THE SMALL SAMPLES STATISTICAL ANALYSIS IN MORPHOLOGICAL STUDIES: RESULTS OF SIMULATION MODELING AND PRACTICAL RECOMMENDATIONS

INTRODUCTION. In morphological studies, sample size is often extremely limited due to rarity of pathological processes, ethical constraints, and high cost of methods. Correct statistical analysis of small samples remains an understudied problem, and researchers often apply inappropriate tests.

OBJECTIVE. To present the results of a simulation study of statistical methods for comparing independent groups under small sample sizes and to formulate practical recommendations for test selection considering the specifics of morphological data.

MATERIALS AND METHODS. A simulation study was performed with 10,000 replications per scenario (sample size from 2 to 12 per group). Categorical, ordinal, and metric data were evaluated. Test performance criteria: type I error control, robustness (Bradley’s criterion), statistical power, and minimum sample size to reject the null hypothesis.

RESULTS. For 2×2 tables, the Williams-corrected G-test and Boschloo’s test showed the best performance (minimum n = 4 – 5). For 3×2 tables, the Fisher–Freeman–Halton test with mid-p correction was preferred. Among ordinal tests, the asymptotic Brunner–Munzel test performed best (from 5 observations in total). For metric data, beta regression (with bias reduction) and Welch’s test were the most robust (from 4–5 observations in total).

CONCLUSION. For small samples in morphology, exact and permutation tests should be preferred. Reducing the measurement scale without a strong biological reason is strongly discouraged. The proposed algorithms for selecting criteria take into account the specifics of morphological data and can provide acceptable control of type I errors in the considered scenarios with a limited amount of material. 

1. Button KS, Ioannidis JPA, Mokrysz C et al. Power failure: why small sample size undermines the reliability of neuroscience. Nat Rev Neurosci. 2013;14:5(365–376). https://doi.org/10.1038/nrn3475

2. Morgan CJ. Use of proper statistical techniques for research studies with small samples. Am J Physiol Lung Cell Mol Physiol. 2017;313:5(873–877). https://doi.org/10.1152/ajplung.00238.2017

3. Jenkins DG, Quintana-Ascencio PF. A solution to minimum sample size for regressions. PLoS One. 2020;15:2(e0229345). https://doi.org/10.1371/journal.pone.0229345

4. Winter JCF de. Using the Student's t-test with extremely small sample sizes. Pract Assess Res Eval. 2013;18:10. https://doi.org/doi:10.7275/E4R6-DJ05

5. Tomchuk ON, Sen'chukova MA, Stadnikov AA i dr. Morfologicheskie osobennosti mikrososudov i limfoidnoy tkani v regionarnykh limfaticheskikh uzlakh pri rake zheludka. Morfologicheskie vedomosti. 2015;23(4):25–31. https://doi.org/10.20340/mvmn.2015.0(4):25-31. In Russian

6. Govorukhina OA, Prokopenya NS, Kharitonchik VA i dr. Osobennosti morfologicheskoy diagnostiki bolezni Girshprunga. Morfologicheskie vedomosti. 2024;32:1(24–35). https://doi.org/10.20340/mv-mn.2024.32(1).818. In Russian

7. Bradley JV. Robustness? Br J Math Stat Psychol. 1978;31:2(144–152). https://doi.org/10.1111/j.2044-8317.1978.tb00581.x

8. Demirhan H. rTableICC: an R package for random generation of 2×2×K and R×C contingency tables. R J. 2016;8:1(48–62). https://doi.org/10.32614/RJ-2016-004

9. Hothorn T, Hornik K, van de Wiel MA, Zeileis A. Implementing a class of permutation tests: the coin package. J Stat Softw. 2008;28:8(1–23). https://doi.org/10.18637/jss.v028.i08

10. Pham TV, Piersma SR, Warmoes M, Jimenez CR. On the beta-binomial model for analysis of spectral count data in label-free tandem mass spectrometry-based proteomics. Bioinformatics. 2010;26:3(363–369). https://doi.org/10.1093/bioinformatics/btp677

11. Cribari-Neto F, Zeileis A. Beta regression in R. J Stat Softw. 2010;34:2(1–24). https://doi.org/10.18637/jss.v034.i02

12. Karch JD. Psychologists should use Brunner-Munzel's instead of Mann-Whitney's U test as the default nonparametric procedure. Adv Methods Pract Psychol Sci. 2021;4:2(2515245921999602) e-pub. https://doi.org/10.1177/2515245921999602

13. Brunner E, Munzel U. The Nonparametric Behrens-Fisher Problem: Asymptotic Theory and a Small-Sample Approximation. Biometrical Journal. 2000;42:1(17–25). https://doi.org/10.1002/(SICI)1521-4036(200001)42:1<17::AID-BIMJ17>3.0.CO;2-U

14. Marateb HR, Mansourian M, Adibi P, Farina D. Manipulating measurement scales in medical statistical analysis and data mining: a review of methodologies. J Res Med Sci. 2014;19:1(47–56)

15. Liddell TM, Kruschke JK. Analyzing ordinal data with metric models: What could possibly go wrong? J Exp Soc Psychol. 2018;79(328–348). https://doi.org/10.1016/j.jesp.2018.08.009

16. Gvozdetskiy AN, Prokopovich GA, Dobrovol'skaya AE i dr. Primenimost' regressionnykh modeley dlya analiza poryadkovykh dannykh PANSS pri shizofrenii: kogortnoe issledovanie. Consortium Psychiatricum. 2026;7:2(10–21). https://doi.org/10.17816/CP15731

17. Fagerland MW, Lydersen S, Laake P. Statistical Analysis of Contingency Tables. 1-st ed. New York: CRC Press, 2017. https://doi.org/10.1201/9781315374116

18. Grzhibovskiy AM, Unguryanu TN, Savvina NV. Analiz kategorial'nykh dannykh s ispol'zovaniem programmnogo obespecheniya SPSS i STATA. Narkologiya. 2018;12:2(39–55). https://doi.org/10.25557/1682-8313.2018.02.39-55. In Russian

19. Indrayan A, Mishra A. The importance of small samples in medical research. J Postgrad Med. 2021;67:4(219–223).