How To Use Gpower To Calculate Sample Size

J Educ Eval Health Prof. 2021; 18: 17.

Sample size determination and power analysis using the Thou*Power software

Hyun Kang

Section of Anesthesiology and Pain Medicine, Chung-Ang University College of Medicine, Seoul, Korea

Sun Huh, Editor

Received 2021 Jun 27; Accepted 2021 Jul 12.

Abstruse

Appropriate sample size calculation and power analysis have go major issues in research and publication processes. All the same, the complexity and difficulty of calculating sample size and power require broad statistical knowledge, in that location is a shortage of personnel with programming skills, and commercial programs are often also expensive to use in practice. The review article aimed to explicate the bones concepts of sample size calculation and ability analysis; the process of sample estimation; and how to calculate sample size using G*Power software (latest ver. 3.1.9.7; Heinrich-Heine-Universität Düsseldorf, Düsseldorf, Germany) with v statistical examples. The null and alternative hypothesis, upshot size, power, blastoff, type I error, and type II mistake should be described when calculating the sample size or power. G*Power is recommended for sample size and power calculations for various statistical methods (F, t, χⁱⁱ, Z, and exact tests), because information technology is easy to use and costless. The process of sample estimation consists of establishing research goals and hypotheses, choosing appropriate statistical tests, choosing one of 5 possible ability assay methods, inputting the required variables for analysis, and selecting the "calculate" push button. The G*Power software supports sample size and power calculation for various statistical methods (F, t, χ², z, and exact tests). This software is helpful for researchers to estimate the sample size and to conduct power analysis.

Keywords: Biometry, Correlation of data, Research personnel, Sample size, Software

Introduction

Background/rationale

If research tin be conducted among the entire population of interest, the researchers would obtain more accurate findings. All the same, in most cases, conducting a study of the entire population is impractical, if not incommunicable, and would exist inefficient. In some cases, it is more accurate to conduct a study of accordingly selected samples than to conduct a study of the entire population. Therefore, researchers use various methods to select samples representing the entire population, to analyze the data from the selected samples, and to estimate the parameters of the entire population, making it very important to determine the appropriate sample size to answer the enquiry question [1,2]. However, the sample size is often arbitrarily chosen or reflects limits of resource allotment. However, this method of conclusion is often not scientific, logical, economical, or even ethical.

From a scientific viewpoint, inquiry should provide an accurate gauge of the therapeutic effect, which may lead to evidence-based decisions or judgments. Studies with inappropriate sample sizes or powers do not provide accurate estimates and therefore written report inappropriate data on the treatment effect, making evidence-based decisions or judgments difficult. If the sample size is also modest, fifty-fifty if a large therapeutic event is observed, the possibility that it could be caused by random variations cannot be excluded. In contrast, if the sample size is too big, besides many variables—beyond those that researchers want to evaluate in the written report—may go statistically meaning. Some variables may prove a statistically meaning deviation, even if the difference is not meaningful. Thus, information technology may be difficult to decide which variables are valid.

From an economic point of view, studies with as well large a sample size may lead to a waste of fourth dimension, money, try, and resources, especially if availability is express. Studies with too small a sample size provide low ability or imprecise estimates; therefore, they cannot answer the research questions, which also leads to a waste of fourth dimension, money, try, and resource. For this reason, considering express resources and budget, sample size calculation and power analysis may crave a merchandise-off between cost-effectiveness and power [iii,4].

From an ethical bespeak of view, studies with too big a sample size cause the research subjects to waste material their effort and time, and may also expose the research subjects to more than risks and inconveniences.

Considering these scientific, economic, and ethical aspects, sample size calculation is critical for research to take adequate power to show clinically meaningful differences. Some investigators believe that underpowered research is unethical, except for small trials of interventions for rare diseases and early stage trials in the development of drugs or devices [5].

Although there have been debates on sample size calculation and power analysis [3,iv,6], the need for an appropriate sample size calculation has become a major trend in research [1,two,7-11].

Objectives

This study aimed to explain the bones concepts of sample size adding and ability analysis; the process of sample estimation; and how to calculate sample size using the G*Power software (latest ver. 3.ane.9.7; Heinrich-Heine-Universität Düsseldorf, Düsseldorf, Frg; http://www.gpower.hhu.de/) with five statistical examples.

Basic concept: what to know earlier performing sample size calculation and ability analysis

In general, the sample size adding and ability analysis are adamant by the following factors: effect size, power (1-β), significance level (α), and type of statistical analysis [ane,7]. The International Committee of Medical Journal Editors recommends that authors describe statistical methods with sufficient detail to enable a knowledgeable reader with admission to the original data to verify the reported results [12], and the same principle should be followed for the clarification of sample size adding or power assay. Thus, the following factors should be described when calculating the sample size or power.

