Statistical comparison of chromosomes and ERs

To statistically assess differences in GC content and replicon size between ERs and chromosomes, we applied the Brunner–Munzel test, a robust nonparametric alternative to the t-test that does not assume normality or equal variances. Analyses were performed separately for each phylum, class, family, and genus containing at least 100 ERs.

In all comparisons, ERs were defined as group 1 and chromosomes as group 2, such that positive test statistics indicate higher values in ERs relative to chromosomes.

In addition to statistical significance, we quantified effect size using Cliff’s delta (δ), which measures stochastic dominance between two distributions. Because large datasets can yield statistically significant results for small differences, Cliff’s delta provides a complementary measure of the magnitude of separation.

Interpretation of test statistics:

• Brunner–Munzel statistic: positive values indicate ERs > chromosomes, negative values indicate ERs < chromosomes
• p̂ (p-hat): probability that a randomly chosen ER has a higher value than a randomly chosen chromosome (null hypothesis p̂ = 0.5)
• p_adj: Benjamini–Hochberg FDR-adjusted p-value (p_adj < 0.05 considered significant)
• Cliff’s delta (δ): magnitude of stochastic dominance

Effect size thresholds (|δ|):
negligible (< 0.147), small (< 0.33), medium (< 0.474), large (≥ 0.474)

Together, the Brunner–Munzel test and Cliff’s delta allow simultaneous evaluation of statistical significance and effect size for differences between ERs and chromosomes across bacterial taxa.

References:
Brunner E, Munzel U. 2000. The nonparametric Behrens–Fisher problem. Biometrical Journal 42:17–25.
Meissel K, Yao ES. 2024. Using Cliff’s Delta as a Non-Parametric Effect Size Measure: An Accessible Web App and R Tutorial. Practical Assessment, Research, and Evaluation 29(1):2. DOI: https://doi.org/10.7275/pare.1977.