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Discussion of statistical analysis of vehicle fires as it relates to Model S

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One more thing that is very important we don't forget when we are discussing these statistics:

When we make a comparison between two groups - two sets of data - we have to be careful how we conclude. Let's say we can agree on a confidence interval (95%) commonly used, what units to use for the calculation in the nominator and denominator (fires? accidents? fires/accident? caryears? miles driven? etc.) and what types of Statistical tests are approriate for the data (since this is nominal continuus data which we don't have any reason wouldn't be normally distributed I agree we can use a binominal Method, I Guess you are using the chi-squared test luvb2b?). Let's further say that we do find a "statistically significant difference" (i.e. we can reject the null hypothesis of there being no difference With a 95% probability) between the groups. From this thread I understand that luvb2b has found that Teslas are in general less likely to Catch fire in general but perhaps more likely to Catch fire after collisions.

There are two fundamentally different ways to interprete this significant difference that we find:

1. Tesla cars are in fact less likely in general to burn, but more prone to burning after collisions, than "other cars".

2A. The difference is due to the fact that we are comparing two sets of data that are not from the same pool. I.e. even though we have the same units in the nominator/denominator in both sets of data the cars/individuals/singe data Points in the two groups come from two different "populations" to begin With.

2B (a variant if you will of 2A). The observation Method and Reporting Method between the two groups is fundamentally different. For the Tesla data it's basically battery fires reported in the News. What is the NTHSA data Collection Method? Definitions? Cut-off for what gets reported and not? What is classified as a fire, a collission, etc in the NTHSA data?

For conclusion 1 above to be the correct one you would have to first know that the two groups are otherwise very similar - ideally the same - when it comes to possible confounders such as age, vehicle size, typical driver background, where in the country/world it has been driven, etc. etc. etc. I would argue that it is very very unlikely that the two data pools have similar enough background characteristics to make for a good comparison.

I think conclusion number 2 above is a lot more likely - we are comparing two very different data sets and we can not apply comparative Statistical tests.

Again to make a parallell to medicine you would never use a chi-squared test, Student's T-test, Wilcox rank test or similar unless you first had one big pool of individuals/test subjects that you than randomized to two groups, made an intervention and followed them afterwards. When you do something like that you know that to begin With both groups came from the one and same population before you divided them. After that any difference you find is likely to be a "true" difference.

Applying Statistical probability testing to observational data With different roots makes for poor statistics without value.

All this being said I believe that there is a real issue here which is that 3 Tesla cars have caught fire after collisions. Even if the fires have been contained and haven't caused injuries or Death this warrants further study. It really doesn't matter if Teslas burn more or less than ICEs to me. I don't like Teslas to burn at all, period. Big batteries under the Whole of the car is a completely New Technology and this potential problem needs to be studied further and adressed regardless of how safe the car is compared to regular ICE cars.

As I said before, I think Elon did it all wrong when he started this stastics argument as a defence. He should have just said: We still believe that the Model S is a very safe car, there have not been Deaths or injuries from these fires, they have been contained, the car informed the driver to pull over and gave them plenty of time. However safety is Our number one priority and we will use all Resources to look in to this matter. If there is a way to make Model S even more safe we will. We welcome a NTHSA investigation.
 
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yep, back to post #37.

The calculation is not the odds of Tesla having a fire, its the odds of any population of cars with 13,300 car-years. You could go out on January 1st and put a red X on 13,300 cars at random and the odds of one of those cars catching fire in the next year (due to a highway collision) are:

0: 59.37%
1 or less: 90.33%
2 or less: 98.39%

so the first fire was a 60/40 split, so pretty random, the second one put it to 90/10 which becomes significant, and if you include the third you get to 98/2 which is very unlikely. (it means that there is something odd about your data-set if you have somehow got three out of your 13,300 sample)

yeah- thanks for reposting that by the way- I think it was buried for a lot of readers
 
I'd like to see that analysis. Will you being doing it?

- - - Updated - - -

Remember that the NHTSA is only considering the U.S. fires and still feels the need to investigate.

