Category Archives: Model Building in R

Simulations in R Part 3: Group comparisons via simulation (simulating t-tests)

To review where we are at so far:

  • Part 1 discussed the basic functions for simulating and sampling data in R.
  • Part 2 walked us through how to perform bootstrap resampling and then simulate bivariate and multivariate distributions.

In Part 3, we are ready to put this info to use and start to simulate data and construct models. To start things off, this tutorial will focus on simulating data for comparison of group means — a t-test.

As always, all code is freely available in the Github repository.

Simulating Two Groups for Comparison

We begin by simulating two groups using the rnorm() function to make a random draw from a normal distribution. Group 1 will have a mean of 50 and a standard deviation of 10 while Group 2 will have a mean of 60 with a standard deviation of 6.25. Both groups have a sample size of 10 (so this is a small study!).

In addition to simulating the data, we will store summary statistics for each group so that we can use them later.

grp1 <- rnorm(n = 10,
              mean = 50,
              sd = 10)

grp2 <- rnorm(n = 10,
              mean = 60,
              sd = 6.25)

## get the summary statistics for both groups
# sample size
n_grp1 <- length(grp1)
n_grp2 <- length(grp2)

# means
mu_grp1 <- mean(grp1)
mu_grp2 <- mean(grp2)

# variances
var_grp1 <- var(grp1)
var_grp2 <- var(grp2)

# standard deviation
sd_grp1 <- sd(grp1)
sd_grp2 <- sd(grp2)

Next, we calculate the t-statistic, which is the difference between the two groups means divided by the pooled standard deviation of the two groups.

## pooled SD
sd_pool <- sqrt(((n_grp1 - 1) * var_grp1 + (n_grp2 - 1) * var_grp2) / (n_grp1 + n_grp2 - 2))

## Compute t-statistic
t_stat <- (mu_grp1 - mu_grp2) / (sd_pool * sqrt(1/n_grp1 + 1/n_grp2))

We can use this t-statistic to determine if the difference is significant or not at a desired alpha level, say p < 0.05, by using a t-distribution (the qt() function, which we were introduced to in Part 1 of this series). We then check our work against R’s build in t.test() function.

alpha <- 0.05
df <- n_grp1 + n_grp2 - 2 if(abs(t_stat) > qt(1 - alpha / 2, df, lower.tail = TRUE)){
  "significant difference"} else {"not a significant difference"} 

# Get p-value
p_value <- 2*pt(abs(t_stat), df, lower=FALSE)

# check work
t.test(grp1, grp2)

The group difference is barely significant at the p < 0.05 level. We calculated the t-statistic by hand and got the exact same value as that which was produced by the t.test() function.

To make this approach more streamlined, let’s create our own t-test function so that all we need to do in the future is pass it a vector of values representing each group’s data and get returned the t-statistic.


## t-test function
t_stat_func <- function(x, y){
  n_grp1 <- length(x)
  n_grp2 <- length(y)
  sd_pool <- sqrt(((n_grp1 - 1) * sd(x)^2 + (n_grp2 - 1) * sd(y)^2) / (n_grp1 + n_grp2 - 2))
  t_stat <- (mean(x) - mean(y)) / (sd_pool * sqrt(1/n_grp1 + 1/n_grp2))

## try out the function
t_stat_func(x = grp1, y = grp2)

We only have 10 observations in each of our groups. With such a small sample size, it may be challenging to really know if the difference we observed is real or just some sort of chance/luck occurrence (we will explore sample size issues via simulation in a later blog post). What we can do is use the function we just created and build simulated distributions using the data generating process (mean and SD) of each group to run a Monte Carlo simulations and explore how often we might reject or accept the null hypothesis at an alpha level of p < 0.05.

# Let's set alpha to 0.05
alpha <- 0.05

# We will run 10,000 simulations
N_sims <- 1e4

# create an empty vector for storing t-statistics and p-valies
t_stat <- rep(NA, N_sims)
p_values <- rep(NA, N_sims)

for(i in 1:N_sims){
  # simulate population 1
  grp1_sim <- rnorm(n = n_grp1, mean = mu_grp1, sd = sd_grp1)
  # simulate group 2
  grp2_sim <- rnorm(n = n_grp2, mean = mu_grp2, sd = sd_grp2)
  # compute the t-statistic with our function
  t_stat[i] <- t_stat_func(grp1_sim, grp2_sim)

  # get degrees of freedom for calculating the p-value
  df <- n_grp1 + n_grp2 - 2
  # obtain the p-value
  p_value[i] <- 2*pt(abs(t_stat[i]), df, lower=FALSE)

par(mfrow = c(1, 2))
     main = "T-Statistic Distribution")
abline(v = mean(abs(t_stat)),
       col = "red",
       lwd = 3,
       lty = 2)
     main = "p-value Distribution")
abline(v = 0.05,
       col = "red",
       lwd = 3,
       lty = 2)

# What percentage of times did we reject the null?
mean(p_value < 0.05)

You will get slightly different results, since I didn’t set a seed, but you will find that we end up rejecting the null about 52-53% of the time, meaning we probably wouldn’t want to be too confident about our “statistically significant” finding.

We could have saved a lot of lines of code and instead used the replicate() function and run 10,000 simulations of the t-test between the two groups (additionally, replicate() will run faster than the for() loop).


t_test <- function(){
  grp1 <- rnorm(n = n_grp1, mean = mu_grp1, sd = sd_grp1)
  grp2 <- rnorm(n = n_grp2, mean = mu_grp2, sd = sd_grp2)
  t_stat_func(grp1, grp2)

# Instead of a for loop, we will use the replicate() function to run this 10,000 times
t_stat_vector <- replicate(n = 10000,

     main = "T-Test Simulation using replicate()",
     xlab = "T-Statistic")

Simulating Data from Studies

Why just create fake data and play around when we can use a similar approach to simulating data to help us further explore data contained in studies we read?! Most studies do not provide full data sets but do provide necessary summary statistics (e.g., sample size, mean, standard deviation, standard errors, confidence intervals, etc.), allowing us to use the data generating processes to simulate the study.

I was reading a 2010 paper from Saenz et al., Knee isokinetic test-retest: A multicenter knee isokinetic test-retest study of a fatigue protocol (Eur J Phys Rehabil Med), where they authors were conducing a series of test-retest protocols using a biodex. The tables in the paper lay out the information required to simulate the study. For example, in Table 1, the authors’ provide the mean, median, and standard deviation for all of the extension tests that were performed. I’ll use the data provided for the test, WRepMax.

