This function performs a correlation test between two variables.
You can easily visualize the result using `plot()`

(see examples **here**).

## Usage

```
cor_test(
data,
x,
y,
method = "pearson",
ci = 0.95,
bayesian = FALSE,
bayesian_prior = "medium",
bayesian_ci_method = "hdi",
bayesian_test = c("pd", "rope", "bf"),
include_factors = FALSE,
partial = FALSE,
partial_bayesian = FALSE,
multilevel = FALSE,
ranktransform = FALSE,
winsorize = FALSE,
verbose = TRUE,
...
)
```

## Arguments

- data
A data frame.

- x, y
Names of two variables present in the data.

- method
A character string indicating which correlation coefficient is to be used for the test. One of

`"pearson"`

(default),`"kendall"`

,`"spearman"`

(but see also the`robust`

argument),`"biserial"`

,`"polychoric"`

,`"tetrachoric"`

,`"biweight"`

,`"distance"`

,`"percentage"`

(for percentage bend correlation),`"blomqvist"`

(for Blomqvist's coefficient),`"hoeffding"`

(for Hoeffding's D),`"gamma"`

,`"gaussian"`

(for Gaussian Rank correlation) or`"shepherd"`

(for Shepherd's Pi correlation). Setting`"auto"`

will attempt at selecting the most relevant method (polychoric when ordinal factors involved, tetrachoric when dichotomous factors involved, point-biserial if one dichotomous and one continuous and pearson otherwise). See below the**details**section for a description of these indices.- ci
Confidence/Credible Interval level. If

`"default"`

, then it is set to`0.95`

(`95%`

CI).- bayesian
If

`TRUE`

, will run the correlations under a Bayesian framework.- bayesian_prior
For the prior argument, several named values are recognized:

`"medium.narrow"`

,`"medium"`

,`"wide"`

, and`"ultrawide"`

. These correspond to scale values of`1/sqrt(27)`

,`1/3`

,`1/sqrt(3)`

and`1`

, respectively. See the`BayesFactor::correlationBF`

function.- bayesian_ci_method, bayesian_test
See arguments in

`model_parameters()`

for`BayesFactor`

tests.- include_factors
If

`TRUE`

, the factors are kept and eventually converted to numeric or used as random effects (depending of`multilevel`

). If`FALSE`

, factors are removed upfront.- partial
Can be

`TRUE`

or`"semi"`

for partial and semi-partial correlations, respectively.- partial_bayesian
If partial correlations under a Bayesian framework are needed, you will also need to set

`partial_bayesian`

to`TRUE`

to obtain "full" Bayesian partial correlations. Otherwise, you will obtain pseudo-Bayesian partial correlations (i.e., Bayesian correlation based on frequentist partialization).- multilevel
If

`TRUE`

, the factors are included as random factors. Else, if`FALSE`

(default), they are included as fixed effects in the simple regression model.- ranktransform
If

`TRUE`

, will rank-transform the variables prior to estimating the correlation, which is one way of making the analysis more resistant to extreme values (outliers). Note that, for instance, a Pearson's correlation on rank-transformed data is equivalent to a Spearman's rank correlation. Thus, using`robust=TRUE`

and`method="spearman"`

is redundant. Nonetheless, it is an easy option to increase the robustness of the correlation as well as flexible way to obtain Bayesian or multilevel Spearman-like rank correlations.- winsorize
Another way of making the correlation more "robust" (i.e., limiting the impact of extreme values). Can be either

`FALSE`

or a number between 0 and 1 (e.g.,`0.2`

) that corresponds to the desired threshold. See the`winsorize()`

function for more details.- verbose
Toggle warnings.

- ...
Additional arguments (e.g.,

`alternative`

) to be passed to other methods. See`stats::cor.test`

for further details.

