# Statistical power

Statistical power

Statistical power is a probability of not rejecting of null hypotesis when in reality that hypotesis is false. It's concept deriving from statistical hypothesis testing. There are two primary types of errors when verifing binary hypotesis:

• type I error - it's so called false positive error. It happens when a test is positive, but in reality the hypotesis is false. For example, when antivirus recognizes file as healty, but in reality it's infected by virus.
• type II error - it's so called false negative error. It happens when a test is negative, but in reality the hypotesis is true. For example, when pregnancy test shows negative value, but in reality patient is pregnant.

Statistical power is equal to 1 - β, where β is probability of occurrence type II error. The greater statistical power in research means more reliable and thrustworthy results.

## Direct factors

Factors that has direct influence on statistical power are:

• sample size used in research study,
• size of bias,
• measurement errors - some tests are much easier to measure than the others. It's much easier determine result of flipping a coin than verify medical test result.

## Mammography problem

Statistical factors such as statistical power or significance threshold have major impact on many areas. One of the known problems is mammography problem. Let's assume that 0.8% of women that are examined by mammograms have breast cancer. Estimated value of statistical power during mammogram test is 90% (it's estimated value, because there is very hard to tell how many cancers are not detected during examination). On the other hand, about 7% of tests gives false positive result. We can ask question: if mammogram test have a positive result what is the probability of actual breast cancer? In group of 1000 women - based on initial assumptions - only 8 have breast cancer. Considering number of false positive results the mammogram test will be false positive in 7% cases which is 70 tests. In total during examination 1000 women mammogram test result will be 77 times positive (one case will not be detected due to statistical power). On 77 positive tests there is 7 actual breast cancers which gives only 9% effectiveness. Based on these studies many countries developed recommendations regarding minimal age of examined women. They started recommending mammography exam only for women which are older than 50. In that population risk of breast cancer is significantly higher that's why tests gives results that more reflects reality.

## Footnotes

1. Banerjee A., Chitnis U.B., Jadhav S.L., Bhawalkar J.S., Chadhury S., Hypothesis testing, type I and type II errors, 2009, p. 127 - 131
2. Banerjee A., Chitnis U.B., Jadhav S.L., Bhawalkar J.S., Chadhury S., Hypothesis testing, type I and type II errors, 2009, p. 127 - 131
3. Reinhart A., Statistics done wrong, 2015, p. 17
4. Reinhart A., Statistics done wrong, 2015, p. 42 - 43