Fisher Freeman Halton Test Model Overview

The Fisher Freeman Halton Test is a conditional exact test of association for an r × c contingency table. It generalizes the fixed-margin logic of Fisher's exact test beyond 2 × 2 tables by evaluating all permissible tables that retain the observed row totals and column totals.

What Question Does It Answer?

The method tests whether two categorical variables are independent after conditioning on the observed margins. In this worked example, it asks whether school membership—GP versus MS—is independent of four G3 final-grade bands: Low, Medium, High and Very high.

How the Exact Test Is Built

For every table with the same row and column totals, the Fisher Freeman Halton Test calculates a multivariate-hypergeometric probability. A two-sided probability-ordered result sums the probabilities of tables whose conditional probability is less than or equal to the probability of the observed table. Complete enumeration is exact; fixed-margin Monte Carlo sampling estimates the same conditional tail when enumeration is computationally demanding.

How It Differs from Pearson Chi-Square

Pearson's chi-square test uses an asymptotic reference distribution derived from observed and expected counts. The Fisher Freeman Halton Test instead uses the finite set of tables permitted by the margins. The two procedures can agree strongly, as they do here, while still relying on different inferential reference distributions.

Worked Example at a Glance

The analysis contains 649 students, with GP = 423 and MS = 226. The four grade-band totals are Low = 100, Medium = 355, High = 148 and Very high = 46. The observed table probability is 4.8603767 × 10⁻¹⁶. Among 50,000 fixed-margin Monte Carlo tables, none was at least as improbable as the observed table; the add-one estimate is therefore 1/50,001 = 0.0000199996.

Quick Answer: Fisher Freeman Halton Test Result

The Excel workbook, Python report, R report and SPSS output reproduce the same 2 × 4 table and the same primary fixed-margin result.

Table of Contents

1. Research question, hypotheses and fixed-margin design
2. When to use the Fisher Freeman Halton Test
3. Formula, table probability and Monte Carlo calculation
4. Variables, observed table, expected counts and residuals
5. Complete results and interpretation
6. Fisher Freeman Halton versus related tests
7. Calculator workflow and verification checks
8. Python and R chart interpretations
9. Python, R, SPSS, SAS, Stata and Excel workflows
10. Expandable software code
11. Assumptions, limitations and advanced interpretation
12. APA reporting for the Fisher Freeman Halton Test
13. Common mistakes and how to correct them
14. Reports and worked Excel download
15. Related Salar Cafe guides
16. Frequently asked questions
17. Conclusion

Research Question, Hypotheses and Fixed-Margin Design

The Fisher Freeman Halton Test frames this research question as a conditional comparison of the complete table rather than a collection of isolated pairwise contrasts.

Why the margins matter

The row totals are GP = 423 and MS = 226. The column totals are Low = 100, Medium = 355, High = 148 and Very high = 46. The Fisher Freeman Halton Test conditions on all six of those totals. The only random component under the conditional null is how the 649 cases are allocated among the eight interior cells while every margin remains unchanged.

Why this is not a goodness-of-fit test

A chi-square goodness-of-fit test compares one categorical distribution with prespecified expected proportions. The current problem has two categorical variables and a 2 × 4 table. It is therefore an association or homogeneity problem, not a one-variable goodness-of-fit problem.

Independence versus homogeneity language

The same table can be described as an independence question when one sample is classified by two variables, or as a homogeneity question when separate school populations are compared on grade-band proportions. See chi-square test of independence and chi-square test of homogeneity. The fixed-margin exact calculation is mathematically the same once the table and conditioning scheme are defined.

When to Use the Fisher Freeman Halton Test

Small expected counts are common, but not required

The Fisher Freeman Halton Test is often introduced as a remedy for sparse expected cells. That is a useful application, but it is not a formal requirement. In this worked example the minimum expected count is 16.0185 and all expected counts exceed 5, so chi-square assumptions are comfortable. The exact procedure is presented because the fixed-margin analysis is the topic, not because Pearson's approximation fails.

Nominal versus ordered categories

The grade bands are ordered, but the Fisher Freeman Halton Test treats Low, Medium, High and Very high as nominal labels. It tests any departure from independence, not a specifically monotonic shift. If the order is scientifically central, supplement the exact association result with an ordinal trend method rather than claiming that the Fisher Freeman Halton Test itself tests trend.

Fisher Freeman Halton Test Formula and Calculation

The Fisher Freeman Halton Test calculation begins with the conditional probability of the observed table and then expands to the probability-ordered reference tail.