Null and alternative hypotheses

A hypothesis is a testable statement of what researchers predict will exist the outcome of a trial. At that place are ii basic types of hypotheses: the null hypothesis and the culling hypothesis. H0: The null hypothesis is a statement that at that place is no difference betwixt groups in terms of a mean or proportion. H1: The alternative hypothesis is contradictory to the zero hypothesis.

Effect size

The consequence size shows the difference or strength of relationships. It too represents a minimal clinically meaningful divergence [12]. Every bit the size, distribution, and units of the effect size vary between studies, standardization of the effect size is unremarkably performed for sample size calculation and power assay.

The choice of the result size may vary depending on the study design, outcome measurement method, and statistical method used. Of the many dissimilar suggested effect sizes, the K*Power software automatically provides the conventional effect size values suggested past Cohen by moving the cursor onto the blank region of "effect size" in the "input parameters" field [12].

Power, alpha, blazon I fault, and type II error

A type I fault, or false positive, is the error of rejecting a nil hypothesis when it is truthful, and a blazon II fault, or false negative, is the error of accepting a null hypothesis when the alternative hypothesis is true.

Intuitively, type I errors occur when a statistically significant departure is observed, despite at that place beingness no difference in reality, and blazon II errors occur when a statistically meaning difference is not observed, even when there is truly a divergence (Table i). In Table i, the significance level (α) represents the maximum allowable limit of type I mistake, and the power represents the minimum commanded limit of accepting the culling hypothesis when the alternative hypothesis is true.

Table 1.

Types of statistical mistake and ability and confidence levels

Aught hypothesis	Decision
Aught hypothesis	Accept H₀	Reject H_o
H₀ is true	Correct (confidence level, 1-α)	Type I error (α)
H₀ is false	Type II error (β)	Correct (power, 1-β)

If the results of a statistical analysis are non-pregnant, there are 2 possibilities for the not-meaning results: (1) correctly accepting the cipher hypothesis when the zip hypothesis is true and (two) erroneously accepting the null hypothesis when the alternative hypothesis is truthful. The latter occurs when the research method does not take plenty power. If the ability of the study is not known, it is not possible to interpret whether the negative results are due to possibility (1) or possibility (ii). Thus, it is important to consider power when planning studies.

Process of sample size adding and power assay

In many cases, the process of sample size adding and ability analysis is too circuitous and hard for common programs to be feasible. To summate sample size or perform power analysis, some programs require a broad knowledge of statistics and/or software programming, and other commercial programs are likewise expensive to utilise in practice.

To avoid the need for extensive knowledge of statistics and software programming, herein, we demonstrate the process of sample size and power calculation using the G*Power software, which has a graphical user interface (GUI). The G*Power software is piece of cake to use for computing sample size and ability for various statistical methods (F, t, χ², Z, and verbal tests), and tin can be downloaded for gratis at www.psycho.uni-duesseldorf.de/abteilungen/aap/gpower3. G*Power as well provides effect size calculators and graphics options. Sample size and ability calculations using G*Power are generally performed in the following order.

First, institute the inquiry goals and hypotheses

The research goals and hypotheses should be elucidated. The null and culling hypotheses should exist presented, every bit discussed higher up.

Second, choose appropriate statistical tests

M*Power software provides statistical methods in these 2 ways.

Distribution-based arroyo

Investigators tin select the distribution-based approach (verbal, F, t, χ^two, and z tests) using the "test family" drib-down card.

Design-based approach

Investigators can select the blueprint-based approach using the "statistical test" drop-downwards menu. This tin also be carried out past selecting the variable (correlation and regression, means, proportions, variance, and generic) and the report design for which statistical tests are performed from the test menu located at the summit of the screen and sub-carte du jour.

Third, cull 1 of 5 possible power analysis methods

This choice tin be made by considering the variables to be calculated and the given variables. Researchers can select i of the v following types in the "type of ability analysis" drop-down menu (Table 2).

Table 2.

Power analysis methods

Type	Independent variable	Dependent variable
one. A priori	Power (ane-β), significance level (α), and effect size	North
2. Compromise	Effect size, Due north, q=β/α	Power (1-β), significance level (α)
iii. Criterion	Ability (1-β), effect size, N	Significance level (α), criterion
4. Mail-hoc	Significance level (α), effect size, N	Power (i-β)
5. Sensitivity	Significance level (α), power (ane-β), Due north	Effect size

An a priori analysis is a sample size adding performed earlier conducting the study and before the blueprint and planning stage of the study; thus, it is used to calculate the sample size Due north, which is necessary to determine the effect size, desired α level, and power level (1-β). As an a priori analysis provides a method for decision-making blazon I and Ii errors to testify the hypothesis, it is an platonic method of sample size and ability calculation.