## Details

### Correlation Types

**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 non-parametric 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 percentage 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 median-based, instead of the traditional mean-based, 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 non-linear 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 down-weight 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 median-based non-parametric correlation that has some advantages over measures such as Spearman's or Kendall's estimates (see Shmid & Schimdt, 2006).**Hoeffding’s D**: The Hoeffding’s D statistics is a non-parametric rank based measure of association that detects more general departures from independence (Hoeffding 1948), including non-linear 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 non-parametric 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.**Point-Biserial and biserial correlation**: Correlation coefficient used when one variable is continuous and the other is dichotomous (binary). Point-Biserial 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 Goodman-Kruskal 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 well-performing 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 Correlation

**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 p-values, 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 partialled-out 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). The model is a random intercept model, i.e. the multilevel
correlation is adjusted for `(1 | groupfactor)`

.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 back-transformation from partial to non-partial
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 *p*-values based on the new *r* scores,
and will drop the Bayes factors (as they are not relevant anymore). To keep
Bayesian scores, set `partial = TRUE`

.

## Examples

```
library(correlation)
cor_test(iris, "Sepal.Length", "Sepal.Width")
#> Parameter1 | Parameter2 | r | 95% CI | t(148) | p
#> -------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | -0.12 | [-0.27, 0.04] | -1.44 | 0.152
#>
#> Observations: 150
cor_test(iris, "Sepal.Length", "Sepal.Width", method = "spearman")
#> Parameter1 | Parameter2 | rho | 95% CI | S | p
#> -----------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | -0.17 | [-0.32, 0.00] | 6.56e+05 | 0.041*
#>
#> Observations: 150
# \donttest{
cor_test(iris, "Sepal.Length", "Sepal.Width", method = "kendall")
#> Parameter1 | Parameter2 | tau | 95% CI | z | p
#> ------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | -0.08 | [-0.18, 0.03] | -1.33 | 0.183
#>
#> Observations: 150
cor_test(iris, "Sepal.Length", "Sepal.Width", method = "biweight")
#> Parameter1 | Parameter2 | r | 95% CI | t(148) | p
#> -------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | -0.13 | [-0.29, 0.03] | -1.65 | 0.100
#>
#> Observations: 150
cor_test(iris, "Sepal.Length", "Sepal.Width", method = "distance")
#> Parameter1 | Parameter2 | r | 95% CI | t(11024) | p
#> ------------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | 0.08 | [-0.08, 0.24] | 8.43 | < .001***
#>
#> Observations: 150
cor_test(iris, "Sepal.Length", "Sepal.Width", method = "percentage")
#> Parameter1 | Parameter2 | r | 95% CI | t(148) | p
#> ---------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | -0.19 | [-0.34, -0.03] | -2.40 | 0.018*
#>
#> Observations: 150
if (require("wdm", quietly = TRUE)) {
cor_test(iris, "Sepal.Length", "Sepal.Width", method = "blomqvist")
}
#> Parameter1 | Parameter2 | r | 95% CI | t(148) | p
#> ----------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | -0.23 | [-0.37, -0.07] | -2.83 | 0.005**
#>
#> Observations: 150
if (require("Hmisc", quietly = TRUE)) {
cor_test(iris, "Sepal.Length", "Sepal.Width", method = "hoeffding")
}
#>
#> Attaching package: ‘Hmisc’
#> The following objects are masked from ‘package:base’:
#>
#> format.pval, units
#> Parameter1 | Parameter2 | r | 95% CI | t(148) | p
#> ------------------------------------------------------------
#> Sepal.Length | Sepal.Width | 0.01 | | | 0.011*
#>
#> Observations: 150
cor_test(iris, "Sepal.Length", "Sepal.Width", method = "gamma")
#> Parameter1 | Parameter2 | r | 95% CI | t(148) | p
#> -------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | -0.08 | [-0.24, 0.08] | -0.99 | 0.323
