No statistical evidence regarding amazon here.

Previous title: Less than .02366865 probability that Amazon does not censor 1-star reviews? This is stored here in lieu of deleting it totally because I don't like the idea of removing posts people have commented on, even posts I don't think have interest or value. [Further update: As I originally mentioned in the piece, the statistical calc was "for our amusement." I didn't take it very seriously and had no idea it would get this much attention. The (very appropriate) criticism emerged in discussions that at least some of the 10 Amazon reviews I was counting as the basis for the stats were undoubtedly the result of the original reddit attention to the Brian Powell issue, invalidating the calcs. I had explicitly noted that possibility in the original piece, but people didn't apparently didn't notice that very much and still took it seriously. As the discussion went on someone suggested that as few as 2 of the posts might have been legit. So I redid the calcs with 2, resulting in the .02366865 result in now the title: still statistically significant, and I did take it more seriously at that point. But now I'm trying to confirm that there actually were 2 "legit" reviews of the book in question on the day in question on Amazon. So far, one of the reviews I thought was likely to be independent has turned out to be in response to the reddit post. It's legit in the sense that the reviewer really did look at the book and write intelligently about it, but he did so in response to the reddit attention, which invalidates it from a statistical point of view. I have a couple more queries out there and if it does turn out that there are 2 independent reviews, I'll note it here. (If YOU wrote one on March 30 for The Final Theory, let me know.) For now, however, my feeling is that this was a not-very-serious and not-very-careful thing that got too much attention. Move on, there's nothing happening here.] [Update: Since this piece was posted there has been a fair amount of discussion about the calculations. I was really doing the most pessimistic possible calculation originally, since the actual number was more for my amusement than anything else, and I wasn't going to the trouble of actively screening out Amazon reviews that were very likely (or certainly) posted due to the attention this issue received on reddit. The probability was .0000000074279, and that number was in the title of this post. But since then there has been a lot of community input and I think the calculation should be taken more seriously. The most optimistic (from Amazon's point of view) version of the calculation still arrives at a probability of .02366864 -- which is still statistically significant. Accordingly, I've changed the title of the piece, but other than that and the updates in bold below, the piece is the same. Partly because I just don't want to spend more time on it, and partly due to a possibly misguided sense of "journalistic integrity". I may at some point right ANOTHER piece covering the same material and link to it from here. Also, if you are seriously interested in this, I strongly recommend that you read the discussion in the comments to get an idea of how seriously to take this result.] This morning I noticed this story by Brian Powell on reddit in which he contends that Amazon rejects one-star reviews. Brian wrote a one-star review for a popular science book. The book, called The Final Theory, apparently claims claims that most physics prior its publication was wrong, and that it advances the first overall theory of physics to be correct. As a grad student in physics, Brian unsurprisingly took exception to many of the book's assertions. His review was rejected repeatedly for different reasons. Finally he gave up in frustration, although at one point it was up on Amazon for two hours before being removed. After reading the reviewer's story I decided to investigate for myself. As of this morning (March 31, 2006) there are a total of 62 reviews. A total of 18 of them gave one star; most of the rest gave five. The question is: is Amazon censoring the negative reviews, as Brian contends? On the surface it looks like there is a mix of different reviews (albeit oddly bipolar). First of all, it seems odd that the overall review Amazon gives the book is four stars given that 18 of the reviews give one-star. I added up the reviews and found there were 18 one's, 5 two's, 0 3's, 4 fours, and 35 fives. The average review according to the simple calculation any elementary school student would do turns out to be 3.3 stars. I understand that Amazon has to round the number because they show the review as a graphic, but they do round to half-stars, so why are they rounding 3.3 up to 4? Something seems fishy here. But let's look a little deeper. Every one of the 10 reviews posted on March 30 (yesterday) gave one star. Hmm. That seems a trifle odd given that there are only 8 previous one star reviews since July 22, 2003 when the first (five-star) review was posted, among the other 52 previous reviews. I don't believe it takes a background in statistics to get a hint that those numbers don't add up. But for our amusement, let's look at it from a statistical point of view. Brian says one of his reviews was visible for two hours, and then removed. So apparently there can be a brief period when a one-star review is visible, before it is removed. Of course, that's a policy that would lead to their being a tendency for one star-reviews to be unusually prevalent among the very newest reviews (in this case, it's 100% of the reviews since yesterday morning). Another possibility is that Amazon is responding to the high visibility of Brian's post on reddit by suddenly letting the bad reviews through. But let's assume that Amazon would claim nothing is wrong: there is no underlying pattern that might lead to a higher propensity of one-star reviews since yesterday. That would imply that the any special preponderance of one-star reviews since yesterday would be a mere coincidence. So let's assume it is, and see where that takes us. More