Calculating pairwise scores using bullseye.

bullseye is an R package which calculates measures of correlation and other association scores for pairs of variables in a dataset and offers visualisations of these measures in different layouts. The package also calculates and visualises the pairwise scores for different levels of a grouping variable.

This vignette gives an overview of how these pairwise variable measures are calculated. Visualisations of these calculated measures are provided in the accompanying vignette.

# install.packages("palmerpenguins")

library(bullseye)
library(palmerpenguins)
library(dplyr)

Table 1 lists the different measures of association provided in the package with the variable types they can be used with, the package used for calculation, the information on whether the measure is symmetric, and the minimum and maximum value of the measure.

List of the functions available in the package for calculating different association measures along with the packages used for calculation.
name nn ff fn from range ordinal
pair_cor TRUE FALSE FALSE cor [-1,1] NA
pair_dcor TRUE FALSE FALSE energy::dcor2d [0,1] NA
pair_mine TRUE FALSE FALSE minerva::mine [0,1] NA
pair_ace TRUE TRUE TRUE acepack::ace [0,1] FALSE
pair_cancor TRUE TRUE TRUE cancor [0,1] FALSE
pair_nmi TRUE TRUE TRUE linkspotter::maxNMI [0,1] FALSE
pair_polychor FALSE TRUE FALSE polycor::polychor [-1,1] TRUE
pair_polyserial FALSE FALSE TRUE polycor::polyserial [-1,1] TRUE
pair_tauB FALSE TRUE FALSE DescTools::KendalTauB [-1,1] TRUE
pair_tauA FALSE TRUE FALSE DescTools::KendalTauA [-1,1] TRUE
pair_tauC FALSE TRUE FALSE DescTools::StuartTauC [-1,1] TRUE
pair_tauW FALSE TRUE FALSE DescTools::KendalW [-1,1] TRUE
pair_gkGamma FALSE TRUE FALSE DescTools::GoodmanKruskalGamma [-1,1] TRUE
pair_gkTau FALSE TRUE FALSE DescTools::GoodmanKruskalTau [0,1] TRUE
pair_uncertainty FALSE TRUE FALSE DescTools::UncertCoef [0,1] FALSE
pair_chi FALSE TRUE FALSE DescTools::ContCoef [0,1] FALSE
pair_scag TRUE FALSE FALSE scagnostics::scagnostics [0,1] NA

Calculating correlation and other association measures

Each of the functions in the first column of Table 1 calculates pairwise scores for a dataset.

sc_dcor <- pair_dcor(penguins)
str(sc_dcor)
#> pairwise [10 × 6] (S3: pairwise/tbl_df/tbl/data.frame)
#>  $ x        : chr [1:10] "bill_depth_mm" "bill_length_mm" "bill_depth_mm" "body_mass_g" ...
#>  $ y        : chr [1:10] "bill_length_mm" "flipper_length_mm" "flipper_length_mm" "flipper_length_mm" ...
#>  $ score    : chr [1:10] "dcor" "dcor" "dcor" "dcor" ...
#>  $ group    : chr [1:10] "all" "all" "all" "all" ...
#>  $ value    : Named num [1:10] 0.387 0.666 0.704 0.867 0.587 ...
#>   ..- attr(*, "names")= chr [1:10] "" "" "" "" ...
#>  $ pair_type: chr [1:10] "nn" "nn" "nn" "nn" ...

For example, we see that pair_dcor calculates the distance correlation for every pair of numeric variables in the penguins dataset. There are missing values in this dataset, all the pair_ functions use pairwise complete observations by default.

sc_dcor is a tibble of class pairwise, with the two variables in columns x and y (arranged in alphabetical order), calculated values in the column value, and the name of the score calculated in the column score. All of the variables are numeric, hence “nn” in the pair_type column.

Similarly, one can use pair_nmi to calculate normalised mutual information for numeric, factor and mixed pairs of variables.

sc_nmi <- pair_nmi(penguins)
sc_nmi
#> # A tibble: 28 × 6
#>    x                 y                 score group  value pair_type
#>    <chr>             <chr>             <chr> <chr>  <dbl> <chr>    
#>  1 bill_depth_mm     bill_length_mm    nmi   all   0.225  nn       
#>  2 bill_length_mm    flipper_length_mm nmi   all   0.375  nn       
#>  3 bill_depth_mm     flipper_length_mm nmi   all   0.470  nn       
#>  4 body_mass_g       flipper_length_mm nmi   all   0.581  nn       
#>  5 bill_length_mm    body_mass_g       nmi   all   0.303  nn       
#>  6 bill_depth_mm     body_mass_g       nmi   all   0.443  nn       
#>  7 bill_length_mm    year              nmi   all   0.0517 nn       
#>  8 bill_depth_mm     year              nmi   all   0.0387 nn       
#>  9 flipper_length_mm year              nmi   all   0.0707 nn       
#> 10 body_mass_g       year              nmi   all   0.0445 nn       
#> # ℹ 18 more rows

The main difference here is that factor variables are included. In the pair_type column, “ff” and “fn” indicate factor-factor and factor-numeric pairs.

