Related Subjects:
|Basic Statistics
|Sampling in Medical Statistics
|Reading a Medical paper
|Different Forms of Medical Trials and Studies
|Hierarchy of Evidence-Based Trials
|Bayes' Theorem
|Comparing Groups
Comparing Populations in Statistics – Updated Feb 2026
📊 Comparing populations involves using statistical tests to determine if observed differences between two or more groups are likely due to chance or reflect true population differences. This is fundamental in evidence-based medicine, epidemiology, social sciences, economics, and quality improvement.
Modern practice (2026): Focus on **effect size + confidence intervals** alongside p-values; report practical/clinical significance; use non-parametric tests liberally when normality fails; correct for multiple comparisons.
🧮 Types of Comparisons
- Means / Central Tendency (continuous data): Compare averages (e.g., blood pressure, HbA1c, pain scores).
- Proportions / Percentages (categorical/binary): Compare rates (e.g., response rates, disease prevalence, adverse events).
- Variances / Dispersion: Compare spread (e.g., variability in response to treatment; often prerequisite for mean tests).
- Distributions (non-parametric): Compare entire shapes when normality violated (e.g., Mann-Whitney, Kolmogorov-Smirnov).
📈 Parametric Tests for Comparing Means
| Test | When to Use | Key Assumptions | Effect Size | Example |
|---|
| Independent (unpaired) t-test | Two independent groups | Normality, homogeneity of variances (Levene), independence | Cohen's d (0.2 small, 0.5 medium, 0.8 large) | Mean BP in drug vs placebo group |
| Paired (dependent) t-test | Two related samples (before-after, matched pairs) | Differences normally distributed | Cohen's d (paired) | Pre- vs post-treatment cholesterol |
| One-way ANOVA | Three or more independent groups | Normality, homogeneity of variances, independence | η² (partial) or ω² (0.01 small, 0.06 medium, 0.14 large) | Three drug doses on pain score |
| Two-way ANOVA | Two factors (e.g., drug × gender) | As above + no significant interaction unless hypothesised | Partial η² per factor/interaction | Treatment effect by age group |
- Post-hoc tests (after ANOVA): Tukey HSD (balanced), Bonferroni, Sidak, Dunnett (vs control).
- Welch t-test / Welch ANOVA: Use when variances unequal (robust to heteroscedasticity).
📊 Tests for Comparing Proportions
| Test | When to Use | Assumptions / Notes | Effect Size | Example |
|---|
| Chi-square (χ²) test of independence | 2×2 or larger contingency table | Expected counts ≥5 in ≥80% cells; independence | Phi (2×2), Cramér’s V (larger) | Vaccine response yes/no by group |
| Fisher’s exact test | Small samples / expected <5 | No assumptions on counts; exact p-value | Phi or Odds ratio | Rare adverse event comparison |
| McNemar’s test | Paired nominal data (before-after) | Dichotomous repeated measures | Odds ratio | Pre- vs post-treatment symptom presence |
| Z-test for proportions | Two large independent proportions | np, n(1-p) ≥5–10 | Risk difference / NNT | Mortality rate A vs B |
📉 Tests for Comparing Variances
- F-test: Two groups; sensitive to normality violation.
- Levene’s test (or Brown-Forsythe): Robust to non-normality; preferred before t-test/ANOVA.
- Bartlett’s test: Sensitive to non-normality; less used now.
⚠️ Key Assumptions & Checks
- Normality: Shapiro-Wilk (small n), Kolmogorov-Smirnov, Q-Q plots, histograms.
- Independence: No clustering/pairing unless accounted for.
- Homogeneity of variances: Levene’s p > 0.05 → assume equal; else use Welch.
- Random sampling: Representative groups.
- Interval/ratio data for parametric tests (or ordinal with caution).
📊 Non-Parametric Alternatives (When Assumptions Fail)
- Mann-Whitney U test: Independent groups (replaces independent t-test).
- Wilcoxon signed-rank test: Paired/related samples (replaces paired t-test).
- Kruskal-Wallis test: 3+ independent groups (replaces one-way ANOVA).
- Friedman test: 3+ related groups (non-parametric repeated measures ANOVA).
- Effect size: r = Z / √N (Mann-Whitney), Kendall’s W (Friedman).
📏 Effect Size & Practical Significance
- Cohen’s d (means): 0.2 small, 0.5 medium, 0.8 large.
- η² / partial η² (ANOVA): 0.01 small, 0.06 medium, 0.14 large.
- Cramér’s V / Phi (chi-square): 0.1 small, 0.3 medium, 0.5 large.
- Odds ratio / Risk ratio: Clinical meaningfulness depends on baseline risk.
- Modern emphasis: Report effect size + 95% CI; p-value alone insufficient (2026 consensus: avoid over-reliance on p < 0.05).
🏥 Clinical & Epidemiological Relevance
- Treatment comparison: RCT – t-test/ANOVA for continuous outcomes (e.g., HbA1c reduction), chi-square for binary (response rate).
- Epidemiology: Cohort/case-control – compare incidence/prevalence (proportions), risk factors (means of biomarkers).
- Health policy: Compare outcomes across regions/populations (e.g., mortality rates, vaccination coverage).
- Quality improvement: Before-after (paired t/Wilcoxon) or group comparisons (e.g., new protocol vs standard).
Teaching Point 🩺
Choose test by: data type (continuous/categorical), groups (2 vs >2, independent vs paired), assumptions (normality, equal variance).
Parametric (t, ANOVA) → powerful if assumptions met.
Non-parametric (Mann-Whitney, Kruskal-Wallis) → robust when normality fails.
Always report **effect size + CI** (not just p-value); correct for multiple comparisons (Bonferroni, FDR).
Clinical significance > statistical significance: is the difference meaningful to patients?
📚 References (Feb 2026)
- Altman DG et al. Statistics with Confidence (3rd ed., BMJ 2025 reprint).
- Lakens D. Effect size guidelines. Psychol Methods 2025 update.
- CONSORT & STROBE guidelines (2025 revisions): Emphasise effect sizes/CIs.
- Recent: Moving beyond p < 0.05 (Am Stat 2026 consensus).
