## 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: A2.2 Power calculations for a difference in means

**Video A2.2 Power Calculation for a Difference in Means (8 minutes)**

## A2.2 PRACTICAL: R

**Power calculations for two means**

Here is an example:

Estimate the sample size needed to compare the mean systolic blood pressure (SBP) in two populations. From a pilot study, you think that the group with lower blood pressure will have a mean SBP of 120 mm Hg, and the standard deviation (SD) of both groups will be 15 mm Hg. You have decided that you are interested in a minimum difference of 5 mm Hg, and you want 90% power, and a 5% significance level.

In R, we need to calculate two statistics to estimate sample size: delta (i.e. the expected difference between groups) and sigma (variance, which in this case is the pooled standard deviation). Once we have delta and sigma, we can calculate the effect size we expect to see, which is Cohenâ€™s d. We can guess delta and sigma from looking at past studies or by running a pilot study. Cohenâ€™s d is estimated by dividing the delta by the sigma.

d <- 5/15

d

[1] 0.3333333

Then we can use the â€˜pwr.t.testâ€™ command (from the power package) to assess the sample size needed to detect this effect size.

*### d = Cohen’s d
### power = 0.9
### alpha = 0.05*

power1<-pwr.t.test(d=d, power=0.9, sig.level =0.05 )

power1

Two-sample t test power calculation

Â Â Â Â Â Â Â Â Â n = 190.0991

d = 0.3333333

sig.level = 0.05

power = 0.9

alternative = two.sided

NOTE: n is number in *each* group

You need approximately 190 participants in each group, and 380 participants overall.

If we want to estimate the power of a given sample size, we omit the â€˜powerâ€™ option, and instead use the â€˜n=â€™ option:

>power2<-pwr.t.test(n=190, d=d, sig.level =0.05 )

> power2

Two-sample t test power calculation

Â Â Â Â Â Â Â Â Â n = 190

d = 0.3333333

sig.level = 0.05

power = 0.8998509

alternative = two.sided

NOTE: n is number in *each* group

We can see here that recruiting 190 participants in each blood pressure group would enable our study to have 90% power.

**Question A2_2: **Using the same study outlined above, how much power would we have ended up with in our study if we only managed to recruit 150 participants in each group, but the variance of our study sample was smaller than what we anticipated (so SD=12)?

**Answer**

We first need to recalculate our Cohenâ€™s d (effect size):

d2 <- 5/12

d2

[1] 0.4166667

power3<-pwr.t.test(n=150, d=d2, sig.level =0.05 )

>power3

Two-sample t test power calculation

Â Â Â Â Â Â Â Â Â n = 150

d = 0.4166667

sig.level = 0.05

power = 0.9491662

alternative = two.sided

NOTE: n is number in *each* group

We recruited fewer participants, which would decrease our power, but since our variance was lower our power actually increased overall to 95%.

## A2.2 PRACTICAL: Stata

**Power calculations for two means**

Here is an example:

Estimate the sample size needed to compare the mean systolic blood pressure (SBP) in two populations. From a pilot study, you think that the group with lower blood pressure will have a mean SBP of 120 mm Hg, and the standard deviation (SD) of both groups will be 15 mm Hg. You have decided that you are interested in a minimum difference of 5 mm Hg, and you want 90% power, and a 5% significance level.

The command and output is as follows:

power twomeans 120, power(0.9) alpha(0.05) diff(5) sd(15)

*– Estimated sample sizes:

Â Â Â Â Â Â Â Â Â Â Â N =Â Â Â Â Â Â 382

Â N per group =Â Â Â Â Â Â 191

*– Estimated sample size: 382 (191 per group).

You need approximately 382 participants overall.

If we want to estimate the power of a given sample size, we omit the â€˜powerâ€™ option, and instead use the â€˜n( )â€™ option:

power twomeans 120, alpha(0.05) diff(5) sd(15) n(382)

* Question A2.2: *Using the same study outlined above, how much power would we have ended up with in our study if we only managed to recruit 300 participants in total, but the variance of our study sample was smaller than what we anticipated (so SD=12)?

**Answer**

power twomeans 120, alpha(0.05) diff(5) sd(12) n(300)

We recruited fewer participants, which would decrease our power, but since our variance was lower our power actually increased overall to 95%.

## A2.2 PRACTICAL: SPSS

__Power calculations for two means__

Here we want to estimate the sample size needed to compare the mean systolic blood pressure (SBP) in two populations. From a pilot study, you think that the group with lower blood pressure will have a mean SBP of 120 mm Hg, and the standard deviation (SD) of both groups will be 15 mm Hg. You have decided that you are interested in a minimum difference of 5 mm Hg, and you want 90% power, and a 5% significance level.

Select

Analyze >> Power Analysis >> Means >> Independent Samples T Test

In the Power Analysis window, you need to enter the following:

- Estimate: Sample size (because this is what we want to calculate)
- Single power value: 0.9 (we are looking for 90% power, and this is shown as a decimal)
- Population mean difference: 5 (the difference we are looking for)
- Population standard deviation: 15 (use the equal for two groups option, as we would not expect the groups to differ from each other.
- Significance level (Î±): 0.05

If we want to estimate the power of a given sample size, we open the power analysis window in the same way, but select power from the drop-down menu at the top instead of sample size. Input the sample size for each of your two groups at the top of the box, then input all the rest of the values as before. Then press OK to run the test.

**Using the same study outlined above, how much power would we have ended up with in our study if we only managed to recruit 300 participants in total, but the variance of our study sample was smaller than what we anticipated (so SD=12)?**

**Answer**

For the first part of the question the output table would look like this.

Estimated sample size is 382 (191 per group), so you need approximately 382 participants overall for the study to have your desired power of 90%

For the second part of the question the output table would like like this.

We recruited fewer participants, which would decrease our power, but since our variance was lower our power actually increased overall to 95%.