## What is Power and How Do We Calculate Sample Size?

One of the favourite sections of the Statistics for Non-Statisticians Course is the section on Sample Size and Power. If you've never done sample sizing yourself before or would just like a little more insight into how it is done then read on.

In a protocol you usually see a sample size/power statement that is often scanned over with minimal interest apart from the final value that states how many subjects are needed in the trial. If we look at this in more detail the components are fairly straight forward. These are Assumptions, Power, Significance, Withdrawals/Non-evaluable subjects and the final number being how many subjects. Let's pick these up one by one and explain what they all mean.

### Assumptions:

In order to perform sample size calculations the first thing that is required is a set of assumptions. The major assumption is what the actual 'real underlying' results are, i.e. what the outcomes to the treatments would be if an entire population was included in the trial. For simple binary endpoint data (e.g. response/no response type) this would be written like the following, e.g. The expected response rate for Treatment A is 65% and the expected response rate for Treatment B is 40%.

The assumptions (in our case 65% and 40%) would come from previous trials and publications. These are usually easier to find for the comparator product than the test product.

### Significance Level:

This is usually set to be 5%. In simple terms it is the predefined value that the p-value at the end of the trial would need to be lower than to declare a significant result. We'll discuss p-values in a future Blog.

### Power:

If the assumptions are correct (in our case the real response rates are 65% and 40%) we want the trial to have a good chance of giving us a positive result (i.e. statistical significance). The percentage chance of the trial giving us a significant result if our assumptions are correct is called POWER. Most trials have either 80% or 90% power.

If a trial has 80% power then, if the assumptions are correct, this still has a 20% (or 1 in 5 chance) of failure. This 20% is referred to as type II error. If a trial has 90% power then the chance of failure is 1 in 10. Therefore moving from 80% power to 90% power does not sound a big change but in effect it is halving the chance of failure (from 20% to 10%).

At this point it is worth noting that the power that is stated in the protocol is only true if the assumptions are correct. If the assumptions are not correct the power will be different as demonstrated later in this Blog.

### Calculations

Once you have the assumptions, significance level and desired power level the next thing to do is to calculate the sample size. This requires a sample size calculator. These calculators are reasonably straightforward, the only complicating factor is that there are several of them. The reason that there are different ones is that each data type and each comparison type require a different formula (and hence calculator). In our example we are looking at detecting a difference between two groups for a binary outcome variable so we would use the appropriate calculator. The link below will take you to this:

http://www.pharmaschool.co/size5.asp

In our example we would include 65% in box (a), 40% in box (b), 5% significance level in box (c), and for 80% power we would put 80 in box (d). If we set the withdrawal rate to be 0 and click Calculate Sample Size this gives us 59 a sample size requirement of 59 subjects per group and a total of 118 subjects.

### Withdrawal/Non-evaluable Rate

In most trials there will be a number of subjects who do not complete the trial or for some reason are classed as non-evaluable. The initial sample size that is calculated is the number of subjects who are required for the final analysis, therefore this needs to be increased to allow for subjects who are excluded from the analysis. This can be done with the sample size calculator by adding a % of withdrawals into box (e).

### What if the initial assumptions are incorrect?

The stated power will be incorrect, this maybe higher or lower. Let us look at an example:

With the assumption of response rates of 65% vs 40% our trial required 118 subjects.

If the actual response rates were, say, 55% vs 40% (so the test drug is less effective than assumed) then for 80% power our trial would require 170 subjects per group, 340 in total. This is nearly 3 times more than initially assumed. (Replace the 65 in box (a) with 55 to calculate this)

If the trial had gone ahead with 118 subjects and the real response rates were 55% and 40% the trial would not have had the stated 80% power and instead would have had 38% power. This equates to a more than 6 in 10 chance of failure.

### Conclusion

The power stated in the protocol is dependent on the assumptions. Any change from the assumptions will affect the power. A power of 80% means that even if the assumptions hold there is still a 1 in 5 chance that the trial will fail to give a positive result. If the assumptions do not hold then it is likely that the chance of failure will increase as the power will be reduced and therefore when looking at 'failed' trial results it is always a very sensible approach to go back and look at the power of the trial, the assumptions that were made, the withdrawal rate and see whether the results are similar to these or vastly different. Trials fail sometimes through bad design and lack of proper sample sizing, not because the drug is ineffective.

I've simplified this by using binary data for illustrative purposes. There are additional assumptions required when using continuous data or time to event data that I can discuss in future blogs if the interest is there.

For more sample size calculators visit:  http://www.pharmaschool.co/size.asp

Please feel free to post thoughts and opinions to this Blog and suggest future topics.

Adrian M Parrott BSc MSc MBA

Adrian is the PharmaSchool Subject Expert for Statistics and delivers the "Statistics for Non-Statisticians" Course. The Course has been delivered across the World for a wide variety of Pharma, Healthcare, CRO, Medical Communications and Academic Organisations in locations ranging from UK, US, Europe, India, China, UAE, South Africa, Thailand.