I think that NHTSA was compelled to open a preliminary investigation to quell the noise. Unfortunately, they were getting pressure to do so from all sides.
I think that they want this out of their hair and the sooner, the better, so that they can get back to business.
 
Could someone clarify why the incident in Mexico is included in so many of the data sets in this thread? That was (by any sane measure) a deliberate act.

It wasn't a random act of 'oh hey, look!, I'm drunk but I'm pretty sure I just found a huge concrete wall in the middle of the road'.

I'd liken it to driving a car deliberately off the road or spilling gasoline around it and lighting a match. It's not random. It's a foreseeable result. Why do people keep talking about three incidents?
 
Could someone clarify why the incident in Mexico is included in so many of the data sets in this thread? That was (by any sane measure) a deliberate act.

It wasn't a random act of 'oh hey, look!, I'm drunk but I'm pretty sure I just found a huge concrete wall in the middle of the road'.

I'd liken it to driving a car deliberately off the road or spilling gasoline around it and lighting a match. It's not random. It's a foreseeable result. Why do people keep talking about three incidents?

Because such incidents are probably included in the large data set People (including Elon) are trying to make a comparison to.
 
This Honda situation was apparently spontaneous combustion. If you have not yet read, the Tesla fires are caused by high energy impacts with unusually shaped ultra-hard metal (trailer hitch, etc), which create significant force in a small area. If you haven't read, people have died from such impacts in ICE, and have had cabin penetration in ICE cars.

So I'm not seeing what Honda 2002 has to do with Tesla in 2013 other than to show Tesla doesn't have a spontaneous fire problem.

Picky, picky!! So one car has dozens of cases of *spontaneously* catching fire (like after an oil change), and another car has two examples of a car striking a metal object at freeway speeds and then having a part of the car catch fire several minutes later after the occupant had been warned to pull over. Gee, seems like apples to apples to me. And I would like to see an analysis of how any ICE car and its occupants would have fared if it had driven through a concrete wall at 40 mph or more.
 

yep - and here we have the major problem with this :
he says "Because only 4 percent of vehicle fires are caused by collisions, Tesla’s Model S sedan, with a rechargeable lithium-ion battery, is statistically more likely to catch fire than are cars with gasoline tanks, wrote Kevin Bullis, senior editor for energy for MIT Technology Review."

well that isn't true either, as all of a sudden (and I knew this would happen), the reader sees "Tesla’s Model S sedan, with a rechargeable lithium-ion battery, is statistically more likely to catch fire than are cars with gasoline tanks" - that's only for collisions, but the inference now is that the ModS has the same propensity for non-collision fires as ICE and so the overall statement now applies to the over all car and the over all safety of the car. This is exactly why Elon can only win the PR battle on this at the safety and overall level
 
...Since the Tesla group - study group - sample size is so small it really doesn't help that the control group - "ICE cars" - is so huge: there will be a lot of innate statistical insecurity in the data, hence wide confidence intervals/high p-numbers.

The next and really biggest problem is how to define the control group and how to get good data to use for the control group. ... Hence we have to do with a control group made up of a very mixed group of vehicles with large variation in age, type of vehicle, etc. etc. The fact that the data of the control group is extensive when it comes to number of incidents, miles driven, time of observation etc. doesn't really help much IMO since no matter how you look at it it will be an apples to pears comparison that gives little meaning.

I think the increased car-years is a good possible answer. I'm still at a loss to reconcile to the testing which should have simulated these kids of car-years.

How would the new car-years basis (linear intra-quarter treatment) form to predict the time we would expect to see the next collision-fire? Seems like with the high certainty of the statistical model, that would be an easy and reliable prediction to make. Might be a good test for the model, if nothing else to register possible effects of the recent height clearance

first, hello johan! nice to see you again. wow you're going to be a phd!

well obviously if i've lost you there is much confusion here. in the course of trying to answer ken's questions, i've come up with a different way to look at the same data (so those who aren't happy with the data, sorry!). let me see if that helps clear up any confusion.

we have two populations, and we can consider them distinct as even 2012 nfpa data is going to include very few tesla model s vehicles. the question of overall safety i feel has been adequately settled in favor of model s. what we're looking at is the risk of collision-fires.

