clinical-biostatistics-effect-measures

name	clinical-biostatistics-effect-measures
description	Computes and interprets treatment effect measures (OR, RR, RD, HR, NNT) with calibrated confidence intervals (Wilson, Newcombe, Miettinen-Nurminen, MOVER, profile likelihood, Bender NNT) and reports marginal vs conditional estimands per FDA 2023 covariate adjustment guidance. Use when reporting treatment effects in confirmatory trials, comparing effect sizes across studies, or constructing forest plots.
tool_type	python
primary_tool	statsmodels

Version Compatibility

Reference examples tested with: statsmodels 0.14+, numpy 1.26+, pandas 2.1+, matplotlib 3.8+, marginaleffects (Python) 0.0.13+ / (R) 0.20+. R packages cited: ratesci, exact2x2, marginaleffects, riskCommunicator, RobinCar.

Before using code patterns, verify installed versions match. If versions differ:

• Python: pip show <package> then help(module.function) to check signatures
• R: packageVersion('<pkg>') then ?function_name

If code throws ImportError, AttributeError, or TypeError, introspect the installed package and adapt the example to match the actual API rather than retrying.

Treatment Effect Measures for Clinical Trials

"Compute treatment effect sizes" -> Estimate the population-level treatment contrast (OR, RR, RD, HR, NNT) with a confidence interval calibrated to sample size and a clear declaration of whether the estimand is marginal or conditional under ICH E9(R1).

Algorithmic Taxonomy

Measure	Scale	Collapsible?	Best CI method	When to use	Fails when
OR	Log-odds ratio	NO (non-collapsible)	Profile likelihood; Wald acceptable for n>100 per arm	Case-control (only valid measure); logistic regression default	Outcome prevalence > 10% (OR overstates RR); Hauck-Donner pathology near boundary
RR	Log-risk ratio	YES	Miettinen-Nurminen score; MOVER-R	Cohort, RCT with common outcomes	Sparse strata; one or both p near 0 (Wald log-RR breaks)
RD (absolute risk difference)	Linear probability	YES	Newcombe-Wilson hybrid; Miettinen-Nurminen	Clinically interpretable absolute scale; FDA-preferred for binary	Predictions outside [0,1] from linear models
HR (hazard ratio)	Log-hazard ratio	NO	Wald with profile likelihood for small n	Time-to-event with PH	PH violation (see clinical-biostatistics/survival-analysis)
NNT/NNH	1/RD	n/a (derived from RD)	Bender 2001 CCT 22:102 (Altman 1998 base)	Communicating absolute benefit to clinicians	RD CI crosses zero (NNT becomes NNTB-infinity-NNTH)
Difference in RMST	Time scale	YES	Wald with delta method; pseudo-observation regression	Time-to-event with PH violation	Different max follow-up across arms (truncation tau ambiguous)

Postdoc reading: Permutt 2020 Stat Biopharm Res 12:45 ("Do covariates change the estimand?") established that the conditional OR from a multivariable logistic regression is a different parameter than the marginal OR -- and the FDA May 2023 Final Guidance "Adjusting for Covariates in RCTs" requires the marginal estimand for primary reporting (see clinical-biostatistics/logistic-regression for g-computation/standardisation machinery).

Decision Tree by Scenario

Scenario	Recommended estimand + CI	Why
RCT, binary outcome, prevalence <10%	OR with Wald CI; report as primary	OR ~= RR at low prevalence; standard regulatory currency
RCT, binary outcome, prevalence >=10%	RR or RD via modified Poisson with HC1/HC3 sandwich SE; OR as secondary	OR substantially overstates RR; modified Poisson directly estimates RR (Zou 2004 AJE 159:702)
RCT, binary outcome, primary endpoint per FDA 2023	Marginal RD via g-computation with sandwich SE; conditional OR as supportive	FDA 2023 final guidance: marginal estimand for primary; cite Permutt 2020
Case-control study	OR only (RR unidentifiable); profile likelihood CI	OR is the only measure estimable from case-control design
Noninferiority on absolute scale	Miettinen-Nurminen score CI for RD	Regulatory standard for RD CIs; consistent with Pearson chi-square
Noninferiority on relative scale	Miettinen-Nurminen score CI for RR; Koopman 1984 acceptable	Wald log-RR has poor coverage near boundary; MN is the regulatory expectation
Stratified design with site/region strata	MH pooled OR with stratified MN CI (R ratesci::scoreci(..., stratified=TRUE))	Preserves stratification; cite Kahan-Morris 2012
Reporting NNT for clinicians	Bender 2001 CCT 22:102 method; report as NNTB(lower)..infinity..NNTH(upper) when CI crosses zero	Standard in BMJ/Lancet/Cochrane
Time-to-event with PH violation	RMST difference; cite Royston-Parmar 2013, Uno 2014	HR is a misleading single-number summary under non-PH (see survival-analysis)

