mileage_man wrote:
It seems that most of Mike's defenders on this forum are mostly ignoring the arguments and trying to deflect attention away from the evidence rather than engaging with it. Honestly, that's probably their best bet. However, to try to keep things focused, I'm going to bring up a new perspective - statistics - on one of our earliest and most important pieces of evidence: the lack of photographs.
You all remember that Rojo created this spreadsheet to distribute the work of looking through the photos along the course:
https://goo.gl/krbo5s. Well, an army of letsrunners (among whom I'm proud to count myself) sifted through the photos of the 50 runners who finished in front of Mr. Rossi, and the 50 runners who finished after him, to determine at how many locations on the course each runner was seen. As you've surely heard, every one of these runners was seen at least three times on the course (at least two places besides the finish line), while Mr. Rossi was only seen once (only at the finish line).
This is already pretty damning by itself - out of the 100 runners nearest Mr. Rossi, every single one was photographed at at least two intermediate locations, while Mr. Rossi was not photographed at any. It is just common sense that this could not happen by chance.
But we don't need to rely on common sense, which after all can go astray. How strong is this evidence, when it really comes down to it? What are the chances that Mr. Rossi was coincidentally missed by all the photographers, assuming that he really did run the race?
Well, apparently about 1 in 11 thousand.
Let me preface this by saying that I am not an expert on statistics. I'm sure that there are better ways to do this. However, I am confident that what I've done makes sense, and that a more careful analysis would only make things worse for Mr. Rossi. (I hope that someone with more expertise takes it on!) Also, the precise value of this number shouldn't be taken too literally, since it would vary somewhat as we modify our statistical assumptions. (For example, in this case we are assuming that the number of locations follows a normal distribution - an assumption which I actually use further statistics to validate.)
So: I calculated the mean and standard deviation of the "number of different locations on the course where a given runner was seen" for the 100 runners closest to Mr. Rossi. The lowest number of locations was 3, the highest was 7, and the average was 4.66 locations. The standard deviation was 0.97 locations.
I then performed a z-test, which means I assumed the data were normally distributed, and then computed the probability that Mr. Rossi would only have been pictured at the finish line assuming that he actually ran the race. (See 'statistical analysis' tab of the spreadsheet for more details). The result was about 1 in 11,190.
Now, is the assumption that the data is normally distributed a valid one? Yes. I computed the Pearson's chi-squared test statistic from these data. This means that I counted how many runners had three locations, how many had four, etc, and compared that to how many should have had these numbers assuming that the distribution were normal. Then I computed the probability that the data would look as normal as it did, or less normal, assuming that it actually was normal. The answer is 80%, a strong indication that the data is normal, or least very close to it.
(More technically, I found that the pearson's chi-squared = 2.34, then integrated the probability density for the chi-squared distribution with 5 degrees of freedom starting at this value. Five degrees of freedom because there are seven frequencies that we are comparing, and because the model we are comparing two has two estimated parameters (mean and standard deviation). We obtain P(chi^2 > 2.34) = 0.8, from
https://www.fourmilab.ch/rpkp/experiments/analysis/chiCalc.html. To recap, this means there is an 80% chance that if the # of locations were normally distributed, we'd get a value of chi^2 at least as high as we did.)
So, in summary, that chance that Mike would simply get unlucky enough to miss all the pictures at intermediate locations is less than 1 in 10 thousand.