For a table with fixed row totals r_i, fixed column totals c_j, grand total N and observed cells n_ij, the conditional probability of one table is:

Worked factorial components

Why logarithms are used

Direct factorials such as 649! are far beyond ordinary floating-point range. Reliable software calculates log-factorials with the gamma function, adds and subtracts those values, then exponentiates the final log probability. The worked Excel file uses GAMMALN(n+1) for the same reason.

Two-sided probability ordering

For a two-sided Fisher Freeman Halton Test, a common definition sums the conditional probabilities of all permissible fixed-margin tables whose table probability is less than or equal to the observed table probability. This rule generalizes the familiar probability-ordering version of Fisher's exact test. It is not the same as doubling a one-sided tail, and it is not based on the Pearson statistic unless software explicitly selects a statistic-based ordering.

Monte Carlo P-Value, Resolution and Precision

Complete enumeration of every 2 × 4 table with the observed margins can be computationally demanding. The verified workflow therefore draws random tables from the exact fixed-margin distribution and counts how many are at least as improbable as the observed table.

Add-one Monte Carlo estimate

Here B = 50,000 and b = 0, meaning none of the 50,000 sampled tables met the probability-ordering extremeness criterion. The reported estimate is therefore:

What the resolution floor means

At 50,000 resamples, 1/50,001 is the smallest add-one estimate that can be reported. The result strongly supports rejection, but the simulation cannot distinguish a true p-value of 10⁻⁶ from 10⁻¹². Reporting many digits does not create information beyond the resample resolution.

Why the convergence chart falls as 1/(B+1)

The reported sequence—0.001996 at 500, 0.000999 at 1,000, 0.00039984 at 2,500 and 0.0000199996 at 50,000—equals 1/(B+1) at every sample size. This indicates zero qualifying extreme tables in each independent run. It documents a very small p-value but should not be described as smooth convergence to a known exact decimal.

How to improve precision

• Increase B substantially when the precise tail probability matters.
• Use reproducible seeds and report them.
• Run multiple independent seeds as a stability check.
• Use exact enumeration or a network algorithm when computationally feasible.
• Never report “p = 0”; report a bounded or resolution-aware statement.

Variables and Data Dictionary

For the Fisher Freeman Halton Test, every category definition must be fixed before analysis because recoding changes the cells, margins and exact reference set.

The categories must be mutually exclusive and collectively defined. Every student contributes to exactly one school row and one grade-band column. For category-design basics, see categorical and quantitative variables, frequency and relative frequency tables and two-way tables and relative frequencies.

Observed 2 × 4 School-by-Grade-Band Table

Within-school row profiles

The dominant substantive contrast is the low grade band: 30.09% of MS students are in Low compared with 7.57% of GP students. GP has higher proportions in every band from Medium upward. The Fisher Freeman Halton Test evaluates the entire table simultaneously rather than choosing one cell after seeing the data.

Column composition

Among all Low-band students, 68% are from MS and 32% are from GP. By contrast, GP supplies 69.0% of Medium students, 75.7% of High students and 73.9% of Very-high students. These percentages are descriptive and should not be confused with within-school risks.

Expected Counts, Pearson Residuals and Cell Contributions

Expected counts are not used to compute the exact table probability, but they provide familiar descriptive context and identify where the association is concentrated.

The two Low-band cells contribute 48.4976 of the total Pearson χ² = 59.5340, or roughly 81.5% of the asymptotic statistic. This reveals that the school difference is primarily a low-band concentration, with a secondary deficit of MS students in the High band.

Residuals are descriptive diagnostics. The exact p-value is still based on conditional table probabilities, not on treating Pearson residuals as independent tests. For more context, see studentized residuals, Cramer's V and effect size.

Worked 2 × 4 Table and Complete Results

The Fisher Freeman Halton Test result should be read with the school profiles, residuals and Cramer's V rather than as a stand-alone threshold decision.

Primary inference

The Fisher Freeman Halton Test indicates that school and final-grade band are not conditionally independent. The exact-reference Monte Carlo estimate is far below .05. The observed pattern is therefore too improbable to attribute to fixed-margin random allocation alone.

Effect-size interpretation

Cramer's V = 0.3029 is calculated from the Pearson statistic and is best described as a moderate descriptive association in this 2 × 4 table. The Fisher Freeman Halton Test itself does not automatically produce a unique confidence interval for Cramer's V. A bootstrap interval can be added as a separate analysis, but it should not be presented as part of the exact conditional test unless the method is stated.

