# Statistical Power

OK, so John Lennon didn’t really write this , but statistical power is a very abstract concept and the ability to “imagine” really helps.

**Power is the probability that you will find a statistically significant difference in your study SAMPLE if it truly exists in the larger POPULATION.**

**Beta is the probability that you will not be able to detect a difference if it is truly there in the population.**

## Hypothetical Dilemma

**The study you can’t afford:** There are 72,000,000 people with hypertension in the U.S. If you could study them all you would find that the new drug Lowpressure® lowers blood pressure by 7mm more than Whocaresapine.

To test the difference on an affordable scale, you need:

## Power Calculations Before the Study !!

We will describe the process in four simple steps.

Decide how big a difference you consider clinically important.

For our Hypothetical Dilemma Example: You think a 7mm difference or more is clinically important.

How variable is the outcome we are testing? (this is a guess, based on available facts)

**Fact: **The measure of variability used in power calculations is variance or (standard deviation)^{2}

**Hypothetical Dilemma:** From previous studies, we know that standard deviation of the mean blood pressure has been 5mmg Hg ( so variance for the calculation would be 25 (5)^{2})

**Fact:** The more variable the outcome, the more difficult it is to be statistically confident that the difference you observe is real and not due to random chance (or variation)

**Fact: ** the more variable the data , the more people you have to study to get statistical significance.

How sure do we need to be?

The usual beta is 0.20 (giving a power of 80%)

If you haven’t picked this up yet, One minus beta = Power.

The Dreaded Calculation

Because the concept is the important thing, we will spare you the headache of the power equation and just tell you that these assumptions yield a calculated N required of approximately 100 in each group.

**Hypothetical Dilemma Example**: You will need to study 200 people, randomized to the drug “Lowpressure” or the drug “Whocaresapine” to have an 80 percent power to detect a 7mm difference or more.

**However, studies often don’t enroll exactly the number of people they need so you may have to do ….**

## Power Calculations After the Study !!

Don’t despair, there are only three steps for this part.

If you found a statistically significant difference (p less than 0.05)…**You had enough POWER. **You don’t need power calculations. Really.

If you found a statistically non-significant difference (p greater than 0.05). There are two __main__ possibilities.

A. There really is no difference in the population

B. You didn’t have enough power. (Congratulations! You have succeeded in making a Type II error

Since you are the insatiably curious type, we can now calculate **The Power we had to detect the difference we said was significant.**

We will use:

- N: the number of people you actually enrolled
- Sigma: the
**measured**variance of the blood pressures in your study population (not**estimated**as before) - The difference you decided before the study was clinically important (7 mm Hg in this case)

The power you calculate from this is **the probability you had of detecting a clinically important difference if it is present in the larger population.**