How Do You Determine if a Literature Review Sample Size Is Adequate
J Hum Reprod Sci. 2012 Jan-April; 5(1): 7–13.
Sample size estimation and ability analysis for clinical research studies
KP Suresh
Section of Biostatistics, National Institute of Animate being Nutrition and Physiology, Bangalore, India
S Chandrashekara
1Section of Immunology and Reumatology, ChanRe Rheumatology and Immunology Eye and Enquiry, Bangalore, Bharat
Received 2012 Feb 16; Revised 2012 Feb 16; Accepted 2012 Mar 7.
Abstract
Determining the optimal sample size for a written report assures an adequate power to notice statistical significance. Hence, it is a critical step in the blueprint of a planned enquiry protocol. Using too many participants in a study is expensive and exposes more number of subjects to procedure. Similarly, if study is underpowered, it will be statistically inconclusive and may make the whole protocol a failure. This paper covers the essentials in computing power and sample size for a variety of practical study designs. Sample size ciphering for single grouping mean, survey type of studies, 2 group studies based on means and proportions or rates, correlation studies and for case-control for assessing the categorical result are presented in detail.
KEY WORDS: Correlation, odds ratio, power, prevalence, survey, proportions, sample size
INTRODUCTION
Clinical research studies tin can be classified into surveys, experiments, observational studies etc. They demand to be carefully planned to accomplish the objective of the study. The planning of a good inquiry has many aspects. Starting time pace is to define the problem and it should be operational. Second footstep is to define the experimental or observational units and the appropriate subjects and controls. Meticulously, one has to ascertain the inclusion and exclusion criteria, which should have care of all possible variables which could influence the observations and the units which are measured. The report design must be articulate and the procedures are defined to the best possible and available methodology. Based on these factors, the study must have an adequate sample size, relative to the goals and the possible variabilities of the study. Sample must be 'big enough' such that the effect of expected magnitude of scientific significance, to be also statistically significant. Same time, It is important that the study sample should non exist 'too big' where an effect of little scientific importance is withal statistically detectable. In add-on, sample size is of import for economic reasons: An nether-sized study can be a waste of resource since it may not produce useful results while an over-sized study uses more than resource than necessary. In an experiment involving human or brute subjects, sample size is a critical upstanding issue. Since an ill-designed experiment exposes the subjects to potentially harmful treatments without advancing knowledge.[1,2] Thus, a fundamental step in the design of clinical research is the computation of ability and sample size. Power is the probability of correctly rejecting the nothing hypothesis that sample estimates (eastward.g. Hateful, proportion, odds, correlation co-efficient etc.) does not statistically differ betwixt report groups in the underlying population. Large values of power are desirable, at least 80%, is desirable given the bachelor resource and ethical considerations. Power proportionately increases equally the sample size for study increases. Accordingly, an investigator can command the study power by adjusting the sample size and vice versa.[3,4]
A clinical study will be expressed in terms of an estimate of effect, appropriate conviction interval, and P value. The confidence interval indicates the likely range of values for the true outcome in the population while the P value determines the how probable that the observed effect in the sample is due to chance. A related quantity is the statistical power; this is the probability of identifying an exact deviation between two groups in the study samples when one genuinely exists in the populations from which the samples were drawn.
Factors that affect the sample size
The calculation of an appropriate sample size relies on selection of certain factors and in some instances on crude estimates. In that location are three factors that should be considered in calculation of appropriate sample size- summarized in Table ane. The each of these factors influences the sample size independently, but information technology is important to combine all these factors in order to arrive at an advisable sample size.
Table 1
Factors that affect sample size calculations
The Normal deviates for different significance levels (Blazon I error or Alpha) for one tailed and 2 tailed alternative hypothesis are shown in Table ii.
Table two
The normal deviates for Type I mistake (Blastoff)
The normal deviates for different ability, probability of rejecting null hypothesis when it is not true or ane minus probability of type 2 error are in shown Tabular array 3.
Tabular array 3
The normal deviates for statistical ability
Written report design, outcome variable and sample size
Study design has a major impact on the sample size. Descriptive studies need hundreds of subjects to give acceptable confidence interval for small furnishings. Experimental studies generally need lesser sample while the cross-over designs needs one-quarter of the number required compared to a control group because every subject field gets the experimental handling in cross-over written report. An evaluation studies in unmarried grouping with pre-post blazon of blueprint needs half the number for a similar report with a control grouping. A study design with 1-tailed hypothesis requires 20% bottom subjects compared to two-tailed studies. Non-randomized studies needs 20% more subjects compared to randomized studies in order to adapt misreckoning factors. Additional 10 - xx% subjects are required to allow adjustment of other factors such as withdrawals, missing data, lost to follow-up etc.