In contrast, a mail service-hoc analysis is typically conducted after the completion of the study. As the sample size N is given, the power level (one-β) is calculated using the given North, the upshot size, and the desired α level. Post-hoc power assay is a less ideal type of sample size and power adding than a priori analysis every bit information technology merely controls α, and non β. Post-hoc power analysis is criticized because the type II fault calculated using the results of negative clinical trials is ever loftier, which sometimes leads to incorrect conclusions regarding power [xiii,14]. Thus, post-hoc power analysis should be charily used for the disquisitional evaluation of studies with large type Ii errors.

4th, input the required variables for assay and select the "summate" button

In the "input parameters" surface area of the primary window of G*Power, the required variables for assay can be entered. If information is available to calculate the effect size from a airplane pilot study or a previous report, the event size calculator window can be opened by checking the "decide→" push button, and the effect size can be calculated using this information.

V statistical examples of using One thousand*Power

G*Power shows the following carte bars at the height of the main window when the plan starts upwards: "file," "edit," "view," "tests," "calculator," and "help" (Fig. 1A). Under these menu bars, at that place is another row of tabs, namely, "central and noncentral distributions" and "protocol of power analyses." The "cardinal and noncentral distribution" tab shows the distribution plot of null and culling hypotheses and α and β values (Fig. 1B). Moreover, checking the "protocol of ability analyses" tab shows the results of the calculation, including the name of the test, type of power analysis, and input and output parameters, which tin be cleared, saved, and printed using the "articulate," "save," and "impress" buttons, respectively, which are located at the correct side of the results (Fig. 1C).

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Using the G*Power software for the 2-independent sample t-test. (A) Chief window earlier a priori sample size calculation using an consequence size, (B) main window after a priori sample size calculation using an effect size, (C) main window showing the protocol for power analyses, (D) event size calculator window, (E) plot window, (F) chief window earlier a priori sample size adding not using an effect size, (Grand) primary window later a priori sample size adding non using an event size, and (H) main window before mail service-hoc ability analysis.

In the center of the main screen, drib-down menus named "test family," "statistical examination," and "type of power analysis" are located, where the appropriate statistical exam and type of ability analysis tin can be selected.

In the lower part, in the "input parameters" field, information regarding the sample size adding or ability assay can be entered, and in the "output parameters" field, the results for sample size adding or power analysis volition announced. On the left side of the "input parameters" field, at that place is a "make up one's mind→" push button. Clicking the "determine→" button volition lead to the effect size calculator window, where the consequence size can be calculated by inputting data (Fig. 1D).

In the lowest part of the main screen, at that place are 2 buttons named "Ten-Y plot for a range of values" and "calculate." Checking the "X-Y plot for a range of values" push leads to the plot window, where the graphs or tables for the α error probability, power (1-β error probability), result size, or full sample size tin can be obtained (Fig. 1E). Checking the "calculate" button enables the calculation of the sample size or power.

Examples one. Two-sample or independent t-examination: t-examination

The 2-sample t-examination (also known as the independent t-test or Student t-test) is a statistical exam that compares the mean values of 2 independent samples. The nothing hypothesis is that the difference in grouping means is 0, and the alternative hypothesis is that the deviation in grouping means is different from 0.

H₀: μ_one-μ_ii=0
H_one: μ_one-μ₂≠0
H₀: nix hypothesis
H_i: alternative hypothesis
μ₁, μ₂: means of each sample

Example: a priori

Let us presume that researchers are planning a report to investigate the analgesic efficacy of 2 drugs. Drug A has traditionally been used for postoperative pain control and drug B is a newly developed drug. In order to compare the efficacy of drug B with that of drug A, a pain score using a Visual Analog Scale (VAS) will be measured at 6 hours postoperation. The researchers want to make up one's mind the sample size for the null hypothesis to exist rejected with a ii-tailed test, α=0.05, and β=0.2. The number of patients in each group is equal.

When the issue size is adamant: If the result size to be plant is adamant, the procedure for calculating the sample size is very easy. For the sample size adding of a statistical test, Grand*Ability provides the event size conventions as "small," "medium" and "large," based on Cohen'due south suggestions [12]. These provide conventional upshot size values that are different for different tests.

Moving the cursor onto the blank of "effect size" in the "input parameters" field will testify the conventional upshot size values suggested by G*Power. For the sample size calculation of the t-exam, Thousand*Power software provides the conventional event size values of 0.2, 0.five, and 0.eight for minor, medium, and big effect sizes, respectively. In this case, we attempted to calculate the sample size using a medium effect size (0.5).