from the nfpa data set of millions of car-years of experience we have an estimate of the proportion of vehicles that have collision fires. this is almost going to be a point estimate, in the sense that the number of observations is so high that the sample mean's standard deviation will be very low and the confidence interval very tight. in the picture below, you can see an example picture of that as the spiky narrow distribution on the left.

from the tesla data set of 10,820 car-years of experience, we would have a different distribution for the sample mean, with a much wider confidence interval. for illustrative purposes i showed that as a wider distribution on the right. it's important to mention here that there's no way the distribution of the tesla sample mean is going to have that shape. that's simply a picture not to scale for illustration purposes.

i hope we agree that we are trying to determine to what degree this two distributions are separated. if they overlap a lot, it means we can't tell much of anything. if they don't overlap, well that means the model s has a much higher risk of collision fires.

View attachment sample mean.bmp

in my original post, i had the point estimate of probability of a collision fire for an ice at .0000392 as follows:

5,020 collision related car fires per year / 128.1 million cars = 0.0000392 collision related fires per car-year. this number is probably too high for an apples-to-apples comparison because the 5,020 collision related fires includes many older cars.

a number of people have questioned the data set, but it's the best available to my knowledge. you can use the normal distribution to construct a confidence interval around this if you want, i can just tell you that .0000392 would have an error of +/-10% and in all likelihood it's much too high because it includes all these older cars which are 2-5 times as likely to catch fire as new ones. so what i'm saying here is that the reference point of interest is .0000392 collision-fires per car year (or less). this is the left hand side of my picture.

the second part, if you approach the problem this way, is that you want to construct a confidence interval around the sample mean of the probability of collision-fire per car year of a model s. that's the right hand side of the picture.

bringing back my "car-years" model,
quarterdeliveriescum vehiclescar-yearscum car-years
9/30/2012250250 31 31
12/31/201224002650 363 394
3/31/201349007550 1,275 1,669
6/30/2013515012700 2,531 4,200
9/30/2013550018200 3,863 8,063
11/20/2013307421274 2,758 10,820
what we're dealing with is 3 collision fires in 10,820 model years. the unbiased estimate for the sample mean of the probability of a collision fire in a model year is simply 3/10,820 = .0002773. i don't think we have any argument about that?

the problem now is how to do a confidence interval, and this is where a bunch of people are getting lost. let's just say you try to do the old textbook normal distribution approximation to the binomial sample proportion, as shown here:
http://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval#Normal_approximation_interval

this results in a formulaic answer that looks like this for a 95% confidence interval.
.00027773 +/- 1.96*sqrt((.0002773)*(1-.0002773)/10820)

this answer is nonsense of course, because it gives a confidence interval of [
-0.000036, 0.000591] which is clearly impossible because you can't have negative (and obviously not even zero in this case) probabilities. i believe this is why so many people keep saying a sample size of 3 is not enough - because they're trying to use normal distribution approximations which are invalid with such low probabilities and outcomes.

but you can construct a two-tailed distribution with the binomial distribution. the upper and lower bounds of a two-tailed 95% confidence interval will be defined by the answers to these questions:

given a binomial distribution of collision-fires, what is the probability p such that i would have a 97.5% chance of observing 3 or fewer car fires in a sample of 10,820 car-years of experience?
given a binomial distribution of collision-fires, what is the probability p such that i would have a 2.5% chance of observing 3 or fewer car fires in a sample of 10,820 car-years of experience?

these questions can be answered quite easily, you can just use goal seek with excel and this formula:
=binomdist(3,10820,<< insert estimate here >>,true)

use goal seek and you'll find the upper and lower ends of the confidence interval are [.000101, .000810]. the interpretation here is as follows:

given that we have observed 3 model s collision fires in 10,820 car-years of experience, the estimated probability of a model s collision fire in a car-year could be as low as .000101 we'd still have a 97.5% chance of seeing this outcome or less in our dataset.
given that we have observed 3 model s collision fires in 10,820 car-years of experience, the estimated probability of a model s collision fire in a car-year could be as high as .000810 we'd still have a 2.5% chance of seeing this outcome or less in our dataset.

