5 Important Non-Parametric Tests for Correlated Data and Uncorrelated Data

Introduction

Non-parametric tests provide robust alternatives to parametric methods when data violate assumptions such as normality or homogeneity of variances. These tests are invaluable when dealing with ordinal data, small samples, or skewed distributions.

Parametric Vs Non-Parametric Statistics

They are broadly categorized into tests for correlated (paired) data where measurements come from the same subjects or matched pairs and uncorrelated (independent) data where samples are independent of each other.

Read More- Correlation

Non-Parametric Tests for Correlated Data

1. Spearman’s Rank Difference Correlation (Spearman’s rho)

Spearman’s rank correlation coefficient (ρ) measures the strength and direction of the monotonic relationship between two variables measured at least on an ordinal scale (Spearman, 1904). Unlike Pearson’s correlation, which assumes linearity and interval-level data, Spearman’s rho is based on ranks, making it suitable for ordinal or non-normally distributed data.

Formula:

Spearman Correlation

Assumptions:

• • Data are paired, with each pair independent of others.
  • Variables are at least ordinal.
  • The relationship is monotonic.

Procedure:

1. 1. Assign ranks to both variables.
   2. Calculate the differences in ranks (D).
   3. Square and sum the differences.
   4. Compute ρ using the formula.

Interpretation:

• • ρ = +1 = strong positive monotonic relationship.
  • ρ = -1 = strong negative monotonic relationship.
  • ρ = 0 = no monotonic association.

Applications: Psychology (e.g., ranking non-normal test scores), education (performance rankings), medicine (non-linear biomarker associations).

2. Sign Test

The Sign Test is a simple, distribution-free method to test whether the median difference between paired observations is significantly different from zero (Conover, 1999). It uses only the direction (+/-) of differences, not their magnitude.

Assumptions:

• • Paired data with interpretable directional differences.
  • No assumption of normality.

Procedure:

1. 1. Determine if each pair’s second measurement is higher (+), lower (-), or equal (0).
   2. Exclude ties.
   3. Count + and – signs.
   4. Use the binomial distribution (p=0.5) to evaluate if one sign significantly exceeds the other.

Interpretation: If one sign significantly outnumbers the other, the null hypothesis of no median difference is rejected.

Applications: Pre/post intervention studies, preference testing, quality control.

3. Wilcoxon Signed-Rank Test

The Wilcoxon Signed-Rank Test improves on the Sign Test by considering both the sign and the magnitude of differences (Wilcoxon, 1945). It tests whether the median of paired differences differs from zero.

Assumptions:

• • Paired differences are symmetrically distributed around the median.
  • Paired data on at least an ordinal scale.

Procedure:

1. 1. Compute differences between paired observations.
   2. Rank the absolute differences (ignoring signs).
   3. Reapply signs to ranks.
   4. Sum positive and negative ranks separately.
   5. The smaller rank sum (W) is the test statistic.

Interpretation: A small W suggests systematic differences in one direction; significance is determined by comparing W to critical values or calculating p-values.

Applications: Repeated measures in psychological or clinical studies, when normality cannot be assumed.

Comparing Tests for Correlated Data

• • Spearman’s rho assesses association strength, not differences.
  • Sign Test is simple but insensitive to magnitude.
  • Wilcoxon Signed-Rank uses both rank and sign, increasing statistical power.

Non-Parametric Tests for Uncorrelated Data

3. Mann-Whitney U Test

The Mann-Whitney U Test (Wilcoxon rank-sum test) compares two independent samples to test whether they come from the same distribution (Mann & Whitney, 1947).

Assumptions:

• • Two independent samples.
  • Data measured on ordinal, interval, or ratio scales.
  • Distributions of groups have similar shapes.

Procedure:

1. 1. Combine both groups’ observations and rank all data.
   2. Sum ranks for each group.
   3. Compute U statistics for both groups:
   4. The smaller of U1 and U2 is compared to critical values or converted to p-values.

Interpretation: A small U suggests a significant difference between groups.

Applications: Comparing treatment vs. control groups, analyzing gender differences, product comparisons.

5. Kruskal-Wallis H Test

The Kruskal-Wallis Test generalizes the Mann-Whitney U Test to more than two independent groups, similar to a non-parametric one-way ANOVA (Kruskal & Wallis, 1952).

Assumptions:

• • Independent samples from k groups.
  • Ordinal or higher-level data.
  • Distributions across groups have similar shapes.

Procedure:

1. 1. Rank all data combined across groups.
   2. Compute the sum of ranks for each group.
   3. Calculate H statistic:
   4. H approximates a chi-square distribution with k-1 degrees of freedom.

Interpretation: If H exceeds the chi-square critical value, the null hypothesis of identical distributions is rejected. Post hoc tests like Dunn’s test can identify which groups differ.

Applications: Comparing multiple treatments, socioeconomic groups, or teaching methods.

Comparing Tests for Uncorrelated Data

• • Mann-Whitney U is for two independent samples.
  • Kruskal-Wallis extends to two or more groups.
  • Both rely on rank sums, not means, and are robust to non-normal data.

Parametric and Non-Parametric Alteratives

Strengths and Limitations of Non-Parametric Tests

Strengths:

• • No assumptions about normality or homogeneity of variance.
  • Work with ordinal and skewed data.
  • More resistant to outliers than parametric tests.

Limitations:

• • Less statistical power than parametric tests when assumptions for parametric methods hold.
  • Assumptions about similar distribution shapes are often overlooked.
  • Interpretation focuses on medians or rank distributions rather than means.

Conclusion

Non-parametric tests for correlated data Spearman’s rho, Sign Test, and Wilcoxon Signed-Rank Test—and for uncorrelated data—the Mann-Whitney U and Kruskal-Wallis tests equip researchers with robust tools to analyze non-normal, ordinal, or small-sample data. Mastery of these tests allows valid inference across diverse fields, including psychology, education, medicine, and social sciences, ensuring reliable conclusions when traditional parametric assumptions fail.

References

Conover, W. J. (1999). Practical nonparametric statistics (3rd ed.). Wiley.

Kruskal, W. H., & Wallis, W. A. (1952). Use of ranks in one-criterion variance analysis. Journal of the American Statistical Association, 47(260), 583–621.

Mann, H. B., & Whitney, D. R. (1947). On a test of whether one of two random variables is stochastically larger than the other. Annals of Mathematical Statistics, 18(1), 50–60.

Siegel, S., & Castellan, N. J. (1988). Nonparametric statistics for the behavioral sciences (2nd ed.). McGraw-Hill.

Spearman, C. (1904). The proof and measurement of association between two things. The American Journal of Psychology, 15(1), 72–101.

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