Hi, my name is Brian Caffo. And this is the lecture on Power. Power is the probability of rejecting the null hypothesis, when in fact the null hypothesis is false. So, like its name would suggest, power is a good thing you want more power. And interestingly, power comes more into play, when you fail to reject an hypothesis than when you reject, and let me try to explain. So, imagine if you were to conduct a study, that has low power. For example, you want to compare treatment A versus treatment B. And you've only randomized three people who received Treatment A, and only three people who received Treatment B. Then when you make the comparison to see if Treatment A results in a better outcome than Treatment B, you would not be surprised to get a null result, because there were only three people in each group. You had very little power, to detect a meaningful difference between the groups. So, your null result was kind of expected. Or kind of what you would think might happen just due to chance alone. On the other hand, if you had 300 people in treatment A and 300 people in treatment A. And you failed to reject, then that would be meaningful, because you would expect to see a difference, since you collected so much data. And that' s why, power actually tends to come more into play for null results, than it does for non null results. And the way the power gets used most often, as at the time of designing the study. You want to design the study, so that you have a reasonable chance of detecting the alternative hypothesis, if the alternative hypothesis is true. So, let's also cover a couple more details. So a type II error is a bad, a bad thing, as its name will suggest, an error. Is failing to reject a null hypothesis, when in fact a null hypothesis is false. So, the probability of a type II error is usually called beta, and Power is just 1 minus beta. So, the two most meaningful quantities for hypothesis testing, in terms of the two error rates for hypothesis testing are alpha, the type I error rate, and beta the type II error rate. But, we tend not to talk so much beta instead, we tend to talk about 1 minus beta, this concept of power. So let's go through a conceptual example, and then we'll go through some specific numerical examples. So, remember we were interested in talking about the hypothesis test for the mean respiratory disturbance index, RDI, in a particular population of obese subjects. So, we wanted to test whether mu was 30, versus the alternative that mu was greater than 30. We did a t statistic, which is, in this case, our sample mean. Minus the hypothesised value 30 divided by the standard error of the mean. That statistic, we're assuming, follows a t distribution under the null hypothesis. So, if we calculate the probability that, that statistic is bigger than the upper one minus alpha quantile of the t distribution. That probability will be alpha, if we calculate under the null hypothesis when mu is 30. So in this case then under, I would write that mu a in this case equals 30. Now power, is simply the same calculation, but instead of plugging in mu a equal to 30. For the probability that we're calculating, we plug in mu a for some value greater than 30. That winds up being our power. Now imagine if I were to plug in a mu a of say 60, then this would be a large number. And that makes sense, right, because if we want to detect whether or not the mean for this population is 30 or not, and it's really 60. Then we should have a lot of power to detect that, because when we collect our data there'll be values around 60 instead of values around 30, okay? Now, imagine instead if our true alternative mean was 30.00001. Then we wouldn't have a lot of power to detect that, because it's quite close to the null value. We're going to have lower power. So, power, this power, is a function that depends on the mean under the null hypothesis. If we plug in values close to 30, it's going to look a lot like our type I error rate. And if we plug in values way away from 30, it should get bigger and bigger, to the point where it's nearly 100% if it's far enough away from 30.
