Sample Size

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Atlanta (www.openepi.com). Epi Info, a free software for statistical analysis and power calculation has been developed by the US Centers for Disease Control ...
Chapter 11

Sample Size Shivali Suri

INTRODUCTION A question that investigators, epidemiologists and clinicians often ask themselves when they plan to conduct a study is regarding the determination of appropriate sample size. Sample size is the number of individuals or patients or other experimental units that should be included so as to answer the research question successfully. Sample size determination must be planned carefully owing to the fact that the reliability of the results and outcome of research is dependent on it.1 Choosing the right sample size is necessary because if it is too small, it will not answer the research question leading to inaccurate results. Studies with insufficient sample size may produce unreliable answers to important research questions. Additionally, it would be unethical to recruit patients or individuals into a study that fails to deliver meaningful information.4 On the other hand, studies with large samples often lead to wastage of time, money and other precious resources.2

INFORMATION REQUIRED FOR SAMPLE SIZE DETERMINATION Prior to sample size calculations, it is essential to have knowledge regarding certain vital concepts and components on which this number is dependent and this has been explained below.3,4 Research is usually conducted in a study sample instead of the whole population; and therefore two fundamental errors are likely to arise, which are termed as type I and type II errors; and determining their values is important in sample size calculations. Type I error or ‘a’ or level of significance is the probability of detecting a treatment effect when no effect exists, that is, rejecting null hypothesis

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when actually it is true (false positive result) and is also known as ‘p value’. The type I error is decided by the investigator before the initiation of study and is most commonly expressed at 5% (or 1%) which means that s/he is prepared to accept a 5% (or 1%) chance of falsely reporting a significant effect. Confidence level is the complement of Type I error, i.e. 1–α, and is the probability that an estimate of a population parameter is within certain decided limits of true value. Type II error or ‘β’ is the probability of failing to reject the null hypothesis when in reality it is false. It occurs when in reality the two treatments are different but we fail to find the difference in our sample. A related value, 1-β is often described as power and is the probability of correctly rejecting the null hypothesis when it is not true. Power is the ability to detect a true difference in outcome. It is usually chosen to be 80%, which means that it accepts 20% likelihood or chances of missing a real difference. The effect size is the biologically or clinically significant difference between two treatments. It is determined from review of literature, discussion with experts or through a pilot study. This is also known as minimum detectable relative risk or odds ratio or treatment effect. Another important determinant of sample size is the precision. It measures how close an estimate (+ or – 5%/10%) is to the true value of a population parameter. Sample size calculation is dependent on the population variance (for continuous outcome variable, it is expressed as standard deviation) of the outcome variable. Greater the variability of outcome, larger the sample size would be required to assess an observed effect. The variance is mostly unknown, and hence researchers use an estimate obtained from pilot study or already performed study. A brief overview of the important steps for determining the study sample required in conducting a successful research have been described below.5,6 1. Objective of the study: The primary objective of the study is the central factor in selecting the sample size and hence should be specified very clearly. 2. Design of study: The study design (case control/experimental/crosssectional study) makes the most important difference in choosing the sample size formula and must be decided as a priority.

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3. Study variables: Delineate the major study variables, for instance, for a study on prevalence of hypertension in the rural areas, the presence or absence of hypertension is the main variable. 4. Type of estimate: Identify the type of estimate to be calculated for the study. For example, it has to be defined if one is going to study mean or ratio or percentage or proportion. 5. Expected frequency of factor of interest: Indicate the proportion of the study variable, because, if one plans to study a rare disease, one needs to have a large sample. 6. Desired precision of estimate: The researcher needs to decide beforehand if s/he wants the estimate to be within 5 percent or 10 percent on either side. 7. Acceptable risk: It must be explained prior to the study, how much of risk is the researcher willing to accept? One should specify how much of the estimate (for instance, + or - 10%) would fall outside its real population value. 8. Adjustment for population size: It must be specified if the researcher would be taking the sample from a very large population or a small one since sample size formulae ordinarily assume that the sample is extracted from a very large population. If the population is not large, an adjustment factor needs to be incorporated. 9. Design effect: For a clustered study design, one would be required to multiply the sample size by a factor called design effect so that it takes care of correlation within the subjects in a cluster. The value of the design effect may be obtained from review of literature. When it is not available, a value between 1.5 and 2.0 may be used; and after the study is completed, investigators should evaluate the actual design effect. 10. Adjustment for expected response rate/attrition: An estimation of the attrition or nonresponse rate suspected in a study is vital in deciding the sample size. For example, if one expects a nonresponse of 10%, one should add 10% to the final sample size. 11. Confounders: When we look for association between two variables, there could be a third factor which could be affecting this association known as confounder. One should increase the sample size by 10 percent for every confounder variable that one can think of. 12. Nonstatistical considerations: The availability of funds, time, ethical issues, similar research being done elsewhere, number of patients

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(in rare diseases), and novelty of the research topic may also play a role in sample size determination.

