Correlation p-values

While we often look at the r value to determine whether two values are correlated, we can also perform a significance test. This test gives us a P-value which we can use to determine if the variables are statistically significantly correlated. The code below allows us to do this in base R. We use the cor.test function which generates a correlation object. We can then use the $ operator to extract a range of parameters from this object: including the p-value.

correlation <- cor.test(news_data$syst, news_data$dias) # using the base R cor.test function
correlation$p.value # using the $ operator to get the p-value from the list

## [1] 1.618988e-98

Multiple correlations

Whilst the above is useful, what if we want to know the correlation between all variables in our dataset? The code below allows us to do that.

#examine the correlation between all variables in the dataset
#the where function tells select to look across all columns for the condition specified within it 
news_data %>% 
  select(where(is.numeric)) %>% 
  cor()

##                age         syst         dias          temp        pulse
## age    1.000000000  0.148244004 -0.119165429 -0.0519958732 -0.005109752
## syst   0.148244004  1.000000000  0.599256892  0.0039228062 -0.150733371
## dias  -0.119165429  0.599256892  1.000000000 -0.0287346006 -0.042963451
## temp  -0.051995873  0.003922806 -0.028734601  1.0000000000  0.229100108
## pulse -0.005109752 -0.150733371 -0.042963451  0.2291001075  1.000000000
## resp   0.170941449 -0.024446803 -0.013886878 -0.0004084707  0.296901622
## sat   -0.237289766 -0.034747455  0.002651182 -0.1181545716 -0.216306805
##                resp          sat
## age    0.1709414489 -0.237289766
## syst  -0.0244468029 -0.034747455
## dias  -0.0138868782  0.002651182
## temp  -0.0004084707 -0.118154572
## pulse  0.2969016219 -0.216306805
## resp   1.0000000000 -0.222280786
## sat   -0.2222807860  1.000000000

This is my preferred way to calculate multiple correlations within a dataframe. It uses the select function from dplyr alongside the tidyselect helper function where and within this where function we use the base R function is.numeric. This line of code only returns columns that are numeric and therefore a correlation can be calculated. We then use the pipe operator to supply this to the cor function to generate a correlation matrix.

We can see from the output we have now generated a standard correlation matrix where there is a line of perfect correlations (1) between the same variables stretching diagonally across the matrix. We can then see all the values above the diagonal line of perfect correlations is a mirror of the values below this line. However, the output is not particularly easy to read. The excessive number of decimal places alongside the monochrome colouring makes identifying any trends or patterns difficult.

P-values from multiple correlations

We can also get P-values from multiple correlations at the same time. To do this, we will use the rcorr package from the Hmisc package. We provide this function our dataframe converted to a matrix (using the as.matrix function).

#as.matrix creates a matrix from the dataframe which is run through the rcorr function
library(Hmisc)
rcorr(as.matrix(news_data)) 

##        male   age  NEWS  syst  dias  temp pulse  resp   sat   sup alert  died
## male   1.00  0.03 -0.01 -0.01 -0.03 -0.04 -0.04  0.00  0.01  0.04  0.06  0.01
## age    0.03  1.00  0.18  0.15 -0.12 -0.05 -0.01  0.17 -0.24  0.09  0.07  0.15
## NEWS  -0.01  0.18  1.00 -0.24 -0.16 -0.08  0.45  0.62 -0.37  0.54  0.34  0.22
## syst  -0.01  0.15 -0.24  1.00  0.60  0.00 -0.15 -0.02 -0.03 -0.12 -0.01  0.00
## dias  -0.03 -0.12 -0.16  0.60  1.00 -0.03 -0.04 -0.01  0.00 -0.05 -0.04 -0.08
## temp  -0.04 -0.05 -0.08  0.00 -0.03  1.00  0.23  0.00 -0.12  0.05 -0.02 -0.05
## pulse -0.04 -0.01  0.45 -0.15 -0.04  0.23  1.00  0.30 -0.22  0.13  0.08  0.10
## resp   0.00  0.17  0.62 -0.02 -0.01  0.00  0.30  1.00 -0.22  0.30  0.07  0.11
## sat    0.01 -0.24 -0.37 -0.03  0.00 -0.12 -0.22 -0.22  1.00 -0.10 -0.09 -0.04
## sup    0.04  0.09  0.54 -0.12 -0.05  0.05  0.13  0.30 -0.10  1.00  0.29  0.11
## alert  0.06  0.07  0.34 -0.01 -0.04 -0.02  0.08  0.07 -0.09  0.29  1.00  0.19
## died   0.01  0.15  0.22  0.00 -0.08 -0.05  0.10  0.11 -0.04  0.11  0.19  1.00
## 
## n= 1000 
## 
## 
## P
##       male   age    NEWS   syst   dias   temp   pulse  resp   sat    sup   
## male         0.3881 0.8678 0.7952 0.2986 0.1647 0.2269 0.8899 0.7277 0.2139
## age   0.3881        0.0000 0.0000 0.0002 0.1003 0.8718 0.0000 0.0000 0.0029
## NEWS  0.8678 0.0000        0.0000 0.0000 0.0096 0.0000 0.0000 0.0000 0.0000
## syst  0.7952 0.0000 0.0000        0.0000 0.9014 0.0000 0.4400 0.2723 0.0001
## dias  0.2986 0.0002 0.0000 0.0000        0.3640 0.1746 0.6609 0.9333 0.1265
## temp  0.1647 0.1003 0.0096 0.9014 0.3640        0.0000 0.9897 0.0002 0.1228
## pulse 0.2269 0.8718 0.0000 0.0000 0.1746 0.0000        0.0000 0.0000 0.0000
## resp  0.8899 0.0000 0.0000 0.4400 0.6609 0.9897 0.0000        0.0000 0.0000
## sat   0.7277 0.0000 0.0000 0.2723 0.9333 0.0002 0.0000 0.0000        0.0013
## sup   0.2139 0.0029 0.0000 0.0001 0.1265 0.1228 0.0000 0.0000 0.0013       
## alert 0.0564 0.0317 0.0000 0.6506 0.2316 0.4678 0.0151 0.0324 0.0045 0.0000
## died  0.6771 0.0000 0.0000 0.9088 0.0175 0.1449 0.0017 0.0004 0.2662 0.0004
##       alert  died  
## male  0.0564 0.6771
## age   0.0317 0.0000
## NEWS  0.0000 0.0000
## syst  0.6506 0.9088
## dias  0.2316 0.0175
## temp  0.4678 0.1449
## pulse 0.0151 0.0017
## resp  0.0324 0.0004
## sat   0.0045 0.2662
## sup   0.0000 0.0004
## alert        0.0000
## died  0.0000