If you want more control over the measure calculated, the function pairwise_scores calculates a different score depending on variable types.

pairwise_scores(penguins) |> distinct(score, pair_type)
#> # A tibble: 3 × 2
#>   score   pair_type
#>   <chr>   <chr>    
#> 1 cancor  ff       
#> 2 cancor  fn       
#> 3 pearson nn

As you can see, the default uses pearson’s correlation for numeric pairs, and canonical correlation for factor-numeric or factor-factor pairs. In addition polychoric correlation is used for two ordered factors, but there are no ordered factors in this data. Alternative scores may be specified using the control argument to pairwise_scores. The default value for this control argument is given by

pair_control()
#> $nn
#> [1] "pair_cor"
#> 
#> $fn
#> [1] "pair_cancor"
#> 
#> $oo
#> [1] "pair_polychor"
#> 
#> $ff
#> [1] "pair_cancor"
#> 
#> $nnargs
#> NULL
#> 
#> $fnargs
#> NULL
#> 
#> $ooargs
#> NULL
#> 
#> $ffargs
#> NULL

Calculating multiple measures

If you want for instance to compare distance correlation and mutual information measures in a display, two pairwise data structures can be combined:

bind_rows(sc_dcor, sc_nmi) |> arrange(x,y)
#> # A tibble: 38 × 6
#>    x             y                 score group value pair_type
#>    <chr>         <chr>             <chr> <chr> <dbl> <chr>    
#>  1 bill_depth_mm bill_length_mm    dcor  all   0.387 nn       
#>  2 bill_depth_mm bill_length_mm    nmi   all   0.225 nn       
#>  3 bill_depth_mm body_mass_g       dcor  all   0.614 nn       
#>  4 bill_depth_mm body_mass_g       nmi   all   0.443 nn       
#>  5 bill_depth_mm flipper_length_mm dcor  all   0.704 nn       
#>  6 bill_depth_mm flipper_length_mm nmi   all   0.470 nn       
#>  7 bill_depth_mm island            nmi   all   0.282 fn       
#>  8 bill_depth_mm sex               nmi   all   0.356 fn       
#>  9 bill_depth_mm species           nmi   all   0.493 fn       
#> 10 bill_depth_mm year              dcor  all   0.112 nn       
#> # ℹ 28 more rows

We provide another function pairwise_multi which calculates multiple association measures for every variable pair in a dataset. By default this function combines the results of pair_cor, pair_dcor,pair_mine,pair_ace, pair_cancor,pair_nmi,pair_uncertainty, pair_chi, but any subset of the pair_ functions may be supplied as an argument, as in the second example below.

pairwise_multi(penguins)
#> # A tibble: 130 × 6
#>    x             y              score    group  value pair_type
#>    <chr>         <chr>          <chr>    <chr>  <dbl> <chr>    
#>  1 bill_depth_mm bill_length_mm pearson  all   -0.235 nn       
#>  2 bill_depth_mm bill_length_mm dcor     all    0.387 nn       
#>  3 bill_depth_mm bill_length_mm MIC      all    0.313 nn       
#>  4 bill_depth_mm bill_length_mm ace      all    0.585 nn       
#>  5 bill_depth_mm bill_length_mm cancor   all    0.235 nn       
#>  6 bill_depth_mm bill_length_mm nmi      all    0.225 nn       
#>  7 bill_depth_mm bill_length_mm spearman all   -0.222 nn       
#>  8 bill_depth_mm body_mass_g    pearson  all   -0.472 nn       
#>  9 bill_depth_mm body_mass_g    dcor     all    0.614 nn       
#> 10 bill_depth_mm body_mass_g    MIC      all    0.518 nn       
#> # ℹ 120 more rows
dcor_nmi <- pairwise_multi(penguins, c("pair_dcor", "pair_nmi"))

Calculating grouped measures

For each of the pairwise calculation functions, they can be wrapped using pairwise_by to build a score calculation for each level of a grouping variable. Of course, grouped scores could be calculated using dplyr machinery, but it is a bit more work.

