This function performs a correlation test between two variables.

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,
robust = NULL,
winsorize = FALSE,
verbose = TRUE,
...
)

## Arguments

data A data frame. Names of two variables present in the data. 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). Confidence/Credible Interval level. If "default", then it is set to 0.95 (95% CI). If TRUE, will run the correlations under a Bayesian framework. Note that for partial correlations, 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). 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. See arguments in model_parameters for BayesFactor tests. 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. Can be TRUE or "semi" for partial and semi-partial correlations, respectively. If TRUE, the factors are included as random factors. Else, if FALSE (default), they are included as fixed effects in the simple regression model. 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. Old name for ranktransform. Will be removed in subsequent versions, so better to use ranktransform which is more explicit about what it does. 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. 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 % 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 and 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). 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, don't forget to set partial = TRUE.

### Notes

• 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).

## 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
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)
}
#> ************
#>
#> Type BFManual() to open the manual.
#> ************
#> Parameter1   |  Parameter2 |   rho |        95% CI |     pd | % in ROPE |         Prior |    BF
#> -----------------------------------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | -0.12 | [-0.24, 0.01] | 91.65% |    41.38% | Beta (3 +- 3) | 0.509
#>
#> Observations: 150

# Tetrachoric
if (require("psych", 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
#> Warning: The items do not have an equal number of response alternatives, global set to FALSE.
#> Warning: 1 cells were adjusted for 0 values using the correction for continuity. Examine your data carefully.
#> 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

# 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
if (FALSE) {
# Partial
cor_test(iris, "Sepal.Length", "Sepal.Width", partial = TRUE)
cor_test(iris, "Sepal.Length", "Sepal.Width", multilevel = TRUE)
cor_test(iris, "Sepal.Length", "Sepal.Width", partial_bayesian = TRUE)
}