Comparing Groups

t-tests, ANOVA, and Non-parametric Alternatives

The Clinical Problem

An ICMR trial in Jharkhand randomises 90 women with iron-deficiency anaemia to three arms (n = 30 each):

Arm	Mean Hb ± SD (g/dL)
Oral iron	10.8 ± 1.4
IV iron	11.5 ± 1.6
Placebo	9.6 ± 1.5

Questions that cascade:

Is oral iron better than placebo? → t-test
Are all three groups different? → ANOVA
Which pairs differ? → Post-hoc tests
What if data aren’t normal? → Non-parametric alternatives

Roadmap

One-Sample t-Test: When to Use

Scenario: One group vs. a known reference value

Example: 120 MBBS students at KGMU Lucknow — is their mean BMI different from the national average of 22.5?

Hypotheses:

H₀: μ = 22.5 (mean BMI equals national average)
H₁: μ ≠ 22.5 (two-tailed)

Formula:

\[t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}, \quad df = n - 1\]

One-Sample t: Worked Example

t = 2.82, df = 119, p = 0.0057

Paired t-Test: Same Subjects, Two Timepoints

Scenario: 40 hypertensive patients at a PHC in Kolar, Karnataka — systolic BP before and after 6 weeks of amlodipine

\[t = \frac{\bar{d}}{s_d / \sqrt{n}}, \quad df = n - 1\]

where \(\bar{d}\) = mean of within-subject differences

Key: Each patient is their own control — reduces between-subject variability

Paired t: Worked Example

Independent Two-Sample t-Test

Scenario: HbA1c at 6 months — metformin (n = 50) vs. glimepiride (n = 50) at KEM Hospital, Pune

\[t = \frac{\bar{x}_1 - \bar{x}_2}{SE}, \quad SE = s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}\]

Two versions:

Condition	Test
Equal variances (Levene’s p > 0.05)	Student’s t-test
Unequal variances	Welch’s t-test (default in R)

Independent t: Worked Example

Effect Size: Cohen’s d

Statistical significance alone is not enough — how large is the difference?

\[d = \frac{\bar{x}_1 - \bar{x}_2}{s_p}\]

Cohen’s d	Interpretation
0.2	Small
0.5	Medium
0.8	Large

Clinical significance adds context: a difference of 0.5% HbA1c may not meet the MCID of 0.5% — statistically significant but clinically marginal.

Checking Assumptions

Normality

Shapiro-Wilk test (p > 0.05 → normal)
Q-Q plot (points on diagonal → normal)
Histograms
CLT: n > 30 often suffices

Equal Variances

Levene’s test (p > 0.05 → equal)
Rule of thumb: ratio of larger to smaller SD < 2
If unequal → use Welch’s t-test

When assumptions fail badly:

→ Use non-parametric alternatives (Section 7)

One-Way ANOVA: Three or More Groups

Why not multiple t-tests?

With 3 groups → 3 pairwise comparisons → inflated Type I error

\[\alpha_{\text{family}} = 1 - (1 - 0.05)^3 = 0.143\]

ANOVA tests all groups simultaneously:

H₀: μ₁ = μ₂ = μ₃ (all means equal)
H₁: At least one mean differs

\[F = \frac{MS_{\text{between}}}{MS_{\text{within}}} = \frac{\text{Signal (between-group variance)}}{\text{Noise (within-group variance)}}\]

ANOVA: Anaemia Trial Example

The ANOVA Table

Source	df	SS	MS	F
Between groups	k − 1	SS_B	SS_B / (k−1)	MS_B / MS_W
Within groups	N − k	SS_W	SS_W / (N−k)
Total	N − 1	SS_T

Significant F → at least one group differs, but WHICH ones?

→ Post-hoc tests

Post-Hoc Comparisons

ANOVA says “something differs” — post-hoc says what.

Method	When to Use	Correction
Tukey HSD	All pairwise comparisons	Controls familywise error
Bonferroni	Conservative; few planned comparisons	α / number of tests
Dunnett	Compare each treatment to one control	Only vs. control

	Comparison	Mean Diff	Lower CI	Upper CI	Adj. p
Oral Iron-Placebo	Oral Iron-Placebo	0.895	0.056	1.734	0.034
IV Iron-Placebo	IV Iron-Placebo	1.326	0.487	2.165	0.001
IV Iron-Oral Iron	IV Iron-Oral Iron	0.431	-0.408	1.270	0.442

Non-Parametric Alternatives: When to Use

Use when:

Data are ordinal (e.g., pain VAS, Likert scales)
Distribution is heavily skewed or has outliers
Small samples (n < 15–20 per group) and normality uncertain

Parametric	Non-Parametric
Independent t-test	Mann-Whitney U
Paired t-test	Wilcoxon signed-rank
One-way ANOVA	Kruskal-Wallis

Mann-Whitney U Test

Example: Length of stay — laparoscopic vs. open cholecystectomy at Safdarjung Hospital, Delhi

Wilcoxon Signed-Rank Test

Example: Pain VAS (0–10) before and after physiotherapy at CMC Vellore — 25 knee OA patients

Kruskal-Wallis Test

Example: Patient satisfaction (Likert 1–5) across 3 CHCs in Chhattisgarh

Decision Flowchart

Scenario	Parametric	Non-Parametric
One group vs. reference value	One-sample t-test	Wilcoxon signed-rank*
Two groups, same subjects (paired)	Paired t-test	Wilcoxon signed-rank
Two independent groups	Independent t-test	Mann-Whitney U
Three+ independent groups	One-way ANOVA	Kruskal-Wallis
Note:
* One-sample Wilcoxon signed-rank tests median vs. hypothesised value

Reporting Template

t-test:

Mean HbA1c was lower in the metformin group (7.1 ± 0.9%) than in the glimepiride group (7.5 ± 1.1%); t(98) = −2.14, p = 0.035, 95% CI for difference [−0.77, −0.03], Cohen’s d = 0.40.

ANOVA:

One-way ANOVA showed a significant difference in Hb across treatment groups, F(2, 87) = 12.4, p < 0.001. Post-hoc Tukey HSD revealed IV iron was significantly higher than placebo (mean diff = 1.9 g/dL, p < 0.001).

Non-parametric:

Median LOS was significantly shorter in the laparoscopic group (3 days, IQR 2–4) than the open group (6 days, IQR 5–7); Mann-Whitney U, p < 0.001.

Do’s and Don’ts

Do

Check normality and variance before choosing a test
Report effect sizes (Cohen’s d, η²)
Use Welch’s t-test by default for independent samples
Apply post-hoc corrections after ANOVA
Report medians and IQR for non-parametric tests

Don’t

Run multiple t-tests instead of ANOVA
Ignore assumption violations
Report only p-values without effect sizes
Use parametric tests on ordinal data
Forget to state whether the test was one-tailed or two-tailed

Key Takeaways

One-sample t → one group vs. reference value
Paired t → same subjects, two timepoints
Independent t → two separate groups
ANOVA → three or more groups; follow with post-hoc if significant
Non-parametric alternatives for skewed data or ordinal scales
Always report: test statistic, df, p-value, CI, and effect size
Clinical significance ≠ statistical significance — interpret both