Crude Effect Measures from 2x2 Tables

Goal: Compute unadjusted OR, RR, RD from a contingency table with calibrated CIs.

Approach: Use Table2x2 for OR/RR with Wald CIs, but switch to score-based methods (Miettinen-Nurminen, Newcombe-Wilson) for regulatory contexts.

from statsmodels.stats.contingency_tables import Table2x2
from statsmodels.stats.proportion import confint_proportions_2indep
import numpy as np

# Table layout: [[treated_event, treated_no_event], [control_event, control_no_event]]
table = np.array([[a, b], [c, d]])
t = Table2x2(table)
print('OR:', t.oddsratio, t.oddsratio_confint())        # Wald log-OR CI
print('RR:', t.riskratio, t.riskratio_confint())         # Wald log-RR CI

# Modern CI for RD (regulatory preferred):
ci_rd = confint_proportions_2indep(a, a+b, c, c+d, method='newcomb', alpha=0.05)
# 'newcomb' = Newcombe-Wilson hybrid; 'agresti-caffo' is the +1+1 +1+1 adjustment
# For Miettinen-Nurminen stratified CIs, use R `ratesci::scoreci`

Table orientation is critical: Table2x2 interprets the first column as "outcome present." pd.crosstab orders columns alphabetically; if outcome is coded 0/1, the table will have 0 first and the OR will be the reciprocal of intended. Always reorder: cross = cross[[1, 0]] or cross = cross[['Yes', 'No']]. This is a silent direction-reversing error.

For the underlying significance test of a 2x2: Pearson chi-square (no Yates) at n>=40 with adequate expected counts; otherwise Boschloo's exact (uniformly more powerful than Fisher's exact at the same Type-I per Mehta-Senchaudhuri 2003; Lydersen-Fagerland-Laake 2009). See clinical-biostatistics/categorical-tests for the algorithmic taxonomy.

Modern Confidence Intervals -- the Postdoc Toolkit

Wald CIs are the textbook default but have well-documented coverage failures (Brown-Cai-DasGupta 2001 Stat Sci 16:101 -- "chaotic" coverage near 0 and 1 even with moderate n; Newcombe 1998a Stat Med 17:873 documents 11 alternatives for the difference of two proportions).

Single proportion

Method	When	Status
Wald	Never preferred	Defunct for serious work; coverage can drop to 0.0 (Brown-Cai-DasGupta)
Wilson (score)	General default	Brown-Cai-DasGupta recommended; matches Pearson chi-square test
Jeffreys (Beta(1/2,1/2))	Small n	Equal-tailed Jeffreys; Bayesian with reference prior; recommended for small n
Clopper-Pearson	When exact guarantee required	Over-covers by 1-4 percentage points; sometimes regulatory-required
Agresti-Coull	Simple Wald replacement	+2/+2 adjustment; teaching default
Mid-p Clopper-Pearson	When CP over-coverage hurts NI margin	Less conservative than CP; slight under-coverage

from statsmodels.stats.proportion import proportion_confint
ci = proportion_confint(45, 60, alpha=0.05, method='wilson')
# Other methods: 'jeffreys', 'agresti_coull', 'beta' (Clopper-Pearson), 'normal' (Wald)

Difference of two proportions (RD)

