Simulating Correlated Multivariate Data

Simulating bivariate normal data

Ah, simulations. The mere mention conjures images of wily statistical wizards from beyond the beyond, fingers flitting about the keyboard bashing out a frenzied stream of code fashioned to render order from the chaos of numerical nothingness.

kermit keyboard

In reality, it’s not nearly this intense. At least not for you, dear user. Your poor computer however, will have to do most of the heavy lifting. So be sure to thank it for the service.

Anyways, yes… simulating correlated data. A useful but slightly tricky endeavor if you’re new to this sort of stuff. It’s not beyond your means however, so stick with me here and we’ll make some progress. Let’s tackle this one in R.

We’ll use the MASS package’s mvrnorm function to quickly and easily generate multiple variables that are distributed as standard normal variables. Here I’m simulating 100 observations, which I define as the value N below (i.e., I’m going to be drawing values randomly from some normally distributed variable in the population 100 times). Next I’m defining two population means for two variables that I’ll simulate for these 100 observations, which I define in a vector called mu (from here on, we’ll just call these variables variable 1 and variable 2 because, let’s be real, it’s past midnight as I’m writing this, and I’m way too tired to be creative about it right now). Lastly, I’m specifying an underlying 2×2 covariance matrix (that I call sigma), which defines the variances of each variable in the population, as well as defining the covariance of both variables in the population. I’ll explain the numbers below after the jump. For now, just peep the code:

#Libraries we'll need:
library (MASS)


N <-100                           #setting my sample size             
mu <- c(2,3)                      #setting the means
sigma <- matrix(c(9,6,6,16),2,2)  #setting the covariance matrix values. The "2,2" part at the tail end defines the number of rows and columns in the matrix

set.seed(04182019)  #setting the seed value so I can reproduce this exact sim later if need be
df1 <- mvrnorm(n=N,mu=mu,Sigma=sigma)  #simulate the data, as specified above

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Basics: Standardization and the Z score

Many students have a difficult time understanding standardization when starting out in learning statistics. Common questions often include:

  • What does standardized mean?
  • How do you standardize a score?
  • Why should I give a damn?

The answers are fairly straightforward. Here’s a rundown for your statistical woes.

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