Boschloo Test: Formula, Statistic, Calculator, Python, R, SPSS and Excel Guide

Each student should contribute one outcome to one group. Exact probability calculations do not correct for repeated measurements, matched observations, family clustering, classroom clustering or school-level dependence. If observations are correlated, the double-binomial probability model may understate uncertainty and a paired or cluster-aware method is required.

The two binomial sample sizes are 383 and 266. Boschloo Test allows the positive counts in both groups to vary under repeated samples and does not condition on the observed combined positive total of 549. If both margins were deliberately fixed by design, Fisher's conditional exact test has a more direct sampling interpretation.

Some software functions treat columns as the two binomial experiments, while conventional printed tables often place groups in rows. A correct implementation must use sample totals 383 and 266. A transposed table that instead treats the positive and negative totals as the samples changes the null sample space and can produce a misleading p-value.

The positive outcome is G3 ≥ 10. The label “positive” is a coding choice, not a judgment of value. Reversing the outcome definition changes directional statistics and effect measures. The event category must be stated before the table is analysed so that the direction of every estimate is reproducible.

Discrete exact procedures do not have one universal two-sided definition. This article uses twice the smaller one-sided Boschloo p-value, clipped at 1. R packages can also provide central or minimum-likelihood conventions. When software packages use different rules, their p-values should not be expected to match exactly.

The unconditional rejection probability changes across the common null probability π and can contain more than one local feature. A coarse grid can miss the supremum and report a p-value that is too small. Reliable software uses refined grids, global optimization or validated quasi-Monte Carlo sampling and checks stability as resolution increases.

Boschloo Test does not require expected counts above 5. In this table, the minimum expected count exceeds 40, so Pearson's chi-square approximation is numerically plausible. The exact procedure remains valid without relying on the approximation, and the cross-method difference comes from the reference rule rather than from sparse expected cells.

The risk-difference, risk-ratio and odds-ratio intervals shown in this post are conventional effect intervals. They are not exact inversions of the primary Boschloo Test. Consequently, one interval can touch or cross its null value at a slightly different boundary from the exact p-value. The interval construction should be named in formal reporting.

G3 is originally measured on a 0–19 scale. Converting it to G3 ≥ 10 versus G3 < 10 discards distance information: a grade of 10 and a grade of 19 receive the same binary code. The threshold should have a substantive interpretation, and a complete study can supplement the binary comparison with an analysis of the original quantitative grade.

The 2 × 2 table is unadjusted. School, prior grades, study time, failures, absences and other characteristics can be related to both sex and the outcome. Boschloo Test describes a marginal association; it does not remove confounding. Use Binary Logistic Regression when conditional or adjusted comparisons are required.

Trying many G3 cutpoints or many subgroup tables and reporting only the smallest p-value inflates the chance of a false-positive conclusion. The primary outcome threshold, group comparison and alternative should be selected before analysis, or a multiplicity adjustment should be incorporated into the study plan.

An exact p-value is exact for the stated probability model, not for every population claim. Generalization depends on how students entered the dataset, and causal conclusions require a suitable design. The current result should be described as an unadjusted association between sex and the G3 threshold outcome.

import numpy as np
from scipy.stats import boschloo_exact

# Rows are outcomes; columns are the two independent samples.
table = np.array([
    [333, 216],  # G3 >= 10
    [50, 50],  # G3 < 10
], dtype=int)

two_sided = boschloo_exact(
    table,
    alternative="two-sided",
    n=4096,
)

greater = boschloo_exact(
    table,
    alternative="greater",
    n=4096,
)

less = boschloo_exact(
    table,
    alternative="less",
    n=4096,
)

print("Fisher-p ordering statistic:", two_sided.statistic)
print("Two-sided Boschloo p-value:", two_sided.pvalue)
print("Greater one-sided p-value:", greater.pvalue)
print("Less one-sided p-value:", less.pvalue)

import math
import numpy as np
from scipy import stats

table_groups = np.array([
    [333, 50],  # Female
    [216, 50],  # Male
], dtype=int)

p_female = 333 / 383
p_male = 216 / 266

risk_difference = p_male - p_female
risk_ratio = p_male / p_female
odds_ratio = (216 / 50) / (333 / 50)

fisher = stats.fisher_exact(table_groups, alternative="two-sided")
pearson = stats.chi2_contingency(table_groups, correction=False)
yates = stats.chi2_contingency(table_groups, correction=True)

print("Female proportion:", p_female)
print("Male proportion:", p_male)
print("Risk difference M-F:", risk_difference)
print("Risk ratio M/F:", risk_ratio)
print("Odds ratio M/F:", odds_ratio)
print("Fisher p:", fisher.pvalue)
print("Pearson p:", pearson.pvalue)
print("Yates p:", yates.pvalue)

# install.packages("exact2x2")
library(exact2x2)

greater_result <- boschloo(
  x1 = 333, n1 = 383,
  x2 = 216, n2 = 266,
  alternative = "greater",
  midp = FALSE
)

less_result <- boschloo(
  x1 = 333, n1 = 383,
  x2 = 216, n2 = 266,
  alternative = "less",
  midp = FALSE
)

two_sided_p <- min(
  1,
  2 * min(greater_result$p.value, less_result$p.value)
)

print(greater_result)
print(less_result)
cat("Two-sided Boschloo p-value:", two_sided_p, "\n")

R packages may also provide central or minimum-likelihood two-sided methods. The selected method must be named whenever results are compared across software.

* Create the binary outcome.
COMPUTE boschloo_outcome = (G3 >= 10).
VALUE LABELS boschloo_outcome
  0 "G3 below 10"
  1 "G3 at least 10".

* Create the independent two-level group.
RECODE sex ("F"=0) ("M"=1) INTO boschloo_group.
VALUE LABELS boschloo_group
  0 "Female"
  1 "Male".
EXECUTE.

CROSSTABS
  /TABLES=boschloo_group BY boschloo_outcome
  /FORMAT=AVALUE TABLES
  /STATISTICS=CHISQ RISK PHI
  /CELLS=COUNT ROW COLUMN EXPECTED RESID SRESID
  /COUNT ROUND CELL.

* Add the Boschloo exact calculation through validated
* embedded Python. Ordinary Fisher output is not Boschloo.

Observed inputs
B4 = 333   Female positive
C4 = 50   Female negative
B5 = 216   Male positive
C5 = 50   Male negative

For every possible positive-count pair (x1, x2):

1. Calculate the one-sided Fisher p-value with HYPGEOM.DIST.
2. Flag a table when Fisher p <= observed Fisher statistic.
3. For each nuisance probability p, calculate:
   BINOM.DIST(x1,n1,p,FALSE) *
   BINOM.DIST(x2,n2,p,FALSE)
4. Sum joint probabilities for flagged tables.
5. Take the maximum across the nuisance grid.
6. Repeat for both directions.
7. Return:
   MIN(1, 2*MIN(less maximum, greater maximum))

Validation target
Observed Fisher-p statistic = 0.0304927910
Greater one-sided Boschloo p = 0.0256936226
Two-sided Boschloo p = 0.0513872452

Fisher's exact test conditions on the observed outcome total and evaluates a hypergeometric sample space. Boschloo Test uses the one-sided Fisher p-value only as an ordering statistic, then calibrates the rejection region with the unconditional double-binomial model. This construction retains exact size control while often reducing the conservatism caused by fixed-margin conditioning.

Discrete exact procedures require a definition of two-sided extremeness. A central method combines directional tails, while a minimum-likelihood method includes tables whose probability or ordering statistic is no greater than the observed table's value. R packages can offer both options. Results should be compared only after confirming that the same convention is used.

The primary result in this article is twice the smaller one-sided Boschloo p-value, clipped at 1. The greater-direction probability is .025694, so the reported two-sided result is .051387. This definition is simple and reproducible, but it is not the only possible two-sided convention in discrete exact inference.

A mid-p calculation reduces the contribution of the observed table, commonly by subtracting half its probability from a conventional exact tail. Mid-p values can be less conservative and more powerful, but they do not provide the same finite-sample size guarantee as the full exact procedure. A mid-p result should be clearly labelled and not mixed with the exact p-value reported here.

The search phrase boschloo berger boos test r combines two related but distinct unconditional strategies. Boschloo Test orders possible tables through Fisher p-values. A Berger-Boos procedure restricts nuisance maximization to a confidence set and adds a small adjustment parameter. R's exact 2 × 2 frameworks can provide both ideas, but they should not be treated as interchangeable names.

An exact p-value does not automatically provide a matching interval for the risk difference, risk ratio or odds ratio. Unconditional exact frameworks can invert tests to construct intervals, but the target parameter, ordering and nuisance treatment must be specified. The intervals displayed in this article are effect-estimation intervals and are not claimed to be exact Boschloo inversions.

The nuisance profile may contain interior peaks or flat regions. Reporting a maximum from a coarse grid without checking nearby values can underestimate the true supremum. A defensible workflow increases the number of sampling points, checks the stability of the p-value and records the numerical controls used by the software.

Boschloo Test can analyse a table containing a zero cell because the exact probabilities remain defined. Ordinary odds ratios or risk ratios can become zero or infinite, and log-based confidence intervals may fail. Use a stated exact interval or a disclosed continuity correction for effect estimation rather than silently adding an arbitrary constant.

Equal sample sizes are not required. The current totals are 383 and 266. The double-binomial model uses each denominator separately, so unequal allocation changes precision and power but does not invalidate the exact test. Sample-size planning should account for the intended allocation ratio.

Power depends on both sample sizes, the baseline probability, the true group difference, alpha, the alternative direction and the exact ordering rule. Exact power is calculated by summing alternative-distribution probabilities over the rejection region. See Statistical Power before selecting a sample size for a new study.

The Two-Proportion Z-Test standardizes the observed proportion difference and uses a normal reference distribution. Boschloo Test uses the finite double-binomial sample space. With large samples and probabilities away from boundaries, both methods may give similar conclusions, but the exact and asymptotic p-values are not mathematically identical.

A one-predictor logistic regression estimates a group odds ratio and can be extended with covariates, interactions and cluster terms. Boschloo Test provides an unadjusted exact comparison without a regression equation. Use Logistic Regression when explanation, adjustment or prediction is the research goal.

A nonsignificant Boschloo Test does not demonstrate that the two probabilities are practically equivalent. Equivalence requires a prespecified margin and an analysis designed to show that the entire plausible difference lies within that margin. The current risk-difference interval permits both a moderate female advantage and a value close to zero.

If several outcome cutpoints, group definitions or exact methods are tried, the chance of obtaining at least one favourable result increases. A transparent analysis defines the primary outcome and exact procedure before inspecting results and labels any additional calculations as sensitivity or exploratory analyses.