## Saenz (2010) - Knee isokinetic test-retest - simulation
## Table 1: Extension
# We will simulate the test scores provided for WrepMax

# number of subjects
N <- 90

## Get the mean and SD for the test and retest from Table 1 for WrepMax
test1_mu <- 99.3
test1_sd <- 18.34
test2_mu <- 104.44
test2_sd <- 24.90

Next, we calculate the mean difference between test 1 and test 2 and grab the standard deviation for the test, provided in Table 3 of the paper.

## Get the difference and standard deviation of the difference between test 1 and 2 from Table 3 (row 1)
diff <- test1_mu - test2_mu
sd_diff <- 17.37

First, we simulate Test 1 using the above parameters (sample size, mean, and standard deviation).

## Simulate the first test using the summary statistics from Table 1
w_rep_max_1 <- rnorm(n = N, mean = test1_mu, sd = test1_sd)

Next, we simulate Test 2. However, we have to remember consider that, because we are trying to simulate the performance of the 90 participant in the study, Test 2 has a relationship to the performance of Test 1 because the authors’ are analyzing the difference in performance between the two tests. Therefore, to create Test 2 we will use our simulation of Test 1 for each participant and include some random error, which is specific to the mean and standard deviation of the difference between Test 1 and Test 2 reported in the study.

## Simulate test 2 by taking test 1 and applying the observed difference between tests (Table 3) 
w_rep_max_2 <- w_rep_max_1 + rnorm(n = N, mean = abs(diff), sd = sd_diff)

Now that we have Test 1 and Test 2 simulated for each participant, we can calculate the summary statistics and see how well they compare to the observed data reported in the study.

## Get the mean and SD of the simulated test1 and test 2
test1_mu_sim <- mean(w_rep_max_1) 
test1_sd_sim <- sd(w_rep_max_1)

test2_mu_sim <- mean(w_rep_max_2)
test2_sd_sim <- sd(w_rep_max_2)



## Get the mean and SD of the difference between simulations
diff_mu_sim <- mean(w_rep_max_1 - w_rep_max_2)
diff_sd_sim <- sd(w_rep_max_1 - w_rep_max_2)


The results are nearly identical to those presented in the study (see the link, as the study is free). Now that we have a simulated data set of all of the participants we can do other things with the data, such as plot it, conduct additional reliability metrics that weren’t performed in the study, recreate the analysis in the study to get a better understanding of the analysis that was performed, or perhaps try and model the data in different ways to explore its properties and generate new ideas for future research.

For example, let’s run a paired t-test on the simulated data, as the author’s did and look at the results in comparison to simulating this study 1000 times. We re-write our t_test() function from above to re-simulate the data in the study and then conduct a paired t-test, storing the t-statistic so that we can investigate how often we would reject the null hypothesis.

## Run a paired t-test
t.test(w_rep_max_1, w_rep_max_2, paired = TRUE)

## create a function for the paired t-test and extract the t-statistic
t_test_retest <- function(){
  w_rep_max_1 <- rnorm(n = N, mean = test1_mu, sd = test1_sd)
  w_rep_max_2 <- w_rep_max_1 + rnorm(n = N, mean = abs(diff), sd = sd_diff)
  t.test(w_rep_max_1, w_rep_max_2, paired = TRUE)$statistic

# Instead of a for loop, we will use the replicate() function to run this 10,000 times
t_stat_test_retest <- replicate(n = 10000,


# turn the t-statistics into p-values
p <- 2*pt(abs(t_stat_test_retest), df = 90 - 2, lower=FALSE)

# histogram of p-values
     main = "p-values of all simulated data\nreject the null ~79% of the time")

# what percentage of times did we reject the null at the p < 0.05 level?
mean(p < 0.05)

We reject the result approximately 79% of the time at the alpha level of p < 0.05.

Wrapping Up

This tutorial worked through using simulation to understand the difference in group means, as we would commonly do with a t-test. Next, we progress on to simulation linear regression models.

As always, all code is freely available in the Github repository.

Simulations in R Part 2: Bootstrapping & Simulating Bivariate and Multivariate Distributions

In Part 1 of this series we covered some of the basic functions that we will need to perform resampling and simulations in R.

In Part 2 we will now move to building bootstrap resamples and simulating distributions for both bivariate and multivariate relationships. Some stuff we will cover:

  • Coding bootstrap resampling by hand for both a single variable (mean) and regression coefficients.
  • Using the boot() function from the boot package to perform bootstrapping for both a single variable and regression coefficients.
  • Creating a simulation where two variables are in some way dependent on each other (correlate).
  • Creating a simulation where multiple variables are correlated with each other.
  • Finally, to understand what is going on with these simulated distributions we will also work through code that shows us the relationship between variables using both covariance and correlation matrices.

As always, all code is freely available in the Github repository.


The resampling approach we will use here is bootstrapping. The general concept of bootstrapping is as follows:

  • Draw multiple random samples from observed data with replacement.
  • Draws must be independent and each observation must have an equal chance of being selected.
  • The bootstrap sample should be the same size as the observed data in order to use as much information from the sample as possible.
  • Calculate the mean resampled data and store it.
  • Repeat this process thousands of times and summarize the mean of resampled means and the standard deviation of resampled means to obtain summary statistics of your bootstrapped resamples.

Write the bootstrap resampling by hand

The code below goes through the process of creating some fake data and then writing a for() loop that produces 1000 bootstrap resamples. The for() loop was introduced in Part 1 of this series. In a nutshell, we are taking a random sample from the fake data (with replacement), calculating the mean of that random sample, and storing it in the element boot_dat. From there, we calculate the summary statistics or the original sample and the bootstrap resample, which we store in a data frame for comparison purposes, and then produce a histogram of the original sample and the bootstrap resample, which are visualized below the code.