FORMULAE FOR CALCULATING SAMPLE SIZE Sample size calculation, as explained previously, is immensely determined by the study design (cross-sectional/experimental/casecontrol and cohort) and is explained below with the aid of examples.7-10

Cross-sectional Study Designs Sample Size Determination for Estimating a Population Mean The researcher might be interested in computing sample size for estimating population mean from the sample mean. For instance, a researcher wishes to conduct a survey among population of elderly females to determine their average daily protein intake. To estimate this sample size, s/he must provide three essential sets of information, namely, desired width of the confidence interval, level of precision desired and population variance (standard deviation) and impute those values into the following formula: n ≥ Z 1- α/22 σ d 2

2

where, ‘d’ is precision or margin of error and is defined as acceptable deviation between hypothesized value and true value of population parameter. ‘σ’ is population standard deviation which is estimated from pilot study or prior similar study. ‘Z1- α/2’ is table values for alpha error corresponding to standard normal distribution. This is 1.96 at 5% and 2.57 at 1% alpha error. In order to obtain the sample size, the researcher decides that he expects the margin of error or precision to be 5 units on either side. 95% confidence interval is decided upon from which the value of Z1- α/2 is 1.96. Also, from literature review, the population standard deviation was observed to be about 25 grams. After inserting these values into above formula, it was calculated that 97 elderly females would be required to get an estimate of the mean protein intake which would be 95 percent of the time within 5 units this side or that side of the true population mean of protein intake.

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Sample Size Determination for Estimating Proportion A public health specialist wants to estimate the immunization coverage of adolescent girls in a community. To determine the sample size, s/he must provide following items of information, namely, desired width of confidence interval, level of precision desired and proportion of population possessing characteristic of interest and impute those values into following formula: 2

n ≥ Z 1- α/22 pq d where, p is proportion in population possessing characteristic of interest and q=1-p. This expected proportion in population is estimated from review of literature or pilot study. The researcher decides that absolute precision should be within 4 percent of true value. The confidence interval is taken as 95 percent (type 1 error=5%), and therefore its corresponding z value is 1.96. Previous studies reveal that immunization coverage is around 70%. After imputing all values into the formula, the required sample size comes out to be 505.

Experimental/Intervention Study Designs Sample Size Determination for Estimating Difference between Means A researcher might be interested in determining the sample size for a trial for comparison of the effect of treatments X and Y in reducing serum cholesterol in a group of patients with hypercholesterolemia. The following formula could be used to calculate sample size for each treatment group: 2 2 n ≥ ( Z1- α/2 + Z B 2) σ × 2 (µ1 - µ2)

where, n1=n2=n; we assumed σ1 and σ2 are same and equal to ‘σ’ (variance). If they are not equal their average can be taken as an estimate of the standard deviation. ‘Z1- α/2’ is the table value for alpha error, ‘ZB’ is the table value for beta error corresponding to the standard normal distribution. ‘µ 1-µ 2’ is the minimum difference that is likely to be detected between the two treatments.

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For determining the sample size, type I error is assumed to be 0.05, for which Z is 1.96; and a type II error of 0.20, for which Z is 0.84. An estimated variance/standard deviation of cholesterol after treatment determined by review of literature is found to be 25 mg/dL. It is also assumed that difference of 3 mg/dL is the minimum that we would like to be able to detect. Including these numbers into formula, the sample size is calculated to be 1089 for each study arm, or a total of 2178. Sample Size Determination for Estimating Difference between Proportions A researcher might be interested in determining the sample size in a trial for comparison of the difference in the proportion of subjects which show an improvement with surgical treatments A versus medical treatment B. The following formula could be used to calculate sample size for each treatment group: 2

Z – √ 2pm (1 – pm )+ ZB √ p1 (1 – p1 ) + p2 (1 – p2 ) N1 = N2 = 1 α/2 (p2 – p1)2 where, n = N1 = N2 = sample size per group p1 = population proportion with improvement with treatment B p 1 (1 – p 1) = population proportion without improvement with treatment B p2 = population proportion with improvement with treatment A p 2 (1-p 2) = population proportion without improvement with treatment A p2 – p1 = minimum important difference pm = average of p1 and p2 [(p1 + p2)/2]