The "upshot" expected under study should be considered. In that location are 3 possible categories of outcome. The first is a simple case where 2 alternatives exist: Yes/no, expiry/live, vaccinated/non vaccinated, etc. The second category covers multiple, mutually sectional alternatives such as religious beliefs or claret groups. For these 2 categories of outcome, the information are by and large expressed equally percentages or rates[five–7] The third category covers continuous response variables such equally weight, height, blood pressure, VAS score, IL6, TNF-a, homocysteine etc, which are continuous measures and are summarized as means and standard deviations. The statistical methods appropriates the sample size based on which of these outcomes measure is disquisitional for the study, for instance, larger sample size is required to appraise the categorical variable compared to continuous outcome variable.
Alpha level
The definition of blastoff is the probability of detecting a significant departure when the treatments are equally effective or risk of false positive findings. The alpha level used in determining the sample size in most of academic research studies are either 0.05 or 0.01.[7] Lower the alpha level, larger is the sample size. For example, a study with alpha level of 0.01 requires more subjects when compared to a study with alpha level of 0.05 for similar outcome variable. Lower blastoff viz 0.01 or less is used when the decisions based on the research are critical and the errors may cause substantial, financial, or personal harm.
Variance or standard deviation
The variance or standard deviation for sample size calculation is obtained either from previous studies or from airplane pilot written report. Larger the standard deviation, larger is the sample size required in a written report. For example, in a study, with master outcome variable is TNF-a, needs more than subjects compared to a variable of birth weight, ten-point Vas score etc. every bit the natural variability of TNF-a is wide compared to others.
Minimum detectable divergence
This is the expected difference or relationship between 2 independent samples, also known as the event size. The obvious question is how to know the difference in a written report, which is not conducted. If bachelor, it may be useful to use the consequence size institute from prior studies. Where no previous report exists, the issue size is adamant from literature review, logical assertion, and conjecture.
Power
The difference between 2 groups in a study will be explored in terms of estimate of result, appropriate conviction interval, and P value. The confidence interval indicates the likely range of values for the truthful effect in a population while P value determines how probable it is that the observed effect in the sample is due to hazard. A related quantity is the statistical power of the written report, is the probability of detecting a predefined clinical significance. The ideal report is the one, which has high power. This means that the written report has a high chance of detecting a deviation between groups if it exists, consequently, if the study demonstrates no difference between the groups, the researcher can reasonably confident in concluding that none exists. The ideal power for whatsoever written report is considered to be 80%.[8]
In research, statistical ability is generally calculated with 2 objectives. 1) Information technology tin exist calculated before data collection based on data from previous studies to determine the sample size needed for the current written report. 2) It tin can also be calculated after data assay. The 2d state of affairs occurs when the result turns out to be non-significant. In this example, statistical power is calculated to verify whether the non-significance result is due to lack of relationship between the groups or due to lack of statistical power.
Statistical power is positively correlated with the sample size, which means that given the level of the other factors viz. alpha and minimum detectable departure, a larger sample size gives greater power. Notwithstanding, researchers should exist clear to find a divergence between statistical deviation and scientific difference. Although a larger sample size enables researchers to find smaller difference statistically significant, the difference found may not exist scientifically meaningful. Therefore, it is recommended that researchers must have prior idea of what they would expect to be a scientifically meaningful difference earlier doing a power analysis and decide the actual sample size needed. Power analysis is now integral to the health and behavioral sciences, and its use is steadily increasing whenever the empirical studies are performed.
Withdrawals, missing data and losses to follow-up
Sample size calculated is the total number of subjects who are required for the final written report analysis. At that place are few practical problems, which need to be considered while calculating the number of subjects required. Information technology is a fact that all eligible subjects may non be willing to take part and may exist necessary screen more subjects than the final number of subjects inbound the study. In improver, even in well-designed and conducted studies, it is unusual to finish with a dataset, which is complete for all the subjects recruited, in a usable format. The reason could exist subject factor like- subjects may fail or turn down to requite valid responses to particular questions, physical measurements may suffer from technical problems, and in studies involving follow-upwardly (eg. Trials or cohort studies), there will be some degree of attrition. The reason could be technical and the procedural problem- like contagion, failure to get the assessment or test performed in time. Information technology may, therefore, necessary to consider these issues before computing the number of subjects to exist recruited in a study in order to attain the final desired sample size.
