Practical guide to calculate sample size for chi-square test in biomedical research

Abstract

In biomedical research, the calculation of sample size is a critical component of study design. Adequate sample size ensures the reliability of statistical tests, including the chi-square test. This manuscript outlines the use of an online sample size calculator for chi-square tests. The paper includes detailed explanations of the formulas used in the calculations and highlights the importance of power analysis in planning research studies. This tool is designed to assist and guide researchers in determining the optimal sample size for detecting statistically significant differences in categorical data. We describe the theory behind the chi-square test, the statistical principles involved in sample size calculation, and the specific methodology for using the sample size calculator. The calculator is freely available to use at https://hanif-shiny.shinyapps.io/chi-sq/.

Introduction

Chi-square test is particularly useful in epidemiological studies, clinical trials, and health sciences, where researchers often aim to examine associations between categorical variables such as treatment groups and outcomes, exposure and disease status, or demographic characteristics and behavioral patterns [1]. However, the power of the chi-square test depends on several factors, including effect size, significance level, and degrees of freedom [2]. This necessitates a rigorous approach to calculating the appropriate sample size, ensuring that the study has adequate statistical power to detect meaningful associations.

Statistical power analysis is an essential aspect of designing a valid and reliable research study. In many cases, researchers must assess the sample size required to detect meaningful effects, ensuring that their study is adequately powered to identify significant differences when they exist [2]. In particular, the chi-square test is a widely used statistical method for evaluating categorical data, such as comparing observed frequencies to expected frequencies across different groups. A well-designed study requires the determination of an appropriate sample size, which can directly influence the outcome of the test [3].

The chi-square test has a long history in statistics, with its foundations laid by Karl Pearson in 1900. Pearson introduced the chi-square goodness-of-fit test, which was later extended to the chi-square test for independence used in contingency tables [4, 5]. Over time, refinements in the methodology have led to various adaptations of the test, including Yates’ continuity correction, Fisher’s exact test, and likelihood ratio tests, which address specific limitations of the classical chi-square test [3–5].

One key milestone in the development of sample size determination for the chi-square test was the introduction of Cohen’s w, an effect size measure for categorical data [6]. Cohen (1988) proposed thresholds for small, medium, and large effect sizes, which have since been widely adopted in power analysis calculations [6, 7]. The availability of statistical software such as G*Power has further facilitated accurate sample size computations, making these calculations more accessible to researchers [8]. Therefore, the chi-square test sample size calculator serves as an invaluable tool for researchers, allowing them to determine the correct sample size based on critical inputs such as the effect size, degrees of freedom, significance level, and power. This manuscript promotes the use of this tool by explaining its underlying theory and providing practical guidance for its application.

Overview of the chi-square test

The chi-square test is used to determine whether there is a significant association between categorical variables [2]. It tests the hypothesis that the observed frequencies in a contingency table match the expected frequencies under the null hypothesis of no association. The formula for the chi-square statistic $\mathrm{\chi }2$ is as follows:

$${\mathrm{\chi }}^2=\sum \frac{{\left(O_i,-,E_i\right)}^2}{E_i}$$

where:

$$O_i\text{= Observed frequency in cell}i$$

$$E_i\text{= Expected frequency in cell}i$$

Degrees of Freedom ($\mathit{df}$):

The chi-square statistic follows a chi-square distribution with $\mathit{df}$ degrees of freedom that are calculated as:

$$df=\left(r,-,1\right)\left(c,-,1\right)$$

where:

$$r\text{= Number of rows in the contingency table}$$

$$c\text{= Number of columns in the contingency table}$$

Statistical power and sample size calculation

Power analysis helps researchers determine the minimum sample size needed to detect an effect, given a particular significance level (α) and desired power (1—$\mathrm{\beta }$) [8]. The power of a statistical test is defined as the probability of correctly rejecting the null hypothesis when it is false (i.e., detecting a true effect). The formula for power is:

$$\text{Power}=1-\mathrm{\beta }$$

where:

$$\mathrm{\beta }\text{= Probability of committing a Type II error (failing to reject the null hypothesis when it is false).}$$

For the chi-square test, the relationship between effect size, significance level, power, and sample size is critical. The effect size for the chi-square test is commonly measured by Cohen’s w, which quantifies the strength of the association between the categorical variables. Cohen’s w is a measure of effect size for the chi-square test and is calculated as:

$$w=\sqrt{\frac{{\mathrm{\chi }}^2}{N}}$$

where:

$${\mathrm{\chi }}^2\text{= Chi-square statistic}$$

$$N\text{= Total sample size}$$

The interpretation of Cohen's w [7] is as follows:

w = 0.1 (Small effect)

w = 0.3 (Medium effect)

w = 0.5 (Large effect)

For a given significance level (α), degrees of freedom ($\mathit{df}$), and desired power, the sample size ($N$) can be determined using power analysis methods.

