Introductory Hypothesis Testing in R (R Code Examples)

2b. Unpooled (Welch's) t‑Test (Unequal Variances)

# Data for two independent groups:
group1 <- c(10, 12, 11, 13, 12)
group2 <- c(9, 11, 10, 12, 11)

# Sample sizes:
n1 <- length(group1)
n2 <- length(group2)

# Sample means:
x_bar1 <- mean(group1)
x_bar2 <- mean(group2)

# Sample standard deviations:
s1 <- sd(group1)
s2 <- sd(group2)

# Compute the t-statistic:
t <- (x_bar1 - x_bar2) / sqrt(s1^2/n1 + s2^2/n2)

# Calculate degrees of freedom using the Welch-Satterthwaite formula:
df <- ( (s1^2/n1 + s2^2/n2)^2 ) / ( ((s1^2/n1)^2)/(n1 - 1) + ((s2^2/n2)^2)/(n2 - 1) )

# Two-tailed test:
p_two <- 2 * (1 - pt(abs(t), df = df))
critical_value_two_upper <- qt(0.975, df = df)
critical_value_two_lower <- qt(0.025, df = df)

# Left-tailed test:
p_left <- pt(t, df = df)
critical_value_left <- qt(0.05, df = df)

# Right-tailed test:
p_right <- 1 - pt(t, df = df)
critical_value_right <- qt(0.95, df = df)

# Compute Confidence Interval (Two-tailed):
CI_lower <- (x_bar1 - x_bar2) - critical_value_two_upper * sqrt(s1^2/n1 + s2^2/n2)
CI_upper <- (x_bar1 - x_bar2) + critical_value_two_upper * sqrt(s1^2/n1 + s2^2/n2)

# Output the t-statistic, degrees of freedom, p-values, and Confidence Interval:
t
df
p_two
critical_value_two_lower
critical_value_two_upper
p_left
critical_value_left
p_right
critical_value_right
CI_lower
CI_upper
              

Formula:
\( t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}},\quad df = \frac{\left(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1-1}+\frac{(s_2^2/n_2)^2}{n_2-1}} \)
Confidence Interval: \( (\bar{x}_1-\bar{x}_2) \pm t_{\alpha/2,\,df}\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}} \)

Note: Students must round down the degrees of freedom to the nearest integer.

2c. Paired t‑Test

# Paired data (e.g., before and after intervention):
before <- c(200, 195, 210, 205, 198)
after <- c(190, 185, 205, 200, 192)

# Calculate the differences:
diff <- before - after

# Summary statistics for differences:
mean_diff <- mean(diff)
s_diff <- sd(diff)
n <- length(diff)

# Compute the t-statistic:
t <- mean_diff / (s_diff / sqrt(n))

# Degrees of freedom:
df <- n - 1

# Two-tailed test:
p_two <- 2 * (1 - pt(abs(t), df = df))
critical_value_two_upper <- qt(0.975, df = df)
critical_value_two_lower <- qt(0.025, df = df)

# Left-tailed test:
p_left <- pt(t, df = df)
critical_value_left <- qt(0.05, df = df)

# Right-tailed test:
p_right <- 1 - pt(t, df = df)
critical_value_right <- qt(0.95, df = df)

# Compute Confidence Interval (Two-tailed):
CI_lower <- mean_diff - critical_value_two_upper * (s_diff/sqrt(n))
CI_upper <- mean_diff + critical_value_two_upper * (s_diff/sqrt(n))

# Output the t-statistic, mean difference, standard deviation of differences,
# degrees of freedom, p-values, and Confidence Interval:
t
mean_diff
s_diff
df
p_two
critical_value_two_lower
critical_value_two_upper
p_left
critical_value_left
p_right
critical_value_right
CI_lower
CI_upper
              

Formula: \( t = \frac{\bar{d}}{s_d/\sqrt{n}}, \quad df = n-1 \)
Confidence Interval: \( \bar{d} \pm t_{\alpha/2,\,n-1}\frac{s_d}{\sqrt{n}} \) (where \( d \) are the paired differences)

Wilcoxon Signed Rank Test

# One-sample data (or paired differences) compared to a hypothesized median:
data <- c(48, 52, 47, 53, 50)
mu0 <- 50

# Two-tailed test:
result_two <- wilcox.test(data, mu = mu0, alternative = "two.sided")
result_two

# Left-tailed test:
result_left <- wilcox.test(data, mu = mu0, alternative = "less")
result_left

# Right-tailed test:
result_right <- wilcox.test(data, mu = mu0, alternative = "greater")
result_right
              

Note: Nonparametric test for a one-sample median (or paired differences).