

REVIEW ARTICLE 

Year : 2020  Volume
: 6
 Issue : 2  Page : 123131 

Basic principles of sample size calculation
Shivam Pandey
Department of Biostatistics, All India Institute of Medical Sciences, New Delhi, India
Date of Submission  27Jun2020 
Date of Decision  03Jul2020 
Date of Acceptance  25Jul2020 
Date of Web Publication  27Aug2020 
Correspondence Address: Dr. Shivam Pandey Department of Biostatistics, All India Institute of Medical Sciences, New Delhi India
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/jpcs.jpcs_34_20
The calculation of sample size helps a medical researcher to assess cost, time, and feasibility of his project besides scientific justification and validity. Although frequently reported in journals, the details or the elements of sample size calculation are not consistently provided by the authors. Sample size calculations reported do not match with replication of sample size in many studies. Most trials with negative results do not have a large enough sample size. Hence, reporting of sample size and power needs to be improved. The sample size calculation can be guided by previous literature, pilot studies, and past clinical experiences. The collaborative effort of the researcher and the statistician is required at this stage. Estimated sample size is our best guess. Issues such as anticipated loss to followup, large subgroup analysis, and complicated study designs, demand a large sample size to ensure power throughout the trial. The present article will help the reader understand the importance of pilot study in sample size estimation, second understand the relationship between primary objective and sample size of a study, third understand the essential components required in a sample size estimation, and fourth calculate sample sizes using reallife examples using an online software.
Keywords: Loss to followup, sample size, trials
How to cite this article: Pandey S. Basic principles of sample size calculation. J Pract Cardiovasc Sci 2020;6:12331 
Introduction   
The calculation of sample size helps a medical researcher to assess cost, time, and feasibility of his project besides scientific justification and validity.^{[1],[2],[3],[4]} Although frequently reported in journals, the details or the elements of sample size calculation are not consistently provided by the authors. The sample size calculations reported do not match with replication of sample size in many studies.^{[5]} Most trials with negative results do not have a large enough sample size. Hence, reporting of sample size and power needs to be improved.^{[6],[7]} There is a belief that studies with small sample sizes are unethical if they do not ensure adequate power. However, the truth is for a study to be ethical in its design, its predicted value must outweigh the projected risks to the participants. In studies, where the risks and inconvenience borne by participants outweigh the benefits received as a result of participation, it is viewed as a burden. A study may still be valid if the projected benefit to the society outweighs the burden to the society. If there is no burden, then any sample size may be ideal.^{[8]} Many different approaches of sample size design exist depending on the study design and research question. Moreover, each study design can have multiple subdesigns resulting in different sample size calculation.^{[9]} Addressing a sample size is a practical issue that has to be solved during planning and designing stage of the study. It may be an important issue in approval or rejection of clinical trial results irrespective of the efficacy.^{[10]} By the end of this article, the reader will be able to understand the importance of pilot study in sample size estimation, secondly understand the relationship between primary objective and sample size of a study, thirdly understand the essential components required in a sample size estimation and fourth calculate sample sizes using reallife examples using an online software.
Importance of Pilot Study in Sample Size Estimation   
In published literature, relevant data for calculating the sample size can be estimated from prevalence estimates or event rates, standard deviation (SD) of the continuous outcome, sample size of similar studies with similar outcomes. The idea of approximate effect estimates can be obtained by reviewing metaanalysis and clinically meaningful effect. Small pilot study, personal experience, expert opinion, educated guess, hospital registers, unpublished reports support the researcher when we have insufficient information in the existing/available literature. A pilot study not only helps in the estimation of sample size but its primary purpose is also to check the feasibility of the study.