Method	Citation	Verdict
Wald	textbook	Poor; can produce limits outside [-1,1]
Newcombe-Wilson hybrid / MOVER	Newcombe 1998a	Recommended; balanced of computational simplicity and coverage
Agresti-Caffo (+1+1, +1+1)	Agresti-Caffo 2000 Am Stat 54:280	Add 1 success + 1 failure per arm, then Wald; surprisingly good for small n
Miettinen-Nurminen score	Miettinen-Nurminen 1985 Stat Med 4:213	Regulatory standard for RD; consistent with Pearson chi-square; SAS PROC FREQ riskdiff(method=mn)
Chan-Zhang exact unconditional	Chan-Zhang 1999 Biometrics 55:1202	Guaranteed coverage; needed when MN under-covers near extreme p

Ratio of two proportions (RR)

Method	Citation	Verdict
Wald on log(RR) ("Katz log")	textbook (Katz 1978)	Defunct for serious work; biased when either p small
Koopman 1984 score	Biometrics 40:513	Iterative; original score CI for RR
Miettinen-Nurminen score	Miettinen-Nurminen 1985	Regulatory standard; R ratesci::scoreci(contrast='RR')
MOVER-R	Donner-Zou 2012 Stat Methods Med Res 21:347	Construct CI on theta1 - R*theta2 not on R directly; allows asymmetric CIs

Odds ratio

Method	Citation	Verdict
Wald on log(OR)	textbook	Fast; suffers Hauck-Donner effect when cell counts small
Cornfield exact	Cornfield 1956	Exact reference standard; often over-conservative
Mid-p Cornfield	Berry-Armitage 1995 Statistician 44:417	Less conservative; slight under-coverage
Profile likelihood	Venzon-Moolgavkar 1988 Appl Stat 37:87	Transformation-invariant; no Hauck-Donner pathology; R MASS::confint.glm default for glm

Hauck-Donner effect (1977 JASA 72:851; revived by Yee 2022 JASA 117:1763): the Wald test statistic is non-monotonic in the parameter estimate near the boundary -- a large OR can produce a tiny Wald chi-square so the test fails to reject when it should. Use LR or profile-likelihood inference. VGAM::hdeff() detects this in fitted models.

Number Needed to Treat (NNT) -- the Bender 2001 CCT 22:102 Way

Goal: Convert RD to clinically intuitive NNT with a CI that handles the singularity at RD = 0.

Approach: Compute RD CI first, then transform; when RD CI crosses zero, report NNTB(lower) -> infinity -> NNTH(upper).

import numpy as np

def nnt_with_ci(treated_events, treated_n, control_events, control_n, alpha=0.05):
    """Bender 2001 *CCT* 22:102 method; transforms RD CI to NNT CI handling singularity."""
    p_t = treated_events / treated_n
    p_c = control_events / control_n
    rd = p_c - p_t   # positive = treatment helps
    se_rd = np.sqrt(p_t * (1 - p_t) / treated_n + p_c * (1 - p_c) / control_n)
    z = 1.96 if alpha == 0.05 else None
    rd_ci = (rd - z * se_rd, rd + z * se_rd)

    nnt = 1 / rd if rd != 0 else float('inf')
    if rd_ci[0] > 0 and rd_ci[1] > 0:
        return f'NNTB {1/rd_ci[1]:.0f} to {1/rd_ci[0]:.0f}'
    elif rd_ci[0] < 0 and rd_ci[1] < 0:
        return f'NNTH {1/-rd_ci[1]:.0f} to {1/-rd_ci[0]:.0f}'
    else:
        # CI crosses zero -- Bender convention
        if rd > 0:
            return f'NNTB {nnt:.0f} (NNTB {1/rd_ci[1]:.0f} to inf to NNTH {1/-rd_ci[0]:.0f})'
        return f'NNTH {-nnt:.0f} (similar split)'

Bender 2001 Controlled Clinical Trials 22:102-110 resolves the discontinuity at RD = 0: NNT has a singularity there, so a CI that crosses zero produces a disjoint NNT CI ("NNTB(some) -> infinity -> NNTH(some)"). The Cochrane/BMJ convention is to report exactly this -- the infinity in the middle signals non-significance and is more honest than truncating to one side.

