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Main Non-Parametric Methods and Their Uses

Non-parametric methods provide powerful alternatives to parametric tests, especially useful when data do not meet the assumptions required for parametric analyses. Here’s a concise overview of the main types of non-parametric methods and their typical applications:

1. Mann-Whitney U Test (Wilcoxon Rank-Sum Test)

2. Wilcoxon Signed-Rank Test

3. Kruskal-Wallis Test

4. Friedman Test

5. Spearman’s Rank Correlation

6. Chi-Square Test

Applications and Considerations:

ToothGrowth Dataset Summary

The ToothGrowth dataset in R examines the impact of Vitamin C on tooth growth in guinea pigs. It includes 60 observations across three variables:

This dataset is useful for statistical analysis, exploring how Vitamin C dosage and supplement type influence tooth growth.

# Load the ToothGrowth data
data("ToothGrowth")

# Load necessary libraries
library(ggplot2)
library(dplyr)
library(tidyr)

# Summary statistics
summary(ToothGrowth)
aggregate(len ~ supp + dose, data = ToothGrowth, FUN = function(x) c(mean = mean(x), sd = sd(x)))

      len        supp         dose      
 Min.   : 4.20   OJ:30   Min.   :0.500  
 1st Qu.:13.07   VC:30   1st Qu.:0.500  
 Median :19.25           Median :1.000  
 Mean   :18.81           Mean   :1.167  
 3rd Qu.:25.27           3rd Qu.:2.000  
 Max.   :33.90           Max.   :2.000
A data.frame: 6 × 3
suppdoselen
<fct><dbl><dbl[,2]>
OJ0.513.23, 4.459709
VC0.5 7.98, 2.746634
OJ1.022.70, 3.910953
VC1.016.77, 2.515309
OJ2.026.06, 2.655058
VC2.026.14, 4.797731
 
## Histogram of tooth lengths
ggplot(ToothGrowth, aes(x = len)) +
  geom_histogram(binwidth = 1, fill = "blue", color = "black") +
  ggtitle("Distribution of Tooth Lengths") +
  xlab("Tooth Length") + ylab("Frequency")

svg

# KDE plot
ggplot(ToothGrowth, aes(x = len)) +
  geom_density(fill = "blue", alpha = 0.5) +
  ggtitle("Kernel Density Estimation of Tooth Lengths") +
  xlab("Tooth Length") + ylab("Density")

svg

ggplot(ToothGrowth, aes(x = supp, y = len, fill = supp)) +
  geom_boxplot() +
  ggtitle("Tooth Length by Supplement Type") +
  xlab("Supplement Type") + ylab("Tooth Length")

svg

ggplot(ToothGrowth, aes(x = as.factor(dose), y = len, fill = as.factor(dose))) +
  geom_boxplot() +
  ggtitle("Tooth Length by Dose") +
  xlab("Dose (mg/day)") + ylab("Tooth Length")

svg

library(dplyr)

ToothGrowth_agg <- ToothGrowth %>%
  group_by(supp, dose) %>%
  summarise(mean_len = mean(len), .groups = "drop")

ggplot(ToothGrowth_agg, aes(x=dose, y=mean_len, group=supp, color=supp)) +
  geom_line() +
  geom_point() +
  labs(title="Mean Tooth Length by Supplement and Dose",
       x="Dose (mg/day)", y="Mean Tooth Length") +
  theme_minimal()

svg

# Normality check with QQ plots
qqnorm(ToothGrowth$len)
qqline(ToothGrowth$len)

svg

# Test normality
shapiro.test(ToothGrowth$len)

	Shapiro-Wilk normality test

data:  ToothGrowth$len
W = 0.96743, p-value = 0.1091

1. Comparing Two Independent Groups (Supplement Type)

To compare the tooth length (len) between the two supplement types (supp), you can use the Mann-Whitney U test (also known as the Wilcoxon rank-sum test).

wilcox.test(len ~ supp, data = ToothGrowth)

Warning message in wilcox.test.default(x = DATA[[1L]], y = DATA[[2L]], ...):
“cannot compute exact p-value with ties”




	Wilcoxon rank sum test with continuity correction

data:  len by supp
W = 575.5, p-value = 0.06449
alternative hypothesis: true location shift is not equal to 0

This test compares the median tooth lengths between groups given ascorbic acid (VC) and orange juice (OJ) without assuming the data are normally distributed.

2. Comparing More Than Two Independent Groups (Dose Levels)

For comparing the effects of different dose levels (0.5, 1, and 2 mg/day) on tooth length, you can use the Kruskal-Wallis test.

kruskal.test(len ~ dose, data = ToothGrowth)

	Kruskal-Wallis rank sum test

data:  len by dose
Kruskal-Wallis chi-squared = 40.669, df = 2, p-value = 1.475e-09

This test is an extension of the Mann-Whitney U test for more than two groups. It compares the median tooth lengths across different dose levels without assuming normal distribution.