Case Control and Cohort Study Designs In case control design, in addition to providing information on acceptable alpha and beta error, the proportion of persons without outcome who give a history of exposure (proportion of controls who are exposed) and the minimum risk (odds ratio) needs to be described. Likewise, in cohort studies, proportion of subjects those who are not exposed but develop the outcome and the minimum risk (relative

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risk) needs to be enumerated. The sample size is calculated using the following formula: 2 n ≥ (Z1- α/2 + Z1- β) * p20q0 * p1q1 (p1 – p0)

p0 = proportion of those without exposure who develop outcome (cohort study) or proportion of those without outcome who are likely to have an exposure (case control study). p1 = proportion of those with exposure who develop the outcome (cohort study) or proportion of those with outcome who are likely to be exposed (case control design). q0 = 1 – p0 and q1 = 1 – p1 The value of p0 or p1 should be described by the researcher from literature review, discussions with experts, or conducting a pilot study. Once specified, it is then obtained as follows: In a prospective study, P1 =

P0 × RR 1 + P0 (RR-1)

In a case-control study, p1 =

P0 × OR 1 + P0 (OR – 1)

And then calculate p and q as, P=

P0 + P1 2

q = (1 – p) A researcher wishes to conduct a cohort study of impact of sedentary lifestyle on the risk of obesity among middle aged men. Previous studies indicate that proportion of active men who are at risk of obesity (p0) is 15% (0.15) and the proportion of inactive men who are at risk of obesity (p1) is 25% (0.25). α is taken as 0.05 and β is taken as 20. When we impute all these values into the formula, the required value of n in each group is equal to 247, that is, 247 active men and 247 inactive men would be required to be followed over a period to get a desired result.

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SOFTWARES FOR SAMPLE SIZE Free softwares are available in the open source which are immensely helpful for sample size calculation of various study designs. A very popular and simple software to use is ‘OpenEpi’ supported by CDC Atlanta (www.openepi.com). Epi Info, a free software for statistical analysis and power calculation has been developed by the US Centers for Disease Control and Prevention. A user-friendly-friendly software to compute sample size is ‘PS-Power and Sample size calculation’, by Department of Biostatistics Vanderbilt University. Another example is the Power program, written by Lubin and Garcia-Closas at the US National Cancer Institute. Sample size can also be calculated using almost all commercial statistical software, such as STATA and SAS.

KEY POINTS zz

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Sample size calculation depends on a number of factors which must be considered beforehand so that reliable study results are obtained Select sample size formula based on the objectives of the research, study design and outcome variables. Decide on the probabilities of type I and type II errors. Determine the meaningful minimum important difference. Find information on the outcome measure variation. To collect the information on important determinants, search the literature for accurate evidence, or perform a pilot study. Adjust the calculated sample size for factors such as losses to followup, confounders. Open access, free softwares can be used for sample size determination.

REFERENCES 1. Altman DG. Statistics and ethics in medical research: III How large a sample? Br Med J. 1980;281:1336–8. 2. Dupont WD, Plummer WD, Jr. Power and sample size calculations for studies involving linear regression. Control Clin Trials. 1998;19:589–601. 3. Garcia-Closas M, Lubin JH. Power and sample size calculations in casecontrol studies of gene-environment interactions: comments on different approaches. Am J Epidemiol. 1999;149:689–92. 4. Norman G, Monteiro S, Salama S. Sample size calculations: should the emperor’s clothes be off the peg or made to measure? BMJ. 2012;345:e5278.

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Handbook on Research Methodology 5. Florey CD. Sample size for beginners. BMJ. 1993;306:1181–4. 6. Machin D, Campbell M, Fayers P, Pinol A. Sample size tables for clinical studies. 2nd edn. London, Blackwell Science, 1997. 7. Lemeshow S, Levy PS. Sampling of populations: Methods and applications. 3rd ed. New York, John Wiley and Sons, 1999. 8. Christensen E. Methodology of superiority vs. equivalence trials and noninferiority trials. J Hepatol. 2007;46:947–54. 9. Bland JM. The tyranny of power: is there a better way to calculate sample size? BMJ. 2009;339:1133–5. 10. Bhalwar R. Textbook of public health and community medicine. Pune, Department of Community Medicine:AFMC; 2009.

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