Power of a test

The power of a test (against a specific alternative value) Is the probability that the test will reject the null hypothesis when the alternative is true In practice, we carry out the test in hope of showing that the null hypothesis is false, so high power is important

H0 True H0 False Reject Fail to reject Suppose H0 is false – what if we decide to reject it? Suppose H0 is false – what if we decide to fail to reject it? We correctly reject a false H0! H0 True H0 False Reject Fail to reject Suppose H0 is true – what if we decide to fail to reject it? Type I Correct a Power Suppose H0 is true – what if we decide to reject it? Correct Type II b

What are the hypotheses? A new flu vaccine claims to prevent a certain type of flu in 70% of the people who are vaccinated. A random sample of 100 people are vaccinated and then exposed to the flu. Is this claim too high? Use a = .05. What are the hypotheses? H0: p = .7 Ha: p < .7

A new flu vaccine claims to prevent a certain type of flu in 70% of the people who are vaccinated. A random sample of 100 people are vaccinated and then exposed to the flu. Is this claim too high? Use a = .05. Find mp and sp. mp = .7 sp = .0458

A new flu vaccine claims to prevent a certain type of flu in 70% of the people who are vaccinated. A random sample of 100 people are vaccinated and then exposed to the flu. Is this claim too high? Use a = .05. What is the probability of committing a Type I error? a = .05

So if we get p-hat=.625 or less, we would reject H0. H0: p = .7 Ha: p < .7 a = .05 For what values of the sample proportion would you reject the null hypothesis? .7 So if we get p-hat=.625 or less, we would reject H0. a = .05 p? Invnorm(.05,.7,.0458) =.625

Where did this number come from? H0: p = .7 Ha: p < .7 We reject H0 and decide that p<.7. Suppose that pa is 0.6. What is the probability of committing a Type II error? I selected a number that was less than .7 .7 .6 What is a type II error? How can we find this area? a = .05 Reject failing to reject H0 when the alternative is true What is the standard deviation of this curve? b = ? Normalcdf(.625,∞,.6,.04899) =.3049

What is the definition of power? Is power a conditional probability? What is the power of the test? What is the definition of power? Power - the probability that the test correctly rejects H0, if p = .6, is .6951 .7 .6 a = .05 b =.3049 Is power a conditional probability? The probability that the test correctly rejects H0 Power = ? Power = 1 - .3049= .6951

Suppose we select .55 as the alternative proportion (p). What is the probability of the type II error? b) What is the power of the test? What happened to the power of the test when the difference |p0 – pa| is increased? .7 .6 .55 a = .05 b = normalcdf(.625,∞, .55,.0497) = .0658 Power = 1 - .0658= .934

What happened to the power when the difference |p0-pa| is decreased? Suppose we select .65 as the alternative proportion (p). a) What is the probability of the type II error? b) What is the power of the test? What happened to the power when the difference |p0-pa| is decreased? .7 .6 .65 a = .05 b = normalcdf(.625,∞, .65,.0477) =.6999 Power b Power = 1 - .6999= .3001

Suppose that we change alpha to 10%. Using pa = .6, what would happen to the probability of a type II error and the power of the test? .7 a = .05 a = .1 Z=-1.282 p-hat=.641 b = .1998 Power = .8002 .6 The probability of the type II error (b) decreased and power increased, BUT the probability of a type I error also increased. Power b

What happens to a, b, & power when the sample size is increased? Fail to Reject H0 Reject H0 Power increases when n increases a p0 P(type II) decreases when n increases Power b pa

Fail to Reject H0 Reject H0 a p0 pa Power = 1 - b b

Recap: What affects the power of a test? As |p0 – pa| increases, power increases As a increases, power increases As n increases, power increases

Facts: The researcher is free to determine the value of a. The experimenter cannot control b, since it is dependent on the alternate value. The ideal situation is to have a as small as possible and power close to 1. (Power > .8) As a increases, power increases. (But also the chance of a type I error has increased!) Best way to increase power, without increasing a, is to increase the sample size

A new flu vaccine claims to prevent a certain type of flu in 70% of the people who are vaccinated. In a test, vaccinated people were exposed to the flu. Is this claim too high? Identify the decision: a) You decide that the proportion of vaccinated people who do not get the flu is less than 70% when it really is not. Type I Error

A new flu vaccine claims to prevent a certain type of flu in 70% of the people who are vaccinated. In a test, vaccinated people were exposed to the flu. Is this claim too high? Identify the decision: b) You decide that the proportion of vaccinated people who do not get the flu is less than 70% when it really is. Correct – Power!!