After opening Thousand*Power, go to "exam>ways>two contained groups." So, the main window shown in Fig. 1A appears.

Here, as the sample size is calculated before the conduction of the report, fix "type of power analysis" as "A priori: compute required sample size-given α, ability, and issue size." Since the researchers decided to utilise a medium effect size, 2-sided testing, α=0.05, β=0.2, and an equal sample size in both groups, select "two" for the "tail(s)" drop-downward menu and input 0.five for the bare of "effect size d," 0.05 for the blank of "α err prob," 0.8 for the blank of "power (1-β err prob)," and one for the blank of "Allocation ratio N2/N1." Upon pushing the "calculate" button, the sample size for grouping 1, the sample size for group ii, and the full sample size will exist computed as 64, 64, and 128, respectively, as shown in the output parameters of the chief window (Fig. 1B).

Upon clicking on the "protocol of ability analyses" tab in the upper role of the window, the input and output of the ability calculation tin be automatically obtained. The results tin can be cleared, saved, and printed using the "clear," "save," and "print" buttons, respectively (Fig. 1C).

When the consequence size is not determined: Adjacent, allow us consider the instance in which the effect size is not determined in the study blueprint similar to that described above. In fact, the effect size is non adamant on many occasions and it is debatable whether a adamant result size can be practical when designing a report. Thus, when the effect size value is arbitrarily determined, we should provide a logical rationale for the arbitrarily selected effect size. When the effect size is not determined, the effect size value can only be causeless using the variables from other previous studies, pilot studies, or experiences.

In this example, as in that location was no previous study comparing the efficacy of drug A to that of drug B, we volition presume that the researchers conducted a pilot study with 10 patients in each grouping, and the means and standard deviations (SDs) of the VAS for drug A and B were 7±iii and five±2, respectively.

In the main window, button the "make up one's mind" button; then, the effect size calculator window will appear (Fig. 1D). Here, there are 2 options: "n1!=n2" and "n1=n2." Because we presume an equal sample size for two groups, we bank check the "n1=n2" box.

Since the pilot study showed that the mean and SD of the VAS for drug A and drug B were 7±3 and 5±2, respectively, input vii for "mean group 1," 3 for "SD σ group 1," v for "mean grouping 2," and two for "SD σ group 2," and so click the "calculate and transfer to main window" button.

So, the respective effect size "0.7844645" will be automatically calculated and appear in the blank infinite for "effect size d" in the main window (Fig. 1F).

As described above, we decided to use 2-sided testing, α=0.05, β=0.2, and an equal sample size in both groups. Nosotros selected "two" in the "tail(s)" drop-down bill of fare and entered 0.05 for "α err prob," 0.8 for "power (i-β err prob)," and 1 for "allocation ratio N2/N1." Clicking the "summate" button computes "sample size group 1," "sample size group two," and "total sample size" as 27, 27, and 54, respectively, in the output parameters expanse (Fig. 1G).

Example: postal service hoc

A clinical trial comparing the efficacy of two analgesics, drugs A and B, was conducted. In this study, 30 patients were enrolled in each group, and the means and SDs of the VAS for drugs A and B were vii±3 and v±2, respectively. The researchers wanted to make up one's mind the ability using a 2-tailed exam, α=0.05, and β=0.2.

After opening G*Power, get to "test>means>ii independent groups." In the main window, select "blazon of power analysis" every bit "post hoc: compute achieved ability-given α, sample size and effect size," and then button the "determine" button. In the issue size computer window, since a clinical trial was conducted with equal group sizes, showing that the means and SDs of the VAS for drug A and B were vii±3 and five±ii, respectively, check the "n1=n2" box and input 7 for "mean group 1," three for "SD σ group 1," v for "mean grouping two," and two for "SD σ group ii," and and then click the "calculate and transfer to main window" button. And then, the respective issue size "0.7844645" volition be calculated automatically and appear in the blank infinite for "event size d" in the main window (Fig. 1H). Because nosotros decided to utilize 2-sided testing, α=0.05, and an equal sample size (northward=30) in both groups, we selected "two" in the tail(due south) drop-down carte, and entered 0.05 for "α err prob," xxx for "sample size grouping ane," and 30 for "sample size grouping two." Pushing the "calculate" push will compute "power (i-β err prob)" as 0.8479274.

Examples 2. Dependent t-test: t-examination

The dependent t-examination (paired t-test) is a statistical exam that compares the means of 2 dependent samples. The naught hypothesis is that the departure between the means of dependent groups is 0, and the alternative hypothesis is that the departure betwixt the means of dependent groups is non equal to 0.

H₀: μ₁-μ₂=0
H₁: μ₁-μ₂≠0
H₀: goose egg hypothesis
H₁: culling hypothesis
μ₁, μ₂: means of each sample from dependent groups

Example: a priori

Imagine a study examining the effect of a diet program on weight loss. In this written report, the researchers programme to enroll participants, counterbalance them, enroll them in a diet program, and counterbalance them again. The researchers want to make up one's mind the sample size for the null hypothesis to be rejected with a 2-tailed test, α=0.05, and β=0.2.

When the effect size is determined: After opening Chiliad*Power, go to "test>means>two dependent groups (matched pairs)."

In the main window, set "blazon of power analysis" as "a priori: compute required sample size-given α, power, and result size" (Fig. 2A). Since the researchers decided to utilize 2-sided testing, α=0.05, and β=0.2, select "ii" for the tail(s) drop-downwards bill of fare, and input 0.05 for the blank of "α err prob" and 0.8 for the bare of "ability (1-β err prob)." Unlike the example for the t-exam, as the number of participants is expected to exist equal in the paired t-exam, the blank for "allocation ratio N2/N1" is not provided. In this case, summate the sample size using a modest upshot size (0.two); thus, input 0.2 for "effect size dz." Upon pushing the "calculate" button, the total sample size will exist computed equally 199 in the output parameter area (Fig. 2B).

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Using the G*Ability software for the dependent t-test. (A) Main window earlier a priori sample size calculation using an effect size, (B) main window after a priori sample size adding using an effect size, and (C) effect size computer window.

When the consequence size is non determined: Suppose the researchers conducted a pilot written report to summate the sample size for the study investigating the consequence of the nutrition programme on weight loss. In this airplane pilot study, the hateful and SD before and after the nutrition program were 66±12 kg and 62±11 kg, respectively, and the correlation between the weights earlier and after the diet programme was 0.7.

Click the "determine" push button in the main window and then check the "from grouping parameters" box from the 2 options "from differences" and "from group parameters" in the effect size reckoner window, equally the group parameters are known.

Since the pilot study showed that the ways and SDs of the results before and subsequently the diet program were 66±12 kg and 62±eleven kg, respectively, and the correlation between the results before and later on the nutrition plan was 0.seven, input 66 for "mean group ane," 62 for "mean grouping two," 12 for "SD group 1," eleven for "SD group 2," and 0.7 for "correlation between groups," and and so click the "calculate and transfer to master window" push (Fig. 2C). So, the respective upshot size "0.4466556" volition exist calculated automatically and appear at the blank space for "upshot size dz" in the main screen and in the effect size estimator screen. In the main window, as we decided to utilize 2-sided testing, α=0.05, and β=0.2, nosotros volition select "2" in the "tail(southward)" drop-down menu, and input 0.05 for "α err prob" and 0.8 for "ability (one-β err prob)." Pushing the "calculate" button will compute the "total sample size" as 42.

Case: postal service hoc

Presume that a clinical trial comparing participants' weight before and after the diet program was conducted. In this study, 100 patients were enrolled, the means and SDs earlier and afterwards the diet programme were 66±12 kg and 62±11 kg, respectively, and the correlation between the weights before and later on the diet program was 0.vii. The researchers wanted to decide the power for 2-tailed testing and α=0.05.

After opening Chiliad*Power, get to "test>means>two dependent groups (matched pairs)." In the main screen, select "type of power analysis" as "postal service hoc: compute accomplished power-given α, sample size and effect size," and and then push the "make up one's mind" button. In the outcome size calculator screen, select the "from grouping parameters" check box from the 2 options "from differences" and "from grouping parameters" as the group parameters are known. In this effect size calculator window, as a clinical trial showed that the means and SDs of the results earlier and later on the diet plan were 66±12 kg and 62±11 kg, respectively, and the correlation between the results earlier and after the diet program was 0.7, input 66 for "mean group ane," 62 for "mean group 2," 12 for "SD group 1," 1 for "SD group ii," and 0.7 for "correlation betwixt groups," and and then click the "summate and transfer to main window" push. And then, the corresponding effect size "0.4466556" volition exist calculated automatically and announced in the blank space for "result size dz" in the main window and in the outcome size reckoner window.

In the main window, every bit we decided to use 2-sided testing and α=0.05 and the enrolled number of patients was 100, select "ii" in the "tail(southward)" drop-down carte and input 0.05 for "α err prob" and 100 for "total sample size." Pushing the "calculate" push volition compute "power (one-β err prob)" as 0.9931086.

