McNemar Bowker Test Model Overview

McNemar Bowker Test, also called Bowker's test of symmetry or the McNemar–Bowker extension, evaluates whether paired transitions between every pair of categories are directionally balanced. The same 649 students are classified into four G2 grade bands and again into four G3 grade bands. The test compares each off-diagonal cell with its mirror cell across the main diagonal.

What question does the McNemar Bowker Test answer?

The method asks whether movement from grade band i at G2 to grade band j at G3 occurs as often as movement from j at G2 to i at G3. In the current analysis, the question is whether the complete G2-to-G3 transition matrix is symmetric.

How it extends the ordinary McNemar test

The ordinary McNemar test applies to paired binary classifications and uses one discordant pair. The McNemar Bowker Test applies to a square k × k table and sums a McNemar-style contribution across every informative off-diagonal category pair. For four categories, there can be six possible pairs, although pairs with no transitions in either direction contribute nothing and provide no information.

Worked example at a glance

The observed transition table contains 521 students on the diagonal and 128 students who changed grade band. Upward band transitions total 113, while reverse-direction transitions total 15. The verified Bowker statistic is 76.2815729. Only three category pairs have nonzero discordant totals, so the verified analysis uses df = 3. The verified p-value is 1.9248733 × 10⁻¹⁶.

Quick Answer: McNemar Bowker Test Result

The McNemar Bowker Test rejects the null hypothesis of paired transition symmetry. The result is driven by three informative off-diagonal pairs, all showing substantially more upward than downward movement.

Table of Contents

1. Research question, hypotheses and paired design
2. When to use the McNemar Bowker Test
3. McNemar Bowker Test formula and calculation
4. Variables and data dictionary
5. Worked 4 × 4 transition table and complete results
6. McNemar Bowker versus related paired tests
7. McNemar Bowker Test calculator workflow
8. Six Python and R chart interpretations
9. Python, R, SPSS and Excel workflows
10. Expandable code for the McNemar Bowker Test
11. Assumptions, limitations and advanced interpretation
12. APA reporting for the McNemar Bowker Test
13. Common McNemar Bowker Test mistakes
14. Reports and worked Excel download
15. Related Salar Cafe guides
16. Frequently asked questions
17. McNemar Bowker Test conclusion

Research Question, Hypotheses and Paired Design

Research question

Are G2-to-G3 movements between the four grade bands directionally symmetric, or do some transitions occur more often in one direction than in the reverse direction?

Null and alternative hypotheses

Why this is a paired design

Each row and column classification belongs to the same student. That dependence is essential. An ordinary chi-square test of independence would treat the row and column classifications as if they came from independent observations and would answer a different question. The McNemar Bowker Test uses the mirrored off-diagonal cells created by the paired design.

Grade-band coding used in the workbook

The workbook preserves the category coding as grade bands 1, 2, 3 and 4. Band 1 is the lowest category and band 4 is the highest. Therefore, cells above the main diagonal represent movement to a higher band, and cells below the diagonal represent movement to a lower band.

When to Use the McNemar Bowker Test

Simple test-selection logic

Symmetry is stricter than marginal homogeneity

Full symmetry requires each mirrored cell pair to match. Marginal homogeneity concerns equality of the row and column marginal distributions. Symmetry implies marginal homogeneity, but marginal homogeneity does not necessarily require every individual mirrored pair to be equal.

For background on paired categorical tables and related methods, see Bowker's Test of Symmetry, cross-tabulation and two-way tables and relative frequency.

McNemar Bowker Test Formula and Calculation

The McNemar Bowker Test compares every upper-triangle off-diagonal count with its mirrored lower-triangle count. Each informative pair contributes a squared directional difference divided by the total discordant count for that pair.

Step 1: Identify the mirrored category pairs

With four categories, the possible pairs are 1↔2, 1↔3, 1↔4, 2↔3, 2↔4 and 3↔4. In this transition table, only 1↔2, 2↔3 and 3↔4 contain nonzero discordant totals.

Step 2: Calculate each pair contribution

Step 3: Sum the contributions

Step 4: Determine the degrees of freedom

A four-category square table has a theoretical maximum of 4(4−1)/2 = 6 mirrored pairs. However, three pairs have zero discordant totals and add no information. The verified Python, R and SPSS-aligned analysis therefore uses the number of informative pairs, df = 3.

Step 5: Obtain the p-value

The verified reference p-value is 1.9248733079049484 × 10⁻¹⁶. The Excel formula display reaches the floating-point boundary at 2.220446049250313 × 10⁻¹⁶; the article uses the independently verified reference value.