Formula for sample size calculation

The formula for determining the required sample size for the chi-square test based on the effect size (w), significance level (α), degrees of freedom ($\mathit{df}$), and desired power (1—$\mathrm{\beta }$) can be derived using the noncentral chi-square distribution [2, 9]. The general formula for sample size is:

$$1-\mathrm{\beta }=\text{Pr}\left({\mathrm{\chi }}_{\text{df},\text{ncp}=n{w}^2}^2,>,{\mathrm{\chi }}_{1-\alpha ,\text{df}}^2\right)$$

$$\text{where df}=\left(r,-,1\right)\left(c,-,1\right)$$

$\text{ncp}=n{w}^2$ is the non-centrality parameter of the non-central chi-squared distribution.

${\mathrm{\chi }}_{1-\alpha ,\text{df}}^2\text{is the}\left(1,-,\alpha \right)$—quantile of the central chi-squared distribution with df degrees of freedom. The quantile from the central chi-squared distribution is associated with the null hypothesis, and the non-central chi-squared distribution is associated with the alternative hypothesis.

To calculate the sample size, solving this equation by hand can be complex, especially for noncentral chi-square distributions. This calculator provides an efficient and user-friendly approach to determining the required sample size based on these parameters.

The chi-square test sample size calculator

The Chi-Square Test Sample Size Calculator is an online tool designed to assist researchers in performing power analysis and determining the optimal sample size for chi-square tests. The calculator takes the following inputs: Effect size (w): A measure of the strength of the association between categorical variables, Degrees of freedom (df): The number of categories or levels in the contingency table, Significance level (α): The probability of making a Type I error (rejecting a true null hypothesis), and Power (1—β): The desired probability of detecting an effect when one exists [9].

The tool computes the required sample size based on these inputs, using the formula described above. In addition to calculating the sample size, the tool provides the following: 1) a step-by-step report summarizing the inputs and the resulting sample size, and 2) a downloadable reports in text format for documentation and future reference. This tool is available free to use at https://hanif-shiny.shinyapps.io/chi-sq/

Example of application:

A researcher may wish to conduct a chi-square test for independence between two categorical variables, with the following parameters:

Effect size (w) = 0.3 (medium effect)

Degrees of freedom (df) = 1 (Two categories in each variable)

Significance level (α) = 0.05

Power (1—β) = 0.8

The calculator uses these inputs to compute the minimum sample size required for the study. For this example, the tool calculates the minimum required sample size to be 88 participants.

Benefits of using the sample size calculator

Determining the correct sample size is crucial for study design. The calculator offers time and cost efficiency by automating the sample size calculation process, ensuring that researchers do not waste resources on underpowered studies. In addition, this tool uses established statistical methods to calculate the sample size, ensuring accurate and reliable results. The user-friendly interface with clear instructions of the calculator is designed to be intuitive, requiring only basic input from the user without the need to download any software or knowledge how to code in R. It is suitable for researchers with varying levels of statistical expertise. Furthermore, the calculator generates detailed reports, providing transparency in the sample size determination process. Researchers can easily document and share the calculations in their publications or grant applications.

Limitations of the sample size calculator

Despite the availability of formulae and software, determining the sample size for a chi-square test is not always straightforward where several challenges exist. The final data collected could have small Expected Cell Counts. If expected frequencies are too low, the chi-square test’s validity is compromised, requiring alternative methods such as Fisher’s exact test [10]. A study could have designed higher-dimensional categorical outcome that requires complex contingency tables, which necessitates larger sample sizes, making recruitment challenging in certain study designs [11]. Additionally, a study may achieve the required sample size but the sample was recruited using non-random sampling techniques. If sampling methods introduce bias, calculated sample sizes may not yield valid results [12]. Furthermore, when researchers need to conduct multiple chi-square tests, they must adjust for inflated Type I error rates, which may further increase sample size requirements [13].

Conclusion

The Chi-Square Test Sample Size Calculator is a valuable tool for researchers conducting studies involving categorical data. By automating the sample size calculation process, it helps ensure that studies are adequately powered to detect meaningful differences. The tool is based on sound statistical principles, using formulas derived from the chi-square distribution and power analysis. Researchers can use the calculator to plan their studies efficiently, avoiding the risks associated with underpowered or overpowered studies. With its user-friendly interface and ability to generate detailed reports, the calculator is an essential resource for researchers in biomedical science and related fields.

Authors’ contributions

HAR, AAN, ANK, AZM, NHZ and NET contributed to the conception or design of the paper. HAR and AAN conducted the data analysis. HAR, AAN, ANK, AZM, NHZ and NET contributed to data interpretation, and drafting/editing the manuscript. HAR, AAN, ANK, AZM, NHZ and NET were involved in revising the manuscript, providing critical comments, and agreed to be accountable for all aspects of the work and any issues related to the accuracy or integrity of any part of the work.

Funding

No funding was received for this study.

Data availability

No datasets were generated or analysed during the current study.

Declarations

Ethical approval and consent to participate:

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.