[MUSIC] So what about the power for samples of size or 100? Supposing we wanted to be sure, or they have some idea of how likely it was that we would reject hardly, wonder if the true f was 0.2. A very high number where we would hope to have high power, so now the non-centrality is sample size 100 times the square of the parametric value, 0.2, the non-centrality errors 4. Now, the probability of rejecting data with that test is 1 minus the pchi squared of the value that will cause rejection, so we remember we're going to reject if the test. Statistics are bigger than 3.84, so will put that in as a value 3.84, degrees of freedom is still 1, the non-centrality parameter is 4, so that quantity the probability of rejecting is 0.52. So not especially bigger, little bit about 0.5, but it's about as likely as not that we will reject Hardy Weinberg if the true f value is 0.2, when we have only 100 individuals. With 1000 individuals, the non-centrality parameter goes up to 40 and the power goes up to much larger, much closer to 1, looks like 0.999994, so it's very likely will reject for 1000 individuals. So in practice in our modern studies, we have millions, if not hundreds of millions of SNPs, and we do a great many Hardy Weinberg tests. Generally, just as a quality control measure, other data accurate, do they truly represent the Genotypes at that SNPs? So, the Hardy Weinberg rejection in the first place may indicate a problem with the data. If we continue to adopt the 5% significance level, we're going to be rejecting falsely, false rejection rate, rejecting Hardy Weinberg when it's true 5% of the time. And that's fine, that's the price we pay for conducting the test to maintain a reasonable power of rejecting when we should and not rejecting when we shouldn't. There's a tradeoff between significance level, typically 5% for a single tests, and power, hopefully in the 80 to 90% range. 5% significance in a million tests, however, will give us 50,000, false rejections and that's a lot of SNPs, that's for tremendous loss of data that we might not be willing to take. The simple minded approaches to say, well, let's maintain the 5% significance level, but have it apply overall to the whole experiment, to the whole set of a million SNPs. We want to have some false rejections 5% of the time, an individual test will be assessed with the 5% divided by the million. Then the target significance level divided by the number of tests, so that gives rise to what's become to be called the genome wide significance level. 5% divided by a typical number of a million, 5 times 10 to the -8, it's arbitrary, but as a convenience measure that people can use them and they use that phrase, we have some understanding of what they mean. Another approach which might be better than the Bonferroni correction that dividing 5% by a million, which is very conservative is to construct what we call a QQ plot. If all the tests were testing a true hypothesis, if all the tests were in Hardy Weinberg equilibrium, the p values we get, of course after p between 0 and 1, because they're probabilities. Moreover, they are spread out uniformly on that interval, so doing a million tests, we should expect to get a million p values spread out, more or less uniformly over the range 0 to 1. So what we do in practice is to take our many significance levels, many p values and all of them, smaller to largest and compare that set with the set of ordered numbers uniformly over the range 0 to 1. So each of those uniform points is obtained by dividing closely, dividing 1 by the number of tests, we have 100 tests, we've got various 0.01, 0.02, 0.03 and so forth. A plot of the ordered observed p values against the expected ordered p values is called a QQ plot, and for convenience, we transform the p by taking minus log to base 10. So a p of value of 0.01 correspond to number 1, number 5 correspond to change to the -5.