Practical interpretation

MS has a much larger Low-band share and smaller Medium, High and Very-high shares than GP. The biggest absolute profile difference is 22.52 percentage points in the Low band. Because grade bands are ordered, an applied report can discuss the direction descriptively while noting that the exact test is a general association test rather than an ordinal trend test.

Fisher Freeman Halton Test Versus Related Tests

The Fisher Freeman Halton Test is selected from the table design and conditioning logic, not because its p-value is expected to be smaller than an asymptotic alternative.

Fisher exact test versus Fisher Freeman Halton

The 2 × 2 Fisher exact test and the Fisher Freeman Halton Test use the same conditional principle. The distinction is table size. A 2 × 4 table cannot be reduced to a set of independent 2 × 2 tests without changing the global question and creating multiple-testing problems.

Fisher Freeman Halton versus chi-square

The Pearson result is even smaller numerically, but the methods answer the same null through different reference distributions. Because all expected counts exceed 5, the close qualitative agreement is unsurprising. Method selection should be based on design and inferential preference, not on which p-value is smaller.

Fisher Freeman Halton versus Barnard and Boschloo

Barnard's Exact Test and Boschloo Test are unconditional procedures for 2 × 2 comparisons. They do not directly replace the Fisher Freeman Halton Test for a 2 × 4 table.

Fisher Freeman Halton Test Calculator Workflow

A valid Fisher Freeman Halton Test calculator must disclose whether it uses complete enumeration, a network algorithm or fixed-margin Monte Carlo sampling.

A reliable Fisher Freeman Halton Test calculator must accept the full r × c count matrix, preserve the selected margins, define two-sided extremeness and disclose whether it enumerates every table or estimates the p-value by Monte Carlo.

Calculator inputs for the worked 2 × 4 table

Step-by-step calculator checks

1. Enter counts, not percentages.
2. Confirm the table orientation and all six margins.
3. Select a two-sided probability-ordered test unless a prespecified alternative is available.
4. Choose exact enumeration or a sufficiently large Monte Carlo B.
5. Set and record a random seed for reproducibility.
6. Verify that the observed table probability is about 4.8604 × 10⁻¹⁶.
7. Verify that the 50,000-resample add-one p-value is 1/50,001.
8. Report row profiles and a descriptive association measure beside the p-value.

What a calculator should not claim

A calculator should not label the observed table mass as the p-value, return p = 0, silently switch to Pearson chi-square, or present a confidence interval for the test itself without identifying a separate effect-size or model-based procedure.

Python and R Fisher Freeman Halton Test Chart Interpretations

Each Fisher Freeman Halton Test chart below is tied to the verified table, margins, simulation settings and exact numerical interpretation.

The Python report presents six chart stories with exact values and next-step guidance. The R report presents six complementary figures in three paired rows so observed tables, expected tables, residuals, row profiles, probability scales and the final result can be compared directly.

Python Fisher Freeman Halton Test Charts

The Python charts move from the observed table to expected counts, residual diagnostics, Monte Carlo resolution and a compact result summary. Each chart is interpreted with exact values rather than generic visual commentary.

Chart 1: Observed 2 × 4 tablePython

Pattern

GP is concentrated in Medium and High, while MS has a much larger Low-band cell.

Key Values

GP counts are 32, 245, 112 and 34; MS counts are 68, 110, 36 and 12.

Interpretation

The largest contrast is not a simple sample-size effect. MS has fewer total students but more Low-band students.

Why It Matters

The fixed-margin exact analysis evaluates the entire eight-cell pattern while retaining all row and column totals.

Next step: Confirm the category definitions and margins before interpreting any p-value.

Chart 2: Fixed-margin expected countsPython

Pattern

The expected Low counts are reversed relative to the observed pattern: GP would be expected to have about 65.18 and MS about 34.82.

Key Values

Expected GP counts are 65.177, 231.379, 96.462 and 29.982; MS counts are 34.823, 123.621, 51.538 and 16.018.

Interpretation

All expected cells exceed 16, so the Pearson approximation is not threatened by small expected counts in this example.

Why It Matters

The exact test remains valid; expected counts are included to explain where the association arises.

Next step: Compare observed minus expected and calculate residuals rather than interpreting expected counts alone.

Chart 3: Pearson residual patternPython

Pattern

The most extreme positive residual is MS-Low, and the most extreme negative residual is GP-Low.

Key Values

Residuals are GP: −4.110, 0.895, 1.582, 0.734; MS: 5.622, −1.225, −2.164, −1.004.

Interpretation

MS has far more Low cases than independence predicts, while GP has far fewer. MS also has a notable High-band deficit.

Why It Matters

Residuals identify the cells responsible for the association but do not replace the global exact p-value.