Examples iii. One-mode assay of variance: F-test

One-fashion analysis of variance (ANOVA) is a statistical test that compares the means of 3 or more samples. The null hypothesis is that all m means are identical, and the alternative hypothesis is that at to the lowest degree 2 of the k means differ.

H₀: μ₁=μ₂= ⋯=μ_yard
H_i: the means are not all equal
H₀: zip hypothesis
H_ane: alternative hypothesis
μ₁, μ_two, ⋯ μ_k: means of each sample from the independent groups (1, 2, ⋯, k)

The assumptions of the ANOVA test are as follows: (one) independence of observations, (2) normal distribution of dependent variables, and (three) homogeneity of variance.

Instance: a priori

Assume that a report investigated the effects of 4 analgesics: A, B, C, and D. Pain volition be measured at 6 hours postoperatively using a VAS. The researchers wanted to determine the sample size for the nothing hypothesis to exist rejected at α=0.05 and β=0.2.

When the upshot size is determined: For the ANOVA test, Cohen suggested the event sizes of "small," "medium," and "large" as 0.1, 0.25, and 0.4, respectively [12], and Chiliad*Power provides conventional issue size values when the cursor is moved onto the "effect size" in the "input parameters" field. In this case, we calculated the sample size using a medium effect size (0.25).

After opening Chiliad*Power, go to "test>means>many groups: ANOVA: one-way (one independent variable)." In the main window, set up "type of power assay" as "a priori: compute required sample size-given α, ability, and effect size." Since we decided to utilise a medium result size, α=0.05, and β=0.two, we enter 0.25 for "consequence size f," 0.05 for "α err prob," and 0.8 for "power (1-β err prob)." Moreover, as we compared 4 analgesics, we input 4 for "number of groups." Pushing the "calculate" push button computes the full sample size as 180, as shown in the "total sample size" in the "output parameters" surface area.

When the effect size is not determined: Assume that a pilot study showed the following means and SDs (of VAS) at half-dozen hours postoperation for drugs A, B, C, and D: two±2, 4±1, v±i, and 5±2, respectively, in five patients for each grouping.

Button the "determine" button and the effect size calculator window will appear. Here, nosotros tin can find two options: "effect size from means" and "effect size from variance." Select "effect size from ways" in the "select procedure" drop-downwards menu and select 4 in the "number of groups" drop-down menu.

Here, as One thousand*Power does not provide the common SD, we must calculate this. The formula for the mutual SD is every bit follows:

$s_{pooled} = \sqrt{\frac{(n_{1} - one) {Due south}_{1}^{two} + (n_{2} - 1) {S_{2}}^{2} + (n_{3} - 1) {S_{iii}}^{two} + ({northward}_{4} - 1) {S_{4}}^{two}}{(({due north}_{1} - ane) + (n_{2} - 1) + (n_{3} - i) + (n_{4} - 1) - 4)}}$

S_pooled: common SD

South₁, S_ii, S_three, S_iv: SDs in each group

n_one, n₂, n_iii, n_iv: numbers of patients in each group

Using the above formula, nosotros obtain the following:

${Due south}_{pooled} = \sqrt{\frac{(5 - 1) {two}^{2} + (5 - i) i^{2} + (v - 1) i^{2} + (five - 1) 2^{two}}{((5 - ane) + (v - 1) + (5 - i) + (5 - ane) - 4)}}$

Herein, the mutual SD will be i.58. Thus, input 1.58 into the blank of "SD σ inside each group." As the means and numbers of patients in each grouping are 2, 4, 5, and 5 and 5, 5, five, and v, respectively, input each value, and and so click the "calculate and transfer to main window" push button. So, the respective effect size "0.7751550" will exist automatically calculated and appear in the bare for "outcome size f" in the main window.

Since we decided to utilise iv groups, α=0.05, and β=0.2, input 4 for "number of groups," 0.05 for "α err prob," and 0.eight for "power (1-β err prob)." Pushing the "calculate" button computes the total sample size as 24, as shown in the "total sample size" in the "output parameters" surface area.

Instance: mail service hoc

Assume that a clinical trial showed the post-obit ways and SDs of VAS at 6 hours postoperation in drugs A, B, C, and D : two±ii, 4±ane, 5±one, and v±2, respectively, in twenty patients for each group. The researchers want to determine the power with ii-tailed testing, α=0.05, and β=0.2.

After opening Chiliad*Ability, go to "test>ways>many groups: ANOVA: one-way (one independent variable)." In the main screen, select "blazon of ability analysis" every bit "post hoc: compute achieved power-given α, sample size and event size," and then push the "decide" push to testify the effect size calculator screen.

