Tidy correlation tests in R

Tidy correlation tests in R

When we try to estimate the correlation coefficient between multiple variables, the task is more complicated in order to obtain a simple and tidy result. A simple solution is to use the tidy() function from the {broom} package. In this post we are going to estimate the correlation coefficients between the annual precipitation of several Spanish cities and climate teleconnections indices: download. The data of the teleconnections are preprocessed, but can be downloaded directly from crudata.uea.ac.uk. The daily precipitation data comes from ECA&D.

Packages

In this post we will use the following packages:

Package Description
tidyverse Collection of packages (visualization, manipulation): ggplot2, dplyr, purrr, etc.
broom Convert results of statistical functions (lm, t.test, cor.test, etc.) into tidy tables
fs Provides a cross-platform, uniform interface to file system operations
lubridate Easy manipulation of dates and times
#install the packages if necessary
if(!require("tidyverse")) install.packages("tidyverse")
if(!require("broom")) install.packages("broom")
if(!require("fs")) install.packages("fs")
if(!require("lubridate")) install.packages("lubridate")

#load packages
library(tidyverse)
library(broom)
library(fs)
library(lubridate)

Import data

First we have to import the daily precipitation of the selected weather stations.

  1. Create a vector with all precipitation files using the function dir_ls() of the {fs} package.
  2. Import the data using the map_df() function of the {purrr} package that applies another function to a vector or list, and joins them together in a single data.frame.
    1. Select the columns that interest us, b) Convert the date string into a date object using the ymd() function of the {lubridate} package, c) Create a new column yr with the years, d) Divide the precipitation values by 10 and reclassify absent values -9999 by NA, e) Finally, reclassify the ID of each weather station creating a factor with new labels.

More details about the use of the dir_ls() and map_df() functions can be found in this previous post.

#precipitation files
files <- dir_ls(regexp = "txt")
files
## RR_STAID001393.txt RR_STAID001394.txt RR_STAID002969.txt 
## RR_STAID003946.txt RR_STAID003969.txt
#import all files and join them together
pr <- files %>% map_df(read_csv, skip = 20)
## Parsed with column specification:
## cols(
##   STAID = col_double(),
##   SOUID = col_double(),
##   DATE = col_double(),
##   RR = col_double(),
##   Q_RR = col_double()
## )
## Parsed with column specification:
## cols(
##   STAID = col_double(),
##   SOUID = col_double(),
##   DATE = col_double(),
##   RR = col_double(),
##   Q_RR = col_double()
## )
## Parsed with column specification:
## cols(
##   STAID = col_double(),
##   SOUID = col_double(),
##   DATE = col_double(),
##   RR = col_double(),
##   Q_RR = col_double()
## )
## Parsed with column specification:
## cols(
##   STAID = col_double(),
##   SOUID = col_double(),
##   DATE = col_double(),
##   RR = col_double(),
##   Q_RR = col_double()
## )
## Parsed with column specification:
## cols(
##   STAID = col_double(),
##   SOUID = col_double(),
##   DATE = col_double(),
##   RR = col_double(),
##   Q_RR = col_double()
## )
pr
## # A tibble: 133,343 x 5
##    STAID SOUID     DATE    RR  Q_RR
##    <dbl> <dbl>    <dbl> <dbl> <dbl>
##  1  1393 20611 19470301     0     0
##  2  1393 20611 19470302     5     0
##  3  1393 20611 19470303     0     0
##  4  1393 20611 19470304    33     0
##  5  1393 20611 19470305    15     0
##  6  1393 20611 19470306     0     0
##  7  1393 20611 19470307    85     0
##  8  1393 20611 19470308     3     0
##  9  1393 20611 19470309     0     0
## 10  1393 20611 19470310     0     0
## # ... with 133,333 more rows
#create levels for the factor 
id <- unique(pr$STAID)

#the corresponding labels
lab <- c("Bilbao", "Santiago", "Barcelona", "Madrid", "Valencia")