NNT from OR + baseline risk (when only the OR is published):

def nnt_from_or(odds_ratio, baseline_risk):
    baseline_odds = baseline_risk / (1 - baseline_risk)
    treatment_odds = baseline_odds * odds_ratio
    treatment_risk = treatment_odds / (1 + treatment_odds)
    arr = abs(baseline_risk - treatment_risk)
    return 1 / arr if arr > 0 else float('inf')

OR	Baseline 5%	Baseline 20%	Baseline 50%
0.5	NNT=42	NNT=12	NNT=6
0.7	NNT=70	NNT=20	NNT=12

Always report baseline risk alongside NNT -- the same OR produces dramatically different NNTs.

Marginal RD via G-Computation -- The FDA 2023 Recipe

Goal: Compute the marginal RD (FDA 2023 primary estimand for binary endpoints) from a fitted logistic regression -- without requiring the analyst to switch model class.

Approach: Standardise over the observed covariate distribution. Predict per-subject probability under treatment=1 AND under treatment=0; take the mean of each; the difference is the marginal RD. SE via influence function or bootstrap.

import statsmodels.formula.api as smf
import numpy as np

# Step 1: Fit logistic with covariates
fit = smf.logit('y ~ z + x1 + x2 + x3', data=df).fit()

# Step 2: Predict under each treatment regime (counterfactual prediction)
df_z1 = df.assign(z=1)
df_z0 = df.assign(z=0)
p_z1 = fit.predict(df_z1)
p_z0 = fit.predict(df_z0)

# Step 3: Marginal estimates
marg_p1 = p_z1.mean()
marg_p0 = p_z0.mean()
marg_rd = marg_p1 - marg_p0          # FDA 2023 primary
marg_rr = marg_p1 / marg_p0
marg_or = (marg_p1 / (1 - marg_p1)) / (marg_p0 / (1 - marg_p0))

# Step 4: SE via bootstrap (or analytical influence function in R `marginaleffects`)
n_boot = 1000
boot_rds = np.zeros(n_boot)
rng = np.random.default_rng(42)
for b in range(n_boot):
    idx = rng.choice(len(df), size=len(df), replace=True)
    fit_b = smf.logit('y ~ z + x1 + x2 + x3', data=df.iloc[idx]).fit(disp=0)
    p1_b = fit_b.predict(df_z1.iloc[idx]).mean()
    p0_b = fit_b.predict(df_z0.iloc[idx]).mean()
    boot_rds[b] = p1_b - p0_b

se_marg_rd = boot_rds.std(ddof=1)
ci_marg_rd = (marg_rd - 1.96*se_marg_rd, marg_rd + 1.96*se_marg_rd)

R equivalent (preferred for confirmatory):

library(marginaleffects)
fit <- glm(y ~ z + x1 + x2 + x3, family = binomial, data = df)
avg_comparisons(fit, variables = 'z', vcov = 'HC3')
# Returns marginal RD with HC3 sandwich SE; CIs; deltamethod_inference

Tsiatis et al 2008 robustness guarantee: under randomisation Z ⊥ X, the g-computation marginal estimator is consistent for marginal ATE EVEN IF the outcome model is misspecified. Efficiency depends on model quality; consistency does not. This is why FDA 2023 accepts marginal RD via g-computation without requiring proof of correct logistic mean structure.

Marginal vs Conditional Effects -- The Core ICH E9(R1) Question

For logistic regression, the maximum-likelihood coefficient on Z in glm(Y ~ Z + X, family=binomial) is the conditional log odds ratio -- the OR comparing Z=1 to Z=0 holding X fixed. Because the OR is non-collapsible, this is different from the marginal log OR even under perfect randomisation. Lin & Wei 1989 first proved the ANCOVA-style estimator from a nonlinear model is generally inconsistent for the marginal ATE; this is the result FDA 2023 implicitly relies on.

# Conditional OR (the default; biased toward null vs marginal OR)
import statsmodels.formula.api as smf
import numpy as np

model = smf.logit('y ~ z + x1 + x2', data=df).fit()
cond_or = np.exp(model.params['z'])

# Marginal RD via g-computation / standardisation:
df_z1, df_z0 = df.assign(z=1), df.assign(z=0)
p_z1 = model.predict(df_z1).mean()
p_z0 = model.predict(df_z0).mean()
marg_rd = p_z1 - p_z0
marg_rr = p_z1 / p_z0
marg_or = (p_z1 / (1-p_z1)) / (p_z0 / (1-p_z0))

# Variance via influence function or bootstrap (RobinCar / marginaleffects)
# R: marginaleffects::avg_comparisons(model, variables='z', vcov='HC3')

The reporting standard post-FDA-2023: marginal estimand as primary (RD with HC3 sandwich SE), conditional as supportive. The two are different parameters, not different estimates of the same parameter. See clinical-biostatistics/logistic-regression for full g-computation machinery and Tsiatis et al 2008 Stat Med 27:4658.