## create fake data
dat <- c(5, 10, 44, 3, 29, 10, 16.7, 22.3, 28, 1.4, 25)

### Bootstrap Resamples ###
# we want 1000 bootstrap resamples
n_boots <- 1000

## create an empty vector to store our bootstraps
boot_dat <- rep(NA, n_boots)

# set seed for reproducibility

# write for() loop for the resampling
for(i in 1:n_boots){
  # random sample of 1:n number of observations in our data, with replacement
  ind <- sample(1:length(dat), replace = TRUE)
  # Use the row indexes to select the given values from the vector and calculate the mean
  boot_dat[i] <- mean(dat[ind])

# Look at the first 6 bootstrapped means

### Compare Bootstrap data to original data ###
## mean and standard deviation of the fake data
dat_mean <- mean(dat)
dat_sd <- sd(dat)

# standard error of the mean
dat_se <- sd(dat) / sqrt(length(dat))

# 95% confidence interval
dat_ci95 <- paste0(round(dat_mean - 1.96*dat_se, 1), ", ", round(dat_mean + 1.96*dat_se, 1))

# mean an SD of bootstrapped data
boot_mean <- mean(boot_dat)

# the vector is the mean of each bootstrap sample, so the standard deviation of these means represents the standard error
boot_se <- sd(boot_dat)

# to get the standard deviation we can convert the standard error back
boot_sd <- boot_se * sqrt(length(dat))

# 95% quantile interval
boot_ci95 <- paste0(round(boot_mean - 1.96*boot_se, 1), ", ", round(boot_mean + 1.96*boot_se, 1)) ## Put everything together data.frame(data = c("fake sample", "bootstrapped resamples"), N = c(length(dat), length(boot_dat)), mean = c(dat_mean, boot_mean), sd = c(dat_sd, boot_sd), se = c(dat_se, boot_se), ci95 = c(dat_ci95, boot_ci95)) %&gt;%

# plot the distributions
par(mfrow = c(1, 2))
     xlab = "Obsevations",
     main = "Fake Data")
abline(v = dat_mean,
       col = "red",
       lwd = 3,
       lty = 2)
     xlab = "bootstrapped means",
     main = "1000 bootstrap resamples")
abline(v = boot_mean,
       col = "red",
       lwd = 3,
       lty = 2)

R offers a bootstrap function from the boot package that allows you to do the same thing without writing out your own for() loop. Below is an example of coding the same procedure in the boot package and the outputs the function provides, which are similar to the output we get above, save for slight differences due to random sampling.

# write a function to calculate the mean of our sample data
sample_mean <- function(x, d){

# run the boot() function

# run the boot function
boot_func_output <- boot(dat, statistic = sample_mean, R = 1000)

# produce a plot of the output

# get the mean and standard error

# get 95% CI around the mean, type = "basic", conf = 0.95)

We can bootstrap pretty much anything we want. We don’t have to limit ourselves to producing the distribution around the mean of a population. For example, let’s bootstrap regression coefficients to understand the uncertainty in them.

First, let’s use the boot() function to conduct our analysis. We will fit a simple linear regression predicting miles per gallon from engine weight using the mtcars package. We will then write a function for that uses a random sample of rows to create the same linear regression and store those coefficients from 1000 linear regressions so that we can plot a histogram representing the slope coefficient from the resampled models and also summarize the distribution with confidence intervals.

# load the mtcars data
d <- mtcars d %>%

# fit a regression model
fit_mpg <- lm(mpg ~ wt, data = d)

# Write a function that can perform a bootstrap over the intercept and slope of the model
# bootstrap function
reg_coef_boot <- function(data, row_id){
  # we want to resample the rows
  fit <- lm(mpg ~ wt, data = d[row_id, ])

# run this once on a small subset of the row ids to see how it works
reg_coef_boot(data = d, row_id = 1:20)

# run the boot() function 1000 times
coef_boot <- boot(data = d, reg_coef_boot, 1000) 

# check the output (coefficient and SE) 

# get the confidence intervals, index= 2) 

# all 1000 of the bootstrap resamples can be called coef_boot$t %&gt;%

# plot the first 20 bootstrapped intercepts and slopes over the original data
plot(x = d$wt,
     y = d$mpg,
     pch = 19)
for(i in 1:20){
  abline(a = coef_boot$t[i, 1],
       b = coef_boot$t[i, 2],
       lty = 2,
       lwd = 3,
       col = "light grey")

## histogram of the slope coefficient
hist(coef_boot$t[, 2])

We can do this by hand if we don’t want to use the built in boot() function. (SIDE NOTE: I usually prefer to code my own resampling and simulations as it gives me more flexibility with respect to the things I’d like to add or the values I’d like to store from each iteration.

Below, instead of producing a histogram of just the slope coefficient, I use both the resampled intercept and slope and add 20 (of the 1000) lines to a scatter plot to show the way in which each of these lines represents a plausible regression line for the data. As you can see, the regression line confidence interval is starting to take shape, even with just 20 out of 1000 resamples, and this gives us a good understanding of not only the variability in our possible regression line fit for the underlying data but also, perhaps should make us less overconfident in our research findings knowing that there are many possible outcomes from the sample data we have obtained.

## 1000 resamples
n_samples <- 1000

## N observations
n_obs <- nrow(mtcars)

## empty storage data frame for the coefficients
coef_storage <- data.frame(
  intercept = rep(NA, n_samples),
  slope = rep(NA, n_samples)

for(i in 1:n_samples){
  ## sample dependent and independent variables
  row_ids <- sample(1:n_obs, size = n_obs, replace = TRUE)
  new_df <- d[row_ids, ]
  ## construct model
  model <- lm(mpg ~ wt, data = new_df)
  ## store coefficients
  # intercept
  coef_storage[i, 1] <- coef(model)[1]
  # slope
  coef_storage[i, 2] <- coef(model)[2]

## see results

## Compare the results to those of the boot function
apply(X = coef_boot$t, MARGIN = 2, FUN = mean)
apply(X = coef_storage, MARGIN = 2, FUN = mean)

apply(X = coef_boot$t, MARGIN = 2, FUN = sd)
apply(X = coef_storage, MARGIN = 2, FUN = sd)

## plot first 20 lines
plot(x = d$wt,
     y = d$mpg,
     pch = 19)
for(i in 1:20){
  abline(a = coef_storage[i, 1],
       b = coef_storage[i, 2],
       lty = 2,
       lwd = 3,
       col = "light grey")

Simulating a relationship between two variables

As discussed in Part 1, simulation differs from resampling in that we use the parameters of the observed data to compute a new distribution, versus sampling from the data we have on hand.

For example, using the mean and standard deviation of mpg from the mtcars data set, we can simulate 1000 random draws from a normal distribution.