Using the in a higher place formula, the common SD is one.58. Thus, input 1.58 into the blank of "SD σ within each group." As the means and numbers of patients in each group are 2, 4, five, and 5, and 20, 20, xx, and 20, respectively, input these values into the corresponding blank; side by side, click the "calculate and transfer to main window" button.

Then, the respective effect size "0.7751550" and the total number of patients will be calculated automatically and appear in the blanks for "effect size f" and "full sample size" on the main screen.

Since the clinical trial used 4 groups and we decided to use α=0.05, input 4 for "number of groups" and 0.05 for "α err prob." Pushing the "calculate" button will compute the power as 0.9999856 at "ability (1-β err prob)" in the "output parameters" area.

Examples 4. Correlation–Pearson r

A correlation is a statistic that measures the human relationship between 2 continuous variables. The Pearson correlation coefficient is a statistic that shows the strength of the relationship between 2 continuous variables. The Greek letter ρ (rho) represents the Pearson correlation coefficient in a population, and r represents the Pearson correlation coefficient in a sample. The null hypothesis is that the Pearson correlation coefficient in the population is 0, and the alternative hypothesis is that the Pearson correlation coefficient in the population is not equal to 0.

H₀: ρ=0
H₁: ρ≠0
H₀: zippo hypothesis
H₁: culling hypothesis
ρ: Pearson correlation coefficient in the population

The Pearson correlation coefficient in the population (ρ, rho) ranges from −1 to 1, where −1 represents a perfect negative linear correlation, 0 represents no linear correlation, and ane represents a perfect positive linear correlation. The coefficient of determination (ρ²) is calculated by squaring the Pearson correlation coefficient in the population (ρ) and is interpreted as "the percent of variation in 1 continuous variable explained by the other continuous variable" [fifteen].

For correlations, Cohen suggested the event sizes of "small," "medium," and "large" equally 0.1, 0.iii, and 0.five, respectively [12]. However, Chiliad*Power does non provide a sample size calculation using the effect size for correlations. Therefore, this commodity does not present a sample size adding using the effect size. Instead, an instance is provided of sample size calculation using the expected population Pearson correlation coefficient or coefficient of determination.

Example: a priori

Consider a hypothetical study investigating the correlation between height and weight in pediatric patients. The researchers wanted to decide the sample size for the nada hypothesis to be rejected with ii-tailed testing, α=0.05, and β=0.2. In the airplane pilot study, the Pearson correlation coefficient for the sample was 0.5.

After opening Chiliad*Power, go to "test>correlation and regression>correlation: bivariate normal model." In the master screen, fix "blazon of power analysis" as "a priori: compute required sample size-given α, power, and effect size." Because we decided to use 2-sided testing, α=0.05, and β=0.2, the Pearson correlation coefficient in the null hypothesis is 0, and the Pearson correlation coefficient in the sample was 0.7, in the pilot report, select "two" for the "tail(due south)" drop-down menu, and input 0.5 for "correlation ρ H1," 0.05 for "α err prob," 0.viii for "power (1-β err prob)," and 0 for "correlation ρ H1." Upon pushing the "calculate" button, the full sample size will be computed every bit 29 in the "output parameters" area.

Grand*Power besides provides an option to calculate the sample size using the coefficient of determination. If the coefficient of determination is known, button the "determine" push in the main window, input the value into the blank of "coefficient of determination ρ2," and then click the "calculate and transfer to main window" button. Finally, push button the "summate" button to compute the total sample size.

Example: post hoc

Assume that a clinical trial investigated the correlation betwixt acme and weight in pediatric patients. In this report, 50 pediatric patients were enrolled, and the sample Pearson correlation coefficient was 0.v. The researchers wanted to decide the power at the 2-tailed and α=0.05 levels.

Later opening K*Power, go to "examination>correlation and regression>correlation: bivariate normal model." In the main screen, select "type of ability analysis" as "post hoc: compute achieved power-given α, sample size, and effect size." And so, as the clinical trial showed that the sample Pearson correlation coefficient was 0.v and the number of enrolled patients was 50, input 0.5 for "correlation ρ H1," 0 for "correlation ρ H1," and 50 for "total sample size." Every bit the researchers want to determine the ability at the 2-tailed and α=0.05 levels, select "two" for the tail(s) drop-down bill of fare and input 0.05 for "α err prob."

Then, the respective power "0.9671566" will be calculated automatically and appear at the blank for "ability (1-β err prob)" in the output parameter expanse of the main screen.

