Performs a correlation analysis.
correlation( data, data2 = NULL, select = NULL, select2 = NULL, method = "pearson", p_adjust = "holm", ci = 0.95, bayesian = FALSE, bayesian_prior = "medium", bayesian_ci_method = "hdi", bayesian_test = c("pd", "rope", "bf"), redundant = FALSE, include_factors = FALSE, partial = FALSE, partial_bayesian = FALSE, multilevel = FALSE, ranktransform = FALSE, robust = NULL, winsorize = FALSE, verbose = TRUE, ... )
data  A data frame. 

data2  An optional data frame. If specified, all pairwise correlations
between the variables in 
select, select2  (Ignored if 
method  A character string indicating which correlation coefficient is
to be used for the test. One of 
p_adjust  Correction method for frequentist correlations. Can be one of

ci  Confidence/Credible Interval level. If 
bayesian  If TRUE, will run the correlations under a
Bayesian framework. Note that for partial correlations, you will also need
to set 
bayesian_prior  For the prior argument, several named values are
recognized: 
bayesian_ci_method  See arguments in

bayesian_test  See arguments in

redundant  Should the data include redundant rows (where each given correlation is repeated two times). 
include_factors  If 
partial  Can be 
partial_bayesian  If TRUE, will run the correlations under a
Bayesian framework. Note that for partial correlations, you will also need
to set 
multilevel  If 
ranktransform  If 
robust  Old name for 
winsorize  Another way of making the correlation more "robust" (i.e.,
limiting the impact of extreme values). Can be either 
verbose  Toggle warnings. 
...  Additional arguments (e.g., 
A correlation object that can be displayed using the print
,
summary
or table
methods.
The p_adjust
argument can be used to adjust pvalues for multiple
comparisons. All adjustment methods available in p.adjust
function
stats
package are supported.
Pearson's correlation: This is the most common correlation method. It corresponds to the covariance of the two variables normalized (i.e., divided) by the product of their standard deviations.
Spearman's rank correlation: A nonparametric measure of rank correlation (statistical dependence between the rankings of two variables). The Spearman correlation between two variables is equal to the Pearson correlation between the rank values of those two variables; while Pearson's correlation assesses linear relationships, Spearman's correlation assesses monotonic relationships (whether linear or not). Confidence Intervals (CI) for Spearman's correlations are computed using the Fieller et al. (1957) correction (see Bishara and Hittner, 2017).
Kendall's rank correlation: In the normal case, the Kendall correlation is preferred than the Spearman correlation because of a smaller gross error sensitivity (GES) and a smaller asymptotic variance (AV), making it more robust and more efficient. However, the interpretation of Kendall's tau is less direct than that of Spearman's rho, in the sense that it quantifies the difference between the % of concordant and discordant pairs among all possible pairwise events. Confidence Intervals (CI) for Kendall's correlations are computed using the Fieller et al. (1957) correction (see Bishara and Hittner, 2017).
Biweight midcorrelation: A measure of similarity that is medianbased, instead of the traditional meanbased, thus being less sensitive to outliers. It can be used as a robust alternative to other similarity metrics, such as Pearson correlation (Langfelder \& Horvath, 2012).
Distance correlation: Distance correlation measures both linear and nonlinear association between two random variables or random vectors. This is in contrast to Pearson's correlation, which can only detect linear association between two random variables.
Percentage bend correlation: Introduced by Wilcox (1994), it is based on a downweight of a specified percentage of marginal observations deviating from the median (by default, 20%).
Shepherd's Pi correlation: Equivalent to a Spearman's rank correlation after outliers removal (by means of bootstrapped Mahalanobis distance).
Blomqvist’s coefficient: The Blomqvist’s coefficient (also referred to as Blomqvist's Beta or medial correlation; Blomqvist, 1950) is a medianbased nonparametric correlation that has some advantages over measures such as Spearman's or Kendall's estimates (see Shmid and Schimdt, 2006).
