How to Plot a Chi-Square Distribution in R

The Chi-square distribution is a widely used probability distribution in inferential statistics, especially in the context of goodness-of-fit tests and tests of independence. The distribution is defined in terms of degrees of freedom, which typically represent the number of independent random variables that were squared and summed to form the Chi-square statistic.

In this comprehensive guide, we’ll explore how to plot a Chi-square distribution in R, a powerful tool for statistical analysis and visualization.

1. Understanding Chi-Square Distribution Functions in R

R provides several functions to work with Chi-square distributions:

1. rchisq(n, df): Generates n random variates from a Chi-square distribution with df degrees of freedom.
2. dchisq(x, df): Gives the height of the probability density function at x for a Chi-square distribution with df degrees of freedom.
3. pchisq(q, df): Returns the cumulative probability at q for a Chi-square distribution with df degrees of freedom.
4. qchisq(p, df): Gives the p-quantile for a Chi-square distribution with df degrees of freedom.

In this guide, we’ll mainly focus on the dchisq() function for plotting the probability density function (PDF) of a Chi-square distribution.

2. Plotting a Chi-Square Distribution

Let’s generate and plot a Chi-square distribution with 4 degrees of freedom.

# Define sequence of x-values
x <- seq(0, 20, length.out = 400)

# Define degrees of freedom
df <- 4

# Compute density values
y <- dchisq(x, df)

# Convert to data frame for ggplot
df <- data.frame(x, y)

# Plot the Chi-square distribution
ggplot(df, aes(x = x, y = y)) +
  geom_line(color = "blue") +
  labs(x = "x", y = "Density", title = "Chi-square Distribution with 4 degrees of freedom") +
  theme_minimal()

In the above code, we first define a sequence of x-values using seq(). We then compute the density values for the Chi-square distribution with 4 degrees of freedom using dchisq(). Finally, we use ggplot() and geom_line() to create the plot.

3. Plotting Chi-Square Distributions with Varying Degrees of Freedom

A crucial aspect of the Chi-square distribution is its degrees of freedom parameter. The shape of the distribution changes as the degrees of freedom change. Let’s plot Chi-square distributions with varying degrees of freedom to see how it influences the shape of the distribution.

# Define sequence of x-values
x <- seq(0, 20, length.out = 400)

# Define a vector of degrees of freedom
dfs <- c(2, 4, 6, 9)

# Create an empty data frame to store the density values
df <- data.frame()

# Loop over degrees of freedom, compute density values, and bind to the data frame
for (df_i in dfs) {
  df_temp <- data.frame(x = x, y = dchisq(x, df_i), df = factor(df_i))
  df <- rbind(df, df_temp)
}

# Plot the Chi-square distributions
ggplot(df, aes(x = x, y = y, color = df)) +
  geom_line() +
  labs(x = "x", y = "Density", color = "Degrees of freedom",
       title = "Chi-square Distributions with Varying Degrees of Freedom") +
  theme_minimal()

In this script, we first define a vector of degrees of freedom. We then loop over the degrees of freedom, compute the density values for each, and bind them to a data frame. The factor(df_i) function is used to convert the numeric degrees of freedom to a factor, which is necessary for ggplot2 to properly handle it as a categorical variable. Finally, we plot the Chi-square distributions, where the color = df aesthetic in ggplot() results in different colors for each degree of freedom.

Conclusion

This article provided a comprehensive guide on how to plot a Chi-square distribution in R using the ggplot2 package. We looked at how to plot a single Chi-square distribution and how to plot multiple Chi-square distributions with varying degrees of freedom on the same plot.