#first changes
pr <- select(pr, STAID, DATE, RR) %>% 
        mutate(DATE = ymd(DATE), 
               RR = ifelse(RR == -9999, NA, RR/10), 
               STAID = factor(STAID, id, lab), 
               yr = year(DATE)) 
pr
## # A tibble: 133,343 x 4
##    STAID  DATE          RR    yr
##    <fct>  <date>     <dbl> <dbl>
##  1 Bilbao 1947-03-01   0    1947
##  2 Bilbao 1947-03-02   0.5  1947
##  3 Bilbao 1947-03-03   0    1947
##  4 Bilbao 1947-03-04   3.3  1947
##  5 Bilbao 1947-03-05   1.5  1947
##  6 Bilbao 1947-03-06   0    1947
##  7 Bilbao 1947-03-07   8.5  1947
##  8 Bilbao 1947-03-08   0.3  1947
##  9 Bilbao 1947-03-09   0    1947
## 10 Bilbao 1947-03-10   0    1947
## # ... with 133,333 more rows

We still need to filter and calculate the annual amount of precipitation. Actually, it is not correct to sum up precipitation without taking into account that there are missing values, but it should be enough for this practice. Then, we change the table format with the spread() function, passing from a long to a wide table, that is, we want to obtain one column per weather station.

pr_yr <- filter(pr, DATE >= "1950-01-01", DATE < "2018-01-01") %>%
           group_by(STAID, yr)%>%
             summarise(pr = sum(RR, na.rm = TRUE))
pr_yr
## # A tibble: 324 x 3
## # Groups:   STAID [5]
##    STAID     yr    pr
##    <fct>  <dbl> <dbl>
##  1 Bilbao  1950 1342 
##  2 Bilbao  1951 1306.
##  3 Bilbao  1952 1355.
##  4 Bilbao  1953 1372.
##  5 Bilbao  1954 1428.
##  6 Bilbao  1955 1062.
##  7 Bilbao  1956 1254.
##  8 Bilbao  1957  968.
##  9 Bilbao  1958 1272.
## 10 Bilbao  1959 1450.
## # ... with 314 more rows
pr_yr <- spread(pr_yr, STAID, pr)
pr_yr
## # A tibble: 68 x 6
##       yr Bilbao Santiago Barcelona Madrid Valencia
##    <dbl>  <dbl>    <dbl>     <dbl>  <dbl>    <dbl>
##  1  1950  1342     1800.      345     NA        NA
##  2  1951  1306.    2344.     1072.   798.       NA
##  3  1952  1355.    1973.      415.   524.       NA
##  4  1953  1372.     973.      683.   365.       NA
##  5  1954  1428.    1348.      581.   246.       NA
##  6  1955  1062.    1769.      530.   473.       NA
##  7  1956  1254.    1533.      695.   480.       NA
##  8  1957   968.    1599.      635.   424.       NA
##  9  1958  1272.    2658.      479.   482.       NA
## 10  1959  1450.    2847.     1006    665.       NA
## # ... with 58 more rows

The next step is to import the climate teleconnection indices.

#teleconnections
telecon <- read_csv("teleconnections_indices.csv")
## Parsed with column specification:
## cols(
##   yr = col_double(),
##   NAO = col_double(),
##   WeMO = col_double(),
##   EA = col_double(),
##   `POL-EUAS` = col_double(),
##   `EATL/WRUS` = col_double(),
##   MO = col_double(),
##   SCAND = col_double(),
##   AO = col_double()
## )
telecon
## # A tibble: 68 x 9
##       yr   NAO   WeMO     EA `POL-EUAS` `EATL/WRUS`    MO    SCAND       AO
##    <dbl> <dbl>  <dbl>  <dbl>      <dbl>       <dbl> <dbl>    <dbl>    <dbl>
##  1  1950  0.49  0.555 -0.332     0.0217     -0.0567 0.335  0.301   -1.99e-1
##  2  1951 -0.07  0.379 -0.372     0.402      -0.419  0.149 -0.00667 -3.65e-1
##  3  1952 -0.37  0.693 -0.688    -0.0117     -0.711  0.282  0.0642  -6.75e-1
##  4  1953  0.4  -0.213 -0.727    -0.0567     -0.0508 0.216  0.0233  -1.64e-2
##  5  1954  0.51  1.20  -0.912     0.142      -0.318  0.386  0.458   -5.83e-4
##  6  1955 -0.64  0.138 -0.824    -0.0267      0.154  0.134  0.0392  -3.62e-1
##  7  1956  0.17  0.617 -1.29     -0.197       0.0617 0.256  0.302   -1.63e-1
##  8  1957 -0.02  0.321 -0.952    -0.638      -0.167  0.322 -0.134   -3.42e-1
##  9  1958  0.12  0.941 -0.243     0.138       0.661  0.296  0.279   -8.68e-1
## 10  1959  0.49 -0.055 -0.23     -0.0142      0.631  0.316  0.725   -7.62e-2
## # ... with 58 more rows

Finally we need to join both tables by year.

data_all <- left_join(pr_yr, telecon, by = "yr")
data_all
## # A tibble: 68 x 14
##       yr Bilbao Santiago Barcelona Madrid Valencia   NAO   WeMO     EA
##    <dbl>  <dbl>    <dbl>     <dbl>  <dbl>    <dbl> <dbl>  <dbl>  <dbl>
##  1  1950  1342     1800.      345     NA        NA  0.49  0.555 -0.332
##  2  1951  1306.    2344.     1072.   798.       NA -0.07  0.379 -0.372
##  3  1952  1355.    1973.      415.   524.       NA -0.37  0.693 -0.688
##  4  1953  1372.     973.      683.   365.       NA  0.4  -0.213 -0.727
##  5  1954  1428.    1348.      581.   246.       NA  0.51  1.20  -0.912
##  6  1955  1062.    1769.      530.   473.       NA -0.64  0.138 -0.824
##  7  1956  1254.    1533.      695.   480.       NA  0.17  0.617 -1.29 
##  8  1957   968.    1599.      635.   424.       NA -0.02  0.321 -0.952
##  9  1958  1272.    2658.      479.   482.       NA  0.12  0.941 -0.243
## 10  1959  1450.    2847.     1006    665.       NA  0.49 -0.055 -0.23 
## # ... with 58 more rows, and 5 more variables: `POL-EUAS` <dbl>,
## #   `EATL/WRUS` <dbl>, MO <dbl>, SCAND <dbl>, AO <dbl>

Correlation test

A correlation test between paired samples can be done with the cor.test() function of R Base. In this case between the annual precipitation of Bilbao and the NAO index.

cor_nao_bil <- cor.test(data_all$Bilbao, data_all$NAO,
                        method = "spearman")
## Warning in cor.test.default(data_all$Bilbao, data_all$NAO, method =
## "spearman"): Cannot compute exact p-value with ties
cor_nao_bil
## 
##  Spearman's rank correlation rho
## 
## data:  data_all$Bilbao and data_all$NAO
## S = 44372, p-value = 0.2126
## alternative hypothesis: true rho is not equal to 0
## sample estimates:
##       rho 
## 0.1531149
str(cor_nao_bil)
## List of 8
##  $ statistic  : Named num 44372
##   ..- attr(*, "names")= chr "S"
##  $ parameter  : NULL
##  $ p.value    : num 0.213
##  $ estimate   : Named num 0.153
##   ..- attr(*, "names")= chr "rho"
##  $ null.value : Named num 0
##   ..- attr(*, "names")= chr "rho"
##  $ alternative: chr "two.sided"
##  $ method     : chr "Spearman's rank correlation rho"
##  $ data.name  : chr "data_all$Bilbao and data_all$NAO"
##  - attr(*, "class")= chr "htest"

We see that the result is in an unmanageable and untidy format. It is a console summary of the correlation with all the statistical parameters necessary to get a conclusion about the relationship. The orginal structure is a list of vectors. However, the tidy() function of the {broom} package allows us to convert the result into a table format.

tidy(cor_nao_bil)
## # A tibble: 1 x 5
##   estimate statistic p.value method                          alternative
##      <dbl>     <dbl>   <dbl> <chr>                           <chr>      
## 1    0.153    44372.   0.213 Spearman's rank correlation rho two.sided

Apply the correlation test to multiple variables

The objective is to apply the correlation test to all weather stations and climate teleconnection indices.

First, we must pass the table to the long format, that is, create a column/variable for the city and for the value of the corresponding precipitation. Then we repeat the same for the teleconnections indices.

data <- gather(data_all, city, pr, Bilbao:Valencia)%>%
                     gather(telecon, index, NAO:AO)
data
## # A tibble: 2,720 x 5
##       yr city      pr telecon index
##    <dbl> <chr>  <dbl> <chr>   <dbl>
##  1  1950 Bilbao 1342  NAO      0.49
##  2  1951 Bilbao 1306. NAO     -0.07
##  3  1952 Bilbao 1355. NAO     -0.37
##  4  1953 Bilbao 1372. NAO      0.4 
##  5  1954 Bilbao 1428. NAO      0.51
##  6  1955 Bilbao 1062. NAO     -0.64
##  7  1956 Bilbao 1254. NAO      0.17
##  8  1957 Bilbao  968. NAO     -0.02
##  9  1958 Bilbao 1272. NAO      0.12
## 10  1959 Bilbao 1450. NAO      0.49
## # ... with 2,710 more rows

To apply the test to all cities, we need the corresponding groupings. Therefore, we use the group_by() function for indicating the two groups: city and telecon. In addition, we apply the nest() function of the {tidyr} package ({tidyverse} collection) with the aim of creating lists of tables nested per row. In other words, in each row of each city and teleconnection index we will have a new table that contains the year, the precipitation value and the value of each teleconection, correspondingly.

data_nest <- group_by(data, city, telecon) %>% nest()
data_nest
## # A tibble: 40 x 3
##    city      telecon data             
##    <chr>     <chr>   <list>           
##  1 Bilbao    NAO     <tibble [68 x 3]>
##  2 Santiago  NAO     <tibble [68 x 3]>
##  3 Barcelona NAO     <tibble [68 x 3]>
##  4 Madrid    NAO     <tibble [68 x 3]>
##  5 Valencia  NAO     <tibble [68 x 3]>
##  6 Bilbao    WeMO    <tibble [68 x 3]>
##  7 Santiago  WeMO    <tibble [68 x 3]>
##  8 Barcelona WeMO    <tibble [68 x 3]>
##  9 Madrid    WeMO    <tibble [68 x 3]>
## 10 Valencia  WeMO    <tibble [68 x 3]>
## # ... with 30 more rows
str(slice(data_nest, 1))
## Classes 'tbl_df', 'tbl' and 'data.frame':    1 obs. of  3 variables:
##  $ city   : chr "Bilbao"
##  $ telecon: chr "NAO"
##  $ data   :List of 1
##   ..$ :Classes 'tbl_df', 'tbl' and 'data.frame': 68 obs. of  3 variables:
##   .. ..$ yr   : num  1950 1951 1952 1953 1954 ...
##   .. ..$ pr   : num  1342 1306 1355 1372 1428 ...
##   .. ..$ index: num  0.49 -0.07 -0.37 0.4 0.51 -0.64 0.17 -0.02 0.12 0.49 ...

The next step is to create a function, in which we define the correlation test and pass it to the clean format using the tidy() function, which we apply to each groupings.

cor_fun <- function(df) cor.test(df$pr, df$index, method = "spearman") %>% tidy()

Now we only have to apply our function to the column that contains the tables for each combination between city and teleconnection. To do this, we use the map() function that applies another function to a vector or list. What we do is create a new column that contains the result, a statistical summary table, for each combination.

data_nest <- mutate(data_nest, model = map(data, cor_fun))
data_nest
## # A tibble: 40 x 4
##    city      telecon data              model           
##    <chr>     <chr>   <list>            <list>          
##  1 Bilbao    NAO     <tibble [68 x 3]> <tibble [1 x 5]>
##  2 Santiago  NAO     <tibble [68 x 3]> <tibble [1 x 5]>
##  3 Barcelona NAO     <tibble [68 x 3]> <tibble [1 x 5]>
##  4 Madrid    NAO     <tibble [68 x 3]> <tibble [1 x 5]>
##  5 Valencia  NAO     <tibble [68 x 3]> <tibble [1 x 5]>
##  6 Bilbao    WeMO    <tibble [68 x 3]> <tibble [1 x 5]>
##  7 Santiago  WeMO    <tibble [68 x 3]> <tibble [1 x 5]>
##  8 Barcelona WeMO    <tibble [68 x 3]> <tibble [1 x 5]>
##  9 Madrid    WeMO    <tibble [68 x 3]> <tibble [1 x 5]>
## 10 Valencia  WeMO    <tibble [68 x 3]> <tibble [1 x 5]>
## # ... with 30 more rows
str(slice(data_nest, 1))
## Classes 'tbl_df', 'tbl' and 'data.frame':    1 obs. of  4 variables:
##  $ city   : chr "Bilbao"
##  $ telecon: chr "NAO"
##  $ data   :List of 1
##   ..$ :Classes 'tbl_df', 'tbl' and 'data.frame': 68 obs. of  3 variables:
##   .. ..$ yr   : num  1950 1951 1952 1953 1954 ...
##   .. ..$ pr   : num  1342 1306 1355 1372 1428 ...
##   .. ..$ index: num  0.49 -0.07 -0.37 0.4 0.51 -0.64 0.17 -0.02 0.12 0.49 ...
##  $ model  :List of 1
##   ..$ :Classes 'tbl_df', 'tbl' and 'data.frame': 1 obs. of  5 variables:
##   .. ..$ estimate   : num 0.153
##   .. ..$ statistic  : num 44372
##   .. ..$ p.value    : num 0.213
##   .. ..$ method     : chr "Spearman's rank correlation rho"
##   .. ..$ alternative: chr "two.sided"

How can we undo the list of tables in each row of our table?

First we eliminate the column with the data and then simply we can apply the unnest() function.

corr_pr <- select(data_nest, -data) %>% unnest()
corr_pr
## # A tibble: 40 x 7
##    city    telecon estimate statistic  p.value method           alternative
##    <chr>   <chr>      <dbl>     <dbl>    <dbl> <chr>            <chr>      
##  1 Bilbao  NAO       0.153     44372. 0.213    Spearman's rank~ two.sided  
##  2 Santia~ NAO      -0.181     61902. 0.139    Spearman's rank~ two.sided  
##  3 Barcel~ NAO      -0.0203    53460. 0.869    Spearman's rank~ two.sided  
##  4 Madrid  NAO      -0.291     64692. 0.0169   Spearman's rank~ two.sided  
##  5 Valenc~ NAO      -0.113     27600. 0.422    Spearman's rank~ two.sided  
##  6 Bilbao  WeMO      0.404     31242. 0.000706 Spearman's rank~ two.sided  
##  7 Santia~ WeMO      0.332     35014. 0.00594  Spearman's rank~ two.sided  
##  8 Barcel~ WeMO      0.0292    50862  0.813    Spearman's rank~ two.sided  
##  9 Madrid  WeMO      0.109     44660  0.380    Spearman's rank~ two.sided  
## 10 Valenc~ WeMO     -0.252     31056. 0.0688   Spearman's rank~ two.sided  
## # ... with 30 more rows

The result is a table in which we can see the correlations and their statistical significance for each city and teleconnection index.

Heatmap of the results

Finally, we make a heatmap of the obtained result. But, previously we create a column that indicates whether the correlation is significant with p-value less than 0.05.

corr_pr <- mutate(corr_pr, sig = ifelse(p.value <0.05, "Sig.", "Non Sig."))
ggplot()+
  geom_tile(data = corr_pr,
            aes(city, telecon, fill = estimate),
            size = 1,
            colour = "white")+
  geom_tile(data = filter(corr_pr, sig == "Sig."),
            aes(city, telecon),
            size = 1,
            colour = "black",
            fill = "transparent")+
  geom_text(data = corr_pr,
            aes(city, telecon, label = round(estimate, 2),
            fontface = ifelse(sig == "Sig.", "bold", "plain")))+
  scale_fill_gradient2(breaks = seq(-1, 1, 0.2))+
  labs(x = "", y = "", fill = "", p.value = "")+
  theme_minimal()+
  theme(panel.grid.major = element_blank(),
        panel.border = element_blank(),
        panel.background = element_blank(),
        axis.ticks = element_blank())

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