Sample size

To conduct a research, researcher needs to select sample out of the target population to which he wants to generalize his/her findings. Once samples are studied, findings will be extrapolated to the population of interest.

Sample size is nothing but the number of participants in a study. It can be people, objects or any other units which are being studied.

The estimation of sample size for a study is a crucial step and any negligence in its estimation may lead to rejection of an efficacious drug and approval of an ineffective drug.

Importance of correct sample size

It is always better to take large sample size as it can be more representative of the population. Large sample size can also limit the influence of extremes.

Let us understand this with an example. Suppose, researcher is interested in studying a population of a society where half of the population are males and half of the population are females. Now, researcher needs to choose the sample out of the target population. So, for fair result, researcher should study half males and half females. Researcher sets the sample size 3. With sample size 3, there are chances of picking up 4 possible combinations.

From the above example, it is clear that small sample size may elicit extreme combinations (in the above example, extreme combinations are all females or all males as participant in the study) which may not be the true representative of the population. Taking a large sample size can minimize the risk of accidentally selecting “extreme” sample combinations.

Thus, a study with a small sample size may produce statistically insignificant results (Statistical significance is a measure of meaningfulness of research findings) which may be inconclusive. A study conducted with small sample size may fail to answer the research question appropriately. They may also fail to detect the associations/effects in correct manner.

On the other hand, a study with a too larger sample size may waste resources and may expose more of participants which is not required.

Hence, the selection of appropriate sample size for a study is a critical step. The estimation of appropriate sample size is required for the following:

To achieve the desired accuracy

To do analysis properly

To allow validity of significance test

Components of Sample size calculation

Sample size calculation is one of the important steps required for designing a study. It is based on prior information of certain elements.

In a Randomized Controlled Trial, hypothesis assumptions are created in the initial stage. The assumptions are as follows:

Null hypothesis - No difference exists between the intervention and control groups in terms of the primary outcome of interest.

Alternative hypothesis - Difference exists between the intervention and control groups in terms of the primary outcome of interest.

Thus, null hypothesis and alternative hypothesis are assumptions (statements) which represent the differences, association or effects that are present in the population. Study sample helps the researcher to find out which assumption (the null hypothesis or alternative hypothesis) is most likely to occur.

As discussed above that in clinical research, treatment effects are usually studied in a study sample instead of in the whole population. This may lead to generation of some errors. Two fundamental errors are type I and type II errors. The values of these type I and type II errors are important components in sample size calculations. Furthermore, knowledge of the results expected in a study can also be helpful in calculating the sample size.

Determination of sample size is the mathematical calculation of the number of the participants to be included in the study. The fundamental elements required for sample size calculation are as follows:

Type 1 error

In this type of error, the probability is that the study does find a difference between the intervention and control groups where no true difference exists i.e., the chance of a false-positive conclusion. The probability of making a type I error is denoted by “alpha”.

The probability of committing a type I error is equal to the level of significance that has been set for the hypothesis testing. Level of significance is the measurement of the statistical significance (significance in statistics is “probably true”) i.e., fixed probability of wrong rejection of null hypothesis when it is actually true. Therefore, if the level of significance is 0.05, there is a 5% chance of type I error to occur. The sample size increases as the level of significance decreases.

Type II error

In this type of error, the probability is that the study will not find a difference when a true difference between the control and intervention arms exists i.e., the chance of false-negative conclusion. The probability of making a type II error is denoted by “beta”.

The probability of committing a type II error is equal to the power of the study. The power of study can be increased by increasing the sample size, which decreases the risk of committing a type II error.

Statistical Power

There are chances that researcher draws a false-negative i.e., type II error. Statistical power is the probability that a statistical analysis will be able to identify and reject the false null hypotheses and will not make a type II error. Sample size is directly proportional to the power of the study. With large sample size, study will have greater power to detect significant difference, effects or associations. The power is the complement of beta, i.e. 1 – beta.

Conventionally, beta is set at a level of 0.20, which means that a researcher desires a less than 20% chance of a false-negative conclusion.

Hence, the power is 0.80 or 80% when beta is 0.20. Knowledge of the association between sample size and power is an important factor for interpretation of conclusions elicited from the study.

Minimal Clinically Relevant Difference (Effect size)

The minimal clinically relevant difference is the smallest effect between the studied groups that the researcher wishes to identify. It is the difference that the researcher believes to be clinically relevant and biologically plausible.

For continuous outcome variable, the minimal clinically relevant difference is a numerical difference. For instance, if weight of the body is the outcome of a study, a researcher can choose a difference of 3 kg as the minimal clinically relevant difference. For a trial with a binary outcome, such as the effect of a drug on the development of a particular disability (yes/no), researcher should estimate a relevant difference between the event rates in both treatment groups. For example, he/she could choose a difference of 10% between the treatment group and the control group as minimal clinically relevant difference.

The sample size is inversely proportional to the square of the difference. Hence a small change in the expected difference with the treatment has a major effect on the estimated sample size.

Variability

The sample size calculation is also based on the population variance of the given outcome variable. If variability of the outcome variable is large, larger sample size is required to assess whether an observed effect is a true effect. Generally, more precise inference can be drawn on a population parameter when the sample is taken from a homogeneous sample.

For a continuous outcome variable, the variability is estimated by means of the standard deviation (SD). The variance is usually unknown. Systematic review of published documents and pilot study can be useful in projecting population variance.

Prior calculation of sample size can be helpful in minimizing the risk of an under-powered (false-negative) results. There are different approaches for the calculation of sample size for different study designs and different outcome measures. Sample size can be calculated by different means such as mathematical formulae, tables and computer calculators.