Non-Parametric Tests

When to Use Non-Parametric Tests

Non-parametric tests are alternatives to t-tests and ANOVA when your data violate assumptions (especially normality). They work with ranks instead of raw values.

🎯 Why Use Non-Parametric Tests?

When Assumptions Fail

Parametric tests (t-tests, ANOVA) assume: - Normal distribution - Continuous data - Equal variances (sometimes)

If these fail: Use non-parametric alternatives!

Advantages

✓ No normality assumption
✓ Robust to outliers
✓ Work with ordinal data
✓ Work with small samples

Disadvantages

✗ Less statistical power (when assumptions are met)
✗ Can't calculate effect sizes as easily
✗ More conservative

🔀 Non-Parametric Alternatives

Parametric Test	Non-Parametric Alternative	Use When
Independent t-test	Mann-Whitney U	2 independent groups, non-normal
Paired t-test	Wilcoxon Signed-Rank	2 paired samples, non-normal
One-Way ANOVA	Kruskal-Wallis	3+ independent groups, non-normal
Repeated Measures ANOVA	Friedman Test	3+ paired samples, non-normal
Pearson correlation	Spearman correlation	Non-linear or ordinal data

📊 Mann-Whitney U Test

Purpose: Compare two independent groups when data are non-normal

R Code:

# Perform test
wilcox.test(outcome ~ group, data = data)

# Descriptive statistics by group
tapply(data$outcome, data$group, median)
tapply(data$outcome, data$group, IQR)

# Visualization
boxplot(outcome ~ group, data = data,
        main = "Comparison by Group",
        ylab = "Outcome")

Interpreting Output: - W statistic: Test statistic (or U) - p-value: If < .05, groups differ significantly - Report medians instead of means

Reporting: "A Mann-Whitney U test showed that Group A (Mdn = [median], IQR = [IQR]) scored significantly higher than Group B (Mdn = [median], IQR = [IQR]), U = [statistic], p = [p-value]."

🔄 Wilcoxon Signed-Rank Test

Purpose: Compare two paired samples when differences are non-normal

R Code:

# Perform test
wilcox.test(data$pre, data$post, paired = TRUE)

# Descriptive statistics
median(data$pre)
IQR(data$pre)
median(data$post)
IQR(data$post)

# Visualization
boxplot(data$pre, data$post,
        names = c("Pre", "Post"),
        main = "Before vs After",
        col = c("lightblue", "lightgreen"))

Interpreting Output: - V statistic: Test statistic - p-value: If < .05, significant change occurred - Report medians

Reporting: "A Wilcoxon signed-rank test showed that scores significantly increased from pre-test (Mdn = [median], IQR = [IQR]) to post-test (Mdn = [median], IQR = [IQR]), V = [statistic], p = [p-value]."

📈 Kruskal-Wallis Test

Purpose: Compare 3+ independent groups when data are non-normal

R Code:

# Perform test
kruskal.test(outcome ~ group, data = data)

# Descriptive statistics
tapply(data$outcome, data$group, median)
tapply(data$outcome, data$group, IQR)

# Visualization
boxplot(outcome ~ group, data = data,
        main = "Comparison Across Groups",
        col = rainbow(length(unique(data$group))))

# Post-hoc tests (if significant)
pairwise.wilcox.test(data$outcome, data$group,
                     p.adjust.method = "bonferroni")

Interpreting Output: - H statistic: Test statistic (chi-square distribution) - p-value: If < .05, at least one group differs - Post-hoc tests: Show which groups differ

Reporting: "A Kruskal-Wallis test showed a significant effect of group, H(df) = [H], p = [p-value]. Post-hoc Wilcoxon tests with Bonferroni correction revealed that Group A (Mdn = [median]) differed significantly from Group B (Mdn = [median]), but not from Group C."

🔁 Friedman Test

Purpose: Compare 3+ paired samples when data are non-normal (repeated measures)

R Code:

# Data must be in wide format (one row per subject)
# Columns: subject, time1, time2, time3, ...

friedman.test(y = as.matrix(data[, c("time1", "time2", "time3")]))

# Descriptive statistics
apply(data[, c("time1", "time2", "time3")], 2, median)
apply(data[, c("time1", "time2", "time3")], 2, IQR)

# Visualization
boxplot(data[, c("time1", "time2", "time3")],
        names = c("Time 1", "Time 2", "Time 3"),
        main = "Changes Over Time")

Reporting: "A Friedman test showed a significant effect of time, χ²(df) = [statistic], p = [p-value]."

📊 Spearman Correlation

Purpose: Measure relationship strength when data are non-normal or ordinal

R Code:

# Perform test
cor.test(data$var1, data$var2, method = "spearman")

# Visualization
plot(data$var1, data$var2,
     main = "Relationship Between Variables",
     xlab = "Variable 1", ylab = "Variable 2",
     pch = 19, col = "steelblue")

Interpreting Output: - rho (ρ): Correlation coefficient (-1 to 1) - p-value: If < .05, correlation is significant - Interpretation same as Pearson r

Reporting: "A Spearman correlation showed a significant positive relationship between Variable 1 and Variable 2, ρ = [rho], p = [p-value]."

🔍 Choosing the Right Test

Decision Guide

How many groups?

2 independent groups → Mann-Whitney U
2 paired samples → Wilcoxon Signed-Rank
3+ independent groups → Kruskal-Wallis
3+ paired samples → Friedman Test

Examining relationship? → Spearman correlation

⚠️ Important Notes

Assumptions Still Exist!

Non-parametric tests still have some assumptions:

Independence of observations
Similar distribution shapes (for some tests)
Symmetric differences (Wilcoxon)

When in Doubt

If your data meet parametric assumptions, use parametric tests (they have more power). Only use non-parametric when assumptions clearly fail.

Effect Sizes

Effect sizes for non-parametric tests are less standardized. You can report:

r = Z / √N (general effect size)
Median differences
Rank-biserial correlation

📚 Quick Reference Table

Test	R Function	Reports	Effect Size
Mann-Whitney U	`wilcox.test()`	U, medians	r = Z/√N
Wilcoxon	`wilcox.test(paired=TRUE)`	V, medians	r = Z/√N
Kruskal-Wallis	`kruskal.test()`	H, medians	η² (approx)
Friedman	`friedman.test()`	χ², medians	Kendall's W
Spearman	`cor.test(method="spearman")`	ρ	ρ itself

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