## Learning Outcomes

By the end of this section, students will be able to:

- Explain the key concept of power and what impacts it
- Estimate the power of a given study
- Estimate the sample size needed to test hypotheses in different study designs

You can download a copy of the slides here: A2.3 Power calculations for a difference in proportions

**Video A2.3 Power Calculation for Two Proportions (10 minutes)**

## A2.3 PRACTICAL: R

**Power calculations for two proportions**

Here is an example:

Estimate the sample size needed to compare the proportion of people who smoke in two populations. From previous work, you think that 10% of the people in population A smoke, and that an absolute increase of 5% in population B (compared to population A) would be clinically significant. You want 90% power, and a 5% significance level.

In this scenario we use the â€˜pwr.2p.testâ€™ command in the power package.

### alpha = sig.level option and is equal to 0.05

### power = 0.80

### p1 = 0.10

### p2 = 0.15

power4<-pwr.2p.test(h=ES.h(p1=0.1, p2=0.15), sig.level=0.05, power=0.9)

With this command, you can specify â€˜h=â€™ for an effect size, or you can ask R to compute an effect size for two propotions with the â€˜ES.h(p1, p2)â€™ option, as we did here.

> power4<-pwr.2p.test(h=ES.h(p1=0.1, p2=0.15), sig.level=0.05, power=0.9)

> power4

Difference of proportion power calculation for binomial distribution (arcsine transformation)

h = 0.1518977

n = 910.8011

sig.level = 0.05

power = 0.9

alternative = two.sided

NOTE: same sample sizes

You estimate that you need 911 participants from each population, with a total sample of around 1,821. If you wanted different sample sizes in each group, you would use the command â€˜pwr.2p2n.testâ€™ instead.

If we type â€˜plot(power4)â€™ we can see how the power level changes with varying sample sizes:

> plot(power4)

**Question A2_3: **Unfortunately, the funding body has informed you, you only have enough resources to recruit a fixed number of people. Can you estimate the power of a study if you only had 500 people in total (with even numbers in each group)? (hint: type ?pwr.2p.test if you need help setting up the command)

**Answer**

> power5<-pwr.2p.test(h=ES.h(p1=0.1, p2=0.15), n=250, sig.level=0.05)

> power5

Difference of proportion power calculation for binomial distribution (arcsine transformation)

Â Â Â Â Â Â Â Â Â h = 0.1518977

n = 250

sig.level = 0.05

power = 0.396905

alternative = two.sided

NOTE: same sample sizes

In this scenario, the power of the study would be only 0.40.Â Most people would regard such a study as under-powered as there is only a 40% chance that the effect will be detected if one truly exists.

## A2.3 PRACTICAL: Stata

**Power calculations for two proportions**

Here is an example:

Estimate the sample size needed to compare the proportion of people who smoke in two populations. From previous work, you think that 10% of the people in population A smoke, and that an absolute increase of 5% in population B (compared to population A) would be clinically significant. You want 90% power, and a 5% significance level.

The command and output is as follows:

power twoproportions 0.1, alpha(0.05) power(0.9) diff(0.05)

*– Estimated sample sizes:

Â Â Â Â Â Â Â Â Â Â Â N =Â Â Â Â Â 1836

Â N per group =Â Â Â Â Â Â 918

*– Estimated sample size: 1836 (two groups of 918 each).

You estimate that you need 1836 participants overall, 918 from each population.

* Question A2.3: *Unfortunately, the funding body has informed you, you only have enough resources to recruit a fixed number of people. Can you estimate the power of a study if you only had 500 people in total?

**Answer**

power twoproportions 0.1, alpha(0.05) diff(0.05) n(500)

*– Estimated power:

Â Â Â Â Â Â Â power =Â Â Â 0.3935

In this scenario, the power of the study would be only 0.39.Â Most people would regard such a study as under-powered as there is only a 39% chance that the effect will be detected if one truly exists.

## A2.3 PRACTICAL: SPSS

__Power Calculations for Proportions__

Estimate the sample size needed to compare the proportion of people who smoke in two populations. From previous work, you think that 10% of the people in population A smoke, and that an absolute increase of 5% in population B (compared to population A) would be clinically significant. You want 90% power, and a 5% significance level.

Select

Analyze >> Power Analysis >> Proportions >> Independent Samples Binomial Test

Then input your data into each of the boxes in the Power Analysis window as in the previous practical. Remember that all percentages are expressed as decimals, for 90% is 0.9, 10% is 0.1 etc. Then press OK to run the test.

**Unfortunately, the funding body has informed you, you only have enough resources to recruit a fixed number of people. Can you estimate the power of a study if you only had 500 people in total?**

**Answer**

In the first part of the question you estimate that you need 1836 participants overall, 918 from each population.

In the second scenario, the power of the study would be only 0.39.Â Most people would regard such a study as under-powered as there is only a 39% chance that the effect will be detected if one truly exists.