bio-clinical-biostatistics-categorical-tests

What this skill does

## Version Compatibility

Reference examples tested with: scipy 1.12+ (Boschloo and Barnard added in 1.7), statsmodels 0.14+, pingouin 0.5+, exact2x2 (R) 1.6+, pandas 2.1+, numpy 1.26+.

Before using code patterns, verify installed versions match. If versions differ:
- Python: `pip show <package>` then `help(module.function)` to check signatures
- R packages cited for reference (exact2x2, Exact, ratesci): use `packageVersion()` then `?function_name` to verify parameters

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

# Categorical Association Tests for Clinical Data

**"Test association between categorical variables"** -> Determine whether treatment and a categorical clinical outcome are statistically independent (or that marginal proportions agree, for paired data) using a test calibrated to the design, the sample size, and the regulatory question.

## Algorithmic Taxonomy

| Test | Design | Asymptotic / exact | Conditioning | Strength | Fails when |
|------|--------|--------------------|--------------|----------|------------|
| Pearson chi-square (no continuity correction) | Independent groups, any RxC | Asymptotic | None | Standard for n>=40 with all expected counts >=5; matches Miettinen-Nurminen score CI | Any expected cell <1; >20% of cells with expected <5 (Cochran 1954) |
| Fisher's exact (conditional) | Independent 2x2 | Exact | Conditions on BOTH margins | Exact small-sample guarantee on level | Conservative (true alpha << nominal); discards information by double conditioning (Mehta-Senchaudhuri 2003) |
| Boschloo's exact | Independent 2x2 | Exact unconditional | Conditions on ONE margin only | Uniformly more powerful than Fisher (Boschloo 1970; Mehta-Senchaudhuri 2003); preserves nominal alpha exactly | Computationally heavier; RxC extensions limited |
| Barnard's exact | Independent 2x2 | Exact unconditional | Conditions on ONE margin only | Maximises nuisance parameter; well-calibrated | Slightly less powerful than Boschloo on average; compute scales O(n^2) |
| CMH (Mantel-Haenszel) | Stratified independent groups | Asymptotic | Conditions within strata | Tests common-OR null across strata; pooled OR estimator | Assumes no qualitative interaction; misleading when ORs reverse direction across strata |
| Breslow-Day | Stratified independent groups | Asymptotic | Within strata | Tests homogeneity of stratum ORs | Underpowered with few strata or sparse strata; non-significance does NOT prove homogeneity |
| McNemar (asymptotic, no continuity correction) | Paired binary | Asymptotic | Conditions on discordant pairs | Fagerland 2013 default; outperforms exact conditional | Discordant pair count b+c < 25 (chi-square approximation breaks) |
| Mid-p McNemar | Paired binary | Quasi-exact | Discordant pairs | Fagerland-Lydersen-Laake 2013 recommended default; less conservative than exact conditional | Slight under-coverage tolerable at small b+c |
| Exact conditional McNemar (Liddell 1983) | Paired binary | Exact | Discordant pairs only | Guaranteed coverage | Over-conservative; loses power vs mid-p or unconditional |
| Suissa-Shuster exact unconditional | Paired binary | Exact unconditional | All N pairs | Uniformly more powerful than exact conditional McNemar; 20-40% smaller n for same power | Implementation only in R `exact2x2::mcnemarExactDP` and SAS macros |

**Postdoc reading:** Lydersen, Fagerland & Laake 2009 *Stat Med* 28:1159 ("Recommended tests for association in 2x2 tables") argues **Fisher's exact should be retired from routine use** in favour of Boschloo or asymptotic Pearson; Fagerland-Lydersen-Laake 2013 *BMC Med Res Methodol* 13:91 makes the parallel case for mid-p or asymptotic McNemar over exact conditional. Regulatory practice (FDA reviewers) is moving in this direction but Fisher and exact conditional McNemar remain entrenched in many SAPs by inertia.