## load the mtcars data set
d <- mtcars

## make a random draw from the normal distribution for mph
mpg_sim <- rnorm(n = 1000, mean = mean(d$mpg), sd = sd(d$mpg))

## plot and summarize


Frequently, we are interested in the relationship between two variables (e.g., correlation, regression, etc.). Let’s simulate two variables, x and y, which are linearly related in some way. To do this, we first simulate the variable x and then simulate y to be x plus some level of random noise.

# simulate x and y
x <- rnorm(n = 10, mean = 50, sd = 10)
y <- x + rnorm(n = length(x), mean = 0, sd = 10)

# put the results in a data frame
dat <- data.frame(x, y)

# how correlated are the two variables
cor.test(x, y)

# fit a regression for the two variables
fit <- lm(y ~ x)

# plot the two variables with the regression line
plot(x, y, pch = 19)
abline(fit, col = "red", lwd = 2, lty = 2)

Simulating a data set with multiple variables

Frequently, we might have a hypothesis regarding how correlated multiple variables are with each other. The example above produced a relationship of two variables with a direct relationship between them along with some noise. We might want to specify this relationship given a correlation coefficient or covariance between them. Additionally, we might have more than two variables that we want to simulate relationships between.

To do this in R we can take advantage of two packages:

  • MASS via the mvrnorm()
  • mvtnorm via the mvrnorm()

Both packages have a function for simulating multivariate normal distributions. The primary difference is that the Sigma argument in the MASS package function, mvrnorm(), accepts a covariance matrix while the sigma argument in the mvtnorm package, rmvnorm() accepts a correlation matrix. I’ll show both examples but I tend to stick with the mtvnorm package because (at least for my brain) it is easier for me to think in terms of correlation coefficients instead of covariances.

First we simulate some data:

## create fake data
fake_dat <- data.frame(
  group = rep(c("a", "b", "c"), each = 5),
  x = rnorm(n = 15, mean = 10, sd = 2),
  y = rnorm(n = 15, mean = 30, sd = 10),
  z = rnorm(n = 15, mean = 75, sd = 20)


Look at the correlation and variance between the three numeric variables.

# correlation
round(cor(fake_dat[, -1]), 3)

# variance
round(var(fake_dat[, -1]), 3)

We can use this information to simulate new x, y, or z variables.

Simulating x and y with the MASS package

Remember, for the MASS package, the Sigma argument is a matrix of covariances for the variables you are simulating from a multivariate normal distribution.

## get a vector of the mean for each variable
variable_means <- apply(X = fake_dat[, c("x", "y")], MARGIN = 2, FUN = mean)

## Get a matrix of the covariance between x and y
variable_sigmas <- var(fake_dat[, c("x", "y")])

## simulate 1000 new x and y variables using the MASS package
new_sim <- MASS::mvrnorm(n = 1000, mu = variable_means, Sigma = variable_sigmas)

### look at the results relative to the original x and y
## column means
apply(X = new_sim, MARGIN = 2, FUN = mean)

## covariance
var(fake_dat[, c("x","y")])


Notice that the variance between x and y (the off diagonal of the matrix) is very similar between the fake data (the observed sample) and the simulated data (new_sim).

Simulating x and y with the mtvnorm package

Different than the MASS package, The rmvnorm() function from the mtvnorm package requires the sigma argument to be a correlations matrix.

Let’s repeat the above process with our fake_dat and simulate a relationship between x and y.

## get a vector of the mean for each variable
variable_means <- apply(X = fake_dat[, c("x", "y")], MARGIN = 2, FUN = mean)

## Get a matrix of the correlation between x and y
variable_cor <- cor(fake_dat[, c("x", "y")])

## simulate 1000 new x and y variables using the mvtnorm package
new_sim <- mvtnorm::rmvnorm(n = 1000, mean = variable_means, sigma = variable_cor)

### look at the results relative to the original x and y
## column means
apply(X = new_sim, MARGIN = 2, FUN = mean)

## correlation
cor(fake_dat[, c("x","y")])

Similar to the previous example, we notice that the off diagonal correlation coefficient between x and y is very similar when comparing the simulated data to the fake data.

So, what is happening here? Both packages produce the same result, one uses a covariance matrix and the other uses a correlation matrix. The kicker here is understanding the relationship between covariance and correlation. Covariance is explaining how two variables vary together, however, its units aren’t on a scale that is directly interpretable to us. But, we can convert the covariance between two variables to a correlation by dividing their covariance by the product of their individual standard deviations.

For example, here is the covariance matrix between x and y in the fake data set.

cov(fake_dat[, c("x", "y")])

The covariance between the two variables is on the off diagonal, 2.389. We can store this in its own element.

cov_xy <- cov(fake_dat[, c("x", "y")])[2,1]

Let’s store the standard deviation of both `x` and `y` in their own elements to make the equation easier to read.

sd_x <- sd(fake_dat$x)
sd_y <- sd(fake_dat$y)

Finally, we calculate the correlation by dividing the covariance by the product of the two standard deviations and check our results by calling the cor() function on the two variables.

## covariance to correlation
cov_to_cor <- cov_xy / (sd_x * sd_y)

## check results with the corr() function
cor(fake_dat[, c("x", "y")])

By dividing the covariance by the product of the standard deviation of the two variables we can see the relationship between covariance and correlation and now understand why the results from the MASS and mtvnorm produce similar results. Understanding this relationship becomes valuable when we move onto simulating more complex relationships, for example when simulating mixed models.

What about three variables?

What if we want to simulate all three variables — x, y, and z?

All we need is a larger covariance or correlation matrix, depending on which of the above packages you’d like to use. Since we usually won’t be creating these matrices from a data set, as I did above, I’ll show how to create your own matrix and run the simulation.

First, let’s store a vector of plausible mean values for x, y, and z.