Hoeffding’s D: The Hoeffding’s D statistics is a nonparametric rank based measure of association that detects more general departures from independence (Hoeffding 1948), including nonlinear associations. Hoeffding’s D varies between 0.5 and 1 (if there are no tied ranks, otherwise it can have lower values), with larger values indicating a stronger relationship between the variables.
Somers’ D: The Somers’ D statistics is a nonparametric rank based measure of association between a binary variable and a continuous variable, for instance, in the context of logistic regression the binary outcome and the predicted probabilities for each outcome. Usually, Somers' D is a measure of ordinal association, however, this implementation it is limited to the case of a binary outcome.
PointBiserial and biserial correlation: Correlation coefficient used when one variable is continuous and the other is dichotomous (binary). PointBiserial is equivalent to a Pearson's correlation, while Biserial should be used when the binary variable is assumed to have an underlying continuity. For example, anxiety level can be measured on a continuous scale, but can be classified dichotomously as high/low.
Gamma correlation: The GoodmanKruskal gamma statistic is similar to Kendall's Tau coefficient. It is relatively robust to outliers and deals well with data that have many ties.
Winsorized correlation: Correlation of variables that have been formerly Winsorized, i.e., transformed by limiting extreme values to reduce the effect of possibly spurious outliers.
Gaussian rank Correlation: The Gaussian rank correlation estimator is a simple and wellperforming alternative for robust rank correlations (Boudt et al., 2012). It is based on the Gaussian quantiles of the ranks.
Polychoric correlation: Correlation between two theorized normally distributed continuous latent variables, from two observed ordinal variables.
Tetrachoric correlation: Special case of the polychoric correlation applicable when both observed variables are dichotomous.
Partial correlations are estimated as the correlation between two
variables after adjusting for the (linear) effect of one or more other
variable. The correlation test is then run after having partialized the
dataset, independently from it. In other words, it considers partialization
as an independent step generating a different dataset, rather than belonging
to the same model. This is why some discrepancies are to be expected for the
t and pvalues, CIs, BFs etc (but not the correlation coefficient)
compared to other implementations (e.g., ppcor
). (The size of these
discrepancies depends on the number of covariates partialledout and the
strength of the linear association between all variables.) Such partial
correlations can be represented as Gaussian Graphical Models (GGM), an
increasingly popular tool in psychology. A GGM traditionally include a set of
variables depicted as circles ("nodes"), and a set of lines that visualize
relationships between them, which thickness represents the strength of
association (see Bhushan et al., 2019).
Multilevel correlations are a special case of partial correlations
where the variable to be adjusted for is a factor and is included as a random
effect in a mixed model (note that the remaining continuous variables of the
dataset will still be included as fixed effects, similarly to regular partial
correlations). That said, there is an important difference between using
cor_test()
and correlation()
: If you set multilevel=TRUE
in correlation()
but partial
is set to FALSE
(as per
default), then a backtransformation from partial to nonpartial correlation
will be attempted (through pcor_to_cor
). However,
this is not possible when using cor_test()
so that if you set
multilevel=TRUE
in it, the resulting correlations are partial one.
Note that for Bayesian multilevel correlations, if partial = FALSE
,
the back transformation will also recompute pvalues based on the new r scores,
and will drop the Bayes factors (as they are not relevant anymore). To keep
Bayesian scores, don't forget to set partial = TRUE
.
Kendall and Spearman correlations when bayesian=TRUE
: These
are technically Pearson Bayesian correlations of rank transformed data,
rather than pure Bayesian rank correlations (which have different priors).
Boudt, K., Cornelissen, J., & Croux, C. (2012). The Gaussian rank correlation estimator: robustness properties. Statistics and Computing, 22(2), 471483.
Bhushan, N., Mohnert, F., Sloot, D., Jans, L., Albers, C., & Steg, L. (2019). Using a Gaussian graphical model to explore relationships between items and variables in environmental psychology research. Frontiers in psychology, 10, 1050.