Next step: Use residuals descriptively and avoid eight unadjusted cell-by-cell significance claims.

Chart 4: Monte Carlo exact p-value convergencePython

Pattern

The plotted estimate decreases almost exactly as 1/(B+1).

Key Values

Values are .001996 at B=500, .000999 at 1,000, .00039984 at 2,500, .00019996 at 5,000, .00009999 at 10,000, .0000399984 at 25,000 and .0000199996 at 50,000.

Interpretation

The sequence indicates zero sampled tables meeting the probability-ordering extremeness rule at each run size.

Why It Matters

The chart documents a resolution floor, not proof that the unknown p-value equals the final decimal exactly.

Next step: Increase B or enumerate exactly if a more precise tail estimate is required.

Chart 5: Monte Carlo precisionPython

Pattern

The displayed uncertainty decreases with B and reaches roughly 2 × 10−5 at 50,000 samples.

Key Values

The final displayed Monte Carlo standard-error quantity is approximately 1.99994 × 10−5.

Interpretation

When no extreme tables are observed, ordinary plug-in precision formulas require cautious interpretation because the estimate sits at the add-one boundary.

Why It Matters

The visual appropriately warns readers not to treat a random Monte Carlo result as infinitely precise.

Next step: Report B, seed and the resolution-aware p-value rather than p = 0.

Chart 6: Exact and asymptotic result summaryPython

Pattern

Both the exact-reference estimate and Pearson p-value indicate strong evidence against independence.

Key Values

Monte Carlo p = 2.0 × 10−5; Pearson p = 7.392 × 10−13; −log10 of the exact estimate is about 4.699.

Interpretation

The two methods differ numerically because they use different reference distributions and because the Monte Carlo p-value is resolution-limited.

Why It Matters

The decision is stable across exact and asymptotic approaches, strengthening the substantive conclusion.

Next step: Report the Fisher Freeman Halton Test as primary and Pearson χ² as supporting context.

R Fisher Freeman Halton Test Charts in Paired Rows

The R report reproduces the table, expected counts, residuals, row profiles and exact-probability concepts. The six R charts are arranged in three responsive pairs, with one explanation card aligned to each chart.

R pair 1: Observed and expected 2 × 4 tables

Pattern and meaning

Observed allocation differs sharply in the Low band

The observed R table confirms 32 GP-Low cases and 68 MS-Low cases, despite GP having the larger total sample. Medium and High counts are concentrated in GP.

Margin check: GP 423, MS 226; Low 100, Medium 355, High 148, Very high 46.

Interpretation and check

Expected counts provide the null benchmark

Under independence, Low counts would be 65.18 for GP and 34.82 for MS. The observed allocation is nearly reversed, creating the dominant departure.

Assumption check: the smallest expected count is 16.02, so Pearson comparison is numerically comfortable.

R pair 2: Pearson residuals and row profiles

Pattern and meaning

Residuals isolate the strongest cells

MS-Low has residual +5.62 and GP-Low has −4.11. MS-High is also lower than expected with residual −2.16.

Interpretation check: residual signs reverse across rows because the margins are fixed.

Interpretation and check

Profiles translate counts into practical percentages

Low is 7.57% in GP and 30.09% in MS. GP exceeds MS in Medium, High and Very high bands.

Key difference: the Low-band gap is 22.52 percentage points, MS minus GP.

R pair 3: Exact probability scales and result summary

Pattern and meaning

Table mass and p-value are fundamentally different

The observed table mass is 4.86 × 10−16, whereas the two-sided Monte Carlo p-value is 2.0 × 10−5. The p-value is larger because it aggregates the probabilities of many extreme tables.

Do not label the observed table mass as the test p-value.

Interpretation and check

Primary and supporting tests agree on rejection

The exact-reference analysis is primary; Pearson χ² = 59.534 provides supporting asymptotic context. Both show that school and grade band are associated.

Reporting check: include B = 50,000 and seed 20260716 with the Monte Carlo result.

Fisher Freeman Halton Test in Python, R, SPSS, SAS, Stata and Excel

A reproducible Fisher Freeman Halton Test workflow should return the same observed table, reference margins and inferential conclusion across software environments.