Case, say in a study, a total of N number of subjects are required in the finish of the written report with all the data existence complete for analysis, just a proportion (q) are expected to pass up to participate or driblet out before the study ends. In this case, the following total number of subjects (Northward1) would have to be recruited to ensure that the final sample size (N) is achieved:
, where q is the proportion of attrition and is generally 10%,
The proportion of eligible subjects who will turn down to participate or provide the inadequate information will exist unknown at the beginning of the report. Approximate estimates is frequently possible using information from similar studies in comparable populations or from an appropriate pilot study.[ix]
Sample size estimation for proportion in survey type of studies
A common goal of survey inquiry is to collect information representative of population. The researcher uses information gathered from the survey to generalize findings from a drawn sample back to a population, within the limits of random mistake. The full general rule relative to acceptable margins of mistake in survey research is v - ten%. The sample size can be estimated using the post-obit formula
Where P is the prevalence or proportion of consequence of interest for the study, E is the Precision (or margin of error) with which a researcher want to measure something. Generally, E will be 10% of P and Zα/2 is normal deviate for two-tailed culling hypothesis at a level of significance; for example, for 5% level of significance, Zα/2 is 1.96 and for 1% level of significance it is two.58 as shown in Tabular array 2. D is the pattern effect reflects the sampling design used in the survey type of report. This is 1 for elementary random sampling and higher values (usually 1 to 2) for other designs such every bit stratified, systematic, cluster random sampling etc, estimated to recoup for deviation from simple random sampling procedure. The design effect for cluster random sampling is taken as 1.five to two. For the purposive sampling, convenience or judgment sampling, D will cantankerous ten. Higher the D, the more than volition exist sample size required for a written report. Elementary random sampling is unlikely to be the sampling method in an bodily filed survey. If some other sampling method such as systematic, stratified, cluster sampling etc. is used, a larger sample size is likely to exist needed considering of the "design outcome".[ten–12] In example of impact study, P may be estimated at fifty% to reflect the assumption that an impact is expected in 50% of the population. A P of fifty% is also a conservative judge; Example: Researcher interested to know the sample size for conducting a survey for measuring the prevalence of obesity in certain community. Previous literature gives the estimate of an obesity at twenty% in the population to be surveyed, and bold 95% confidence interval or 5% level of significance and 10% margin of error, the sample size can be calculated as follow as;
N = (Zα/two)2 P(i-P)*1 / E2 = (1.96)two*0.20*(1-0.20)/(0.1*0.20)2 = 3.8416*0.xvi/(0.02)two = 1537 for a uncomplicated random sampling pattern. Hence, sample size of 1537 is required to deport customs-based survey to estimate the prevalence of obesity. Note-E is the margin of fault, in the present example; it is 10% χ 0.twenty = 0.02.
To find the final adjusted sample size, allowing not-response rate of ten% in the above example, the adapted sample size will be 1537/(one-0.10) = 1537/0.90 = 1708.
Sample size interpretation with single group mean
If researcher is conducting a study in single group such as consequence cess in a group of patients subjected to certain treatment or patients with item blazon of disease and the master outcome is a continuous variable for which the mean and standard difference are expression of results or estimates of population, the sample size can be estimated using the post-obit formula
North = (Zα/ii)ii southward2 / d2,
where south is the standard difference obtained from previous study or pilot study, and d is the accuracy of estimate or how close to the true mean. Zα/2 is normal deviate for two- tailed alternative hypothesis at a level of significance.
Research studies with i tailed hypothesis, higher up formula tin can exist rewritten as
North = (Zα)ii south2 / d2, the Zα values are 1.64 and 2.33 for 5% and 1% level of significance.
Example: In a study for estimating the weight of population and wants the fault of estimation to be less than 2 kg of truthful mean (that is expected deviation of weight to be 2 kg), the sample standard deviation was five and with a probability of 95%, and (that is) at an error charge per unit of 5%, the sample size estimated as N = (1.96)two (5)2/ 22 gives the sample of 24 subjects, if the allowance of 10% for missing, losses to follow-upwardly, withdrawals is assumed, then the corrected sample will be 27 subjects. Corrected sample size thus obtained is 24/(1.0-0.10) ≅ 24/0.9 = 27 and for xx% allowances, the corrected sample size will be 30.