Bishara, A. J., & Hittner, J. B. (2017). Confidence intervals for correlations when data are not normal. Behavior research methods, 49(1), 294309.
Fieller, E. C., Hartley, H. O., & Pearson, E. S. (1957). Tests for rank correlation coefficients. I. Biometrika, 44(3/4), 470481.
Langfelder, P., & Horvath, S. (2012). Fast R functions for robust correlations and hierarchical clustering. Journal of statistical software, 46(11).
Blomqvist, N. (1950). On a measure of dependence between two random variables,Annals of Mathematical Statistics,21, 593–600
Somers, R. H. (1962). A new asymmetric measure of association for ordinal variables. American Sociological Review. 27 (6).
library(correlation) results < correlation(iris) results #> # Correlation Matrix (pearsonmethod) #> #> Parameter1  Parameter2  r  95% CI  t(148)  p #>  #> Sepal.Length  Sepal.Width  0.12  [0.27, 0.04]  1.44  0.152 #> Sepal.Length  Petal.Length  0.87  [ 0.83, 0.91]  21.65  < .001*** #> Sepal.Length  Petal.Width  0.82  [ 0.76, 0.86]  17.30  < .001*** #> Sepal.Width  Petal.Length  0.43  [0.55, 0.29]  5.77  < .001*** #> Sepal.Width  Petal.Width  0.37  [0.50, 0.22]  4.79  < .001*** #> Petal.Length  Petal.Width  0.96  [ 0.95, 0.97]  43.39  < .001*** #> #> pvalue adjustment method: Holm (1979) #> Observations: 150 summary(results) #> # Correlation Matrix (pearsonmethod) #> #> Parameter  Petal.Width  Petal.Length  Sepal.Width #>  #> Sepal.Length  0.82***  0.87***  0.12 #> Sepal.Width  0.37***  0.43***  #> Petal.Length  0.96***   #> #> pvalue adjustment method: Holm (1979) summary(results, redundant = TRUE) #> # Correlation Matrix (pearsonmethod) #> #> Parameter  Sepal.Length  Sepal.Width  Petal.Length  Petal.Width #>  #> Sepal.Length  1.00***  0.12  0.87***  0.82*** #> Sepal.Width  0.12  1.00***  0.43***  0.37*** #> Petal.Length  0.87***  0.43***  1.00***  0.96*** #> Petal.Width  0.82***  0.37***  0.96***  1.00*** #> #> pvalue adjustment method: Holm (1979) # pipefriendly usage if (require("dplyr")) { iris %>% correlation(select = "Petal.Width", select2 = "Sepal.Length") } #> Loading required package: dplyr #> #> Attaching package: ‘dplyr’ #> The following objects are masked from ‘package:Hmisc’: #> #> src, summarize #> The following objects are masked from ‘package:stats’: #> #> filter, lag #> The following objects are masked from ‘package:base’: #> #> intersect, setdiff, setequal, union #> # Correlation Matrix (pearsonmethod) #> #> Parameter1  Parameter2  r  95% CI  t(148)  p #>  #> Petal.Width  Sepal.Length  0.82  [0.76, 0.86]  17.30  < .001*** #> #> pvalue adjustment method: Holm (1979) #> Observations: 150 # Grouped dataframe if (require("dplyr")) { # grouped correlations iris %>% group_by(Species) %>% correlation() # selecting specific variables for correlation mtcars %>% group_by(am) %>% correlation( select = c("cyl", "wt"), select2 = c("hp") ) } #> # Correlation Matrix (pearsonmethod) #> #> Group  Parameter1  Parameter2  r  95% CI  t  df  p #>  #> 0  cyl  hp  0.85  [0.64, 0.94]  6.53  17  < .001*** #> 0  wt  hp  0.68  [0.33, 0.87]  3.82  17  < .01** #> 1  cyl  hp  0.90  [0.69, 0.97]  6.87  11  < .001*** #> 1  wt  hp  0.81  [0.48, 0.94]  4.66  11  < .001*** #> #> pvalue adjustment method: Holm (1979) #> Observations: 1319 # automatic selection of correlation method correlation(mtcars[2], method = "auto") #> # Correlation Matrix (automethod) #> #> Parameter1  Parameter2  r  95% CI  t(30)  p #>  #> mpg  disp  0.85  [0.92, 0.71]  8.75  < .001*** #> mpg  hp  0.78  [0.89, 0.59]  6.74  < .001*** #> mpg  drat  0.68  [ 0.44, 0.83]  5.10  < .001*** #> mpg  wt  0.87  [0.93, 0.74]  9.56  < .001*** #> mpg  qsec  0.42  [ 0.08, 0.67]  2.53  0.222 #> mpg  vs  0.66  [ 0.41, 0.82]  4.86  < .001*** #> mpg  am  0.60  [ 0.32, 0.78]  4.11  < .01** #> mpg  gear  0.48  [ 0.16, 0.71]  3.00  0.097 #> mpg  carb  0.55  [0.75, 0.25]  3.62  < .05* #> disp  hp  0.79  [ 0.61, 0.89]  7.08  < .001*** #> disp  drat  0.71  [0.85, 0.48]  5.53  < .001*** #> disp  wt  0.89  [ 0.78, 0.94]  10.58  < .001*** #> disp  qsec  0.43  [0.68, 0.10]  2.64  0.197 #> disp  vs  0.71  [0.85, 0.48]  5.53  < .001*** #> disp  am  0.59  [0.78, 0.31]  4.02  < .01** #> disp  gear  0.56  [0.76, 0.26]  3.66  < .05* #> disp  carb  0.39  [ 0.05, 0.65]  2.35  0.303 #> hp  drat  0.45  [0.69, 0.12]  2.75  0.170 #> hp  wt  0.66  [ 0.40, 0.82]  4.80  < .01** #> hp  qsec  0.71  [0.85, 0.48]  5.49  < .001*** #> hp  vs  0.72  [0.86, 0.50]  5.73  < .001*** #> hp  am  0.24  [0.55, 0.12]  1.37  > .999 #> hp  gear  0.13  [0.45, 0.23]  0.69  > .999 #> hp  carb  0.75  [ 0.54, 0.87]  6.21  < .001*** #> drat  wt  0.71  [0.85, 0.48]  5.56  < .001*** #> drat  qsec  0.09  [0.27, 0.43]  0.50  > .999 #> drat  vs  0.44  [ 0.11, 0.68]  2.69  0.187 #> drat  am  0.71  [ 0.48, 0.85]  5.57  < .001*** #> drat  gear  0.70  [ 0.46, 0.84]  5.36  < .001*** #> drat  carb  0.09  [0.43, 0.27]  0.50  > .999 #> wt  qsec  0.17  [0.49, 0.19]  0.97  > .999 #> wt  vs  0.55  [0.76, 0.26]  3.65  < .05* #> wt  am  0.69  [0.84, 0.45]  5.26  < .001*** #> wt  gear  0.58  [0.77, 0.29]  3.93  < .05* #> wt  carb  0.43  [ 0.09, 0.68]  2.59  0.205 #> qsec  vs  0.74  [ 0.53, 0.87]  6.11  < .001*** #> qsec  am  0.23  [0.54, 0.13]  1.29  > .999 #> qsec  gear  0.21  [0.52, 0.15]  1.19  > .999 #> qsec  carb  0.66  [0.82, 0.40]  4.76  < .01** #> vs  am  0.26  [0.09, 0.56]  1.50  > .999 #> vs  gear  0.21  [0.15, 0.52]  1.15  > .999 #> vs  carb  0.57  [0.77, 0.28]  3.80  < .05* #> am  gear  0.79  [ 0.62, 0.89]  7.16  < .001*** #> am  carb  0.06  [0.30, 0.40]  0.32  > .999 #> gear  carb  0.27  [0.08, 0.57]  1.56  > .999 #> #> pvalue adjustment method: Holm (1979) #> Observations: 32