Expandable Fisher Freeman Halton Test Code

import numpy as np
from math import lgamma, exp
from scipy.stats import random_table

observed = np.array([[32, 245, 112, 34],
                     [68, 110,  36, 12]], dtype=int)
row = observed.sum(axis=1)
col = observed.sum(axis=0)
N = observed.sum()

def log_mass(table):
    return (sum(lgamma(x + 1) for x in row)
            + sum(lgamma(x + 1) for x in col)
            - lgamma(N + 1)
            - sum(lgamma(x + 1) for x in table.ravel()))

obs_log_mass = log_mass(observed)
rng = np.random.default_rng(20260716)
B = 50000
extreme = 0
for table in random_table.rvs(row, col, size=B, random_state=rng):
    if log_mass(table) <= obs_log_mass + 1e-12:
        extreme += 1
p_mc = (extreme + 1) / (B + 1)

tab <- matrix(c(32,245,112,34,
                68,110,36,12),
              nrow=2, byrow=TRUE)
rownames(tab) <- c("GP","MS")
colnames(tab) <- c("Low","Medium","High","Very high")
set.seed(20260716)
fisher.test(tab, simulate.p.value=TRUE, B=50000)

CROSSTABS
 /TABLES=school_n BY grade3
 /FORMAT=AVALUE TABLES
 /STATISTICS=CHISQ PHI
 /CELLS=COUNT ROW COLUMN EXPECTED RESID
 /COUNT ROUND CELL.

* In the Exact Tests dialog, request the Fisher-Freeman-Halton exact
* or Monte Carlo result and record the sampling settings.

Log numerator = GAMMALN(row1+1)+GAMMALN(row2+1)
              + SUM(GAMMALN(each column total+1))
Log denominator = GAMMALN(N+1)
                + SUM(GAMMALN(each observed cell+1))
Observed table mass = EXP(Log numerator - Log denominator)

Assumptions, Limitations and Interpretation Safeguards

The Fisher Freeman Halton Test remains valid only when independent cases, mutually exclusive categories and the declared fixed-margin reference model match the study design.

Independence of cases

The {P} does not correct repeated observations, clusters, siblings, classrooms or paired measurements. If students are clustered within classes or schools beyond the row grouping, a multilevel or generalized estimating approach may be necessary. See generalized estimating equations and multilevel regression.

Fixed margins are a modeling choice

Conditional inference treats both row and column totals as fixed for the reference distribution. This is mathematically well-defined, but analysts should still explain why that conditioning is reasonable for the study design. An exact method is not automatically the best method for every data-generating process.

Effect size and confidence interval

The {P} is a significance test. It does not supply one universally accepted effect-size confidence interval. Report the observed row profiles and Cramer's V, and add bootstrap or model-based intervals only when the method is clearly named. See confidence interval and margin of error.

APA Reporting for the Fisher Freeman Halton Test

In formal writing, name the Fisher Freeman Halton Test explicitly and report the Monte Carlo implementation rather than using the vague label “an exact test.”

An APA-style report should identify the table, fixed-margin conditional method, two-sided ordering rule, Monte Carlo sample count, random seed, estimated p-value, descriptive profiles and a suitable association measure. The observed table probability should never be mislabeled as the test p-value.

What to Include in an APA-Style Report

APA-Style Methods Sentence

Full APA-Style Results Example

Reusable APA Templates for the Fisher Freeman Halton Test

APA-Style Table Note

APA Rounding Rules

• Report the Monte Carlo result as p = .000020 or p < .001 in prose, while retaining 1/50,001 in technical documentation.
• Report χ² = 59.53 and Cramer's V = .303.
• Give percentages to one or two decimal places and raw counts in the same table or sentence.
• Never report the simulation result as p = 0.

Common Fisher Freeman Halton Test Mistakes and How to Correct Them

Reports and Worked Excel Download

Related Salar Cafe Guides

Fisher Freeman Halton Test Conclusion

The Fisher Freeman Halton Test provides the global fixed-margin conclusion, while row profiles and residuals explain the practical pattern behind that conclusion.

The {P} provides exact conditional logic for contingency tables larger than 2 × 2. In the verified 2 × 4 school-by-grade-band example, the observed table mass was 4.8604 × 10⁻¹⁶, and no table among 50,000 fixed-margin Monte Carlo replicates was at least as improbable as the observed table. The add-one estimate was therefore 1/50,001 = 0.0000199996, providing strong evidence against conditional independence.

The statistical conclusion is reinforced by Pearson χ²(3) = 59.5340 and Cramer's V = 0.3029. The substantive pattern is concentrated in the Low grade band: MS has a 30.09% Low-band rate compared with 7.57% for GP. A complete report should combine the {P}, simulation settings, observed row profiles, effect-size context and a clear distinction between observed table mass and the two-sided p-value.

Continue with Chi Square Test of Independence, Cramer's V, Cochran Mantel Haenszel Test and Categorical Data Analysis in R for related categorical-analysis workflows.