Introduction to power in significance tests

- [Instructor] What we are going to do in this video is talk about the idea of power when we are dealing with significance tests. And power is an idea that you might encounter in a first year statistics course. It's turns out that it's fairly difficult to calculate, but it's interesting to know what it means and what are the levers that might increase the power or decrease the power in a significance test. So just to cut to the chase, power is a probability. You can view it as the probability that you are doing the right thing when the null hypothesis is not true, and the right thing is you should reject the null hypothesis if it's not true. So it's a probability of rejecting, rejecting your null hypothesis given that the null hypothesis is false. So we can view it as a conditional probability like that. But there's other ways to conceptualize it. We can connect it to Type II errors. For example, you could say this is equal to one minus the probability of not rejecting, one minus the probability of not rejecting, not rejecting the null hypothesis given that the null hypothesis is false. And this thing that I just described, not rejecting the null hypothesis given that the null hypothesis is false, this is, that's the definition of a Type II error. So you could view it as just the probability of not making a Type II error, or one minus the probability of making a Type II error. Hopefully that's not confusing. So let me just write it the other way. So you could say it's the probability of not making a Type II error. So what are the things that would actually drive power? And to help us conceptualize that, I'll draw two sampling distributions, one, if we assume that the null hypothesis is true, and one where we assume that the null hypothesis is false, and the true population parameter is something different than the null hypothesis is saying. So for example, let's say that we have a null hypothesis that our population mean is equal to, let's just call it, mu one, and we have an alternative hypothesis, so H sub a, that says, "Hey, no. "The population mean is not equal to mu one." So if you assumed a world where the null hypothesis is true, so I'll do that in blue, so if we assume the null hypothesis is true, what would be our sampling distribution? Remember, what we do in significance tests is we have some form of a population. Let me draw that. You have a population right over here. And our hypotheses are making some statement about a parameter in that population. And to test it, we take a sample of a certain size. We calculate a statistic, in this case, it would be the sample mean, and we say if we assume that our null hypothesis is true, what is the probability of getting that sample statistic? And if that's below a threshold, which we call a significance level, we reject the null hypothesis. And so that world that we have been living in, one way to think about it. In a world where you assume the null hypothesis is true, you might have a sampling distribution that looks something like this. If the null hypothesis is true, then the center of your sampling distribution would be right over here at mu one, and given your sample size, you would get a certain sampling distribution for the sample means. If your sample size increases, this will be narrower. If it decreases, this thing is going to be wider. And you set a significance level, which is essentially your probability of rejecting the null hypothesis even if it is true. You could even view it as, and we've talked about it, you can view your significance level as the probability of making a Type I error. So your significance level is some area. And so let's say it's this area that I'm shading in orange right over here. That would be your significance level. So if you took a sample right over here and you calculated its sample mean and you happened to fall in this area, this area, or this area right over here, then you would reject your null hypothesis. Now if the null hypothesis actually was true, you would be committing a Type I error without knowing about it. But for power, we are concerned with a Type II error. So in this one, it's a conditional probability that our null hypothesis is false. And so let's construct another sampling distribution in the case where our null hypothesis is false. So let me just continue this line right over here, and I'll do that. So let's imagine a world where our null hypothesis is false and it's actually the case that our mean is mu two, and let's say that mu two is right over here, and in this reality, our sampling distribution might look something like this. Once again, it will be for a given sample size. The larger the sample size, the narrower this bell curve would be. And so it might look something like this. So in which situation, so in this world, we should be rejecting the null hypothesis. But what are the samples in which case we are not rejecting the null hypothesis even though we should? Well, we're not going to reject the null hypothesis if we get samples in, if we get a sample here, or a sample here, or a sample here, a sample, where if you assume the null hypothesis is true, the probability isn't that unlikely. And so the probability of making a Type II error when we should reject the null hypothesis but we don't is actually this area right over here. And the power, the probability of rejecting the null hypothesis given that it's false, so given that it's false would be this red distribution, that would be the rest of this area right over here. So how can we increase the power? Well, one way is to increase our alpha, increase our significance level. If we increase our significance level, say from that, well, the significance level is an area. So if we want it to go up, if we increase the area, and it looks something like that, now by expanding that significance area, we have increased the power because now this yellow area is larger. We've pushed this boundary to the left of it. Now you might say, "Oh, well, hey. "If we want to increase the power, "power sounds like a good thing, "why don't we just always increase alpha?" Well, the problem with that is if you increase alpha, so let me write this down, so if you take alpha, your significance level, and you increase it, that will increase the power. That will increase the power, but it's also going to increase your probability of a Type I error, because remember, that's one way to conceptualize what alpha is, what your significance level is. It's a probability of a Type I error. Now what are other ways to increase your power? Well, if you increase your sample size, then both of these distributions will, these sampling distributions are going to get narrower. And so if these sampling distributions, if both of these sampling distributions get narrower, then that situation where you are not rejecting your null hypothesis even though you should is going to have a lot less area. There's gonna be, one way to think about it, there's going to be a lot less overlap between these two sampling distributions. And so let me write that down. So another way is to, if you increase n, your sample size, that's going to increase your power. And this in general is always a good thing if you can do it. Now other things that may or may not be under your control is well, the less variability there is in the dataset, that would also make the sampling distributions narrower, and that would also increase the power. So less variability and you could measure that as by variance or standard deviation of your underlying dataset, that would increase your power. Another thing that would increase the power is if the true parameter is further away than what the null hypothesis is saying. So if you say true parameter, far from null hypothesis what it's saying, that also will increase the power. So these two are not typically under your control but the sample size is and the significance level is. Significance level, there's a trade-off though. If you increase the power through that, you're also increasing the probability of a Type I error. So for a lot of researchers, they might say, "Hey, if a Type II error is worse, "I'm willing to make this trade-off. "I'll increase the significance level. "But if a Type I error is actually what I'm afraid of, "then I wouldn't want top use this lever." But in any case, increasing your sample size, if you can do it, is going to be a good thing.