## Look at the mean values we had in the fake data
apply(X = fake_dat[, c("x", "y", "z")], MARGIN = 2, FUN = mean)

## create a vector of possible mean values for the simulation
mus <- c(9, 26, 63)

## look at the correlation matrix for the three variables in the fake data
cor(fake_dat[, c("x", "y", "z")])

## Create a matrix that stores plausible correlations between the variables you want to simulate
r_matrix <- matrix(c(1, 0.14, -0.24,
                    0.14, 1, -0.35,
                    -0.24, -0.35, 1), 
                   nrow = 3, ncol = 3,
       dimnames = list(c("x", "y", "z"),
                       c("x", "y", "z")))


Next, we create 1000 simulations of a multivariate normal distribution between x, y, and z. We then compare our correlation coefficients between the fake data, which is our observed sample, and the simulated data

## simulate 1000 new x, y, and z variables using the mvtnorm package
new_sim <- mvtnorm::rmvnorm(n = 1000, mean = mus, sigma = r_matrix)

### look at the results relative to the original x, y, and z
## column means
apply(X = fake_dat[, c("x", "y", "z")], MARGIN = 2, FUN = mean)
apply(X = new_sim, MARGIN = 2, FUN = mean)

## correlation
cor(fake_dat[, c("x", "y", "z")])

The results from the simulation are pretty similar to the fake_dat dataset. If you recall from the correlation matrix and the vector of means, we didn’t use exact values from the observed data, so that, along with the random draws from the multivariate normal distribution, leads to the small amount of differences.

Wrapping Up

In this second installment of the Simulations in R series we’ve walked through how to code our own bootstrap resampling for both means and regression coefficients. We then progressed to building simulations of both bivariate and multivariate normal distributions. This, along with the basic info in Part 1, will serve us well as we progress forward in our work and begin to explore using simulations for comparisons between group means (simulated t-tests) and building regression models.

As always, all of the code is available in the Github repository.

Simulations in R Part 1: Functions for Simulation & Resampling

Simulating data is something I find myself doing all the time. Not only to explore uncertainty in data but also to explore model assumptions, understand how models behave under different circumstances, or to try and understand how a future analysis might work given some underlying data generating process. Thus, I decided to put together a series on simulations and resampling using R (I’ll also add a few analog scripts using Python to the GitHub repository).

In Part 1, I’ll provide some thoughts around why you might want to simulate or resample data and then show how you can simply do this in R. Additionally, I’ll walk through several helper functions for conducting and summarizing simulations/resamples as well as some basics around for() and while() loops, as we will use these extensively in our simulation and resampling processes.

My Github repository will contain all of the scripts in this series.

Why do we simulate or resample data?

  • The data generating process is what defines the properties of our data and dictates the type of distribution we are dealing with. For example, the mean and standard deviation reflect the two parameters of the data generating process for a normal distribution. We rarely know what the data generating process of our data is in the real world, thus we must infer it from our sample data. Both resampling and simulation offer methods of understanding the data generating process of data.
  • Sample data represents a small sliver of what might be occurring in the broader population. Using resampling and simulation, we are able to build larger data sets based on information contained in the sample data. Such approaches allow us to explore our uncertainty around what we have observed in our sample and the inferences we might be able to make about that larger population.
  • Creating samples of data allows us to assess patterns in the data and evaluate those patterns under different circumstances, which we can directly program.
  • By coding a simulation, we are able to reflect a desired data generating process, allowing us to evaluate assumptions or limitations of data that we have collected or are going to collect.
  • The world is full of randomness, meaning that every observation we make comes with some level of uncertainty. The uncertainty that we have about the true value of our observation can be expressed via various probability distributions. Resamping and simulation are ways that we can mimic this randomness in the world and help calibrate our expectation about the probability of certain events or observations occurring.

Difference between resampling and simulation

Resampling and simulation are both useful at generating data sets and reflecting uncertainty. However, they accomplish this task in different ways.

  • Resampling deals with techniques that take the observed sample data and randomly draw observations from that data to construct a new data set. This is often done thousands of times, building thousands of new data sets, and then summary statistics are produced on those data sets as a means of understanding the data generating properties.
  • Simulation works by assuming a data generating process (e.g., making a best guess or estimating a plausible mean and standard deviation for the population from previous literature) and then generating multiple samples of data, randomly, from the data generating process features.

Sampling from common distributions

To create a distribution in R we can use any one of the four primary prefixes, which define the type of information we want returned about the distribution, followed by the suffix that defines the distribution we are interested in.

Here is a helpful cheat sheet I put together for some of the common distributions one might use:

Some examples:

# The probability that a random variable is less than or equal to 1.645 has a cumulative density of 95% (CDF)
pnorm(q = 1.645, mean = 0, sd = 1)

# What is the exact probability (PDF) that we flip 10 coins, with 50% chance of heads or tails, and get 1 heads?
dbinom(x = 1, size = 10, prob = 0.5)

# What is the z-score for the 95 percentile when the data is Normal(0, 1)?
qnorm(p = 0.95, mean = 0, sd = 1)

# randomly draw 10 values from a uniform distribution with a min of 5 and max of 10
runif(n = 10, min = 5, max = 10)

We can completely simulate different distributions and properties of those distributions using these function. For several examples of different distributions see the GitHub code. Below is an example of 1,000 random observations from a normal distribution with a mean of 30 and standard deviation of 15 and plot the results..

## set the seed for reproducibility
norm_dat <- rnorm(n = 1000, mean = 30, sd = 15)

     main = "Random Simulation from a Normal Distribution",
     xlab = "N(30, 15^2)")

We can produce a number of summary statistics on this vector of random values:

# sample size

# mean, standard deviation, and variance

# median, median absolute deviation

for & while loops

Typically, we are going to want to resample data more than once or to run multiple simulations. Often, we will want to do this thousands of times. We can use R to help us in the endeavor by programming for() and while() loops to do the heavy lifting for us and store the results in a convenient format (e.g., vector, data frame, matrix, or list) so that we can summarize it later.

for loops

for() loops are easy ways to tell `R` that we want it to do some sort of task for a specified number of iterations.

For example, let’s create a for() loop that adds 5 for every value from 1 to 10, for(i in 1:10).

# program the loop to add 5 to every value from 1:10
for(i in 1:10){
  print(i + 5)

We notice that the result is printed directly to the console. If we are doing thousands of iterations or if we want to store the results to plot and summarize them later, this wont be a good option. Instead, we can allocate an empty vector or data frame to store these values.

## storing values as vector
n <- 10
vector_storage <- rep(NA, times = n)

for(i in 1:n){
  vector_storage[i] <- i + 5


## store results back to a data frame
n <- 10
df_storage <- data.frame(n = 1:10)

for(i in 1:n){
  df_storage$n2[i] <- i + 5


while loops

while() loops differ from for() loops in that they continue to perform a process while some condition is met.