pairwise_by(penguins, by="species", pair_cor)
#> # A tibble: 40 × 6
#>    x             y                 score   group      value pair_type
#>    <chr>         <chr>             <chr>   <fct>      <dbl> <chr>    
#>  1 bill_depth_mm bill_length_mm    pearson Adelie     0.391 nn       
#>  2 bill_depth_mm bill_length_mm    pearson Chinstrap  0.654 nn       
#>  3 bill_depth_mm bill_length_mm    pearson Gentoo     0.643 nn       
#>  4 bill_depth_mm bill_length_mm    pearson all       -0.235 nn       
#>  5 bill_depth_mm body_mass_g       pearson Adelie     0.576 nn       
#>  6 bill_depth_mm body_mass_g       pearson Chinstrap  0.604 nn       
#>  7 bill_depth_mm body_mass_g       pearson Gentoo     0.719 nn       
#>  8 bill_depth_mm body_mass_g       pearson all       -0.472 nn       
#>  9 bill_depth_mm flipper_length_mm pearson Adelie     0.308 nn       
#> 10 bill_depth_mm flipper_length_mm pearson Chinstrap  0.580 nn       
#> # ℹ 30 more rows

Use argument ungrouped=FALSE to suppress calculation of the ungrouped scores.

pairwise_scores has a by argument, and provides pairwise scores for the levels of a grouping variable.

sc_sex <- pairwise_scores(penguins, by="species")

The column group now shows the levels of the grouping variable, along with “all” for ungrouped scores. Use ungrouped=FALSE to suppress calculation of the ungrouped scores.

sc_sex |> distinct(group)
#> # A tibble: 4 × 1
#>   group    
#>   <fct>    
#> 1 Adelie   
#> 2 Chinstrap
#> 3 Gentoo   
#> 4 all

If you want to calculate different scores to the default, specify this via the control argument:

pc <- pair_control(nn="pairwise_multi", nnargs= c("pair_dcor", "pair_ace"), fn=NULL, ff=NULL)
sc_sex <- pairwise_scores(penguins, by="species", control=pc, ungrouped=FALSE) 

Scagnostic measures

The package scagnostics provides pairwise variable scores based on graph-theoretic interestingness measures, for numeric variable pairs only.

pair_scagnostics(penguins[,1:5], scagnostic=c("Stringy", "Clumpy"))
#> # A tibble: 6 × 6
#>   x              y                 score   group  value pair_type
#>   <chr>          <chr>             <chr>   <chr>  <dbl> <chr>    
#> 1 bill_depth_mm  bill_length_mm    Stringy all   0.331  nn       
#> 2 bill_depth_mm  bill_length_mm    Clumpy  all   0.0328 nn       
#> 3 bill_depth_mm  flipper_length_mm Stringy all   0.378  nn       
#> 4 bill_depth_mm  flipper_length_mm Clumpy  all   0.530  nn       
#> 5 bill_length_mm flipper_length_mm Stringy all   0.370  nn       
#> 6 bill_length_mm flipper_length_mm Clumpy  all   0.0388 nn

Note that the first two variables of the penguins data are non-numeric and so are ignored in the above calculation.

For group-wise calculation:

pairwise_by(penguins[,1:5], by="species",function(x) pair_scagnostics(x, scagnostic=c("Stringy", "Clumpy")))
#> # A tibble: 24 × 6
#>    x             y                 score   group      value pair_type
#>    <chr>         <chr>             <chr>   <fct>      <dbl> <chr>    
#>  1 bill_depth_mm bill_length_mm    Stringy Adelie    0.278  nn       
#>  2 bill_depth_mm bill_length_mm    Clumpy  Adelie    0.0477 nn       
#>  3 bill_depth_mm bill_length_mm    Stringy Chinstrap 0.325  nn       
#>  4 bill_depth_mm bill_length_mm    Clumpy  Chinstrap 0.0758 nn       
#>  5 bill_depth_mm bill_length_mm    Stringy Gentoo    0.393  nn       
#>  6 bill_depth_mm bill_length_mm    Clumpy  Gentoo    0.0579 nn       
#>  7 bill_depth_mm bill_length_mm    Stringy all       0.331  nn       
#>  8 bill_depth_mm bill_length_mm    Clumpy  all       0.0328 nn       
#>  9 bill_depth_mm flipper_length_mm Stringy Adelie    0.392  nn       
#> 10 bill_depth_mm flipper_length_mm Clumpy  Adelie    0.0598 nn       
#> # ℹ 14 more rows

Converting symmetric matrices to pairwise and vice-versa.

The conventional way of representing pairwise scores or correlations is via a numeric symmetric matrix. The tidy pairwise structure we use in bullseye is more flexible, and is amenable to multiple measures of association and grouped measures.

