如何在R中创建含指定相关性的正态分布变量数据框？

Let's walk through this clearly—first we'll cover your initial setup, then adjust to get that 0.5 correlation between RootCause and OtherThing.

Step 1: Your Original Independent Variables #

First, let's confirm your initial code for generating independent normal variables. This creates a data frame where RootCause, OtherThing, and Errors are all independent (so their correlations will be close to 0):

# Generate independent normal variables
RootCause <- rnorm(500, mean = 0, sd = 9)
OtherThing <- rnorm(500, mean = 0, sd = 9)
Errors <- rnorm(500, mean = 0, sd = 4)

# Create data frame
df_independent <- data.frame(RootCause, OtherThing, Errors)

# Check correlation (should be near 0)
cor(df_independent$RootCause, df_independent$OtherThing)

Step 2: Generate Variables with Target Correlation (0.5) #

To get RootCause and OtherThing to have a correlation of 0.5, we need to generate them as correlated normal variables instead of independent ones. Here are two reliable methods:

Method 1: Use the MASS Package (Multivariate Normal Distribution) #

The mvrnorm() function from the MASS package lets you specify a covariance matrix, which we can build to get the exact correlation we want.

First, remember that covariance = correlation * sd1 * sd2. For our case:

• Both RootCause and OtherThing have a standard deviation of 9, so variance = 9² = 81
• Their covariance = 0.5 * 9 * 9 = 40.5
• Errors is independent of both, so its covariances with the other two are 0, and its variance is 4² = 16

Here's the code:

# Load the MASS package (install if needed with install.packages("MASS"))
library(MASS)

# Define mean vector: all variables have mean 0
mean_vec <- c(0, 0, 0)

# Define covariance matrix
cov_mat <- matrix(
  c(
    81,  40.5, 0,  # RootCause: variance 81, cov with OtherThing 40.5, cov with Errors 0
    40.5, 81,  0,  # OtherThing: cov with RootCause 40.5, variance 81, cov with Errors 0
    0,   0,   16   # Errors: cov with others 0, variance 16
  ),
  nrow = 3,
  byrow = TRUE,
  dimnames = list(c("RootCause", "OtherThing", "Errors"), c("RootCause", "OtherThing", "Errors"))
)

# Generate correlated variables
correlated_vars <- mvrnorm(n = 500, mu = mean_vec, Sigma = cov_mat)

# Convert to data frame
df_correlated <- as.data.frame(correlated_vars)

# Verify the correlation (should be ~0.5)
cor(df_correlated$RootCause, df_correlated$OtherThing)

Method 2: Manual Linear Combination (No Extra Packages) #

If you don't want to use the MASS package, you can construct OtherThing as a linear combination of RootCause and an independent normal variable. This works because:

• We keep RootCause as is
• We create OtherThing such that part of it depends on RootCause (to get the correlation) and the rest is noise (to maintain the standard deviation of 9)

The formula is:
OtherThing = (correlation) * RootCause + sqrt(1 - correlation²) * independent_normal_variable

Since we want a correlation of 0.5 and a standard deviation of 9, here's how to implement it:

# Generate RootCause first
RootCause <- rnorm(500, mean = 0, sd = 9)

# Generate independent noise for OtherThing
noise <- rnorm(500, mean = 0, sd = 9)

# Construct OtherThing with 0.5 correlation to RootCause
OtherThing <- 0.5 * RootCause + sqrt(1 - 0.5^2) * noise

# Generate independent Errors
Errors <- rnorm(500, mean = 0, sd = 4)

# Create data frame
df_correlated_manual <- data.frame(RootCause, OtherThing, Errors)

# Verify correlation
cor(df_correlated_manual$RootCause, df_correlated_manual$OtherThing)

This works because the linear combination preserves the standard deviation (since 0.5² + (sqrt(0.75))² = 0.25 + 0.75 = 1, so the total variance remains 9² = 81) and sets the exact correlation we want.

Why Your Initial Code Didn't Work #

The rnorm() function generates independent random variables by default, so their correlations will always be close to 0. To get a specific non-zero correlation, you need to use methods that account for the relationship between variables (like multivariate normal or linear combinations).

内容的提问来源于stack exchange，提问作者MGaius