/students show up with automatic weapons and body armour.' These are as much statements as they are calls to action.
Let's take a look at this year's Emmy nominations for the best actor category, these actors play: a ruthless corrupt bootlegging politician-gangster, a high school chemistry teacher turned murdering drug kingpin, a vigilante serial killer working in law enforcement, a terrorist war-hero congressman secretly radicalised in captivity and an alcoholic misogynistic deserter ad executive with a stolen identity. The only leading character that does not belong in prison is from a British production. Perhaps this category should be renamed the 'Best Magnificent Bastard' award.
America has this fascination with revealing its weaknesses to 'enemies - both foreign and domestic' and celebrating monsters. Most people learn early on that, unless one is trying to play and remain the victim, painful memories and weaknesses should not be put on public display for the casual friend, enemy and stranger.
A shotgun is a far better choice.
Fun fact: the reason the military went with 5.56mm is that killing power is less important for the military (wounding a guy takes him out of the fight, as well as 1-2 of his buddies to take him home), while ammunition weight is a big deal (lighter ammo = less for a soldier to carry).
I mean, the military even manages to make 9mm ineffective. LC 115gr 9mm FMJ is maybe 10% as effective as my Speer Gold Dot 124gr +P JHP.
At the same time, tests have shown less overpenetration after encountering drywall than hollowpoint handgun rounds, which tend to fill up the hollowpoint without expanding and then go through several barriers.
We have a problem with guns and crime, not with massacres. Massacres happen, and if there were no other murders to worry about I would say we should focus on massacres. The trend in gang violence is very clear: concealable weapons (handguns) purchased on the black market. If you want a sensible gun policy, you need to focus on handguns; gangs rarely use rifles because rifles are too difficult to conceal.
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...
Normally I'd love to dig a little deeper, but it's time for me to play Santa. Merry Christmas!
The parent comment is providing a better, more specific grouping that helps shed light on where murders are actually happening in the US and points us towards where we need to focus our efforts if we want to actually change the murder rate in the US. Inner city interventions will result in the greatest improvements to the murder rate, whereas focusing efforts elsewhere has literally zero chance of bringing the murder rate into line with the rest of the G8 because that's not where the murders are happening.
And all that is leaving aside the matter of whether the statement is actually true, as questioned by a number of other threads
In slums you have a completely different expectation of law enforcement and law enforcement presence, social service, education, economy, transportation, etc. In every respect except for lines on a map, slums are a completely different world. "Countries" are an obscenely coarse classification, we should always be open to more fine-grained analysis.
And the police itself kills an astonishing number of people each year, compared to a any western european country.
Nope. It is both. From dictionary.com "ton" entry:
"7. Informal a. A large extent, amount, or number. Often used in the plural: has a ton of work; gets tons of fan mail."
Also, please provide a reference.
Since you asked politely, here's one. Manhattan, which I don't consider an "inner city slum" had 88 homicides in 2005, 102 in 2006, 68 in 2007, 62 in 2008, etc.
That's comparable to Ireland, Switcherland, Denmark, Austria, countries of 3 to 6 times the population of Manhattan (1.6), and 3 to 4 times the percentage of murders in Germany (adjusted for the 80 million population).
Not only that, but Manhattan is the entertainment center for the surrounding Burroughs and cities, where a lot of violence happens.
This is true in Toronto as well, our "club district" has one of the highest murder rates from people visiting who don't live in the club district.
I don't think Manhattan makes a fair comparison and doesn't disprove the OP's comment.
http://www.nationalatlas.gov/articles/people/IMAGES/crime_mu...
What the heck is wrong with Northern Idaho? Elko Nevada? So Cal's inland empire? I didn't think there was anyone to kill in western New Mexico. The Mississippi Delta also doesn't look like a healthy place to live.
If we excluded Louisiana, the murder rate would probably plummet to mostly normal levels. And it just isn't New Orleans.
Notice that the many of the places with the lowest murder rates also tend to be sparsely populated. What you're seeing is the increased variance that one should expect to see when dealing with smaller sample sizes, not evidence that small towns are more dangerous than big cities.
http://www.textmap.com/offence/vehicular-homicide.htm
I bet we would see more geographic correlation with not too much variation in city areas (e.g., the Delta is still a dangerous place to live, which would correspond to anecdotal evidence). Unfortunately, I can't google up a decent gun homicide heat map.
Do you mean levels of murder in peer countries where you've decided to not exclude the "innercity slums" that exist there?
This is actually a serious issue - please treat it as such.
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): http://imgur.com/Onoxv . The colours represent the group of how many incidents happened in the 180 days prior to the incident.
Here is the distribution of the number of incidents 180 days prior to an incident: http://imgur.com/9VTjQ
Draw your own conclusions
That said, I think this trend is hidden from this dataset because it relates specifically to mass shootings, including only those incidents in which the shooter took the lives of at least four people.
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.
See this paper for a more detailed discussion:
http://www.crim.upenn.edu/faculty/papers/berk/regression.pdf
Moreover Poisson distributions are generated from events that occur in a very small fraction of all opportunities, but with many opportunities each time period. So, unless the timing of shootings is correlated (again, the thing he is testing for), this is a perfect use case for the Poisson distribution.
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 <insert your word here> that causes these events, report on the impacts of these events. It would, hopefully, stop deifying the perpetrator, which might reduce the likelihood of other perpetrators from doing the same.
What if the goal in large part is to inflict the most possible suffering on the world? Out of anger or hurt or whatever the case may be.
Then cutting out coverage on the person isn't as important, it's the entirety of the coverage that matters.
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.
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.)
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.
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.
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.)
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.
Unfortunately, I'm pretty sure there's a fatal flaw in the analysis.
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].
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.
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.
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.
[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.
"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."
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.
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: http://www.youtube.com/watch?v=KsN0FCXw914 .
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?
We haven't yet defined the goals. Unless "Minimum amount of gun-related deaths" is the goal (clearly, it isn't that simple in this country), in which case his highlighted claim is definitely false - we don't know much about effective metrics. Are the psychological effects of a mass shooting similar to a snowball effect? Maybe in the long run? There's a lot more to research before saying 'insignificant', especially if you don't describe the model.
One more thought: could you define 'mass shooting'? Usually the media is responsible for that definition, whereas a random murder isn't called that.
Mother Jones probably did a pretty decent job of defining 'mass shooting', I assume they errored on the side of inclusion (I only recall 2 this year). The authors analysis seems to make it pretty clear it is only considering that definition.
I'm really not reading this as an attack on discussions of gun violence in general, as it seems you are.
I wouldn't characterize the number of deaths via mass shootings as "quite a lot". It's probably in the same ballpark as death by lightning strike (~50/year, according to Wikipedia).
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.
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.
Enough of this nonsense. It reminds me of climate change deniers pointing out "Yeah but it's snowing now."
Look at the regional areas where the news may not have spread out.
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.
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.
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.
Of course binning the events by year throws away data about specific timing. Having actual event times would allow fitting a hazard rate model.
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?
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)
2) secular trend (the rate of events is changing over time)
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.
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.
Also, chi-square distribution comparison is for large samples. This is a meh-kinda-borderline-midsize sample, to use a technical term.
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).
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).
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.