Question

Explain how number of subjects, effect-size, and statistical power are related.

- Imagine there is a true effect, state how increasing the number of subjects can influence statistical power?

- If the number of subjects are held constant, state how an increase in the size of the true effect influences statistical power.

- Explain how the type II error rate is different from the type I error rate.

- State how increasing statistical power influences the type II error rate.

Answer 1

Part 3: -

In statistical hypothesis testing, a type I error is the rejection of a true null hypothesis (also known as a "false positive" finding), while a type II error is failing to reject a false null hypothesis (also known as a "false negative" finding). More simply stated, a type I error is to falsely infer the existence of something that is not there, while a type II error is to falsely infer the absence of something that is present.

When a test of significance is performed, there are four possible conclusions: -

There is a type I error when H(0) is true, but it is rejected. So, a type I error is the unjustified rejection of the null hypothesis. This error is indicated by the sigma sign α. A type II error occurs when H(0) is not true, but it is not rejected. This means the null hypothesis is unjustly accepted as being correct. This error is indicated by the beta sign β. You are right when H(0) is true and you did not reject it. This can be calculated by means of the formula 1 – α. You are also right when H(0) is not true and you rejected it. This can be calculated by means of the formula 1 – β. This is the statistical power of the test. This statistical power is the rightful rejection of the null hypothesis. You want the value of the statistical power to be as large as possible. When you, for example, find a significant outcome in your research and are rejecting the null hypotheses, you also want to do this rightfully. (To calculate the power, you first calculate the value of mean (X bar) under the wrong null hypothesis. You do this using the z-score. So you look at when you would reject this null hypothesis under the wrong null hypothesis. When you know the value of mean (X bar), you calculate the right probability of rejecting H(0), using the z-score by including the correct value of the average).

Part 4: -

The power of any test of statistical significance is defined as the probability that it will reject a false null hypothesis. Statistical power is inversely related to beta or the probability of making a Type II error. In short, power = 1 – β.

Statistical power is the likelihood that a study will detect an effect when there is an effect there to be detected. If statistical power is high, the probability of making a Type II error, or concluding there is no effect when, in fact, there is one, goes down. Statistical power is affected chiefly by the size of the effect and the size of the sample used to detect it. Bigger effects are easier to detect than smaller effects, while large samples offer greater test sensitivity than small samples.