****note for those who think i should use only 2 fires, the confidence interval is [.0000566, .000663] - you can check yourself simply substitute
=binomdist(2,10820,<< insert estimate here >>,true)

now my point is this:

the point estimate for ice collision-fire in a car-year probability is .0000392. the lower end of the 95% confidence interval for the model s collision-fire in a car-year probability starts at .000101.

that puts the lower bound of the 95% model s confidence interval 2.5x above the point estimate for ice collision-fire in a car-year probability. you see my point here johan? it's not even very close. and we know that the ice figure has a super low standard deviation due to the huge sample size. if you really wanted to, you could work out the sample standard deviation and get a confidence interval for that ice point estimate, and verify how tight it is. even with 2 fires, the lower bound of the confidence interval has good separation from the ice figure.

any differences that are showing up here vs. my first analysis are due partially to a different methodology and partially due to my tweaking of the car-year model (which was much too generous to tesla initially).

i had to do it this other way to answer the question ken's been asking, which is:
can we say anything about when we'll see the next fire?

to get to this i'll answer the following questions:
assuming that the model s collision fire per car-year probability is .000101, how many car-years have to pass before we have a 90% chance of seeing another collision-fire? (lower bound of confidence interval)
assuming that the model s collision fire per car-year probability is .000277, how many car-years have to pass before we have a 90% chance of seeing another collision-fire? (sample mean)
assuming that the model s collision fire per car-year probability is .000810, how many car-years have to pass before we have a 90% chance of seeing another collision-fire? (upper bound of confidence interval)

in this case you can use excel and goal seek again, this time with =binomdist(0,<< insert guess here >>,.000101,FALSE) in excel, try to set the value to 0.10. then replace the .000101 with .000277 and .000810, repeating the process of goal seeking the value to 10% each time.

what you should find is that assuming
p = .000101, it will take 22781 car years
p = .000277, it will take 8295 car years
p = .000810, it will take 2850 car years
before you will have a 90% chance of having seen another model s collision fire.
i acknowledge the people who say this won't be valid with raised air suspension. they may be right, time will tell.

now ken, the next step is to project the model s car-year model forward, using some basic assumptions for future production (5500 and 6000 per quarter).
quarterdeliveriescum vehiclescar-yearscum car-years
9/30/2013550018200 3,863 8,063
12/31/2013550023700 5,238 13,300
3/31/2014600029700 6,675 19,975
6/30/2014600035700 8,175 28,150
9/30/2014600041700 9,675 37,825
12/31/2014600047700 11,175 49,000
remember that at 11/20 we're at 10,820 car years.

if model s true probability is on the low side of my confidence interval, we'll see have a 90% chance of seeing the next fire by 10,820+22,781 = 33,601 total car-years, at or before the end of 2014q3.
if model s true probability is at the sample mean of my confidence interval, we'll see have a 90% chance of seeing the next fire by 10,820+8,295 = 19,115 total car-years, at or before the end of 2014q1.
if model s true probability is on the high side of my confidence interval, we'll see have a 90% chance of seeing the next fire by 10,820+2,850 = 13,670 total car-years, before or near the start 2014q1.

if you put the gun to my head and said, "just shut up and tell me when already!", i'll say the models predict you'll see another collision-fire before the end of 2014q1.

ken, may i ask, are you really able to follow what i mean? even my head hurts now.
 
Well, I have a few concerns with your analysis/creating of the dataset.
First of all, the mexican car fire shouldn't count to the 3 fires. But if you count it in, you have to expand the dataset of ice-cars to Mexico as well. Most probably expanding that dataset to the Mexico wouldn't actually do the MS a favour, but it shows the point how fragile an analysis based on 2/3 events can be. As shown by others, the p-value sinks to around 0.9, which is high but (to me) not statistically significant.

Second, one should consider the impact of correlation: there are quite a few variables I would control for if I would really want to find out which car is more prone for fires.
MS are exciting cars, fast cars, probably get driven faster and longer on highways compared to millions of city cars. As they are new and cheap, people use them as their first car and put much more miles up than their ICE-cars waiting in their garage do over their "car-age". These are all variables that could bias heavily and end up giving you a different picture.

