Correlated vs. Uncorrelated

gistfile1.txt

# EXERCISE TO BUILD INTUITION FOR CORRELATED VS. UNCORRELATED DATA
# PLEASE FOCUS ON UNDERSTANDING THE BELOW
### DO NOT JUST EXECUTE ALL THE CODE IN ONE BATCH--RUN IT LINE BY LINE...

### Simulation of analysis on correlated data

set.seed(1314)

nsims <- 10000

storage_correlated_df <- matrix(NA, ncol = 10000, nrow = 100)

# GO THROUGH THE BELOW, LINE BY LINE, & TRY TO UNDERSTAND EVERYTHING
# FOCUS ON WHATEVER YOU FIND CONFUSING

set.seed(2321)
for(j in 1:nsims) {

  simulated_original_data <- rnorm(100)
  
    for(i in 1:100)
  {

    a_bootstrapped_sample <- sample(simulated_original_data, 100, replace = TRUE)
    summary_statistic_of_boot_sample <- mean(a_bootstrapped_sample)
    storage_correlated_df[i,j] <- summary_statistic_of_boot_sample
  }
  
}

## ANALYSIS WITH THE RESULTS...
correlated_means <- apply(storage_correlated_df, 2, mean)
print(mean(correlated_means))

var_of_correlated_means <- var(correlated_means)
print(var_of_correlated_means)

### NOW, what if our data is not correlated?

storage.NOTcorr.df <- matrix(NA, ncol = 10000, nrow = 100)

# AGAIN, GO THROUGH THE BELOW, LINE BY LINE, & TRY TO UNDERSTAND EVERYTHING
# FOCUS ON WHATEVER YOU FIND CONFUSING

set.seed(2353)
for(j in 1:nsims) {
  for(i in 1:100)
  {
    an_independent_sample_of_the_data <- rnorm(100)
    storage.NOTcorr.df[i,j] <- mean(an_independent_sample_of_the_data)
  }
  
}

## ANALYSIS WITH THE RESULTS...
NOTcorr_means <- apply(storage.NOTcorr.df, 2, mean)
print(mean(NOTcorr_means))
var_of_NOTcorr_means <- var(NOTcorr_means)
print(var_of_NOTcorr_means)

## SYNTHESIZING THE RESULTS (is the var of uncorrelated smaller, and if so, by how much?)
percent_var_redux <- 100*(var_of_correlated_means - var_of_NOTcorr_means)/var_of_correlated_means
print(percent_var_redux)

# LET'S VISUALIZE!!!
par(mfrow = c(4,1))
hist(correlated_means, main = "Distribution of Correlated Means")
## WHEN YOU RUN THE HIST BELOW, COMPARE THE SCALE OF THE X-AXIS TO THE HIST ABOVE
hist(NOTcorr_means, main = "Distribution of NOT Correlated Means")

# HERE IS THE SAME INFO IN ONE FIGURE, WITH LINES SHOWING DENSITY...
plot(density(NOTcorr_means), col = "blue", lwd = 2)
lines(density(correlated_means), col = "red", lwd = 2) # 

# IF YOU HAVE TROUBLE INTERPRETING THIS, HERE IT IS ON A DIFFERENT SCALE...
plot(density(correlated_means), col = "red", lwd = 2)
lines(density(NOTcorr_means), col = "blue", lwd = 2)