Paired Categorical Data

Overview

Paired categorical data arise when two binary measurements are linked within the same analytical unit or matched pair.

Examples include:

• disease status before and after an intervention;
• two diagnostic tests applied to the same participant;
• matched case–control exposure status;
• two raters classifying the same specimen;
• presence or absence of a symptom at two visits.

For two paired binary outcomes, McNemar’s test evaluates whether the marginal proportions are equal by comparing the two types of discordant pair [1].

The method is not an ordinary chi-square test of independence. Agreement and marginal change are different questions.

Why this matters

Suppose 100 participants are classified as controlled or not controlled at baseline and follow-up.

Participants who are:

• not controlled at both times;
• controlled at both times

show agreement but do not indicate the direction of change.

The directional information comes from:

• not controlled → controlled;
• controlled → not controlled.

McNemar’s test compares these discordant counts.

A large number of concordant pairs can produce high agreement while the marginal proportion still changes. Conversely, equal marginal proportions can coexist with poor individual agreement.

When to use this method

Use McNemar’s test when:

• the outcome is binary at both measurements;
• observations are paired correctly;
• pairs are independent of other pairs;
• the question concerns equality of the two marginal proportions;
• and discordant counts support the chosen asymptotic or exact method.

Do not use it to assess:

• agreement between classifications;
• two independent groups;
• more than two repeated occasions;
• multi-category paired outcomes without an extension;
• or causal intervention effects without an appropriate comparison design.

Research question and variables

Define positive as controlled blood pressure.

The baseline controlled proportion is:

$$p_0=\frac{c+d}{n}$$

The follow-up controlled proportion is:

$$p_1=\frac{b+d}{n}$$

The paired risk difference is:

$$RD_{\text{paired}}=p_1-p_0=\frac{b-c}{n}$$

The McNemar null is:

$$H_0:P(0\rightarrow1)=P(1\rightarrow0)$$

Data structure

Paired binary table

simulated data

Table 1. Blood-pressure control at baseline and follow-up

Baseline status	Follow-up uncontrolled	Follow-up controlled	Total
Uncontrolled	18	32	50
Controlled	8	42	50
Total	26	74	100

Note: The discordant cells are 32 and 8. Counts are simulated.

The table orientation must be stated. Reversing rows or columns reverses the paired-difference sign but not the two-sided test.

Statistical assumptions

Participants missing either measurement do not contribute to the complete-pair McNemar table. Excluding them may bias the result if completeness is related to outcome change.

How the method works

Without continuity correction, the asymptotic statistic is:

For small discordant totals, an exact conditional test treats $b$ as a binomial count among $b+c$ discordant pairs under probability 0.5.

Paired proportion difference

The estimated change is:

$$\widehat{RD}_{\text{paired}} = \frac{b-c}{n}$$

An approximate standard error is:

This interval should be interpreted cautiously when discordant counts are small. Improved paired-binomial interval methods are available [2].

Worked example

Reproducible R analysis

Base R implements McNemar’s test using mcnemar.test() [3].

paired_table <- matrix(
  c(
    18, 32,
    8, 42
  ),
  nrow = 2,
  byrow = TRUE,
  dimnames = list(
    Baseline = c("Uncontrolled", "Controlled"),
    Followup = c("Uncontrolled", "Controlled")
  )
)

fit <- mcnemar.test(
  paired_table,
  correct = FALSE
)

b <- paired_table["Uncontrolled", "Controlled"]
c <- paired_table["Controlled", "Uncontrolled"]
n <- sum(paired_table)

baseline_controlled <- sum(paired_table["Controlled", ]) / n
followup_controlled <- sum(paired_table[, "Controlled"]) / n
paired_difference <- (b - c) / n

paired_se <- sqrt(
  (
    b + c - (b - c)^2 / n
  ) / n^2
)

paired_ci <- paired_difference + c(-1, 1) * 1.96 * paired_se

exact_fit <- binom.test(
  x = b,
  n = b + c,
  p = 0.5,
  alternative = "two.sided"
)

output <- data.frame(
  baseline_percent = 100 * baseline_controlled,
  followup_percent = 100 * followup_controlled,
  difference_pp = 100 * paired_difference,
  ci_lower_pp = 100 * paired_ci[1],
  ci_upper_pp = 100 * paired_ci[2],
  mcnemar_chi_square = unname(fit$statistic),
  asymptotic_p = fit$p.value,
  exact_p = exact_fit$p.value
)

round(output, 3)

R output

	McNemar's Chi-squared test

McNemar's chi-squared = 14.4, df = 1, p-value = 0.0001478

Cleaned output

 baseline_percent followup_percent difference_pp ci_lower_pp ci_upper_pp mcnemar_chi_square asymptotic_p exact_p
             50.0             74.0          24.0      12.531      35.469               14.4     0.000148 0.000182

Report both p-values as $p<0.001$, not p=0.000.

Interpreting the output

Presenting the results

Table 2. Paired change in blood-pressure control

McNemar $\chi^2(1)=14.40$, $p<0.001$; exact discordant-pair $p<0.001$.

Suggested reporting

Paired multi-category outcomes MMed extension

For more than two paired nominal categories, categorical-data extensions are required [4]:

• Stuart–Maxwell test of marginal homogeneity;
• Bowker test of symmetry;
• multinomial marginal models.

For repeated binary outcomes at more than two times, consider:

• generalised estimating equations;
• mixed-effects logistic regression;
• transition models.

These methods answer different marginal or subject-specific questions.

Common mistakes

Knowledge check

Which cells determine McNemar's test?

Answer and reasoning

The two discordant cells: participants changing from uncontrolled to controlled and those changing from controlled to uncontrolled.

Why do the 18 and 42 concordant cells not determine the direction of change?

Answer and reasoning

They represent participants whose status stayed the same. They contribute to marginal prevalence but not to the imbalance between the two change directions.

Does McNemar's test assess agreement?

Answer and reasoning

No. It tests equality of paired marginal proportions. Agreement requires measures such as observed agreement and kappa, interpreted in a separate framework.

Why might an exact McNemar test be preferred?

Answer and reasoning

When the total number of discordant pairs is small, the chi-square approximation may be inaccurate. The exact binomial procedure conditions on the discordant total.

Can this before-and-after result establish programme effectiveness?

Answer and reasoning

No. Without an appropriate control group, secular changes, regression to the mean and co-interventions may explain part of the change.

Key take-home messages

References

1. McNemar Q. Note on the sampling error of the difference between correlated proportions or percentages. Psychometrika. 1947;12(2):153–157. https://doi.org/10.1007/BF02295996
2. Newcombe RG. Improved confidence intervals for the difference between binomial proportions based on paired data. Stat Med. 1998;17(22):2635–2650. https://doi.org/10.1002/(SICI)1097-0258(19981130)17:22%3C2635::AID-SIM954%3E3.0.CO;2-C
3. R Core Team. McNemar’s chi-squared test. R documentation, package stats version 4.6.1. Vienna: R Foundation for Statistical Computing; 2026. https://stat.ethz.ch/R-manual/R-devel/library/stats/html/mcnemar.test.html
4. Agresti A. Categorical data analysis. 3rd ed. Hoboken (NJ): Wiley; 2013.