R Hypothesis Testing - Complete Guide

# Understanding p-values and significance
alpha <- 0.05  # Significance level

# If p-value < alpha: Reject H₀ (statistically significant)
# If p-value >= alpha: Fail to reject H₀ (not significant)

# Type I Error (False Positive): Reject H₀ when it's true
# Type II Error (False Negative): Fail to reject H₀ when it's false

# Power: 1 - Type II Error rate (ability to detect true effects)

# One-sample t-test
data <- c(23, 25, 24, 26, 25, 23, 24, 25, 26, 24)

# Test if mean differs from 25
result <- t.test(data, mu = 25)
print(result)
# t = -0.66667, df = 9, p-value = 0.5189
# 95 percent confidence interval: [23.71 25.89]
# sample estimates: mean of x = 24.8

# Interpretation
if (result$p.value < 0.05) {
  print("Statistically significant difference from 25")
} else {
  print("No statistically significant difference from 25")
}

# One-tailed test (mean > 25)
result_greater <- t.test(data, mu = 25, alternative = "greater")

# One-tailed test (mean < 25)
result_less <- t.test(data, mu = 25, alternative = "less")

# Wilcoxon signed-rank test
result <- wilcox.test(data, mu = 25)
print(result)
# W = 24, p-value = 0.5469

# Extract components
p_value <- result$p.value
statistic <- result$statistic

# Create two independent samples
group1 <- c(23, 25, 24, 26, 25)
group2 <- c(20, 22, 21, 19, 23)

# Two-sample t-test (assume equal variances)
result <- t.test(group1, group2)
print(result)

# Welch's t-test (doesn't assume equal variances)
result_welch <- t.test(group1, group2, var.equal = FALSE)

# One-tailed test
result_one_tail <- t.test(group1, group2, alternative = "greater")

# Paired measurements (before/after)
before <- c(120, 125, 130, 118, 122)
after <- c(115, 120, 128, 116, 119)

# Paired t-test
result <- t.test(before, after, paired = TRUE)
print(result)

# Calculate mean difference
mean_diff <- mean(before - after)
print(paste("Mean difference:", mean_diff))

# Mann-Whitney U test (Wilcoxon rank-sum test)
result <- wilcox.test(group1, group2)
print(result)
# W = 23, p-value = 0.0635

# Create data with multiple groups
control <- c(20, 22, 21, 23, 19)
treatment_a <- c(25, 27, 26, 28, 24)
treatment_b <- c(30, 32, 31, 33, 29)

# Combine into data frame
data <- data.frame(
  value = c(control, treatment_a, treatment_b),
  group = rep(c("Control", "Treatment_A", "Treatment_B"), each = 5)
)

# One-way ANOVA
result <- aov(value ~ group, data = data)
print(summary(result))

# Extract F-statistic and p-value
f_stat <- summary(result)[[1]]$`F value`[1]
p_value <- summary(result)[[1]]$`Pr(>F)`[1]
print(paste("F-statistic:", f_stat, "P-value:", p_value))

# Tukey HSD (Honestly Significant Difference)
tukey_result <- TukeyHSD(result)
print(tukey_result)

# Plot Tukey results
plot(tukey_result)

# Pairwise t-tests with Bonferroni correction
pairwise.t.test(data$value, data$group, p.adjust.method = "bonferroni")

# Kruskal-Wallis test (non-parametric ANOVA)
result_kw <- kruskal.test(value ~ group, data = data)
print(result_kw)
# Kruskal-Wallis chi-squared = 9.6, df = 2, p-value = 0.00821

# Create contingency table
contingency <- matrix(c(10, 15, 20, 25), nrow = 2, ncol = 2,
                      dimnames = list(c("Yes", "No"), c("Success", "Failure")))
print(contingency)

# Chi-square test
result <- chisq.test(contingency)
print(result)

# Extract components
chi_stat <- result$statistic
p_value <- result$p.value
print(paste("Chi-square statistic:", chi_stat, "P-value:", p_value))

# Expected frequencies
print(result$expected)

# Two variables
x <- c(1, 2, 3, 4, 5)
y <- c(2, 4, 5, 4, 6)

# Pearson correlation test
result <- cor.test(x, y, method = "pearson")
print(result)

# Extract correlation coefficient and p-value
correlation <- result$estimate
p_value <- result$p.value
print(paste("Correlation:", round(correlation, 3), "P-value:", p_value))

# Spearman correlation test
result_spearman <- cor.test(x, y, method = "spearman")
print(result_spearman)

# Fisher's exact test
contingency <- matrix(c(8, 2, 1, 9), nrow = 2, ncol = 2)
result <- fisher.test(contingency)
print(result)
# Odds ratio and p-value

# Cohen's d (effect size for t-tests)
cohens_d <- function(x1, x2) {
  n1 <- length(x1)
  n2 <- length(x2)
  var1 <- var(x1)
  var2 <- var(x2)
  pooled_sd <- sqrt(((n1-1)*var1 + (n2-1)*var2) / (n1 + n2 - 2))
  (mean(x1) - mean(x2)) / pooled_sd
}

d <- cohens_d(group1, group2)
print(paste("Cohen's d:", round(d, 3)))

# Interpretation: |d| < 0.2 (small), 0.2-0.5 (small-medium),
#                 0.5-0.8 (medium-large), > 0.8 (large)

# Shapiro-Wilk test for normality
result <- shapiro.test(data)
print(result)

# Visual inspection
qqnorm(data)
qqline(data)

# Histogram
hist(data, main = "Histogram with Normal Curve")

# Levene's test (assumes normality)
library(car)
result <- leveneTest(value ~ group, data = data)
print(result)

# Bartlett's test
result_bartlett <- bartlett.test(value ~ group, data = data)
print(result_bartlett)

# Complete hypothesis testing workflow
library(dplyr)

# 1. Load and explore data
data(mtcars)
head(mtcars)

# 2. Define hypothesis
# H₀: Mean MPG for automatic and manual transmissions are equal
# H₁: Means differ

# 3. Check assumptions
# Normality
with(mtcars, {
  print(shapiro.test(mpg[am == 0]))  # Automatic
  print(shapiro.test(mpg[am == 1]))  # Manual
})

# Homogeneity of variance
with(mtcars, bartlett.test(mpg ~ am))

# 4. Perform test
auto <- mtcars$mpg[mtcars$am == 0]
manual <- mtcars$mpg[mtcars$am == 1]
result <- t.test(auto, manual)

# 5. Interpret results
print(result)
print(paste("Mean automatic:", round(mean(auto), 2)))
print(paste("Mean manual:", round(mean(manual), 2)))
print(paste("P-value:", round(result$p.value, 4)))
print(paste("Conclusion:", ifelse(result$p.value < 0.05,
                                   "Significant difference",
                                   "No significant difference")))

# 6. Calculate effect size
d <- cohens_d(auto, manual)
print(paste("Cohen's d:", round(d, 3)))