bio-clinical-biostatistics-logistic-regression

What this skill does

## Version Compatibility

Reference examples tested with: statsmodels 0.14+, scipy 1.12+, numpy 1.26+, pandas 2.1+, firthmodels 0.3+, marginaleffects 0.0.13+ (Python) / 0.20+ (R). R packages cited: RobinCar, marginaleffects, brant, MASS, VGAM.

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.

# Logistic Regression for Clinical Outcomes

**"Model clinical outcomes with logistic regression"** -> Estimate the marginal or conditional treatment effect on a binary or ordinal endpoint using a model that respects randomisation stratification, declares its estimand, and survives covariate misspecification.

**Conditional vs marginal (the non-collapsibility subtlety):** the OR is non-collapsible. The conditional OR from logistic regression is a *different parameter* than the marginal OR, even when there is NO confounding and randomisation is perfect. This is mathematical, not statistical bias. FDA 2023 favours marginal RD (via g-computation) for primary reporting to avoid parameter ambiguity. See clinical-biostatistics/effect-measures and Permutt 2020.

## Algorithmic Taxonomy

| Approach | Estimand | Inference | Strength | Fails when |
|----------|----------|-----------|----------|------------|
| Unadjusted logistic / chi-square | Marginal OR | Wald or LR | Simple; transparent | Loses efficiency vs adjusted (Senn 2013); inflates SE under stratified randomisation (Kahan-Morris 2012) |
| Logistic with covariates (ML, Wald CI) | **Conditional** log-OR | Wald | Standard; widely available | Conditional OR != marginal OR due to non-collapsibility (Permutt 2020); not the FDA 2023 primary estimand |
| Logistic + g-computation / standardisation | **Marginal** RD/RR/OR | Influence function or bootstrap SE | FDA 2023 recommended primary estimand for binary | Requires correct outcome model AND post-fit standardisation; needs robust SE machinery |
| Targeted Maximum Likelihood (TMLE) | Marginal RD/RR/OR | Influence function | Provably efficient; doubly robust in observational | Implementation heavier; mostly R (`tmle`, `tmle3`); rare in confirmatory submissions |
| Modified Poisson with sandwich SE | Marginal RR | HC1/HC3 sandwich | Direct RR estimation when prevalence >10% | Slightly less efficient than log-binomial when log-binomial converges |
| Log-binomial regression | Marginal RR | Wald | Direct RR estimation | Frequent convergence failure when predicted risk near 1 |
| Firth penalised logistic | Conditional OR (penalised) | Penalised LR test preferred | Handles separation, rare events (<5% prevalence) | Wald CI/p liberal; must use PLR test (Heinze-Schemper 2002) |
| Ordinal logistic (proportional odds) | Common conditional OR across cut-points | Wald or LR | Preserves ordering information | Proportional odds assumption violation (Brant test) |
| Partial proportional odds | PO holds for some covariates, not others | Hybrid | Salvages ordinal model when PO fails for one predictor | Increased complexity; harder interpretation |
| Multinomial logistic | Per-category log-OR | Wald | No PO assumption needed | Loses efficiency; harder communication |

**Postdoc reading list:** Permutt 2020 *Stat Biopharm Res* 12:45 (conditional vs marginal estimand); FDA May 2023 Final Guidance "Adjusting for Covariates in RCTs" (the regulatory rulebook); Tsiatis et al 2008 *Stat Med* 27:4658 (robust ANCOVA framework); Senn 2013 *Stat Med* 32:1439 (precision from adjustment even under perfect balance); Kahan-Morris 2012 *Stat Med* 31:328 (must adjust for stratification factors).