3. Analyzing Paired Data

If your data structure included paired observations (which the ToothGrowth dataset does not inherently have, but could if measuring tooth length before and after treatment within the same subjects), you would use the Wilcoxon signed-rank test.

# Hypothetical example if you had pre- and post-treatment measurements
#wilcox.test(preTreatmentLengths, postTreatmentLengths, paired = TRUE)

This test compares the median of the paired differences to zero, suitable for before-after studies or matched pairs.

While non-parametric tests do not assume normality, they still have assumptions (e.g., independence of observations). Always ensure these assumptions are met before proceeding. Additionally, non-parametric tests generally compare medians rather than means, so interpret your results accordingly.

Second dataset: Faithful Dataset

The faithful dataset comprises 272 observations of Old Faithful geyser in Yellowstone National Park, featuring:

data("faithful")
summary(faithful)
hist(faithful$eruptions, breaks = 20, main = "Eruption Duration", xlab = "Duration (minutes)")
hist(faithful$waiting, breaks = 20, main = "Waiting Time to Next Eruption", xlab = "Time (minutes)")


   eruptions        waiting    
 Min.   :1.600   Min.   :43.0  
 1st Qu.:2.163   1st Qu.:58.0  
 Median :4.000   Median :76.0  
 Mean   :3.488   Mean   :70.9  
 3rd Qu.:4.454   3rd Qu.:82.0  
 Max.   :5.100   Max.   :96.0

svg

svg

qqnorm(faithful$eruptions)
qqline(faithful$eruptions)

svg

qqnorm(faithful$waiting)
qqline(faithful$waiting)

svg

# Shapiro-Wilk test for normality
shapiro.test(faithful$eruptions)
shapiro.test(faithful$waiting)

	Shapiro-Wilk normality test

data:  faithful$eruptions
W = 0.84592, p-value = 9.036e-16





	Shapiro-Wilk normality test

data:  faithful$waiting
W = 0.92215, p-value = 1.015e-10

Suppose you want to compare the eruption durations (eruptions) between short and long waiting times. You first need to categorize the waiting times into two groups.

# Categorize 'waiting' into two groups
faithful$waiting_group <- ifelse(faithful$waiting < median(faithful$waiting), "Short", "Long")

# Mann-Whitney U test 
wilcox.test(eruptions ~ waiting_group, data = faithful)


	Wilcoxon rank sum test with continuity correction

data:  eruptions by waiting_group
W = 16952, p-value < 2.2e-16
alternative hypothesis: true location shift is not equal to 0

If you had more than two groups and wanted to compare the distributions across those groups, you’d use the Kruskal-Wallis test. For demonstration purposes, let’s simulate a scenario by dividing waiting times into three groups based on quantiles:

# Divide 'waiting' into three groups
faithful$waiting_quantiles <- cut(faithful$waiting, breaks = quantile(faithful$waiting, probs = c(0, 1/3, 2/3, 1)), include.lowest = TRUE, labels = c("Low", "Medium", "High"))

# Kruskal-Wallis test
kruskal.test(eruptions ~ waiting_quantiles, data = faithful)


	Kruskal-Wallis rank sum test

data:  eruptions by waiting_quantiles
Kruskal-Wallis chi-squared = 182.21, df = 2, p-value < 2.2e-16

If you want to examine the relationship between the two variables without assuming a linear relationship or normality, use Spearman’s rank correlation.

# Spearman correlation
cor.test(faithful$eruptions, faithful$waiting, method = "spearman")


Warning message in cor.test.default(faithful$eruptions, faithful$waiting, method = "spearman"):
“Cannot compute exact p-value with ties”




	Spearman's rank correlation rho

data:  faithful$eruptions and faithful$waiting
S = 744659, p-value < 2.2e-16
alternative hypothesis: true rho is not equal to 0
sample estimates:
      rho 
0.7779721

This means that there’s strong evidence to reject the null hypothesis that there’s no monotonic relationship between the two variables.

ggplot(faithful, aes(x = waiting, y = eruptions)) +
  geom_point() +
  theme_minimal() +
  labs(title = "Correlation between Eruption Duration and Waiting Time",
       x = "Waiting Time to Next Eruption (minutes)",
       y = "Eruption Duration (minutes)")

svg

ggplot(faithful, aes(x = waiting, y = eruptions)) +
  geom_point() + 
  geom_smooth(method = "lm", color = "blue") + 
  theme_minimal() +
  labs(title = "Correlation between Eruption Duration and Waiting Time with Regression Line",
       x = "Waiting Time to Next Eruption (minutes)",
       y = "Eruption Duration (minutes)")

`geom_smooth()` using formula = 'y ~ x'

svg

Weaknesses of Non-Parametric Tests

Non-parametric tests are essential in statistical analysis but come with certain limitations: