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Correlation test

Source: R/cor_test.R
cor_test.Rd

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,
  bootstrap = FALSE,
  iterations = 1000,
  cluster = NULL,
  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 ranktransform 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 parameters::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). Ordered factors are converted to the ranks of their levels. 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 ranktransform=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 datawizard::winsorize() function for more details.

bootstrap

If TRUE, the confidence interval, standard error, and p-value are obtained from a nonparametric percentile bootstrap instead of the analytic formulas: rows are resampled with replacement iterations times and the coefficient is recomputed on each resample. See the 'Bootstrap' section. Not available for Bayesian, partial, or multilevel correlations; the interval and p-value are two-sided, so alternative (passed through ...) must be "two.sided", the default.

iterations

Number of bootstrap resamples (default 1000). Only used when bootstrap = TRUE or cluster is given.

cluster

Name of a column in data identifying clusters of rows (for example, participants with repeated measures). If given, a cluster bootstrap is run (bootstrap is set to TRUE): whole clusters, rather than rows, are resampled with replacement; rows with a missing cluster value are dropped. See the 'Bootstrap' section.

verbose

Toggle warnings.

...

Additional arguments (e.g., alternative) to be passed to other methods. See stats::cor.test for further details.

Value

A data frame of class easycor_test with the coefficient, its confidence interval, the test statistic, and the p-value. Under bootstrap (bootstrap = TRUE or cluster), the object also has an SE column and three attributes: ci_method ("bootstrap" or "cluster-bootstrap"), iterations (the number of replicates kept after dropping the failed ones), and bootstrap_replicates (the kept replicate coefficients, from which CI_low, CI_high, SE, and, except for distance correlation and Hoeffding's D, p are computed). With fewer than three complete observations the bootstrap is skipped and neither the SE column nor these attributes are added.

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). It is computed by wdm::wdm(), which splits each variable at its median by average rank (observations with rank at or below (n + 1) / 2 go in the lower half), so when an observation sits exactly at the split (an odd number of rows, or ties spanning the median) the coefficient is not exactly sign-symmetric.

  • 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.

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).

Bootstrap confidence intervals

With bootstrap = TRUE, or when cluster names a column, the confidence interval, standard error, and p-value of every non-Bayesian method are obtained by resampling instead of from the analytic formulas above. The rows with complete data on both variables (and on cluster, if given) or, with cluster, whole clusters of such rows, are resampled with replacement iterations times and the coefficient is recomputed on each resample. CI_low and CI_high are the (1 - ci) / 2 and 1 - (1 - ci) / 2 quantiles of the replicates (the percentile interval; Efron & Tibshirani, 1993, section 13.3, pp. 170-171); SE is the standard deviation of the replicates (Efron & Tibshirani, 1993, eq. 6.6, p. 47); and p is the two-sided achieved significance level of the same replicate distribution, 2 * min(sum(t <= 0) + 1, sum(t >= 0) + 1) / (B + 1) capped at 1, where t are the B kept replicates (Efron & Tibshirani, 1993, section 15.4, pp. 214-215; the + 1 form is the Monte Carlo p-value of Davison & Hinkley, 1997, eq. 4.11, p. 141). This p-value tests whether the population coefficient is 0. It is approximate, somewhat above nominal in small samples, and can never be smaller than 2 / (iterations + 1); the + 1 makes it slightly conservative relative to the percentile interval, so near the boundary an interval that excludes 0 can go with a p just above 1 - ci. The analytic test statistic and its degrees of freedom are set to NA.

Distance correlation and Hoeffding's D are degenerate under independence, so the bootstrap cannot test whether they are 0: their p is NA under bootstrap (use the analytic test instead), and their interval and standard error are reliable only when an association is present.

Replicates on which the coefficient cannot be computed (for instance, a resample in which a variable is constant or loses a category) are dropped with a warning naming the number kept; if fewer than half survive, an error is raised. In correlation(), this warning is shown for the first pair only, as for every other warning it raises. The winsorize and ranktransform transformations are applied once, to the full data, before resampling.

The cluster bootstrap (cluster) resamples whole clusters, so the dependence within a cluster (repeated measures of one participant, say) is kept intact. Its accuracy is driven by the number of clusters, not the number of rows: a warning is raised below 20 clusters, and very unequal cluster sizes lower the coverage of the interval (Davison & Hinkley, 1997, section 3.8, pp. 100-102; Field & Welsh, 2007, section 3.3, p. 383, whose cluster bootstrap this is). Giving each row its own cluster, numbered in row order, reproduces the plain bootstrap's replicates (the result is still labelled "cluster-bootstrap" and the fewer-than-20-clusters warning applies).

In correlation() with p_adjust = "holm" (the default) or "bonferroni", adjusted p-values cannot fall below 2 * m / (iterations + 1) for m pairs, so with many pairs raise iterations (to at least 2 * m / alpha for a threshold alpha) or use p_adjust = "none"; the other adjustment methods have a lower floor, but never below 2 / (iterations + 1). Hoeffding's D and Shepherd's Pi are slow under bootstrap: each replicate recomputes a statistic quadratic in the number of rows, or an inner 1000-sample outlier bootstrap.

References

  • Davison, A. C., & Hinkley, D. V. (1997). Bootstrap methods and their application. Cambridge University Press.

  • Efron, B., & Tibshirani, R. J. (1993). An introduction to the bootstrap. Chapman & Hall.

  • Field, C. A., & Welsh, A. H. (2007). Bootstrapping clustered data. Journal of the Royal Statistical Society: Series B, 69(3), 369-390.

  • Boudt, K., Cornelissen, J., & Croux, C. (2012). The Gaussian rank correlation estimator: robustness properties. Statistics and Computing, 22(2), 471-483.

  • 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), 294-309.

  • Fieller, E. C., Hartley, H. O., & Pearson, E. S. (1957). Tests for rank correlation coefficients. I. Biometrika, 44(3/4), 470-481.

  • 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).

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{
# Bootstrap confidence interval, standard error, and p-value
cor_test(iris, "Sepal.Length", "Sepal.Width", bootstrap = TRUE, iterations = 200)
#> Parameter1   |  Parameter2 |     r |   SE |        95% CI | t | df |     p
#> --------------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | -0.12 | 0.08 | [-0.27, 0.02] |   |    | 0.109
#> 
#> Confidence intervals: percentile bootstrap (200 iterations)
#> Observations: 150

# Cluster bootstrap: rows of the same cluster are resampled together
data <- iris
data$id <- rep(1:30, 5)
cor_test(data, "Sepal.Length", "Sepal.Width", cluster = "id", iterations = 200)
#> Parameter1   |  Parameter2 |     r |   SE |        95% CI | t | df |     p
#> --------------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | -0.12 | 0.08 | [-0.29, 0.03] |   |    | 0.109
#> 
#> Confidence intervals: percentile cluster bootstrap (200 iterations)
#> 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)
}
#> ************
#> Welcome to BayesFactor 0.9.12-4.8. 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
#> -----------------------------------------------------------------------
#> Sepal.Length | Sepal.Width | -0.11 | [-0.25, 0.05] | 93.17% |    42.85%
#> 
#> Parameter1   |         Prior |    BF
#> ------------------------------------
#> Sepal.Length | 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 2.32.2
#> - 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
if (require("lme4", quietly = TRUE)) {
  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
if (require("rstanarm", quietly = TRUE)) {
  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
# }