Chi-square test applications

Abstract

Background: The chi-squared (x²) test is a fundamental non-parametric statistical method. It is widely employed in clinical, epidemiological, and biomedical research, including ophthalmology and optometry. It is useful for testing hypotheses regarding the independence of categorical data or the goodness-of-fit of the observed data to the expected distributions within contingency tables. In this review, we present a thorough examination of the statistical principles and clinical relevance of the x² test, focusing on its application in vision science and related research domains. Methods: We outline the conceptual framework and methodological steps for conducting the x² test, emphasizing its two primary forms: the goodness-of-fit test and the test of independence. We discuss key assumptions, such as the independence of observations, use of frequency data, and minimum expected cell counts in detail. Moreover, we explain the process of calculating degrees of freedom (df) and interpreting results based on critical values from the x² distribution. Additionally, appropriate measures of effect size, i.e., Phi for 2 × 2 tables and Cramer’s V for larger tables, for assessing association strength, are introduced. To contextualize its clinical relevance, we present four examples from ophthalmology. Results: In the first example, the association between vision impairment (VI) and sex was examined using a 2 × 6 contingency table. The x² statistic was 4.37 with 5 df (P > 0.05), indicating no statistically significant association. Cramer’s V was 0.04, suggesting a very weak effect. The second example tested the association between age category and first-year persistence with antiglaucoma therapy. Here, x² = 5.93 (df = 2, P > 0.05), also showing no significant association, Cramer’s V was weak (0.04). In the third example, a 2 × 2 table was used to analyze the association between sex and the type of anti-vascular endothelial growth factor injection (aflibercept or ranibizumab) used. This yielded a x² = 0.214 (df = 1, P > 0.05) and phi = 0.05, again indicating no statistically significant association and a weak effect. In a goodness-of-fit test assessing the pattern of contact lens usage, the x² exceeded the critical threshold, indicating a significant deviation between the observed and expected frequencies, leading to rejection of the null hypothesis. Conclusions: The x² test is a robust tool for analyzing categorical data, enabling clinicians and researchers to identify potential relationships between variables. However, its reliability depends on its proper application, including verification of assumptions and appropriate interpretation of effect sizes, along with consideration of statistical significance. In clinical disciplines, such as ophthalmology or optometry, understanding and utilizing the x² test enhances research rigor and the validity of research findings, facilitating better-informed decisions in patient care and in program development.