Non-collapsibility -- the standard explanation

A conditional (adjusted) OR typically differs from the marginal (unadjusted) OR even WITHOUT confounding. Conditioning on a prognostic covariate (one that predicts outcome but is unrelated to treatment) changes the OR by a mathematical property -- not bias. In most settings, the conditional OR is further from the null than the marginal OR. This is what postdocs argue about: Frank Harrell argues conditional effects are more clinically meaningful (they apply to individuals); FDA via Permutt argues marginal effects are what regulatory submissions need (the population-level decision). FDA 2023 settles it for regulatory reporting: marginal primary, conditional supportive.

Risk ratios and risk differences are collapsible. Marginal and conditional RR/RD are equal absent confounding. This is why FDA 2023 favours RD as the primary estimand for binary outcomes -- the conditional vs marginal distinction disappears.

Reconciliation: When Methods Disagree

Pattern	Likely cause	Action
Conditional OR (from adjusted logistic) > marginal OR (from g-computation)	Non-collapsibility (Permutt 2020); adjusted ORs further from null than marginal under most covariate structures	Report marginal RD as primary per FDA 2023; conditional OR as supportive with explicit parameter label
Wald CI excludes null but Wilson/Newcombe-Wilson CI overlaps null	Wald has poor coverage near 0 and 1 (Brown-Cai-DasGupta 2001)	Wilson for single proportion; Newcombe-Wilson hybrid or Miettinen-Nurminen for differences; cite as regulatory standard
Crude OR vs stratified MH-OR differ substantially	Confounding by stratification factor OR non-collapsibility OR effect modification	If RCT with stratified randomisation, use stratified analysis (Kahan-Morris 2012); if observational, investigate confounding via stratum-specific OR forest plot
OR vs RR direction agree but magnitudes differ greatly	Outcome prevalence > 10% (OR overstates RR at high baseline risk)	Use modified Poisson (Zou 2004) for direct RR estimation; report both for transparency
NNT from one trial vs another differs despite similar OR	Baseline risk differs across trials (NNT = 1/ARR depends on baseline)	Always report NNT alongside baseline risk; cite Bender 2001 CCT 22:102 for NNTB-inf-NNTH convention
Profile-likelihood CI differs from Wald CI in small-cell scenario	Hauck-Donner effect (Wald non-monotone near boundary; Yee 2022)	Use profile likelihood (R MASS::confint.glm); detect via VGAM::hdeff()
RMST difference and HR give different conclusions on treatment benefit	HR is time-averaged log-HR under PH violation; RMST captures cumulative benefit	Under PH violation, RMST is the more interpretable summary; see clinical-biostatistics/survival-analysis

Modified Poisson Regression for Common Outcomes

When prevalence > 10% and the policy quantity is RR (not OR), modified Poisson with sandwich SE directly estimates the RR (Zou 2004 AJE 159:702):

import statsmodels.api as sm
import numpy as np

poisson_model = sm.GLM(df['outcome'],
                       sm.add_constant(df[['treatment', 'age']]),
                       family=sm.families.Poisson()).fit(cov_type='HC1')
rr_estimates = np.exp(poisson_model.params)
rr_ci = np.exp(poisson_model.conf_int())

cov_type='HC1' (Huber-White sandwich) corrects the over-dispersion that Poisson assumes; without it, SEs are wrong because binary data are NOT Poisson. Use HC3 (jackknife approximation) for n < 250 (Long-Ervin 2000 Am Stat 54:217). HC1 is the Stata default; HC3 is the R sandwich package default -- the difference can flip noninferiority p-values for n < 200.

