[Slartie]

Statistics works exactly like this. What doesn't work is saying "Okay, we have one death in 3 million miles, that extrapolates to 33 deaths in 100 million miles", because it implies a silent addition of "with nearly 100% certainty", which is the part that's wrong here.

But the poster did something different. He took it one level further and attempted to calculate this confidence number for different spans in which the actual "deaths per 100 million miles" number of Uber's current cars would fall into, given an ideal world (from a data perspective) in which they would have driven an infinite amount of miles. But he actually did it the other way round - he modified the confidence variable and calculated the spans, and then he adjusted the confidence until he arrived at a span that would put Uber's cars just on par with human driving in the best case.

The fact that a fatal incident happens that early (at 3 million, and not closer or past the 86 million that a statistical human drives on average until a fatal incident occurs) does not allow us to extrapolate a sound number per 100 million miles, but it tells us something about the probability by which the actual number of fatalities by 100 million miles that we'd get if Uber continued testing just like it did and racked up enough miles (and killed people) for a statistically sound calculation will fall into different margins. Sure, Uber could have been just very, very unlucky - but that's pretty unlikely, and the unlikeliness of Uber's bad luck (and conversely the likeliness of the fact that Uber's tech is just systematically deadly) is precisely what can be calculated with this single incident.

[nmca]

The statement "with 95% confidence" is a classic misinterpretation of what a CI is, the assumption of Poisson is dubious but there's no obvious plausible alternative. Overall seems reasonable.

[pgreenwood]

Hello! I'd be interested to hear what you think the correct interpretation of these CIs are in this case. Failing that can you explain what is wrong with saying something like "with xx% confidence we can conclude that the rate is within these bounds" is?

The assumption of using Poisson seems pretty solid to me, given we are talking about x events in some continuum (miles traveled in this case), but always happy to hear any cogent objections.

[muraiki]

The Poisson distribution assumes equal probability of events occurring. That seems to me to be an oversimplification, given that AV performance varies over time as changes are made, and also given that terrain / environment plays a huge factor here, whether looking at one particular vehicle or comparing to vehicles across companies (and drivers in general). Since AV performance will hopefully be improved when an accident occurs, we also cannot meet the assumption of independence between events. Although if AVs are simply temporarily stopped after an accident, that also breaks the independence assumption as we'd have a time period of zero accidents.

The bigger problem though is what you are doing with your confidence interval. A CI is a statement about replication. A 95% confidence level means that in 100 replications of the experiment using similar data, 5 of the generated CIs -- which will all have different endpoints -- will _not_ contain the population parameter, although IIRC this math is more complicated in practice, meaning that the error rate is actually higher. As such, if you generate a CI and multiply the endpoints by some constant, that's a complete violation of what is being expressed: there is vastly more data with 100m driving miles than 3m miles, which will cause the CI to shrink and the estimate of the parameter to become more accurate. There is absolutely no basis for multiplying the endpoints of a CI!

Ultimately, given that the size of the sample has an effect on CI width, you need to conduct an appropriate statistical test to compare the estimated parameters between the 1 in 3m deaths for Uber and whatever data generated the 1.18 in 100m deaths for sober drivers. There's a lot more that needs to be taken into account here than what a simple Poisson test can do.

For an analysis of how AVs with various safety levels perform in terms of lives saved over time, I recommend https://www.rand.org/blog/articles/2017/11/why-waiting-for-p...

Edit: Note the default values of the T and r parameters when you run poisson.test(1, conf.level = 0.95), and also that the p-value of the one-sample exact test you performed is 1. Also, since this is an exact test, the rate of rejecting true null hypotheses at 0.95 is 0.05, but given my reservations about the use of a Poisson distribution here, I don't think that using an exact Poisson test is appropriate.

[muraiki]

To be more clear, when you run poisson.test(1, conf.level = 0.95) with the default values of T and r (which are both 1) you are performing the following two-sided hypothesis test:

Null hypothesis: The true rate of events is 1 (r) with a time base of 1 (T).

Alternative hypothesis: The true rate is not equal to 1.

The reason that you end up with a p-value of 1 is because you've said that you've observed 1 event in a time base of 1 with a hypothesized rate of 1. So given this data, of course the probability of observing a rate equal to or more extreme than 1 is 1! As such, you're not actually testing anything about the data that you claim you are testing.

I'm not trying to be harsh here, but please be careful when using statistics!

[pgreenwood]

Ok I re-ran setting T properly for both cases. The results were similar:

> poisson.test(c(1, 11800), c(3, 1000000), alternative = c("two.sided"),conf.level = .93)

Comparison of Poisson rates

  data:  c(1, 11800) time base: c(3, 1e+06)
  count1 = 1, expected count1 = 0.035403, p-value = 0.03478
  alternative hypothesis: true rate ratio is not equal to 1
  93 percent confidence interval:
     1.006334 146.142032
  sample estimates:
  rate ratio 
    28.24859

The lower bound of the CI approaches a rate ratio = 1 for a 93% confidence interval.

Interestingly, if you multiply the CI I claimed before by the rate ratio instead of the expected rate, you get almost exactly the same CI as here.

  > ci <- c(0.03562718, 5.17251332)
  > 28.24859 * ci
  [1]   1.006418 146.116208
 

* Note 11800 is about two years of pedestrian deaths and time units are in millions of miles. https://crashstats.nhtsa.dot.gov/Api/Public/ViewPublication/...

[pgreenwood]

Facinating, thank you. Particularly the part about multiplying the CI. I wonder if the analysis could be resuced to some extent? I feel there must be a way to use the information we have do draw some conclusions, at least relative to some explicit assumptions.