The pilot study is a small scale trial run as a pretest, and it tries out for the proposed major trial. It allows preliminary testing of the hypotheses, and may suggest some change, such as dropping some part or developing new hypotheses so that it can be tested more precisely.^{[11]} These studies can address many issues such as checking that instructions are comprehensive, and the investigators are adequately skilled for the trial. The pilot study almost always provides enough data for the researcher to decide whether to go ahead with the main study or to abandon. Many research ideas that seem to show great promise are unproductive when actually carried out. From the findings of the pilot study, the researcher may abandon the main study involving large logistic resources, and thus can save a lot time and money.^{[11]}
Prerequisites for Sample Size Estimation   
At the outset, primary objectives (descriptive/analytical) and primary outcome measure (mean/proportion/rates) should be defined. Often there is a primary research question that the researcher wants to investigate. It is important to choose a primary outcome and lock that for the study. The minimum difference the investigator wants to detect between the groups makes the effect size for the sample size calculation.^{[10]} Hence, if the researcher changes the planned outcome after the start of the study, the reported P value and inference becomes invalid.^{[12]} The level of acceptable Type 1 error (α) and Type II error (β) should also be determined. The error rate of Type 1 error (alpha) is customarily set lower than Type II error (beta). The philosophy behind this is the impact of a false positive error (Type 1) is more detrimental than that of falsenegative (Type II) error. Hence, they are protected against more rigidly. Besides, the researcher needs to know the control arm mean/event rates/proportion, and the smallest clinically important effect that one is trying to detect.
The Relation between Primary Objective and the Sample Size   
The type of primary outcome measure with its clear definition helps in computing the correct sample size as there are definite ways to reach sample size for each outcome measure. It needs special attention as it principally influences how impressively the research question is answered. The type of primary outcome measure also is the basis for mode of estimation regarding population variance. For continuous variable (e.g., mean arterial pressure), population SD is incorporated in the formula whereas the SD needs to be worked out from the proportion of outcomes for binary variables (e.g., hypertensionyes/no). In literature, there can be several outcomes for each study design. It is the responsibility of the researcher to find out the primary outcome of the study. Mostly, the sample size is estimated based on the primary outcome. It is possible to estimate sample size taking into consideration all outcome measures, both primary and secondary at the cost of much larger sample size.
Essential Components of Sample Size Estimation   
The sample size for any study depends on certain factors such as the acceptable level of significance (P value), power (1β) of the study, expected “clinically relevant” effect size, underlying event rate in the population, etc.^{[10]} Primarily, three factors P value (depends on α), power (related with β), and the effect size (clinically relevant assumption) govern an appropriate sample size.^{[12],[13],[14],[15],[16],[17]} The “effect size” means the magnitude of clinically relevant effect under the alternative hypothesis. It quantifies the difference in the outcomes between the study and control groups. It refers to the smallest difference that would be clinical importance. Ideally, the basis of selection of effect size should be on clinical judgment. It varies with different clinical trials. The researcher has to determine this effect size with scientific knowledge and wisdom. Previous publications on the related topic might be useful in this regard. “Minimal clinically important difference is the smallest difference that would be worth testing.” Sample size varies inversely with effect size. The ideal study to make a researcher happy is one where power of the study is high, or in other words, the study has high chance of making a conclusion with reasonable confidence, be it accepting or rejecting null hypothesis.^{[12]} A sample size matrix includes different values of sample sizes using varying dimensions of alpha, power (1β) and effect size. It is often more useful for the research team to choose the sample size that fits conveniently with the need of the researcher [Table 1].  Table 1: The matrix showing changes of sample size with varying values of alpha. Power (1β), and effect size
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Formulae and Software   
Once these three factors are fixed, there are many ways (formulae, nomogram, tables, and software) for estimating the optimum sample size. At present, there are a good number of softwares available in the internet. It is sufficient to be familiar with instructions of any software to get sample size for one arm of the study. Perhaps the most important step is to check with the most appropriate formula to get the correct sample size. Websites of some of the commonly used softwares are summarized in [Table 2].^{[5],[9]} Interested readers may access all relevant sample size estimation formulae using the relevant links.
