0:16
power, specifically for the T-test, and because the T-test involves
maybe a little bit of math that we don't want to
get into, we'll talk about how you can do it
using Monte Carlo, which is a standard way to estimate power.
0:30
So, as it's name would suggest, power is a good thing.
You want more power.
so power is the probability of rejecting
the null hypothesis when, in fact, it's false.
So the opposite of power, 1 minus power is the type two error rate.
and that's the probability of failing to reject a null hypothesis when it's false.
And we usually label that as beta. So, consider our
previous example involving the Respiratory Disturbance Index.
Here, we are testing whether or not mu was
30 versus the alternative, then mu is specifically greater
than 30.Then the power the power is exactly the
probability there are t statistics lies in the rejection region.
Recall that we calculated the rejection region was if the
normalized mean X bar minus 30,the value under the null hypothesis
divided by the standard error of the mean, s over square root n.
If that normalized mean was greater than a t quantile specifically, t1
minus alpha, n minus 1 quantile where alpha is the desired type one error rate.
So if the statistic is larger than that
t quantile, then we reject the null hypothesis.
So then the power is the probability that we reject the null hypothesis.
In other
words, the probability that our test statistic is larger than that quantile
but, instead of calculated under the null hypothesis that mu was 30, calculated
under the assumption of the alternative hypothesis that mu is specifically greater
than 30. So, in the even that
mu a approaches 30,
then this probability approaches alpha, the type one error rate.
but on the other hand, if it is not
specifically 30, if it's actually a power calculation, so the
mu is not 30 then this calculation depends on
the specific value of mu a that we plug in.
So, to do a power calculation, you actually
have to know the value under the null alternative
that you want to plug in and then you can perform the calculation.
Brian Caffo, PhDProfessor, Biostatistics
