ICIKendallTau

You can see the pkgdown site here.

Installation

You can install the current version of ICIKendallTau via GitHub:

remotes::install_github("MoseleyBioinformaticsLab/ICIKendallTau")

You can also install Windows or Mac binaries using our r-universe:

options(repos = c(
    moseleybioinformaticslab = 'https://moseleybioinformaticslab.r-universe.dev',
    CRAN = "https://cloud.r-project.org"))
install.packages("ICIKendallTau")

Problem

• How to handle missing data (i.e. NA’s) in calculating a correlation between two variables.
• Current calculations of correlation are based on having all pairs of observations for two variables.
  • However, whether an observation is present or missing is semi-quantitative information for many analytical measurements with sensitivity limits.
  • i.e. in many cases, missing observations are not “missing-at-random”, but “missing-not-at-random” due to falling below the detection limit.
  • In these cases, NA is informative.
  • Therefore, in most analytical measurements (gene expression, proteomics, metabolomics), missing measurements should be included, and contribute to the correlation.

If you want to read more on how we solve this problem, see the package vignette.

Package Functions

The functions that implement this include:

• ici_kt: the C++ workhorse, actually calculating a correlation between an X and Y.
  • The option perspective will control how the NA values influence ties.
  • When comparing samples, you likely want to use perspective = "global".
• ici_kendallt: Handles comparisons for a large matrix.
  • Rows are features, columns are samples.
  • Implicitly parallel, but have to call:
    • library(furrr)
    • plan(multiprocess)
  • Otherwise will only use a single core.

We’ve also included a function for testing if the missingness in your data comes from left-censorship, test_left_censorship. We walk through creating example data and testing it in the vignette Testing for Left Censorship. In addition to testing, you can also visualize the missing data pattern by feature rank using the rank_order_data function, and use visdat::vis_miss() on the original and reordered missing data.

Examples

The most common case is a large matrix of independent samples (columns) and measured features in each of the samples (i.e. gene expression).

Here we will make some artificial data to show how the correlation changes as we introduce missingness.

set.seed(1234)
library(ICIKendallTau)

s1 = sort(rnorm(1000, mean = 100, sd = 10))
s2 = s1 + 10 

matrix_1 = cbind(s1, s2)

r_1 = ici_kendalltau(matrix_1)
r_1$cor
#>    s1 s2
#> s1  1  1
#> s2  1  1

Now we introduce some missing values at the low end of each one. We will just do the simplest thing and introduce NA values in the bottom set.

s3 = s1
s3[sample(100, 50)] = NA

s4 = s2
s4[sample(100, 50)] = NA

matrix_2 = cbind(s3, s4)
r_2 = ici_kendalltau(matrix_2)
r_2$cor
#>           s3        s4
#> s3 1.0000000 0.9944616
#> s4 0.9944616 1.0000000

Is It Fast?

The C++ code implementation (thanks {Rcpp}!) is based on the SciPy implementation, which uses two merge sorts of the ranks of each vector, and then looks for differences between them. This is the fastest method we know of, and has a complexity of O(nlogn). The naive way of computing it, which explicitly examines all of the pairs, has a complexity of n^2. Our implementation was compared to the {pcaPP::cov.fk} function, and the use of {Rcpp} and our inefficient copying of vectors makes ours 3X slower than that one. Which honestly isn’t too bad.

library(microbenchmark)
x = rnorm(1000)
y = rnorm(1000)

x2 = rnorm(40000)
y2 = rnorm(40000)

microbenchmark(
  cor(x, y, method = "kendall"),
  ici_kt(x, y, "global"),
  ici_kt(x2, y2, "global"),
  times = 5
)
#> Unit: microseconds
#>                           expr       min        lq       mean    median
#>  cor(x, y, method = "kendall") 12299.117 12617.607 13300.3072 13214.135
#>         ici_kt(x, y, "global")   366.796   370.173   530.6206   401.068
#>       ici_kt(x2, y2, "global") 19343.691 19680.732 20578.4926 19799.741
#>         uq       max neval
#>  13767.479 14603.198     5
#>    405.009  1110.057     5
#>  20533.946 23534.353     5

In the case of 40,000 features, the average time on a modern CPU is 14 milliseconds.