Calculating Sample Size by Using Epitools: an Online Software   
Below are some examples to calculate sample size by using Epitools (Sergeant ESG, 2018. Epitools Epidemiological Calculators. Ausvet. Available at (http://epitools.ausvet.com.au), a freely available online application for calculating sample size [Figure 1]. Values of level of significance and power for different scenarios are given in [Table 3].
In the options under the tab “Epidemiological studies,” choose “Sample size calculations.”
Next we will perform some sample size calculations using examples.
Calculating the Sample Size for Comparing Two Means   
Example 1: A randomized trial was conducted by the nutrition department to see whether a dietary supplement given to pregnant women will reduce the systolic blood pressure (SBP) level during the time of delivery. One group received the null supplement whereas the other group received the normal diet. From a pilot study, the SD of SBP was 40 mm/hg and is expected to be the same for both the groups. What is the required sample size if we expect a difference of 20 mm/hg SBP at 5% level of significance with 90% power?
We select the option “Two means with equal sample sizes and equal variances” under “Epidemiological studies” [Figure 2]a.
Enter the value 70 under mean in population 1. Enter the value 50 under mean in population 2. This accounts for a difference of 20. Enter the variance as 1600 (40 × 40) [Figure 2]b. Leave the confidence level as 0.95 and the enter the desired power as 0.9. Select the option of 2 tailed under “use one tailed or two tailed.” Press “submit” [Figure 2]c.
Ans: We require 85 individuals per group.
If we make the power as 80% by changing the desired power to 0.8 in [Figure 2]b, the sample size reduces to 63 per group [Figure 2]d and [Figure 2]e.
Example 2. A placebocontrolled randomized trial proposes to assess the effectiveness of Drug A in preventing the stress response to laryngoscopy. A previous study showed that there is an average rise of 20 mmhg in SBP during laryngoscopy (with SD of 15 mmhg), alpha = 0.05, power = 90% [Figure 3]a.
We select the option “Two means with equal sample sizes and equal variances” under “Epidemiological studies” [Figure 2]a.
Enter the value 70 under mean in population 1. Enter the value 50 under mean in population 2. This accounts for a difference of 20. Enter the variance as 225 (15 × 15) [Figure 2]f. Leave the confidence level as 0.95 and the enter the desired power as 0.9. Select the option of 2 tailed under “use one tailed or two tailed.” Press “submit” [Figure 2]g.
Ans: We require 12 individuals per group.
If we change the power to 80% by changing the desired power to 0.8 in [Figure 2]f, the sample size reduces to 9 per group [Figure 2]h and [Figure 2]i.
Example 3: Chanmicket al. studied the effects of the dual endothelinreceptor antagonist basentan in patients with pulmonary hypertension in a randomized controlled trial.^{[18]} They proposed to detect a mean difference of 50 m in the 6 min walk test given a common SD of 50 m between the two groups at 80% power and 5% level of significance, using a one sided test [Figure 4]a.
We select the option “Two means with equal sample sizes and equal variances” under “Epidemiological studies” [Figure 2]a as in Examples 1 and 2. Enter the value 100 under mean in population 1. Enter the value 50 under mean in population 2. This accounts for a difference of 50. Enter the variance as 2500 (50 × 50) [Figure 2]j. Leave the confidence level as 0.95 and the enter the desired power as 0.8. Select the option of 2 tailed under “use one tailed or two tailed.” Press “submit” [Figure 2]k.
Ans: We require 16 individuals per group.
If we change the power to 90% by changing the desired power to 0.9, the sample size increases to 22 per group [Figure 2]k and [Figure 2]l.
Example 4. Suppose a researcher wants to see the effects of a potential antihypertensive drug and he wants to compare the new drug with placebo. Researcher thinks that if this new drug reduces blood pressure by 10 mmhg as compared to placebo then it should be considered as clinically significant. Let us assume SD found in previous studies was 25 mmhg, considering significance level of 5% and 80% power, what is the required sample size?
