Calculating Correlations

Testing the Assumption

You can easily check this by creating a scatter plot. You can use the base function plot() for this. We will learn the more powerful graphics library ggplot2 in the last learning block.

plot(
  pss$trstprl, 
  pss$trstplt
)

Since the data is only pseudo-metric and many data points (integer) overlap, you can’t see much from the plot.

Solution: Use the jitter() function to spread the points more widely:

plot(
  jitter(
    pss$trstprl, 
    3
  ) ~ 
    jitter(
      pss$trstplt, 
      3
    )
)

This graphical analysis takes some getting used to: If you don’t see clear patterns like in the Anscombe quartet, you can assume a linear relationship.

In conclusion, we can state that the conditions are met:

• Sample of paired values \(\checkmark\)

• both variables are metric \(\checkmark\)

• Relationship between variables is linear \(\checkmark\)

\(\Rightarrow\) You can now calculate Pearson’s r!

Calculating the Coefficient

To calculate the correlation coefficient, you use the cor() function (for both Pearson’s r and Spearman’s \(\rho\)). First, you need to name the two variables in the function. Then you should choose the correlation coefficient and finally select how to deal with NA's in the variables. Here, you delete any row that has an NA value in either of the two variables.

cor(
  pss$trstprl, 
  pss$trstplt, 
  method = "pearson",  # alternativ hier "spearman"  
  use = "complete.obs"
)      

## [1] 0.2318401

The output shows a correlation coefficient of \(r \approx 0.232\). In this output, there is no p-value included, and you cannot make a statement about significance.

Calculating the coefficient with the library psych

With the library psych, you can use the function corr.test(), which also provides the significance test:

install.packages("psych")
library("psych")

corr.test(
  pss$trstprl, 
  pss$trstplt,
  method = "pearson",      
  use = "complete.obs"
)     

## Call:corr.test(x = pss$trstprl, y = pss$trstplt, use = "complete.obs", 
##     method = "pearson")
## Correlation matrix 
## [1] 0.23
## Sample Size 
## [1] 4954
## These are the unadjusted probability values.
##   The probability values  adjusted for multiple tests are in the p.adj object. 
## [1] 0
## 
##  To see confidence intervals of the correlations, print with the short=FALSE option

This test generates three matrices (correlation matrix, sample size matrix, p-value matrix) that you can later use for visualization.

The additional information needed for the p-value is found at the last position. Here, a p-value of \(0\) is present.

Calculating multiple correlations

With both functions, you can calculate not only the correlation between two variables but also specify more than two variables directly. It will then calculate the correlation pairwise between all variables. To do this, use the function c() to specify between which variables you want to get pairwise correlation values:

cor(
  pss[
    ,
    c(
      "trstprl", 
      "trstplt",
      "trstprt",
      "trstlgl"
    )
  ],
  method = "pearson",
  use = "complete.obs"
)  

##           trstprl    trstplt   trstprt    trstlgl
## trstprl 1.0000000 0.22712300 0.3824902 0.22536686
## trstplt 0.2271230 1.00000000 0.3992831 0.05203906
## trstprt 0.3824902 0.39928307 1.0000000 0.24868786
## trstlgl 0.2253669 0.05203906 0.2486879 1.00000000

corr.test(
  pss[
    ,
    c(
      "trstprl",
      "trstplt",
      "trstprt",
      "trstlgl"
    )
  ],
  method = "pearson",
  use = "complete.obs"
)  

## Call:corr.test(x = pss[, c("trstprl", "trstplt", "trstprt", "trstlgl")], 
##     use = "complete.obs", method = "pearson")
## Correlation matrix 
##         trstprl trstplt trstprt trstlgl
## trstprl    1.00    0.23    0.38    0.23
## trstplt    0.23    1.00    0.40    0.05
## trstprt    0.38    0.40    1.00    0.25
## trstlgl    0.23    0.05    0.25    1.00
## Sample Size 
##         trstprl trstplt trstprt trstlgl
## trstprl    4965    4954    4948    4953
## trstplt    4954    4989    4972    4977
## trstprt    4948    4972    4983    4971
## trstlgl    4953    4977    4971    4988
## Probability values (Entries above the diagonal are adjusted for multiple tests.) 
##         trstprl trstplt trstprt trstlgl
## trstprl       0       0       0       0
## trstplt       0       0       0       0
## trstprt       0       0       0       0
## trstlgl       0       0       0       0
## 
##  To see confidence intervals of the correlations, print with the short=FALSE option

Both outputs show a correlation matrix. The variable names are given in the columns and rows. The diagonal is always \(1\) because the relationship between the variable and itself is \(1\) (perfect!).

How do we graphically represent correlations now?