Depending on how you put that regression model together, you will probably with two different answers.
 

thanks for the link. you can see by the muted stock reaction the smart money had figured this out already. i think if i had done this analysis immediately after the 2nd or 3rd fire there would have been a chance to make some money on the short side.

the most interesting piece to me is "And that’s what the NHTSA wants to know–whether the Model S is particularly vulnerable to catching on fire when it collides with something," this gives me and others before me who said the same things some validation.

here's the mit article
http://www.technologyreview.com/vie...-are-more-frequent-in-the-tesla-model-s-than/
 
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[...]
i hope we agree that we are trying to determine to what degree this two distributions are separated. if they overlap a lot, it means we can't tell much of anything. if they don't overlap, well that means the model s has a much higher risk of collision fires.

View attachment 36156

[...]

if model s true probability is on the low side of my confidence interval, we'll see have a 90% chance of seeing the next fire by 10,820+22,781 = 33,601 total car-years, at or before the end of 2014q3.
if model s true probability is at the sample mean of my confidence interval, we'll see have a 90% chance of seeing the next fire by 10,820+8,295 = 19,115 total car-years, at or before the end of 2014q1.
if model s true probability is on the high side of my confidence interval, we'll see have a 90% chance of seeing the next fire by 10,820+2,850 = 13,670 total car-years, before or near the start 2014q1.

if you put the gun to my head and said, "just shut up and tell me when already!", i'll say the models predict you'll see another collision-fire before the end of 2014q1.

ken, may i ask, are you really able to follow what i mean? even my head hurts now.

Thanks for your detailed reply, for visualizing it the way you did and for actually doing the math on the boundries of the confidence interval!

I agree with the way your calculating it. I agree with your conclusion and I think your prediction in to the future is actually quite good however we need to take in to account any possible effect of the winter season when it comes to the risk of hitting debris on road (higher risk? lower?). Also will the disabling of low setting in the 5.8 firmware have a positive effect on this ocurrence?

My only gripe with the whole thing, and to me it is a really big issue, is the fact that we don't have a good hold on what's hiding behind the numbers in the two groups. Is the definition of a fire the same? I would guess that in the NFPA data the car-years figure is quite accurate, as it is in the Tesla data. In the Tesla data the number of fires is also probably accurate, as we would have probably heard of any other fires occuring. However, do we really know enough to judge the quality of the NFPA data when it comes to the "5,020 collision related car fires per year"??? If that number is too low due to definitions, incomplete reporting, biased reporting etc. then there is a problem. And there is no way to be quite sure of the quality of the data. I do agree with you though that it's the best data available so you just have to work with it.

Thanks anyway for a great answer and a great discussion. Now I have to think about selling my TSLA stock if I don't see another tactic soon by Elon and Tesla on handling this matter, PR-wise.
 
all data is flawed in some ways. if you're waiting for a perfect set of data, well it'll be like waiting for the perfect man (or woman)... you'll wait a long time.

When you are aware of flaws you don't make conclusions from those flaws.

can you show me a better set of data and better analysis?

No. The absence of better data does not make the inferior data magically true.

3 tesla batteries have caught fire shortly after collisions. that is a fact.

True, but it is very probable that in another more common vehicle two of the fires would have been categorized as mechanical or electrical failure because no investigation would have been done to find the road debris.

the best data set available indicates that this is anomalous number given how many teslas have been on the road and for how long.

No it does not. The best data set available indicates an upper bound of 113,000 vehicle fires that may have been caused by road debris or collision. There is no way to know how many of them were caused by road debris. The data collection method does not distinguish that category. You are guessing that it does, and there is plenty of reason and evidence to believe that it does not.
 
However, do we really know enough to judge the quality of the NFPA data when it comes to the "5,020 collision related car fires per year"??? If that number is too low due to definitions, incomplete reporting, biased reporting etc. then there is a problem. And there is no way to be quite sure of the quality of the data. I do agree with you though that it's the best data available so you just have to work with it.