Variables and Data Dictionary

Observed 4 × 4 transition table

Diagonal and off-diagonal structure

Worked 4 × 4 Transition Table and Complete Results

Primary McNemar Bowker Test result

Contribution shares

Marginal distribution change

The G2 marginal counts are 145, 352, 117 and 35, while the G3 marginal counts are 100, 355, 148 and 46. Band 1 decreases by 45 students, band 2 increases by 3, band 3 increases by 31 and band 4 increases by 11. These marginal changes align with the directional off-diagonal evidence.

Statistical and practical interpretation

The p-value is far below any conventional significance threshold, so the statistical conclusion is unambiguous. The substantive pattern is also clear: 113 students moved upward compared with only 15 moving downward. However, the McNemar Bowker Test does not itself provide a standardized effect-size measure. Report the transition counts, direction and percentages rather than inventing a generic effect size.

McNemar Bowker Test Versus Related Paired Tests

Why the ordinary chi-square test is not a substitute

An ordinary chi-square test treats the two classifications as independent variables. The McNemar Bowker Test uses the paired nature of the table and evaluates mirrored transition counts. The tests answer different questions and use different reference structures.

Why high agreement can coexist with significant asymmetry

Agreement is determined largely by the diagonal, whereas symmetry is determined by the balance of off-diagonal pairs. In this analysis, 80.28% stayed in the same band, yet the remaining changes were overwhelmingly upward. Therefore, high diagonal agreement and strong directional asymmetry occur together.

For a separate agreement-focused analysis, see Cohen's Kappa. For the closely related symmetry method, see Bowker's Test of Symmetry.

McNemar Bowker Test Calculator: Step-by-Step Workflow

A reliable McNemar Bowker Test calculator must accept a square paired transition matrix, compare every mirrored off-diagonal pair, exclude zero-total pairs from informative degrees of freedom when that is the selected implementation, and return the global statistic together with pair-level contributions.

Calculator inputs for the worked example

Calculator step 1: verify the matrix is square

The row and column categories must match in number and meaning. A rectangular table cannot be evaluated with the standard McNemar Bowker symmetry formula.

Calculator step 2: inspect the diagonal and off-diagonal counts

The diagonal counts describe unchanged classifications. The inferential statistic uses only off-diagonal mirrored pairs. In the worked matrix, the diagonal sum is 521 and the off-diagonal sum is 128.

Calculator step 3: compare mirrored pairs

The calculator should display 56 versus 11, 45 versus 3 and 12 versus 1 as the three informative pairs. The three zero-versus-zero pairs should contribute zero.

Calculator step 4: confirm the statistic and df

The expected target is χ²_B = 76.2815729 with 3 informative-pair degrees of freedom. A calculator that automatically reports df = 6 should be checked to determine how it handles zero-total pairs.

Calculator step 5: interpret the direction

The global p-value indicates whether symmetry fails, but pair directions explain how it fails. Here, all informative pairs show more upward than downward movement.

Six Python and R McNemar Bowker Test Chart Interpretations

The Python and R charts cover the same paired transition structure from complementary visual angles. Each chart pair is interpreted once through Pattern, Key Values, Interpretation, Why It Matters and Next Step.

Chart pair 1: Transition table and transition matrix

Pattern

Most observations lie on the diagonal

The largest cells are 89, 296, 102 and 34 on the main diagonal, showing substantial stability between G2 and G3 bands.

Key value: 521 of 649 students, or 80.28%, remained in the same band.

Meaning

Off-diagonal direction is still highly uneven

Despite strong diagonal stability, the most visible off-diagonal cells are 1→2, 2→3 and 3→4, not their reverse cells.

Next step: isolate mirrored cells to assess symmetry rather than agreement.

Chart pair 2: Asymmetry matrices

Pattern

Three adjacent-pair imbalances dominate

The signed differences are +45 for 1→2, +42 for 2→3 and +11 for 3→4 when upward movement is treated as positive.

All nonzero directional differences favor movement to the higher band.

Meaning

The asymmetry has a coherent direction

The result is not produced by random mixed-direction discrepancies. The informative imbalances all point upward.

Why it matters: coherent direction strengthens the substantive interpretation.

Chart pair 3: Pair contributions

Pattern

The 2↔3 pair contributes the most

The contributions are 30.2239, 36.7500 and 9.3077 for the three informative pairs.

The 2↔3 pair accounts for 48.18% of the total χ².

