I think the bravo might be premature. In my understanding of sample-size bias in a statistic, it is simply this: the mean of the statistic in samples of a given size is not equal to the population (very large sample) value of the statistic. That is, the sense of magnitude you would get by looking at a lot of samples would be different from the true, population, or very-large-sample magnitude. Hence, for example, the sample standard deviation is biased low, because the mean of the SD of samples of a given size is less than the population SD. With SDs derived from small samples, you can actually get the impression that the SD is a bit too small, with really small sample sizes. Surely exactly the same thing applies to the Pearson correlation coefficient? It's not a question of what transformation you apply to it before you then consider whether the transformed value is biased. I can transform the SD by squaring it. The resulting variance is unbiased: the mean of a lot of small-sample variances is unbiased. When I back transform the mean of the variances, I am back to a biased statistic, but there is much less bias, because the sample size is much bigger. The bottom line is that the Pearson correlation coefficient, as observed in samples of a finite size, is biased low. Isn't that the end of the story? When SAS shows a "correlation estimate" that is less than the sample correlation, it is quite simply wrong. I submit that the authors of the papers that have been cited here have actually misunderstood what small-sample bias is all about. The original authors, Olkin & Pratt (1958), got it right.
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