Sample size estimation with ii ways
In a study with inquiry hypothesis viz; Aught hypothesis H o: m1 = chiliad2 vs. alternative hypothesis H a: m1 = yardii + d where d is the difference between two means and n1 and n2 are the sample size for Group I and Group II such that Due north = n1 + n2. The ratio r = n1/n2 is considered whenever the researcher needs unequal sample size due to various reasons, such every bit ethical, cost, availability etc.
And so, the full sample size for the study is as follows
Where Zα is the normal deviate at a level of significance (Zα is 1.96 for 5% level of significance and 2.58 for 1% level of significance) and Zi-β is the normal deviate at i-β% power with β% of type II mistake (0.84 at 80% power and i.28 at xc% statistical power). r = n1/n2 is the ratio of sample size required for 2 groups, mostly it is one for keeping equal sample size for 2 groups If r = 0.five gives the sample size distribution as 1:2 for 2 groups. σ and d are the pooled standard divergence and deviation of means of 2 groups. These values are obtained from either previous studies of like hypothesis or conducting a pilot report. Allow`s us say a clinical researcher wanting to compare the result of 2 drugs, A and B, on systolic claret pressure (SBP). On literature search, researcher found the mean SBP in two groups were 120 and 132 and common standard deviation of fifteen. The total sample size for the study with r = 1 (equal sample size), a = 5% and power at eighty% and ninety% were computed as 
and for xc% of statistical power, the sample size will be 32. In unequal sample size of 1: 2 (r = 0.5) with ninety% statistical power of 90% at 5% level significance, the total sample size required for the study is 48.
Sample size estimation with two proportions
In study based on event in proportions of issue in 2 populations (groups), such as percentage of complications, mortality comeback, awareness, surgical or medical outcome etc., the sample size estimation is based on proportions of outcome, which is obtained from previous literature review or conducting airplane pilot study on smaller sample size. A study with null hypothesis of H o: πi = π2 vs. H a: π1 = π2 + d, where π are population proportion and p1 and p2 are the corresponding sample estimates, the sample size can exist estimated using the following formula
Where p1 and p2 are the proportion of outcome of interest (outcome) for group I and grouping 2, and p is 
Zα/2 is normal deviate at a level of significance and Z1-β is the normal deviate at 1-β% power with β% of blazon 2 error, normally type II error is considered xx% or less.
If researcher is planning to acquit a study with diff groups, he or she must calculate N as if nosotros are using equal groups, and then calculate the modified sample size. If r = n1/n2 is the ratio of sample size in two groups, then the required sample size is N i = N(i+r)two/fourr, if n1 = 2n2 that is sample size ratio is 2:1 for grouping 1 and group 2, and then N 1 = ixDue north/8, a fairly pocket-sized increase in total sample size.
Case: It is believed that the proportion of patients who develop complications afterwards undergoing i blazon of surgery is 5% while the proportion of patients who develop complications afterwards a second type of surgery is fifteen%. How big should the sample exist in each of the 2 groups of patients if an investigator wishes to detect, with a ability of 90%, whether the 2d procedure has a complications rate significantly higher than the commencement at the 5% level of significance?
In the example,
-
a)
Test value of departure in complication rate 0%
-
b)
Anticipated complication charge per unit v%, 15% in 2 groups
-
c)
Level of significance 5%
-
d)
Power of the test ninety%
-
e)
Alternative hypothesis(1 tailed) (p1-p2) < 0%
The full sample size required is 74 for equal size distribution, for unequal distribution of sample size with 1.v:1 that is r = i.5, the total sample size will be 77 with 46 for group I and 31 for group 2.
Sample size estimation with correlation co-efficient
In an observational studies, which involves to gauge a correlation (r) between 2 variables of interest say, X and Y, a typical hypothesis of form H0: r = 0 against Ha:r ≠ 0, the sample size for correlation study can be obtained by computing
where Zα/ii and Z1-β are normal deviates for type I error (significance level) and Power of report [Tables 2 and 3].
Example: According to the literature, the correlation between salt intake and systolic blood force per unit area is effectually 0.30. A study is conducted to attests this correlation in a population, with the significance level of 1% and ability of 90%. The sample size for such a written report can exist estimated every bit follows:
the sample size for 90% power at 1% level of significance was 99 for two-tailed culling examination and 87 for one-tailed test.