specifically, let's test the assumption that whatever factors might result in one-star reviews being published for a book like this one are not different in the period since yesterday morning than they were in the previous period. Any propensity for recent one-star reviews would therefore be mere innocent coincidence. In the previous period, we had a total of 8 one-star reviews, out of 52 reviews: a proportion of 0.15384615. What is the probability that our assumption is true? Using the assumption that underlying probability of .15384615, we can easily calculate the probability that all 10 out of 10 reviews since yesterday might be one-star. It's the same calculation as calculating the probability that tossing a coin 10 times in a row would be heads. As most people know, the way to do that is to multiply .5*.5*.5*.5*.5*.5*.5*.5*.5*.5. This is also denoted with the expression .5^10 (".5 to the 10th power"). In our case, we need to calculate .15384615^10. The answer is .0000000074279. In other words, there is a .00000074279% chance that the unusual preponderance of one-star reviews is due to coincidence. Note: this is not a rigorous calculation. It's an estimate. For instance, one might point out from the earlier 52 reviews, we can only get an estimate of the proportion of one-star reviews that Amazon would usually let through for a book like this. The upshot is that our computed probability is also an estimate. That's OK. It is standard in statistics to reject an assumption when the probability of its being true is .05. .0000000074279 is a very long way from .05. In addition, there may be other explanations to the propensity of recent one-star reviews. For instance, they could all be due to the reddit posting. However, reading the reviews, this does not appear to be the case. Most of them give ample evidence of direct knowledge of the book and thoughtful criticism. [UPDATE: This is in response to the comments I'm getting that I ignored the possibility mentioned in the paragraph above. I didn't ignore it -- I explicitly discussed it. Right there. Above. See it? About two inches above this sentence? I also said above that the calc is "for our amusement". It is NOT rigorous. The thing that I'll be interested to see is how many of those reviews remain over time. The hypothesis is that they are like Brian's review that was up for two hours before it was taken down. Will these reviews STAY up? I'll post something here when we know.] [FURTHER UPDATE: Someone over at reddit notes that there are now 11 neg reviews, and guesses that between 3 and 9 of the one-star reviews are due to the reddit posting. As the most pessimistic case assume that there are only 2 legit ones. That gives a result of .15384615^2 or 0.02366864; still below the usual statistical limit of .05 that leads the rejection an assumption (null hypothesis). So, as a statistical test, the conclusion stands even under most pessimistic of the guesstimates that appear in the the harshest numerical criticism I've received. However, I do now wish I had counted the suspicious ones instead of forming a rough impression before doing my initial calculation. It's a nuisance that people are focusing on the extreme number I calculated, when it really isn't necessary to be that extreme for deriving the statistically valid conclusion, and it IS probable that at least a couple of those reviews were due to reddit. I'd change the calc now, but there are already a lot of links out there so I think I should just leave it. This isn't a wiki such that it's supposed to be improved under community input; it's a post and is therefore more subject to journalistic standards.] One other thing should be mentioned. Brian's story quotes Amazon's responses to his reviews. On the surface, they sound like they could reflect a legitimate policy. But as Brian points out, the standards Amazon expresses there don't seem to be enforced for positive reviews. But even if they are, any policy which has the effect of screening out negative reviews more than positive ones, even if it is consistently applied, must be considered very, very suspect. On a personal note, I have made a lot of use of Amazon's reviews. I have thought that Amazon's reviews are one of the great Internet resources. But recently I've occasionally felt suspicion about them. It's seemed that there were too many five-star reviews for items that haven't seemed that good, and the summary star rating has sometimes seemed higher than my informal guesstimate of the average of the published ratings. This is the first time I've done a real analysis. Unfortunately it seems to confirm my suspicions: unless I'm missing something, Amazon now has policies in place that slant their reviews toward the positive. It's not hard to figure out why they might want to make the items they sell look better than they are! But I am very disappointed because I have always thought that Amazon reviews had integrity; I thought that they had realized that by having integrity, they earn the trust of buyers, and can use that trust to build their business. It always seemed to me that that was the smart approach, as well as the Right Thing, and it seemed to be what they were doing. But the fact is that Amazon is now in such a dominant position that their dominance probably wouldn't be threatened just because they lost some integrity. But, losing some integrity could increase revenues, as long as that dominant position remained. So it is arguable that their underlying financial pressures are now different than they were in earlier times, when they were still rapidly growing. And in fact, it appears that they may be responding to their new situation by having a new policy toward reviews. I very much hope that that is not true, or that if it is, that it's a temporary thing that Jeff Bezos somehow isn't aware of and that will be fixed. Right now, however, I'm feeling pessimistic. I notice that Brian says that Barnes & Noble did accept his review. I'll bet you can guess where I'm going to go next time I am considering purchasing a book.

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