It is straightforward to convert from a symmetric matrix to pairwise:

x <- cor(penguins[, c("bill_length_mm", "bill_depth_mm" ,"flipper_length_mm" ,"body_mass_g")], 
         use= "pairwise.complete.obs")
pairwise(x, score="pearson", pair_type = "nn")
#> # A tibble: 6 × 6
#>   x              y                 score   group  value pair_type
#>   <chr>          <chr>             <chr>   <chr>  <dbl> <chr>    
#> 1 bill_depth_mm  bill_length_mm    pearson all   -0.235 nn       
#> 2 bill_length_mm flipper_length_mm pearson all    0.656 nn       
#> 3 bill_depth_mm  flipper_length_mm pearson all   -0.584 nn       
#> 4 body_mass_g    flipper_length_mm pearson all    0.871 nn       
#> 5 bill_length_mm body_mass_g       pearson all    0.595 nn       
#> 6 bill_depth_mm  body_mass_g       pearson all   -0.472 nn

And for the reverse, converting a pairwise to a symmetric matrix:

as.matrix(sc_dcor)
#>                   bill_depth_mm bill_length_mm body_mass_g flipper_length_mm
#> bill_depth_mm                NA     0.38720211   0.6141631         0.7039636
#> bill_length_mm        0.3872021             NA   0.5871319         0.6664558
#> body_mass_g           0.6141631     0.58713186          NA         0.8674122
#> flipper_length_mm     0.7039636     0.66645577   0.8674122                NA
#> year                  0.1117057     0.07842516   0.0790560         0.1643876
#>                         year
#> bill_depth_mm     0.11170568
#> bill_length_mm    0.07842516
#> body_mass_g       0.07905600
#> flipper_length_mm 0.16438763
#> year                      NA

Converting structures from package correlation:

correlation package calculates different kinds of correlations, such as partial correlations, Bayesian correlations, multilevel correlations, polychoric correlations, biweight, percentage bend or Sheperd’s Pi correlations, distance correlation and more. The output data structure is a tidy dataframe with a correlation value and correlation tests for variable pairs for which the correlation method is defined.

correlation::correlation(penguins)
#> # Correlation Matrix (pearson-method)
#> 
#> Parameter1        |        Parameter2 |     r |         95% CI | t(340) |         p
#> -----------------------------------------------------------------------------------
#> bill_length_mm    |     bill_depth_mm | -0.24 | [-0.33, -0.13] |  -4.46 | < .001***
#> bill_length_mm    | flipper_length_mm |  0.66 | [ 0.59,  0.71] |  16.03 | < .001***
#> bill_length_mm    |       body_mass_g |  0.60 | [ 0.52,  0.66] |  13.65 | < .001***
#> bill_length_mm    |              year |  0.05 | [-0.05,  0.16] |   1.01 | 0.797    
#> bill_depth_mm     | flipper_length_mm | -0.58 | [-0.65, -0.51] | -13.26 | < .001***
#> bill_depth_mm     |       body_mass_g | -0.47 | [-0.55, -0.39] |  -9.87 | < .001***
#> bill_depth_mm     |              year | -0.06 | [-0.17,  0.05] |  -1.11 | 0.797    
#> flipper_length_mm |       body_mass_g |  0.87 | [ 0.84,  0.89] |  32.72 | < .001***
#> flipper_length_mm |              year |  0.17 | [ 0.06,  0.27] |   3.17 | 0.007**  
#> body_mass_g       |              year |  0.04 | [-0.06,  0.15] |   0.78 | 0.797    
#> 
#> p-value adjustment method: Holm (1979)
#> Observations: 342

The default calculation uses Pearson correlation. Other options are available via the method argument.

As there is an as.matrix method provided for the results of correlation::correlation, it is straightforward to convert this to a pairwise tibble.

x <- correlation::correlation(penguins)
pairwise(as.matrix(x)) 
#> # A tibble: 10 × 6
#>    x                 y                 score group   value pair_type
#>    <chr>             <chr>             <chr> <chr>   <dbl> <chr>    
#>  1 bill_depth_mm     bill_length_mm    <NA>  all   -0.235  <NA>     
#>  2 bill_length_mm    flipper_length_mm <NA>  all    0.656  <NA>     
#>  3 bill_depth_mm     flipper_length_mm <NA>  all   -0.584  <NA>     
#>  4 body_mass_g       flipper_length_mm <NA>  all    0.871  <NA>     
#>  5 bill_length_mm    body_mass_g       <NA>  all    0.595  <NA>     
#>  6 bill_depth_mm     body_mass_g       <NA>  all   -0.472  <NA>     
#>  7 bill_length_mm    year              <NA>  all    0.0545 <NA>     
#>  8 bill_depth_mm     year              <NA>  all   -0.0604 <NA>     
#>  9 flipper_length_mm year              <NA>  all    0.170  <NA>     
#> 10 body_mass_g       year              <NA>  all    0.0422 <NA>