We select the option “Two means with equal sample sizes and equal variances” under “Epidemiological studies” [Figure 2]a as in the previous examples. Enter the value 100 under mean in population 1. Enter the value 90 under mean in population 2. This accounts for a difference of 10. Enter the variance as 625 (25 × 25) [Figure 2]m. Leave the confidence level as 0.95 and the enter the desired power as 0.8. Select the option of 2 tailed under “use one tailed or two tailed.” Press “submit” [Figure 2]n.
We require 99 individuals in each group.
If we increase the power to 90% by changing the desired power to 0.9 in [Figure 2]m, the sample size increases to 132 per group [Figure 2]o and [Figure 2]p.
Calculating the Sample Size by Comparing Two Proportions   
Below are some examples to calculate sample size for comparing two proportions using Epitools.
Example 1: A study is planned to check the effectiveness to dexmedtomidine in preventing postoperative shivering. From previous studies, it was found that incidence of postoperative shivering is 60%. Reduction in the incidence of shivering up to 20% will be significant. Calculate sample size at 5% level significance and 95% power.
We select the option “Two proportions” under “Epidemiological studies” [Figure 3]a. Enter the value 0.6 under proportion in population 1. Enter the value 0.2 under proportion in population 2. Leave the ratio of the sample sizes as 1 [Figure 3]b. Leave the confidence level as 0.95 and the enter the desired power as 0.95. Select the option of 2 tailed under “use one tailed or two tailed.” Press “submit” [Figure 3]c.
Ans: We require 41 individuals per group.
If we change the power to 80% by changing the desired power to 0.8, the sample size reduces to 28 per group [Figure 3]d and [Figure 3]e.
Example 2: In the bypass, venous angioplasty in severe ischemia of the leg study on bypass venous angioplasty for leg ischemia. The statistical calculations were based at 5% level of significance with 90% power on the 3year survival value of 50% in the angioplasty and 65% in the bypass group.
We select the option “Two proportions” under “Epidemiological studies” [Figure 3]a as in the previous example. Enter the value 0.5 under proportion in population 1. Enter the value 0.65 under proportion in population 2. Leave the ratio of the sample sizes as 1 [Figure 3]f. Leave the confidence level as 0.95 and the enter the desired power as 0.90. Select the option of 2 tailed under “use one tailed or two tailed.” Press “submit” [Figure 3]g.
Ans: We require 240 individuals per groupWe require 240 individuals per group.
If we change the power to 80% by changing the desired power as 0.8, the sample size reduces to 183 per group [Figure 3]h and [Figure 3]i.
Example 3: Suppose the researcher is interested in knowing protective effect of a drug on mortality in patients of myocardial infarction (MI). Two groups of MI patients were selected. One was given that drug other was given placebo. Difference in the proportion of the event was 0.1. In the above example, let us assume that in a previous study says that 20% of patients of MI die within a specified time. If the drug being tested increases survival to 30% then this finding can be considered clinically significant, alpha = 0.05, power = 80%. What is the sample size?
We select the option “Two proportions” under “Epidemiological studies” [Figure 3]a as in the previous examples. Enter the value 0.2 under proportion in population 1. Enter the value 0.3 under proportion in population 2. Leave the ratio of the sample sizes as 1 [Figure 3]j. Leave the confidence level as 0.95 and the enter the desired power as 0.80. Select the option of 2 tailed under “use one tailed or two tailed.” Press “submit” [Figure 3]k.
Ans: We require 313 individuals per group.
If we change the power to 90% by changing the desired power to 0.9, the sample size increases to 412 per group [Figure 3]l and [Figure 3]m.
Summary
Sample size calculation can be guided by previous literature, pilot studies, and past clinical experiences. The collaborative effort of the researcher and the statistician is required at this stage. Estimated sample size is our best guess. Issues such as anticipated loss to followup, large subgroup analysis and complicated study designs, demands a large sample size to ensure power throughout the trial. A change in sample size is proportional to variance (square of SD) and inversely proportional to the detected difference.
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Conflicts of interest
There are no conflicts of interest.
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[Figure 1], [Figure 2], [Figure 3], [Figure 4]
[Table 1], [Table 2], [Table 3]