Thanks anyway for a great answer and a great discussion. Now I have to think about selling my TSLA stock if I don't see another tactic soon by Elon and Tesla on handling this matter, PR-wise.

just to be clear, the 5,020 figure is an estimate i derived for 2012 - not an nfpa number. i posted the derivation in the first post. the mit article that lola just posted has a similar discussion:
Early Data Suggests Collision-Caused Fires are More Frequent in the Tesla Model S than Conventional Cars | MIT Technology Review

regarding selling tesla stock, well that's a judgement of the stock's value, and whether or not it has already discounted the collision fire issue. i think judging by the response to today's bloomberg / mit articles you can see the smart investors already knew what i've been trying to explain.

had i gotten on this earlier there were probably a good 40-50 points to be made after that second fire was reported (because as i see now, even 2 fires was enough). and possibly the information would have saved a bunch of people on the board a bunch of dollars.

well good to see you again. i'll check back in a bit later to see if there's anything new here but i think this horse as been beaten pretty hard for now. time to start getting back to vacationing!
 
When you are aware of flaws you don't make conclusions from those flaws.

are you going to tell me you've never made an investment decision from a set of imperfect data? give me a break!

you have no other analysis to present. and you don't want to believe what i've posted, which fyi is now being affirmed directly by mit and at least indirectly by the nhtsa.

nothing else i can say except good luck.

- - - Updated - - -

They made the same mistake. There is no data on road debris.

road debris is a subset of collision. if you really have more doubts just call marty ahrens, the gal who writes all these nfpa reports.
 
Personal opinion: this thread has gotten painfully tedious. FCOL people, you're dissecting 2 freak events (and throwing in a drunk driving incident in a foreign country for good measure) then extrapolating all that to compare it to a high number situation in totally different circumstances. BTW, the accident in Mexico would only count if you included Mexican ICE statistics in all your calculations and how many Model "car-years" have been driven in Mexico? My gosh, it's probably less than 1 car-year and 1 fire occurred, that must mean there's a 100% chance of Model S catching fire in Mexico!

The only true and certain findings are:

1. The Model S battery pack can be pierced if hit by a heavy steell object with a force of ~25 tons.
2. A battery pack compromised by said metal object could catch fire after the fact.

If we say that the battery pack is the potential weak point of Model S, then we also have to admit that gasoline and oil are the potential weak point of all ICEs. In fact we have a whole thread on how dangerous gas stations are. You do yourselves a disservice trying to compare those 2 freak accidents to the thousands of fires in ICE cars which undoubtedly all had different causes and many of which caused lasting injuries or even death to the vehicle occupants.

[/rant]
 
are you going to tell me you've never made an investment decision from a set of imperfect data? give me a break!

Of course I have. But I try to know the quality of the data and use that as a factor.

you have no other analysis to present. and you don't want to believe what i've posted, which fyi is now being affirmed directly by mit and at least indirectly by the nhtsa.

It's not a question of what I believe, it's a question of evidence. Two people making the same mistake doesn't sway me in the least.
The evidence doesn't support the claim. What the evidence does show is a lower bound and an upper bound. It is possible that the ratio is governed by the lower bound, and also possible that it is governed by the upper bound. There is no way to know exactly where it falls.
An appropriate analysis is to present the full range and say "it is somewhere in here but I don't know where on this spectrum it lies".
If a value anywhere in that range is cause for alarm, then be alarmed, because it is possible. If a value anywhere in that range should cause you to make or change an investment decision, then determine the decisions you would make from all the values in the range. But then you still have to mitigate the fact that you dont know exactly which it is.

road debris is a subset of collision. if you really have more doubts just call marty ahrens, the gal who writes all these nfpa reports.

How road debris is defined academically is meaningless. How the people on the ground gather the data that goes into the study is what matters.
Even if the first responders were told to gather such data ( and I don't believe they are ) they have no method or inclination to do so.
They don't do it.
 
Apparently, folks don't read past the headlines all the often these days. Probably the most salient part of Bullis' article was this assessment:

That said, we’re only talking about three car fires—that could still be in the realm of bad luck. Based on the limited data, Musk probably isn’t justified in making a strong claim that the Model S is less likely to catch fire. It’s also probably too early to make the reverse claim—that the Model S is more likely to catch on fire–based on the numbers I give above.

O