Forest Plots

import matplotlib.pyplot as plt
import numpy as np

def forest_plot(labels, effects, lower_cis, upper_cis, ref_line=1.0, figsize=(8, 6), log_scale=True):
    fig, ax = plt.subplots(figsize=figsize)
    y_pos = range(len(labels))
    ax.errorbar(effects, y_pos,
                xerr=[np.array(effects) - np.array(lower_cis),
                      np.array(upper_cis) - np.array(effects)],
                fmt='D', color='black', capsize=3, markersize=5)
    ax.axvline(x=ref_line, color='gray', linestyle='--', linewidth=0.8)
    ax.set_yticks(list(y_pos))
    ax.set_yticklabels(labels)
    ax.set_xlabel('Effect estimate (95% CI)')
    if log_scale:
        ax.set_xscale('log')
    plt.tight_layout()
    return fig

Log scale ensures reciprocal effects (OR 0.5 and OR 2.0) appear equidistant from the null. For RD or RMST difference, use a linear scale.

Quantitative Thresholds

Threshold	Source	Rationale
Prevalence > 10% -> prefer RR over OR	Zhang-Yu 1998 JAMA 280:1690	OR overstates RR materially; modified Poisson or log-binomial recovers RR directly
n > 100 per arm for Wald CI on log-OR	Brown-Cai-DasGupta 2001; Newcombe 1998a	Below this, Wald coverage is poor; profile likelihood or score-based preferred
HC1 (Stata default) vs HC3 (R default) for small n	Long-Ervin 2000 Am Stat 54:217	HC3 (jackknife approximation) recommended for n <=250; difference can flip NI p-values
Marginal estimand for primary regulatory report	FDA 2023 Final Guidance	Marginal RD/RR/OR via g-computation; conditional OR is a different parameter (Permutt 2020)
Miettinen-Nurminen CI for RR/RD	Miettinen-Nurminen 1985; FDA/EMA NI margin practice	Regulatory standard; consistent with Pearson chi-square
Bender NNT convention	Bender 2001 Controlled Clinical Trials 22:102-110; Cochrane/BMJ style	When RD CI crosses zero, NNT CI is disjoint; report NNTB(lower) -> inf -> NNTH(upper)

Common Errors

Error / symptom	Cause	Solution
Adjusted OR much larger than unadjusted	Non-collapsibility, not confounding	Cite Permutt 2020; report both with explicit estimand labels (conditional vs marginal)
NNT reported as "NNT=25, CI 12 to -200"	Sign-confused output when RD CI crosses zero	Use Bender 2001 CCT 22:102 convention: "NNTB 25 (NNTB 12 to inf to NNTH 200)"
OR reported without baseline risk for clinical translation	Common in published papers	Always report event rates per arm alongside OR; provide NNT at observed baseline
Table2x2 returns reciprocal OR	Column ordering puts outcome=0 first	Reorder: crosstab[[1, 0]]
Poisson SE much smaller than expected	Forgot cov_type='HC1' or 'HC3'	Always specify sandwich SE for modified Poisson; without it, SEs are wrong
Wald log-RR CI extends below 0 or implausibly	Katz log RR fails for small p	Switch to Miettinen-Nurminen or Koopman score; R ratesci::scoreci(contrast='RR')
Hauck-Donner: huge OR with non-significant Wald p	Wald non-monotonic near boundary	Use profile likelihood: R MASS::confint.glm; detect with VGAM::hdeff()
"Adjusted" model gives smaller effect than expected	Mediator adjustment	Check causal DAG; adjusting for mediators attenuates effect toward null

