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Compare Two Related Measurements Without Assuming Normality

Source: R/wilcoxon_test.R
wilcoxon_test.Rd

wilcoxon_test() compares two paired measurements from the same subjects when your data isn't normally distributed. It's the non-parametric alternative to the paired t-test.

Think of it as:

  • A way to test whether scores changed between two time points

  • Comparing ratings of two items from the same respondents

  • A robust paired comparison that works with any data shape

The test tells you:

  • Whether scores are significantly different between the two measurements

  • How many subjects increased, decreased, or stayed the same

  • The strength of the effect (effect size r)

Usage

wilcoxon_test(data, x, y, weights = NULL, conf.level = 0.95)

Arguments

data

Your survey data (a data frame or tibble) in wide format, with one row per subject and the two measurements in separate columns

x

The first measurement variable (e.g., pre-test, trust in government)

y

The second measurement variable (e.g., post-test, trust in media). The difference is computed as y - x

weights

Optional survey weights for population-representative results

conf.level

Confidence level for intervals (Default: 0.95 = 95 percent)

Value

Test results showing whether the two measurements differ, including:

  • Z statistic (standardized test statistic, normal approximation)

  • P-value (is there a significant difference?)

  • Effect size r (how strong is the difference?)

  • Rank statistics (negative ranks, positive ranks, ties)

Details

Understanding the Results

P-value: If p < 0.05, the two measurements are significantly different

  • p < 0.001: Very strong evidence of a difference

  • p < 0.01: Strong evidence of a difference

  • p < 0.05: Moderate evidence of a difference

  • p > 0.05: No significant difference found

Effect Size r (How strong is the difference?):

  • < 0.1: Negligible effect

  • 0.1 - 0.3: Small effect

  • 0.3 - 0.5: Medium effect

  • 0.5 or higher: Large effect

Rank Categories:

  • Negative Ranks: subjects where y < x (decreased)

  • Positive Ranks: subjects where y > x (increased)

  • Ties: subjects where y = x (no change)

When to Use This

Use Wilcoxon signed-rank test when:

  • Comparing two related measurements from the same subjects

  • Your data is not normally distributed

  • You have ordinal data (ratings, rankings)

  • Sample size is small

  • You want a robust alternative to the paired t-test

Relationship to Other Tests

  • For normally distributed paired data: Use paired t-test instead

  • For independent groups: Use mann_whitney() instead

Weighted variants

SPSS NPAR TESTS ignores WEIGHT BY, so weighted results have no SPSS reference. The weighted variant is an R-only frequency-weight extension that reduces exactly to the unweighted test when all weights equal 1 (enforced by an internal invariance suite); see vignette("spss-compatibility") for validation status.

References

Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics Bulletin, 1(6), 80-83.

Fritz, C. O., Morris, P. E., & Richler, J. J. (2012). Effect size estimates: current use, calculations, and interpretation. Journal of Experimental Psychology: General, 141(1), 2.

See also

wilcox.test for the base R Wilcoxon test.

mann_whitney for comparing two independent groups.

Other hypothesis_tests: ancova(), binomial_test(), chi_square(), chisq_gof(), factorial_anova(), fisher_test(), friedman_test(), kruskal_wallis(), mann_whitney(), mcnemar_test(), oneway_anova(), t_test()

Examples

# Load required packages and data
library(dplyr)
data(survey_data)

# Compare trust in government vs trust in media
survey_data %>%
  wilcoxon_test(x = trust_government, y = trust_media)
#> Wilcoxon Signed-Rank Test: trust_media - trust_government
#>   Z = -5.097, p < 0.001 ***, r = 0.123 (small), N = 2227
#> Use summary() for detailed output.

# Weighted analysis
survey_data %>%
  wilcoxon_test(x = trust_government, y = trust_media,
                weights = sampling_weight)
#> Wilcoxon Signed-Rank Test: trust_media - trust_government [Weighted]
#>   Z = -5.035, p < 0.001 ***, r = 0.121 (small), N = 2242
#> Use summary() for detailed output.

# Grouped analysis (separate test per region)
survey_data %>%
  group_by(region) %>%
  wilcoxon_test(x = trust_government, y = trust_media)
#> [region = 1]
#> Wilcoxon Signed-Rank Test: trust_media - trust_government
#>   Z = -2.727, p = 0.006 **, r = 0.147 (small), N = 435
#> [region = 2]
#> Wilcoxon Signed-Rank Test: trust_media - trust_government
#>   Z = -4.346, p < 0.001 ***, r = 0.117 (small), N = 1792
#> Use summary() for detailed output.