## Decision Tree by Scenario

| Scenario | Recommended approach | Why |
|----------|---------------------|-----|
| RCT, binary primary endpoint, no covariate adjustment in SAP | Unadjusted logistic with marginal OR; chi-square supportive | Simple regulatory case; consider whether covariate adjustment would gain power (Senn 2013) |
| RCT, binary primary, covariates pre-specified, FDA 2023 compliant | **Logistic adjusted + g-computation for marginal RD**; conditional OR supportive | FDA 2023 final: marginal estimand for primary; mention HC3 sandwich SE |
| RCT, stratified randomisation (sex/region/severity) | Include strata as covariates in logistic; the analytic decision is non-optional | Kahan-Morris 2012: ignoring strata inflates Type-I error up to 30% |
| Common outcome (prevalence >10%) where RR is the policy quantity | Modified Poisson + HC1/HC3 OR log-binomial regression | Avoid OR misinterpretation; Zou 2004; cite EMA 2015 on covariate adjustment |
| Rare event (<5% prevalence) or separation observed | Firth penalty + penalised LR test for p-value | Heinze-Schemper 2002; Wald p from Firth is liberal |
| Ordinal outcome, PO assumption supportable | Proportional odds; verify with Brant test (R `brant::brant`) | Most efficient when PO holds; common in toxicity grading and PROs |
| Ordinal outcome, PO fails on one or two predictors | Partial PO model (R `VGAM::vglm(..., cumulative(parallel=FALSE~X))`) | Salvages most efficiency; only the offending coefficient gets per-cut-point estimate |
| Ordinal outcome, PO fails widely | Multinomial logistic | Worst case; loses ordering info but valid |
| Single arm observational with strong confounding | Logistic adjusted + g-computation + propensity weighting | Doubly robust; cite Hernan-Robins; consider TMLE |
| Binary endpoint, longitudinal repeated measures | GEE or generalised linear mixed model | Sandwich SE for GEE; mixed-model OR is subject-specific not marginal |

## Standard Workflow

**Goal:** Fit a covariate-adjusted logistic regression for a binary clinical endpoint with explicit reference category and pre-specified covariates.

**Approach:** Use the formula API for automatic categorical handling and explicit reference; report both the conditional log-OR and the marginal RD via g-computation.

```python
import statsmodels.formula.api as smf
import pandas as pd
import numpy as np

# CRITICAL: set reference category explicitly. statsmodels default is alphabetical,
# so 'Active' sorts before 'Placebo' and the OR direction silently flips.
model = smf.logit(
    'outcome ~ C(ARM, Treatment(reference="Placebo")) + age + C(sex) + baseline_score',
    data=df
).fit()

# Conditional ORs (Wald)
or_table = pd.DataFrame({
    'OR': np.exp(model.params),
    'Lower_CI': np.exp(model.conf_int()[0]),
    'Upper_CI': np.exp(model.conf_int()[1]),
    'p_value': model.pvalues
})

# Marginal RD via g-computation (the FDA 2023-recommended primary estimand):
df_active = df.assign(ARM='Active')
df_placebo = df.assign(ARM='Placebo')
risk_active = model.predict(df_active).mean()
risk_placebo = model.predict(df_placebo).mean()
marginal_rd = risk_active - risk_placebo
# For SE, use influence-function or bootstrap; see RobinCar / marginaleffects in R
```

**The reference-category trap is the single most common silent bug.** `smf.logit('y ~ C(ARM)')` uses alphabetical ordering, so `ARM='Active'` becomes 1 and `ARM='Placebo'` becomes 0 -- but with `ARM=['Active','Placebo','Control']`, 'Active' is the reference and 'Placebo' is the comparison. Always pass `Treatment(reference="Placebo")` or equivalent. R `glm(y ~ relevel(ARM, ref='Placebo'))`.

## Marginal vs Conditional Estimand -- The FDA 2023 Pivot

**The single most important methodological shift in clinical biostatistics 2020-2025.** Permutt 2020 established and FDA 2023 codified: the maximum-likelihood coefficient on Z in `glm(Y ~ Z + X, family=binomial)` is the **conditional log-OR** -- the OR comparing Z=1 to Z=0 holding X fixed. Due to OR non-collapsibility, this is a *different parameter* than the **marginal log-OR** (the OR comparing the entire treated population to the entire control population), even und