For example, if we start with a count of 0 observations and continually add 1 observation we want to perform this process as long as the observations are below 10.

observations <- 0

while(observations < 10){
	observations <- observations + 1

We can also use while() loops to test logical arguments.

For example, let’s say we have five coins in our pocket and want to play a game with a fried where we flip a fair coin and every time it ends on heads (coin_flip == 1) we get a coin and every time it ends on tails we lose a coin. We are only willing to continue playing the game as long as retain between 3 and 10 coins.

## starting number of coins
coins <- 5 

## while loop 
while(coins >= 3 && coins <= 10){
  # flip a fair coin (50/50 chance of heads or tails)
	coin_flip <- rbinom(1,1,0.5)
	# If the coin leads on heads (1) you win a coin and if it lands on tails (0) you lose a coin
	if(coin_flip == 1){
		coins <- coins + 1
			coins <- coins - 1
	## NOTE: we only play while our winnings are between 3 and 10 coins
	# print the result

You can run the code many times and find out, on average, how many flips you will get!

Finally, we can also use while() loops if we are building models to minimize error. For example, lets say we have an error = 30 and we want to continue running the code until we have minimized the error below 1. So, the code will run while(error > 1).

error <- 30 while(error > 1){
 error <- error / 2

Helper functions for summarizing distributions

There are a number of helper functions in base R that can assist us in summarizing data.

  • apply() will return your results in a vector
  • lapply() will return your results as a list
  • sapply() can return the results as a vector or a list (if you set the argument `simplify = FALSE`)
  • tapply() will return your results in a named vector based on whichever grouping variable you specify
## create fake data
fake_dat <- data.frame(
  group = rep(c("a", "b", "c"), each = 5),
  x = rnorm(n = 15, mean = 10, sd = 2),
  y = rnorm(n = 15, mean = 30, sd = 10),
  z = rnorm(n = 15, mean = 75, sd = 20)


#### apply ####
# get the column averages
apply(X = fake_dat[,-1], MARGIN = 2, FUN = mean)

# get the row averages
apply(X = fake_dat[,-1], MARGIN = 1, FUN = mean)

#### lapply ####
# Get the 95% quantile interval for each column
lapply(X = fake_dat[,-1], FUN = quantile, probs = c(0.025, 0.975))

#### sapply ####
# Get the standard deviation of each column in a vector
sapply(X = fake_dat[,-1], FUN = sd)

# Get the standard deviation of each column in a list
sapply(X = fake_dat[,-1], FUN = sd, simplify = FALSE)

#### tapply ####
# Get the average of x for each group
tapply(X = fake_dat$x, INDEX = fake_dat$group, FUN = mean)

We can alternatively do a lot of this type of data summarizing using the convenient R package {tidyverse}.


## get the mean of each numeric column
fake_dat %>%
  summarize(across(.cols = x:z,
                   .fns = ~mean(.x)))

## get the mean across each row for the numeric columns
fake_dat %>%
  rowwise() %>%
  mutate(AVG = mean(c_across(cols = x:z)))

## Get the mean of x for each grou
fake_dat %>%
  group_by(group) %>%
  summarize(avg_x = mean(x),
            .groups = "drop")

Finally, another handy base R function is replicate(), which allows us to replicate a task n number of times.

For example, let’s say we want to draw from a random normal distribution, rnorm() with a mean = 0 and sd = 1 but, we want to run this random simulation 10 times and get 10 different data sets. replicate()` allows us to do this and stores the results in a matrix with 10 columns, each with 10 rows of the random sample.

replicate(n = 10, expr = rnorm(n = 10, mean = 0, sd = 1))

Wrapping Up

In this first part of my simulation and resampling series we went through some of the key functions in R that will help us build the scaffolding for our future work. In Part 2, we we dive into bootstrap resampling and simulating bivariate and multivariate normal distributions.

All code is available in both rmarkdown and html format on my Github page.

Different ways of calculating intervals of uncertainty

I’ve talked a lot in this blog about making predictions (see HERE, HERE, and HERE) as well as the difference between confidence intervals and prediction intervals and why you’d use one over the other (see HERE). Tonight I was having a discussion with a colleague about some models he was working on and he was building some confidence intervals around his predictions. That got me to thinking about the various ways we can code confidence intervals, quantile intervals, and prediction intervals in R. So, I decided to put together this quick tutorial to provide a few different ways of constructing these values (after all, unless we can calculate the uncertainty in our predictions, point estimate predictions are largely useless on their own).

The full code is available on my GITHUB page.

Load packages, get data, and fit regression model

The only package we will need is {tidyverse}, the data will be the mtcars dataset and the model will be a linear regression which attempts to predict mpg from wr and carb.

## Load packages


## Get data
d <- mtcars d %>%

## fit model
fit_lm <- lm(mpg ~ wt + carb, data = d)

Get some data to make predictions on

We will just grab a random sample of 5 rows from the original data set and use that to make some predictions on.

## Get a few rows to make predictions on
d_sample <- d %>%
  sample_n(size = 5) %>%
  select(mpg, wt, carb)


Confidence Intervals with the predict() function

Using preidct() we calculate the predicted value with 95% Confidence Intervals.

## 95% Confidence Intervals
d_sample %>%
    predict(fit_lm, newdata = d_sample, interval = "confidence", level = 0.95)

Calculate confidence intervals by hand

Instead of using the R function, we can calculate the confidence intervals by hand (and obtain the same result).

## Calculate the 95% confidence interval by hand
level <- 0.95
alpha <- 1 - (1 - level) / 2
t_crit <- qt(p = alpha, df = fit_lm$df.residual) 

d_sample %>%
  mutate(pred = predict(fit_lm, newdata = .),
         se_pred = predict(fit_lm, newdata = ., se = TRUE)$,
         cl95 = t_crit * se_pred,
         lwr = pred - cl95,
         upr = pred + cl95)

Calculate confidence intervals with the qnorm() function

Above, we calculated a 95% t-critical value for the degrees of freedom of our model. Alternatively, we could calculate 95% confidence intervals using the standard z-critical value for 95%, 1.96, which we obtain with the qnorm() function.

d_sample %>%
  mutate(pred = predict(fit_lm, newdata = .),
         se_pred = predict(fit_lm, newdata = ., se = TRUE)$,
         lwr = pred + qnorm(p = 0.025, mean = 0, sd = 1) * se_pred,
         upr = pred + qnorm(p = 0.975, mean = 0, sd = 1) * se_pred)

Calculate quantile intervals via simulation

Finally, we can calculate quantile intervals by simulating predictions using the predicted value and standard error for each of the observations. We simulate 1000 times from a normal distribution and then use the quantile() function to get our quantile intervals.