Sample size estimation with odds ratio
In case-control study, data are unremarkably summarized in odds ratio, rather than difference between two proportions when the outcome variables of interest were chiselled in nature. If P1 and P2 are proportion of cases and controls, respectively, exposed to a risk gene, then:
if we know the prevalence of exposure in the general population (P), the total sample size Due north for estimating an OR is 
where Zα/2 and Z1-β are normal deviates for blazon I error (significance level) and Power of study [Tables 2 and iii].
Example: The prevalence of vertebral fracture in a population is 25%. When the written report is interested to estimate the issue of smoking on the fracture, with an odds ratio of 2, at the significance level of five% (i-sided test) and ability of 80%, the total sample size for the report of equal sample size can be estimated by:
DISCUSSION
The equations in this newspaper presume that the choice of private is random and unbiased. The decisions to include a subject in the report depend on whether or not that subject area has the characteristic or the consequence studied. 2nd, in studies in which the mean is calculated, the measurements are assumed to have normal distributions.[13,14]
The concept of statistical power is more than associated with sample size, the power of the study increases with an increment in sample size. Ideally, minimum power of a study required is eighty%. Hence, the sample size calculation is disquisitional and fundamental for designing a study protocol. Even after completion of study, a retrospective power assay volition be useful, especially when a statistically not a significant results are obtained.[15] Here, actual sample size and alpha-level are known, and the variance observed in the sample provides an gauge of variance of population. The analysis of power retrospectively re-emphasizes the fact negative finding is a true negative finding.
The ideal study for the researcher is one in which the power is high. This means that the study has a high chance of detecting a difference between groups if one exists; consequently, if the written report demonstrates no deviation between groups, the researcher tin can be reasonably confident in final that none exists. The Power of the study depends on several factors, but every bit a general dominion, higher power is accomplished by increasing the sample size.[16] Many apparently null studies may exist nether-powered rather than genuinely demonstrating no difference between groups, absence of evidence is not evidence of absence.[ix]
A Sample size calculation is an essential step in inquiry protocols and is a must to justify the size of clinical studies in papers, reports etc. Nevertheless, 1 of the most common mistake in papers reporting clinical trials is a lack of justification of the sample size, and it is a major business organization that important therapeutic effects are existence missed because of inadequately sized studies.[17,18] The purpose of this review is to make available a collection of formulas for sample size calculations and examples for variety of situations probable to be encountered.
Oftentimes, the research is faced with various constraints that may force them to utilise an inadequate sample size because of both practical and statistical reasons. These constraints may include upkeep, time, personnel, and other resources limitations. In these cases, the researchers should report both the advisable sample size along with sample size actually used in the written report; the reasons for using inadequate sample sizes and a give-and-take of the effect of inadequate sample size may take on the results of the study. The researcher should practise caution when making pragmatic recommendations based on the research with an inadequate sample size.
Decision
Sample size determination is an important major step in the pattern of a research study. Appropriately-sized samples are essential to infer with confidence that sample estimated are reflective of underlying population parameters. The sample size required to reject or accept a study hypothesis is adamant by the power of an a-test. A written report that is sufficiently powered has a statistical rescannable chance of answering the questions put along at the beginning of research study. Inadequately sized studies often results in investigator'south unrealistic assumptions about the effectiveness of study treatment. Misjudgment of the underlying variability for parameter estimates wrong judge of follow-up menstruation to notice the intended effects of the treatment and disability to predict the lack of compliance of the written report regimen, and a loftier drib-rate rates and/or the failure to account for the multiplicity of study endpoints are the common error in a clinical research. Conducting a study that has niggling risk of answering the hypothesis at mitt is a misuse of fourth dimension and valuable resources and may unnecessarily expose participants to potential impairment or unwarranted expectations of therapeutic benefits. As scientific and ethical issue become hand-in-hand, the sensation of determination of minimum required sample size and application of appropriate sampling methods are extremely important in achieving scientifically and statistically audio results. Using an adequate sample size along with high quality information collection efforts will issue in more reliable, valid and generalizable results, it could also result in saving resources. This paper was designed as a tool that a researcher could use in planning and conducting quality research.
Footnotes
Source of Support: Nil
Conflict of Interest: None declared.
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