Example v. Two independent proportions: chi-foursquare examination

The chi-foursquare test, also known as the χ² test, is used to compare two proportions of independent samples. Thousand*Power provides the options to compare 2 proportions of independent samples, namely "two independent groups: inequality, McNemar test," "ii contained groups: inequality, Fisher's exact exam," "two contained groups: inequality, unconditional exact," "two independent groups: inequality with outset, unconditional exact," and "two independent groups: inequality, z-exam." In this article, we innovate the "two independent groups: inequality, z-test" because many statistical software programs provide like options.

The null hypothesis is that the difference in proportions of independent groups is 0, and the alternative hypothesis is that the difference in proportions of independent groups is not equal to 0.

H₀: π₁-π₂=0
H₁: π_i-π_two≠0

As G*Power does not provide sample size adding using the upshot size for this option, this is not provided in this commodity; instead, an example of sample size calculation using the expected proportions of each group is provided.

Example: a priori

Assume that a report examined the effects of 2 treatments, for which the mensurate of the effect is a proportion. Treatment A has been traditionally used for the prevention of post-herpetic neuralgia, and treatment B is a newly developed treatment. The researchers wanted to determine the sample size for the zilch hypothesis to be rejected using 2-tailed testing, α=0.05, and β=0.2. The number of patients was equal in both groups. Suppose the researchers conducted a airplane pilot study examining the effects of treatments A and B. In the pilot study, the proportions of post-herpetic neuralgia development were 0.iii and 0.1, respectively, for treatments A and B.

Later opening Thou*Power, go to "test>proportions>2 independent groups: inequality, z-test." In the primary window, prepare "type of power analysis" as "a priori: compute required sample size-given α, ability, and effect size."

Because nosotros decided to use ii-sided testing, α=0.05, and β=0.two, there was an equal sample size in both groups, and the proportions of post-herpetic neuralgia development were 0.three and 0.ane for treatments A and B, respectively, in the pilot study, select "2" for the "tail(s)" drib-down carte, and input 0.05 for "α err prob," 0.viii for "ability (1-β err prob)," 1 for "allocation ratio N2/N1," and 0.3 and 0.1 for "proportion p2" and "proportion p1," respectively.

Pushing the "summate" button will compute "sample size group ane," "sample size group 2," and "full sample size" as 62, 62, and 124, respectively.

Example: mail service hoc

Assume that a clinical trial compared the result of 2 treatments, A and B, on the incidence of post-herpetic neuralgia. In this study, researchers enrolled 101 and 98 patients, respectively, and post-herpetic neuralgia occurred in 31 and ix patients later treatments A and B, respectively. The incidence of post-herpetic neuralgia in the groups that received treatments A and B was 0.307 (31/101) and 0.092 (9/98), respectively.

The researchers wanted to make up one's mind the ability using ii-tailed testing and α=0.05. Later opening K*Power, become to "examination>proportions>ii contained groups: inequality, z-test." In the main screen, select "type of ability analysis" equally "mail service hoc: compute achieved ability-given α, sample size and outcome size." Then, as a clinical trial showed that the incidence of post-herpetic neuralgia in the groups that received treatments A and B was 0.307 and 0.092, respectively, and the number of enrolled patients was 101 and 98, respectively, input 0.307 for "proportion p2," 0.092 for "proportion p1," 101 for "sample size group ii," and 98 for "sample size grouping 1."

As researchers wanted to determine the ability for 2-tailed testing and α=0.05, select "ii" for the tail(s) drop-downwardly bill of fare and input 0.05 for "α err prob." Pushing the "summate" button will compute "power (1-β err prob)" as 0.9715798 in the "output parameter" expanse of the main window.

Sample size calculation considering the drop-out rate

When conducting a written report, drib-out of study subjects or non-compliance to the study protocol is inevitable. Therefore, we should consider the drop-out rate when computing the sample size. When computing the sample size considering the driblet-out charge per unit, the formula for the sample size adding is as below:

$N^{D} = \frac{N}{(i - d)}$
N: sample size before because drop-out
d: expected drop-out rate
ND: sample size because drop-out

Permit us assume that a sample size for a study was calculated every bit 100. If the drop-out rate during the report process is expected to be 20% (0.2), the sample size considering drib-out will be 125.

Decision

Appropriate sample size calculation and power assay have get major issues in research and analysis. The Thousand*Power software supports sample size and ability adding for various statistical methods (F, t, χ², z, and verbal tests). Thousand*Power is easy to apply because information technology has a GUI and is free. This article provides guidance on the application of G*Power to calculate sample size and power in the design, planning, and analysis stages of a study.

Footnotes

Authors' contributions

Conceptualization, data curation, formal assay, writing–original draft, writing–review & editing: HK.

Conflict of interest

No potential conflict of interest relevant to this article was reported.

Funding

None.

Information availability

None.

Supplementary materials

Supplement one. Audio recording of the abstract.

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