If all we care about is a predicted value and the lower and upper intervals, we can use the rowwise() function to indicate that we are going to do a simulation for each row and then store the end result (our lower and upper quantile intervals) in a new column.

## 95% Quantile Intervals via Simulation
d_sample %>%
  mutate(pred = predict(fit_lm, newdata = .),
         se_pred = predict(fit_lm, newdata = ., se = TRUE)$ %>%
  rowwise() %>%
  mutate(lwr = quantile(rnorm(n = 1000, mean = pred, sd = se_pred), probs = 0.025),
         upr = quantile(rnorm(n = 1000, mean = pred, sd = se_pred), probs = 0.975))

While that is useful, there might be times where we want to extract the full simulated distribution. We can create a simulated distribution (1000 simulations) for each of the 5 observations using a for() loop.

## 95% quantile intervals via Simulation with full distribution
N <- 1000
pred_sim <- list()

for(i in 1:nrow(d_sample)){
  pred <- predict(fit_lm, newdata = d_sample[i, ])
  se_pred <- predict(fit_lm, newdata = d_sample[i, ], se = TRUE)$
  pred_sim[[i]] <- rnorm(n = N, mean = pred, sd = se_pred)

sim_df <- tibble( sample_row = rep(1:5, each = N), pred_sim = unlist(pred_sim) ) 

sim_df %>%

Next we summarize the simulation for each observation.

# get predictions and quantile intervals
sim_df %>%
  group_by(sample_row) %>%
  summarize(pred = mean(pred_sim),
         lwr = quantile(pred_sim, probs = 0.025),
         upr = quantile(pred_sim, probs = 0.975)) %>%
  mutate(sample_row = rownames(d_sample))

We can then plot the entire posterior distribution for each observation.

# plot the predicted distributions
sim_df %>%
  mutate(actual_value = rep(d_sample$mpg, each = N),
         sample_row = case_when(sample_row == 1 ~ "Hornet 4 Drive",
                                sample_row == 2 ~ "Toyota Corolla",
                                sample_row == 3 ~ "Honda Civic",
                                sample_row == 4 ~ "Ferrari Dino",
                                sample_row == 5 ~ "Pontiac Firebird")) %>%
  ggplot(aes(x = pred_sim)) +
  geom_histogram(color = "white",
                 fill = "light grey") +
  geom_vline(aes(xintercept = actual_value),
             color = "red",
             size = 1.2,
             linetype = "dashed") +
  facet_wrap(~sample_row, scale = "free_x") +
  labs(x = "Predicted Simulation",
       y = "count",
       title = "Predicted Simulation with actual observation (red line)",
       subtitle = "Note that the x-axis are specific to that simulation and not the same")

Prediction Intervals with the predict() function

Next we turn attention to prediction intervals, which will be wider than the confidence intervals because they are incorporating additional uncertainty.

The predict() function makes calculating prediction intervals very convenient.

## 95% Prediction Intervals
d_sample %>%
    predict(fit_lm, newdata = d_sample, interval = "predict", level = 0.95)

Prediction Intervals from a simulated distribution

Similar to how we simulated a distribution for calculating quantile intervals, above, we will perform the same procedure here. The difference is that we need to get the residual standard error (RSE) from our model as we need to add this additional piece of uncertainty (on top of the predicted standard error) to each of the simulated predictions.

## 95% prediction intervals from a simulated distribution 
# store the model residual standard error
sigma <- summary(fit_lm)$sigma

# run simulation
N <- 1000
pred_sim2 <- list()

for(i in 1:nrow(d_sample)){
  pred <- predict(fit_lm, newdata = d_sample[i, ])
  se_pred <- predict(fit_lm, newdata = d_sample[i, ], se = TRUE)$
  pred_sim2[[i]] <- rnorm(n = N, mean = pred, sd = se_pred) + rnorm(n = N, mean = 0, sd = sigma)

# put results in a data frame
sim_df2 <- tibble( sample_row = rep(1:5, each = N), pred_sim2 = unlist(pred_sim2) ) 

sim_df2 %>%

We summarize our predictions and their intervals.

# get predictions and intervals
sim_df2 %>%
  group_by(sample_row) %>%
  summarize(pred = mean(pred_sim2),
            lwr = quantile(pred_sim2, probs = 0.025),
            upr = quantile(pred_sim2, probs = 0.975)) %>%
  mutate(sample_row = rownames(d_sample))

Finally, we plot the simulated distributions for each of the observations.

Wrapping Up

Uncertainty is important to be aware of and convey whenever you share your predictions. The point estimate prediction is one a single value of many plausible values given the data generating process. This article provided a few different approaches for calculating uncertainty intervals. The full code is available on my GITHUB page.

Plotting Mixed Model Outputs

This weekend I posted two new blog articles about building reports that contained both data tables and plots on the same canvas (see HERE and HERE). As a follow up, James Baker asked if I could do some plotting of mixed model outputs. That got me thinking, I’ve done a few blog tutorials on mixed models (see HERE and HERE) and this got me thinking. Because he left it pretty wide open (“Do you have any guides on visualizing mixed models?”) I was trying to think about what aspects of the mixed models he’d like to visualize. R makes it relatively easy to plot random effects using the {lattice} package, but I figured we could go a little deeper and customize some of our own plots of the random effects as well as show how we might plot future predictions from a mixed model.

All of the code for this article is available on my GITHUB page.

Loading Packages & Data

As always we begin by loading some of the packages we require and the data. In this case, we will use the sleepstudy dataset, which is freely available from the {lme4} package.

## Load packages


## load data
dat <- sleepstudy dat %>%

Fit a mixed model

We will fit a mixed model that sets the dependent variable as Reaction time and the fixed effect as days of sleep deprivation. We will also allow both the intercept and slope to vary randomly by nesting the individual SubjectID within each Day of sleep deprivation.