## Decision Tree by Experimental Scenario

| Scenario | Recommended test | Why |
|----------|------------------|-----|
| Independent 2x2, all expected >=5, n>=40 | Pearson chi-square, `correction=False` | Standard asymptotic; Yates' continuity correction is overly conservative and now discouraged |
| Independent 2x2, expected <5 in any cell OR n<40 | Boschloo's exact (`scipy.stats.boschloo_exact`) | Uniformly more powerful than Fisher; preserves exact Type-I control |
| Independent RxC, expected <5 in >20% of cells | Permutation chi-square or Fisher-Freeman-Halton (R `coin::chisq_test(distribution = approximate())`) | Exact RxC asymptotic invalid; permutation preserves level |
| Stratified design (multi-site, multi-stratum randomisation) | CMH for the pooled test + Breslow-Day for homogeneity + per-stratum ORs | Stratification factor in randomisation MUST appear in analysis (Kahan-Morris 2012 *Stat Med* 31:328: ignoring inflates Type-I error by up to 30%) |
| Stratified with sign-reversing effect (Simpson's paradox suspected) | Always report stratum-specific ORs + visual diagnostic; consider logistic regression with interaction | CMH pooled OR can mask sign reversal; homogeneity test underpowered |
| Paired binary (pre/post on same subjects; matched case-control) | Asymptotic McNemar (`mcnemar(table, exact=False, correction=False)`) when b+c >= 25; mid-p when b+c < 25 | Fagerland 2013 simulations show mid-p and asymptotic outperform exact conditional |
| Matched-pair non-inferiority (especially diagnostics) | Suissa-Shuster exact unconditional via R `exact2x2` | 20-40% smaller n than exact conditional for same power |
| Composite endpoint (any of several events) | Logistic regression with covariate adjustment, not chi-square | Composite changes the estimand under ICH E9(R1); see clinical-biostatistics/effect-measures |

## Chi-Square Test (Pearson, no continuity correction)

**Goal:** Test whether treatment group and outcome category are independent under asymptotic Type-I control.

**Approach:** Construct a contingency table, verify expected cell counts, compute the Pearson chi-square statistic without Yates' continuity correction.

```python
from scipy.stats import chi2_contingency
import pandas as pd

table = pd.crosstab(df['treatment'], df['outcome'])
chi2, p, dof, expected = chi2_contingency(table, correction=False)
if (expected < 5).any():
    print('WARNING: switch to Boschloo (2x2) or permutation chi-square (RxC)')
```

**Cochran 1954 rule (the precise version, not the textbook caricature):** "no expected cell should be <1 AND no more than 20% of cells should have expected <5." The textbook "all >=5" rule is the conservative simplification. With well-balanced 2x2 trials this rarely matters; with sparse RxC tables it materially expands the asymptotic range. The R `chisq.test` issues a warning under the strict Cochran rule; Python users must check manually.

**Why Yates' correction is now discouraged:** continuity correction was introduced to approximate the exact distribution under H0 but inflates Type-II error by ~10% (D'Agostino, Casagrande, Pike 1988 *Stat Med* 7:347). Modern computing makes Boschloo's exact test cheap; the correct fix for sparse 2x2 is Boschloo, not chi-square + continuity.

## Fisher's Exact -- and why Boschloo is usually better

**Goal:** Test 2x2 association with exact Type-I control.

**Approach:** Use Fisher's exact only when historical SAP requires it; otherwise prefer Boschloo's test.

```python
from scipy.stats import fisher_exact, boschloo_exact

odds_ratio, p_fisher = fisher_exact(table.values, alternative='two-sided')

# Boschloo (uniformly more powerful than Fisher):
# n= controls Sobol sampling resolution for the null distribution (scipy 1.12+; default 32);
# higher = more precise p-value at higher CPU cost. NOT the sample size per arm.
result = boschloo_exact(table.values, alternative='two-sided', n=64)
p_boschloo = result.pvalue
```

**The conditioning critique (Mehta-Senchaudhuri 2003):** Fisher's exact conditions on both