Of course, if you want to use it to calculate Kendall-tau-b without incorporating missingness, it can do that just fine as well.

k_tau = ici_kt(x, y, "global")
all.equal(k_tau[[1]] ,cor(x, y, method = "kendall"))
#> [1] TRUE

We also provide the kt_fast function, if you want something that treats NA values similarly to stats::cor.

k_tau_fast = kt_fast(x, y)
k_tau_fast
#> $tau
#>              x            y
#> x  1.000000000 -0.003411411
#> y -0.003411411  1.000000000
#> 
#> $pvalue
#>           x         y
#> x 0.0000000 0.8716723
#> y 0.8716723 0.0000000
#> 
#> $run_time
#> [1] 0.02207708

P-Values

ICI-Kt functions only calculates the tau-b variant that handles ties. P-value calculations use the asymptotic approximation in all cases, and thus may vary slightly from the p-values returned by R’s cor.test and Python’s scipy.stats.kendalltau depending on the number of values in x and y.

Parallelism

If you have {future} and the {furrr} packages installed, then it is also possible to split up the a set of matrix comparisons across compute resources for any multiprocessing engine registered with {future}.

library(furrr)
future::plan(multicore, workers = 4)
r_3 = ici_kendalltau(matrix_2)

Many Many Comparisons

In the case of hundreds of thousands of comparisons to be done, the result matrices can become very, very large, and require lots of memory for storage. They are also inefficient, as both the lower and upper triangular components are stored. An alternative storage format is as a data.frame, where there is a single row for each comparison performed. This is actually how the results are stored internally, and then they are converted to a matrix form if requested (the default). To keep the data.frame output, add the argument return_matrix=FALSE to the call of ici_kendalltau.

r_4 = ici_kendalltau(matrix_2, return_matrix = FALSE)
r_4
#> $cor
#>   s1 s2 core       raw pvalue   taumax completeness       cor
#> 1 s3 s4    1 0.9924359      0 0.997963        0.921 0.9944616
#> 2 s3 s3    0 1.0000000      0 1.000000        0.950 1.0000000
#> 3 s4 s4    0 1.0000000      0 1.000000        0.950 1.0000000
#> 
#> $run_time
#> [1] 0.02029967

Other Correlations

ici_kendalltau and ici_kt calculate the p-value of the correlation as part of the overall calculation. stats::cor does not, and stats::cor.test can only calculate the p-value for a single comparison of two vectors. It is sometimes advantageous to obtain p-values for a large number of correlations. We provide cor_fast, which works analogously to kt_fast, with the ability to choose pearson or spearman as the method. Note that if a matrix is provided, the columns must be named.

r_5 = cor_fast(x, y, method = "pearson")
r_5
#> $rho
#>            x          y
#> x 1.00000000 0.00720612
#> y 0.00720612 1.00000000
#> 
#> $pvalue
#>           x         y
#> x 0.0000000 0.8199608
#> y 0.8199608 0.0000000
#> 
#> $run_time
#> [1] 0.02324367

m_3 = cbind(x, y, x)
colnames(m_3) = c("s1", "s2", "s3")
r_6 = cor_fast(m_3)
r_6
#> $rho
#>            s1         s2         s3
#> s1 1.00000000 0.00720612 1.00000000
#> s2 0.00720612 1.00000000 0.00720612
#> s3 1.00000000 0.00720612 1.00000000
#> 
#> $pvalue
#>           s1        s2        s3
#> s1 0.0000000 0.8199608 0.0000000
#> s2 0.8199608 0.0000000 0.8199608
#> s3 0.0000000 0.8199608 0.0000000
#> 
#> $run_time
#> [1] 0.0229435

Code of Conduct

Please note that the ICIKendallTau project is released with a Contributor Code of Conduct. By contributing to this project, you agree to abide by its terms.