Meaning

The global test is not equally distributed

Nearly half of the asymmetry evidence comes from the strong imbalance between bands 2 and 3.

Next step: report the dominant pair instead of only the global p-value.

Chart pair 4: Pair directions

Pattern

Forward counts exceed reverse counts

The comparisons are 56 versus 11, 45 versus 3 and 12 versus 1.

Upward transitions total 113; downward transitions total 15.

Meaning

The result reflects systematic upward movement

The paired table indicates more progression to higher grade bands from G2 to G3 than regression to lower bands.

Why it matters: direction is central to the educational interpretation.

Chart pair 5: Margins and diagonal agreement

Pattern

The marginal distribution shifts upward

Band 1 falls from 145 to 100, while bands 3 and 4 increase from 117 to 148 and from 35 to 46.

Marginal changes: −45, +3, +31 and +11.

Meaning

Stability and change coexist

The diagonal remains dominant, but the off-diagonal flow changes the overall grade-band distribution.

Next step: report both 80.28% unchanged and the strong asymmetric change pattern.

Chart pair 6: Result summary

Pattern

The final decision is decisive

The summary panels combine the statistic, informative degrees of freedom, p-value and rejection decision.

χ²_B = 76.2816, df = 3, p = 1.92 × 10⁻16.

Meaning

Paired transition symmetry is rejected

The grade-band change pattern cannot be explained as balanced movement in opposite directions.

APA core: χ²_B(3, N = 649) = 76.28, p < .001.

McNemar Bowker Test in Python, R, SPSS and Excel

Expandable Code for the McNemar Bowker Test

table = [
    [89, 56, 0, 0],
    [11, 296, 45, 0],
    [0, 3, 102, 12],
    [0, 0, 1, 34]
]
statistic = 0.0
df = 0
for i in range(len(table)):
    for j in range(i + 1, len(table)):
        nij, nji = table[i][j], table[j][i]
        if nij + nji > 0:
            statistic += (nij - nji) ** 2 / (nij + nji)
            df += 1

x <- matrix(c(
  89,56,0,0,
  11,296,45,0,
  0,3,102,12,
  0,0,1,34
), nrow=4, byrow=TRUE)
stat <- 0
df <- 0
for (i in 1:3) for (j in (i+1):4) {
  total <- x[i,j] + x[j,i]
  if (total > 0) {
    stat <- stat + (x[i,j] - x[j,i])^2 / total
    df <- df + 1
  }
}

Pair contribution = (n_ij - n_ji)^2 / (n_ij + n_ji)
Bowker statistic = SUM(all informative pair contributions)
Informative df = COUNT(pair totals greater than zero)

Assumptions, Limitations and Advanced Interpretation

How zero-total pairs affect the analysis

Pairs 1↔3, 1↔4 and 2↔4 contain no transitions in either direction. Their contribution is mathematically zero. The verified workflow counts only nonzero discordant pairs as informative degrees of freedom, giving df = 3. Analysts should state this convention because some software may display the theoretical maximum df instead.

Asymptotic versus exact inference

The standard Bowker statistic uses a chi-square approximation. When informative discordant totals are very small, an exact or permutation approach may be preferable. In this workbook, the informative pair totals are 67, 48 and 13; the first two are substantial, while the third is smaller but still nonzero.

Ordered-category interpretation

Although Bowker's test is valid as a nominal symmetry test, the ordered grade bands make direction meaningful. The strong excess of upward transitions gives a clearer substantive story than a nominal-label application would provide.

APA Reporting for the McNemar Bowker Test

Common McNemar Bowker Test Mistakes and How to Correct Them

McNemar Bowker Test Reports and Worked Excel Download

Related Salar Cafe Guides

McNemar Bowker Test Conclusion

McNemar Bowker Test is the appropriate extension of McNemar's paired binary method when the same participants are classified into three or more matching categories at two occasions. It tests whether each transition direction is balanced by its reverse direction and therefore reveals systematic change that a simple agreement percentage can miss.

In this worked analysis of 649 students, 521 remained in the same grade band, but the 128 changes were highly directional: 113 upward and only 15 downward. The verified result, χ²_B(3, N = 649) = 76.28, p < .001, rejects transition symmetry decisively. The main evidence came from the 1↔2 and 2↔3 pairs, with an additional contribution from 3↔4.

For the next step, compare the symmetry result with Cohen's Kappa for agreement magnitude, Bowker's Test of Symmetry for related worked interpretation and Categorical Data Analysis in Python for a broader software workflow.