Confidence intervals & tests

Interval estimation for trial results — means, proportions, differences, ratios — plus 2×2-table hypothesis tests.

2.1 · CI for a mean (t-based)

Two-sided confidence interval for a single sample mean.

Sample mean x̄ (endpoint units)

Sample SD s (endpoint units)

Sample size n

integer ≥ 2

Confidence level

Lower bound	93.81
Upper bound	106.2
Margin of error	6.192

Formula

x̄ ± t₁₋α/₂,ₙ₋₁ · s/√n

Reference

Altman, D.G., Practical Statistics for Medical Research (1991), Ch. 6.

Assumptions & notes

Assumes approximately normal data or large n.

Note

Assumes approximately normal data or large n.

Exploratory / educational use only — not validated for regulatory submissions. See footer.

2.2 · CI for a proportion

Confidence interval for a binomial proportion — Wilson score (default) or Clopper–Pearson exact.

Successes x

integer, 0 ≤ x ≤ n

Trials n

integer ≥ 1

Confidence level

Method

Lower bound	0.2366
Upper bound	0.7634
Method	Wilson score

Formula

Wilson: [p̂ + z²/2n ± z·√(p̂(1−p̂)/n + z²/4n²)] / (1 + z²/n)

Reference

Wilson, E.B. (1927) J. Amer. Statist. Assoc. 22:209–212; Clopper, C. & Pearson, E.S. (1934) Biometrika 26:404–413; Newcombe, R.G. (1998) Stat. Med. 17:873–890 (recommends Wilson over Wald).

Assumptions & notes

Clopper–Pearson is conservative (actual coverage ≥ nominal); Wilson is the preferred default.

Exploratory / educational use only — not validated for regulatory submissions. See footer.

2.3a · CI for the difference of means

Confidence interval for the between-group mean difference — the standard primary-efficacy summary.

Mean, group 1 (endpoint units)

SD, group 1 (endpoint units)

n, group 1

integer ≥ 2

Mean, group 2 (endpoint units)

SD, group 2 (endpoint units)

n, group 2

integer ≥ 2

Confidence level

Variance assumption

Difference (group 1 − group 2)	2
Lower bound	0.9663
Upper bound	3.034
Method	Welch (unequal variances)

Formula: (x̄₁−x̄₂) ± t₁₋α/₂,df · √(s₁²/n₁ + s₂²/n₂); Welch–Satterthwaite df (default) or pooled df = n₁+n₂−2
Reference: Altman (1991), Ch. 6–8.

Exploratory / educational use only — not validated for regulatory submissions. See footer.

2.3b · CI for the difference of proportions

Confidence interval for the between-group risk difference.

Events, group 1

integer

n, group 1

integer ≥ 1

Events, group 2

integer

n, group 2

integer ≥ 1

Confidence level

Method

Difference (group 1 − group 2)	0.2
Lower bound	-0.08195
Upper bound	0.4445
Method	Newcombe–Wilson (method 10)

Formula: Newcombe (1998) method 10 from the two Wilson intervals; or Wald: d ± z·√(p̂₁(1−p̂₁)/n₁ + p̂₂(1−p̂₂)/n₂)
Reference: Newcombe, R.G. (1998) Stat. Med. 17:873–890; Altman (1991), Ch. 8.

Exploratory / educational use only — not validated for regulatory submissions. See footer.

2.4 · Risk ratio & odds ratio

Point estimate and CI for RR and OR from a 2×2 table (binary efficacy/safety summaries).

a — events, group 1

integer ≥ 0

b — non-events, group 1

integer ≥ 0

c — events, group 2

integer ≥ 0

d — non-events, group 2

integer ≥ 0

Confidence level

Risk ratio	2 (95 % CI 0.9866–4.054)
Odds ratio	2.25 (95 % CI 0.9943–5.092)

Formula

RR = p̂₁/p̂₂, SE(log RR) = √(1/x₁ − 1/n₁ + 1/x₂ − 1/n₂); OR = [x₁/(n₁−x₁)] / [x₂/(n₂−x₂)], SE(log OR) = √(1/a+1/b+1/c+1/d); CI = exp(log ± z·SE)

Reference

Altman (1991), Ch. 8; Woolf, B. (1955) Ann. Hum. Genet. 19:251–253 (OR log method).

Assumptions & notes

Log method; CIs are exponentiated from the log scale.

Exploratory / educational use only — not validated for regulatory submissions. See footer.

2.5 · 2×2 hypothesis tests

χ² test (optional Yates correction) and Fisher’s exact test (two-sided) for a 2×2 table.

a — events, group 1

integer ≥ 0

b — non-events, group 1

integer ≥ 0

c — events, group 2

integer ≥ 0

d — non-events, group 2

integer ≥ 0

Yates continuity correction

χ² statistic (df = 1)	3.75
χ² p-value	0.05281
Fisher’s exact p-value (two-sided)	0.08094

Formula

χ² = Σ(O−E)²/E (df = 1); Fisher: two-sided hypergeometric tail sum

Reference

Altman (1991), Ch. 8; Fisher, R.A. (1934).

Assumptions & notes

CIs (§2.4) remain the primary reporting tool per CONSORT — tests are supplementary.

Exploratory / educational use only — not validated for regulatory submissions. See footer.