Anticipated Reviewer Pushback

Pushback	Response
"Is the OR collapsible across the included covariates?"	No -- OR is non-collapsible. Cite Permutt 2020. Report marginal RD via g-computation as primary per FDA 2023.
"Why MN over Wald CI?"	Wald has poor coverage for sparse 2x2; MN matches Pearson chi-square test and is the FDA/EMA standard for NI margins.
"Hauck-Donner pathology check?"	For small cell counts: switch to profile likelihood CI (R MASS::confint.glm); cite Yee 2022 JASA 117:1763.
"Why HC3 over HC1?"	n <=250 favours HC3 per Long-Ervin 2000; HC1 (Stata) and HC3 (R sandwich) can disagree at small n.
"Where is the marginal effect?"	Per FDA 2023, marginal RD/RR via g-computation is the primary estimand for binary outcomes; conditional is supportive.
"Why NNT in 'NNTB-infinity-NNTH' notation?"	Bender 2001 CCT 22:102 convention; standard in BMJ/Lancet/Cochrane. Disjoint CI honestly conveys non-significance.
"Adjustment for stratification factors?"	Strata included in modified Poisson or in MN stratified CI via ratesci::scoreci(stratified=TRUE); ignoring inflates Type-I (Kahan-Morris 2012).
"What about effect modification across subgroups?"	Pooled estimate reported as primary; stratum-specific ORs in forest plot; Breslow-Day test for homogeneity; if heterogeneous, do not pool -- see clinical-biostatistics/subgroup-analysis.
"Post-hoc subgroup OR was significant -- can a claim be made?"	Not credible without pre-specification per EMA 2019 / CONSORT 2025; frame as hypothesis-generating; replicate in independent cohort before claiming.

References

• Agresti A, Caffo B. 2000. Simple and effective confidence intervals for proportions and differences of proportions. Am Stat 54:280-288.
• Altman DG. 1998. Confidence intervals for the number needed to treat. BMJ 317:1309.
• Bender R. 2001. Calculating confidence intervals for the number needed to treat. Controlled Clinical Trials 22:102-110.
• Brown LD, Cai TT, DasGupta A. 2001. Interval estimation for a binomial proportion. Stat Sci 16:101-117.
• Chan ISF, Zhang Z. 1999. Test-based exact confidence intervals for the difference of two binomial proportions. Biometrics 55:1202-1209.
• Cornfield J. 1956. A statistical problem arising from retrospective studies. 3rd Berkeley Symp 4:135-148.
• Donner A, Zou GY. 2012. Closed-form confidence intervals for ratios involving population means via MOVER-R. Stat Methods Med Res 21:347-359.
• FDA. 2023. Adjusting for Covariates in Randomized Clinical Trials. Final Guidance, May 2023.
• Hauck WW, Donner A. 1977. Wald's test as applied to hypotheses in logit analysis. JASA 72:851-853.
• Katz D, Baptista J, Azen SP, Pike MC. 1978. Obtaining confidence intervals for the risk ratio in cohort studies. Biometrics 34:469-474.
• Koopman PAR. 1984. Confidence intervals for the ratio of two binomial proportions. Biometrics 40:513-517.
• Long JS, Ervin LH. 2000. Using heteroscedasticity consistent standard errors in the linear regression model. Am Stat 54:217-224.
• Miettinen O, Nurminen M. 1985. Comparative analysis of two rates. Stat Med 4:213-226.
• Newcombe RG. 1998a. Interval estimation for the difference between independent proportions: comparison of eleven methods. Stat Med 17:873-890.
• Permutt T. 2020. Do covariates change the estimand? Stat Biopharm Res 12:45-53.
• Royston P, Parmar MKB. 2013. Restricted mean survival time. BMC Med Res Methodol 13:152.
• Tsiatis AA, Davidian M, Zhang M, Lu X. 2008. Covariate adjustment for two-sample treatment comparisons in randomized clinical trials. Stat Med 27:4658-4677.
• Venzon DJ, Moolgavkar SH. 1988. A method for computing profile-likelihood-based confidence intervals. Appl Stat 37:87-94.
• Yee TW. 2022. On the Hauck-Donner effect in Wald tests: detection, tipping points, and parameter space characterization. JASA 117:1763-1774.
• Zou G. 2004. A modified Poisson regression approach to prospective studies with binary data. AJE 159:702-706.

Related Skills

• clinical-biostatistics/categorical-tests - 2x2 testing infrastructure that produces ORs/RRs/RDs
• clinical-biostatistics/logistic-regression - Adjusted ORs, g-computation, modified Poisson for marginal RR
• clinical-biostatistics/subgroup-analysis - Forest plots and stratified effect estimates
• clinical-biostatistics/survival-analysis - HR, RMST, hazard-free alternatives for time-to-event
• clinical-biostatistics/trial-reporting - CONSORT 2025 and ICH E9(R1) effect reporting
• clinical-biostatistics/missing-data-sensitivity - Effect measure CIs under MI pooling (Rubin's rules)