#>
#> Observations: 150
cor_test(iris, "Sepal.Length", "Sepal.Width", method = "gaussian")
#> Parameter1 | Parameter2 | r | 95% CI | t(148) | p
#> -------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | -0.10 | [-0.26, 0.06] | -1.21 | 0.229
#>
#> Observations: 150
cor_test(iris, "Sepal.Length", "Sepal.Width", method = "shepherd")
#> Parameter1 | Parameter2 | rho | 95% CI | S | p
#> ------------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | -0.23 | [-0.39, -0.07] | 6.02e+05 | 0.005**
#>
#> Observations: 150
if (require("BayesFactor", quietly = TRUE)) {
cor_test(iris, "Sepal.Length", "Sepal.Width", bayesian = TRUE)
}
#> ************
#> Welcome to BayesFactor 0.9.12-4.5. If you have questions, please contact Richard Morey (richarddmorey@gmail.com).
#>
#> Type BFManual() to open the manual.
#> ************
#> Parameter1 | Parameter2 | rho | 95% CI | pd | % in ROPE | Prior | BF
#> -----------------------------------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | -0.11 | [-0.27, 0.04] | 91.90% | 42.70% | Beta (3 +- 3) | 0.509
#>
#> Observations: 150
# Robust (these two are equivalent)
cor_test(iris, "Sepal.Length", "Sepal.Width", method = "spearman")
#> Parameter1 | Parameter2 | rho | 95% CI | S | p
#> -----------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | -0.17 | [-0.32, 0.00] | 6.56e+05 | 0.041*
#>
#> Observations: 150
cor_test(iris, "Sepal.Length", "Sepal.Width", method = "pearson", ranktransform = TRUE)
#> Parameter1 | Parameter2 | r | 95% CI | t(148) | p
#> ---------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | -0.17 | [-0.32, -0.01] | -2.06 | 0.041*
#>
#> Observations: 150
# Winsorized
cor_test(iris, "Sepal.Length", "Sepal.Width", winsorize = 0.2)
#> Parameter1 | Parameter2 | r | 95% CI | t(148) | p
#> ----------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | -0.25 | [-0.39, -0.09] | -3.14 | 0.002**
#>
#> Observations: 150
# Tetrachoric
if (require("psych", quietly = TRUE) && require("rstanarm", quietly = TRUE)) {
data <- iris
data$Sepal.Width_binary <- ifelse(data$Sepal.Width > 3, 1, 0)
data$Petal.Width_binary <- ifelse(data$Petal.Width > 1.2, 1, 0)
cor_test(data, "Sepal.Width_binary", "Petal.Width_binary", method = "tetrachoric")
# Biserial
cor_test(data, "Sepal.Width", "Petal.Width_binary", method = "biserial")
# Polychoric
data$Petal.Width_ordinal <- as.factor(round(data$Petal.Width))
data$Sepal.Length_ordinal <- as.factor(round(data$Sepal.Length))
cor_test(data, "Petal.Width_ordinal", "Sepal.Length_ordinal", method = "polychoric")
# When one variable is continuous, will run 'polyserial' correlation
cor_test(data, "Sepal.Width", "Sepal.Length_ordinal", method = "polychoric")
}
#>
#> Attaching package: ‘psych’
#> The following object is masked from ‘package:Hmisc’:
#>
#> describe
#> The following objects are masked from ‘package:ggplot2’:
#>
#> %+%, alpha
#> This is rstanarm version 2.26.1
#> - See https://mc-stan.org/rstanarm/articles/priors for changes to default priors!
#> - Default priors may change, so it's safest to specify priors, even if equivalent to the defaults.
#> - For execution on a local, multicore CPU with excess RAM we recommend calling
#> options(mc.cores = parallel::detectCores())
#>
#> Attaching package: ‘rstanarm’
#> The following object is masked from ‘package:psych’:
#>
#> logit
#> Parameter1 | Parameter2 | rho | 95% CI | t(148) | p
#> ---------------------------------------------------------------------------
#> Sepal.Width | Sepal.Length_ordinal | -0.14 | [-0.30, 0.02] | -1.77 | 0.079
#>
#> Observations: 150
# Partial
cor_test(iris, "Sepal.Length", "Sepal.Width", partial = TRUE)
#> Parameter1 | Parameter2 | r | 95% CI | t(148) | p
#> ---------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | 0.43 | [0.29, 0.55] | 5.84 | < .001***
#>
#> Observations: 150
cor_test(iris, "Sepal.Length", "Sepal.Width", multilevel = TRUE)
#> Parameter1 | Parameter2 | r | 95% CI | t(148) | p
#> ---------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | 0.43 | [0.29, 0.55] | 5.85 | < .001***
#>
#> Observations: 150
cor_test(iris, "Sepal.Length", "Sepal.Width", partial_bayesian = TRUE)
#> Parameter1 | Parameter2 | r | 95% CI | t(148) | p
#> ---------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | 0.43 | [0.29, 0.55] | 5.84 | < .001***
#>
#> Observations: 150
# }
```