## Fit mixed model
fit_lmer <- lmer(Reaction ~ Days + (1 + Days|Subject), data = dat)

Inspect the random effects

We can see in the model output above that we have a random effect standard deviation for the Intercept (24.84) and for the slope, Days (5.92). We can extract out the random effect intercept and slope for each subject with the code below. This tells us how much each subject’s slope and intercept vary from the population fixed effects (251.4 and 10.5 for the intercept and slope, respectively).

# look at the random effects
random_effects <- ranef(fit_lmer) %>%
  pluck(1) %>%
  rownames_to_column() %>%
  rename(Subject = rowname, Intercept = "(Intercept)") 

random_effects %>%

Plotting the random effects

Aside from looking at a table of numbers, which can sometimes be difficult to draw conclusions from (especially if there are a large number of subjects) we can plot the data and make some observational inference.

The {lattice} package allows us to create waterfall plots of the random effects for each subject with the dotplot() function.

## plot random effects

That’s a pretty nice plot and easy to obtain with just a single line of code. But, we might want to create our own plot using {ggplot2} so that we have more control over the styling.

I’ll store the standard deviation of the random slope and intercept, from the model read out above, in their own element. Then, I’ll use the random effects table we made above, which contains the intercept and slope of each subject, to plot them and add the standard deviation to them as error bars.

## Make one in ggplot2
subject_intercept_sd <- 24.7
subject_days_sd <- 5.92

int_plt <- random_effects %>%
mutate(Subject = as.factor(Subject)) %>%
ggplot(aes(x = Intercept, y = reorder(Subject, Intercept))) +
geom_errorbar(aes(xmin = Intercept - subject_intercept_sd,
xmax = Intercept + subject_intercept_sd),
width = 0,
size = 1) +
geom_point(size = 3,
shape = 21,
color = "black",
fill = "white") +
geom_vline(xintercept = 0,
color = "red",
size = 1,
linetype = "dashed") +
scale_x_continuous(breaks = seq(-60, 60, 20)) +
labs(x = "Intercept",
y = "Subject ID",
title = "Random Intercepts")

slope_plt <- random_effects %>%
mutate(Subject = as.factor(Subject)) %>%
ggplot(aes(x = Days, y = reorder(Subject, Days))) +
geom_errorbar(aes(xmin = Days - subject_days_sd,
xmax = Days + subject_days_sd),
width = 0,
size = 1) +
geom_point(size = 3,
shape = 21,
color = "black",
fill = "white") +
geom_vline(xintercept = 0,
color = "red",
size = 1,
linetype = "dashed") +
xlim(-60, 60) +
labs(x = "Slope",
y = "Subject ID",
title = "Random Slopes")

slope_plt / int_plt

We get the same plot but now we have more control. We can color the dot specific subjects, or only choose to display specific subjects, or flip the x- and y-axes, etc.

Plotting the model residuals

We can also plot the model residuals. Using the residual() function we can get the residuals directly from our mixed model and the plot() function with automatically plot the Residual and Fitted values. These types of plots are useful for exploring assumptions such as normality of the residuals and homoscedasticity.

## Plot Residual

As above, perhaps we want to have more control over the bottom plot, so that we can style it however we’d like. We can extract the fitted values and residuals and build our own plot using base R.

## Plotting our own residual ~ fitted
lmer_fitted <- predict(fit_lmer, newdata = dat, re.form = ~(1 + Days|Subject))
lmer_resid <- dat$Reaction - lmer_fitted

plot(x = lmer_fitted,
     y = lmer_resid,
     pch = 19,
     main = "Resid ~ Fitted",
     xlab = "Fitted",
     ylab = "Residuals")
abline(h = 0,
       col = "red",
       lwd = 3,
       lty = 2)

Plotting Predictions

The final plot I’ll build are the predictions of Reaction time as Days of sleep deprivation increase. This is time series data, so I’m going to extract the first 6 days of sleep deprivation for each subject and build the model using that data. Then, make predictions on the next 4 days of sleep deprivation for each subject and get both a predicted point estimate and 90% prediction interval. In this way, we can observe the next 4 days of sleep deprivation for each subject and see how far outside of what we would expect (from our mixed model predictions) those values fall.


### Plotting the time series on new data
# training set
dat_train <- dat %>%
  group_by(Subject) %>%
  slice(head(row_number(), 6)) %>%

# testing set
dat_test <- dat %>%
  group_by(Subject) %>%
  slice(tail(row_number(), 4)) %>%

## Fit mixed model
fit_lmer2 <- lmer(Reaction ~ Days + (1 + Days|Subject), data = dat_train)

# Predict on training set
train_preds  <- merTools::predictInterval(fit_lmer2, newdata = dat_train, n.sims = 100, returnSims = TRUE, seed = 657, level = 0.9) %>%

dat_train <- dat_train %>% bind_cols(train_preds)

dat_train$group <- "train"

# Predict on test set with 90% prediction intervals
test_preds  <- merTools::predictInterval(fit_lmer2, newdata = dat_test, n.sims = 100, returnSims = TRUE, seed = 657, level = 0.9) %>%

dat_test <- dat_test %>% bind_cols(test_preds)

dat_test$group <- "test"

## Combine the data together
combined_dat <- bind_rows(dat_train, dat_test) %>%
  arrange(Subject, Days)

## Plot the time series of predictions and observed data
combined_dat %>%
mutate(group = factor(group, levels = c("train", "test"))) %>%
ggplot(aes(x = Days, y = Reaction)) +
  geom_ribbon(aes(ymin = lwr,
                  ymax = upr),
              fill = "light grey",
              alpha = 0.8) +
  geom_line(aes(y = fit),
            col = "red",
            size = 1) +
  geom_point(aes(fill = group),
             size = 3,
             shape = 21) +
  geom_line() +
  facet_wrap(~Subject) +
  theme(strip.background = element_rect(fill = "black"),
        strip.text = element_text(face = "bold", color = "white"),
        legend.position = "top") +
  labs(x = "Days",
       y = "Reaction Time",
       title = "Reaction Time based on Days of Sleep Deprivation")

Wrapping Up

Above are a few different plot options we have with mixed model outputs. I’m not sure what James was after or what he had in mind because he left the question very wide open. Hopefully this article provides some useful ideas for your own mixed model plotting. If there are other things you are hoping to see or have other ideas of things to plot from the mixed model output, feel free to reach out!

The complete code for this article is available on my GITHUB page.