We can see this function gives us two dataframes, one with the perfect 1 correlation between matching variables and one where the perfect correlations have been removed. You can see from this matrix that some of our variables have very statistically significant correlations, whilst others do not.

Trying to assess the coefficients and p values within the console can get quite confusing and so using some little tricks from our previous posts, we can save the results as r objects.

#as.matrix creates a matrix from the dataframe which is run through the rcorr function
library(Hmisc)
rcorr <- rcorr(as.matrix(news_data)) 
#saves the correlation matrix as an object

rcorr.coeff <- rcorr$r %>%
  round(digit = 3)
#this piece of code assigns the coefficients from the rcorr list into its own object
#the round() code makes the numbers easier to interpret by rounding to 3 decimal places

rcorr.p <- rcorr$P %>%
  round(digit = 3)
#this achieves the same result but now for the p values

#displays the p results from the correlations
rcorr.p

##        male   age  NEWS  syst  dias  temp pulse  resp   sat   sup alert  died
## male     NA 0.388 0.868 0.795 0.299 0.165 0.227 0.890 0.728 0.214 0.056 0.677
## age   0.388    NA 0.000 0.000 0.000 0.100 0.872 0.000 0.000 0.003 0.032 0.000
## NEWS  0.868 0.000    NA 0.000 0.000 0.010 0.000 0.000 0.000 0.000 0.000 0.000
## syst  0.795 0.000 0.000    NA 0.000 0.901 0.000 0.440 0.272 0.000 0.651 0.909
## dias  0.299 0.000 0.000 0.000    NA 0.364 0.175 0.661 0.933 0.126 0.232 0.018
## temp  0.165 0.100 0.010 0.901 0.364    NA 0.000 0.990 0.000 0.123 0.468 0.145
## pulse 0.227 0.872 0.000 0.000 0.175 0.000    NA 0.000 0.000 0.000 0.015 0.002
## resp  0.890 0.000 0.000 0.440 0.661 0.990 0.000    NA 0.000 0.000 0.032 0.000
## sat   0.728 0.000 0.000 0.272 0.933 0.000 0.000 0.000    NA 0.001 0.005 0.266
## sup   0.214 0.003 0.000 0.000 0.126 0.123 0.000 0.000 0.001    NA 0.000 0.000
## alert 0.056 0.032 0.000 0.651 0.232 0.468 0.015 0.032 0.005 0.000    NA 0.000
## died  0.677 0.000 0.000 0.909 0.018 0.145 0.002 0.000 0.266 0.000 0.000    NA

From the above, you can now view and also export the results from the correlation matrix. This makes the results easier to understand and come back to later, rather than always viewing the results within the console.

kable

kable is a package that allows use to generate a more attractive looking table in R. In the code below, we use two function (kable and kable_styling) to generate a much more attractive version of our correlation matrix.

library(kableExtra)
#kable is a package that allows use to generate a more attractive looking table in R
#kable gives a basic kable table as an HTML table
news_data %>% 
  select(where(is.numeric)) %>% 
  cor() %>%
  kable() %>% 
  kable_styling() 

Hopefully you can appreciate that this output is more visually appealing and easier to read than the default output produced by R. However, even this table still requires significant effort to pick out the patterns and trends in the correlation matrix due to the lack of colour to differentiate values. Luckily, there is another package that can help us with this.

corrplot

The corrplot package provides a number of functions that allow us to generate plots of correlation matrices. These plots can be a highly effective method to communicate the overall trends and patterns regarding the correlations in a dataframe. The code below generates a basic correlation plot.

library(corrplot)
#generates a number corrplot
news_data %>% 
  select(where(is.numeric)) %>% 
  cor() %>%
  corrplot(method = "number")

We can see in this plot we have limited strong correlations in our data. However, systolic and diastolic blood pressure have a strong positive relationship (as one increases, so does the other) whilst blood oxygen saturation and respiration have a weak negative relationship. This visualisation allows us to much more easily determine patterns in our dataset and determine which variables are correlated.

The corrplot function can take a wide range of arguments (to see them all you can run ?corrplot) which can change the visual properties of the visualisation created. In the code below we change the method argument to circle and create a more visually appealing graph whilst sacrificing some finer details.

#generates a circle corrplot
news_data %>% 
  select(where(is.numeric)) %>% 
  cor() %>%
  corrplot(method = "circle")

The corrplot package also offers a function corrplot.mixed which can be used if we want to combine the two visualisations above. This can bring together the benefits of both of the above graphs into one.

#generate a mixed corrplot
news_data %>% 
  select(where(is.numeric)) %>% 
  cor() %>%
  corrplot.mixed()

As mentioned earlier, there are a host of options in the corrplot package which are outlined here including the ability to add p-values as well as other visual changes.