Calculating sample size. Alpha, Beta, Mu and Sigma – it's all Greek

Resident statisticians in a clinical department are often tasked with calculating the number of participants (sample size) required for randomised controlled trials. Since it is neither practical, feasible nor ethical [1, 2] to study the whole population, a number of participants must be selected from the population, which is fewer in number, but nevertheless adequately represents the population so that inferences can be drawn from the study results [2]. With the myriad of statistical software available nowadays, we no longer have to cast our minds back to the intricacies of the integration/ differentiation equations, nor strain our eyes looking at the tiny numbers on the p, z and t tables in order to calculate sample size. But in planning a clinical trial, there are several important factors we have to consider which no statistician or software can help us with. Let us first remind ourselves of the information required before sample size, or power calculations can be attempted. In most statistical software packages, the first variable you will be asked to give is the a, that is the value we will set as our significance cut off, more familiarly known as the p value. With the majority of disciplines, this is traditionally set at 0.05 [3], although sometimes 0.01 is used. In the jargon of hypothesis testing, a is set to address the type-1 error rate, type-1 error being the incorrect rejection of the null hypothesis. The second variable you will be asked to fill in, is the b, or ‘1 – the power’ of the study to detect a true difference. This is otherwise known as the type-2 error, or the failure to reject a false null hypothesis, and just as the a is usually set at 0.05, b is practically defaulted to be 0.2, which corresponds to a power of 80%. In other words, if a is set at 0.05, and b at 0.2, you have an 80% chance of obtaining results of p ≤ 0.05 if there is a difference between the interventions in your trial. Though most clinical research conventionally adhere to the use of 0.05 as the significance level, with a power of 80%, it is proposed here that decisions on a and b levels should be made on a case to case basis. While we want to limit the probability of finding false positives, we also want to limit the probability of missing a real effect. For example, it is not uncommon to detect heart murmurs at pre-operative assessment of patients due to undergo surgery. Most of the time, these murmurs are inconsequential but we often refer patients for echocardiography testing to exclude the possibility of less common but more serious valvular defects which allows us to identify higher risks of the anaesthetic and which may alter the anaesthetic management. Having decided the best balance between making a type-1 and type2 error, we have to consider the effect (of our intervention), and the magnitude of this effect. This is prior knowledge which we must have, or should be able to estimate, before we can plan a trial. Take for example, a trial of drug A, and its ability to reduce morphine consumption, after surgery. It has been shown that the average amount of morphine used after surgery is 70 mg, with a standard deviation (SD) of 13 mg. The third variable we would be expected to fill in the sample size calculation formula is l0, which is the known mean of the response variable in your population, this is followed by r, the SD of this variable in the general population. In this example, we would use 70 for the l0 and 13 for the r. Then comes the final, and perhaps the most important piece of the puzzle. The final variable you will be asked to fill in is l1, or the expected mean of this variable in your experimental population.