In GRousselet/bootcorci: Bootstrap correlation confidence intervals

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.path = "man/figures/README-",
  out.width = "100%"
)

# library(bootcorci)
devtools::load_all()
library(ggplot2)
library(tibble)

bootcorci (0.0.0.9000)

The goal of bootcorci is to provide bootstrap methods to compute confidence intervals for correlation coefficients and their differences. All the functions have been adapted from Rand Wilcox's functions. The functions in the package have been standardised in their names, inputs and outputs. When two versions of the same function existed to handle separately Pearson's correlation and robust correlations, the two versions have been merged. Simplified versions of the correlation functions have been created to speed up the bootstrap analyses. All the bootstrap functions have a new option to return the bootstrap samples (see example below).

Not implemented yet:
- methods to compare dependent Pearson correlations;
- skipped correlations.

Functions

Correlation functions

|name|purpose|syntax| |-----|-----|-----| |pearson.test|Pearson correlation|x, y, alternative = "two.sided"| |spearman.test|Spearman correlation|x, y, alternative = "two.sided"| |pbcor.test|Percentage bend correlation|x, y, beta = 0.2, alternative = "two.sided"| |wincor.test|Winsorized correlation|x, y, tr = 0.2, alternative = "two.sided"|

Simplified version are called by the bootstrap functions: pearson, spearman, pbcor, wincor.

Bootstrap functions

CI = Percentile bootstrap confidence interval.

|name|purpose|syntax| |-----|-----|-----| |corci|CI for a correlation|x, y, method = "pbcor", nboot = 2000, alpha = 0.05, alternative = "two.sided", null.value = 0, saveboot = TRUE, ...| |twocorci|CI for the difference between two independent correlations|x1, y1, x2, y2, method = "pbcor", nboot = 2000, alpha = 0.05, alternative = "two.sided", null.value = 0, saveboot = TRUE, ...| |twocorci.ov|CI for the difference between two overlapping dependent correlations|x1, x2, y, method = "pbcor", nboot = 2000, alpha = 0.05, alternative = "two.sided", null.value = 0, saveboot = TRUE, ...| |twocorci.nov|CI for the difference between two non-overlapping dependent correlations|x1, y1, x2, y2, method = "pbcor", nboot = 2000, alpha = 0.05, alternative = "two.sided", null.value = 0, saveboot = TRUE, ...|

Functions corci and twocorci can be applied to Pearson correlation and to robust regressions. The other functions can only be applied to robust regressions.

Utilities

|name|purpose|syntax| |-----|-----|-----| |mkcord|Make standard normal correlated data (2 conditions)|rho=0, n=50| |mkcord3|Make standard normal correlated data (3 conditions)|rho.x1y=0, rho.x2y=0.5, rho.x1x2=0.2, n=50| |mkcord4|Make standard normal correlated data (4 conditions)|rho.x1y1=0, rho.x2y2=0.5, rho.x1x2=0.2, rho.y1y2=0.2, rho.x1y2=0, rho.x2y1=0, n=50|

Installation

You can install bootcorci from GitHub by using devtools:

install.packages("devtools")
devtools::install_github("GRousselet/bootcorci")

Examples

1 correlation

Let's generate a correlated sample:

# library(bootcorci)
set.seed(21) # reproducible example
d <- mkcord(rho=0.4, n=50)

Note that we set Pearson population correlation using the rho argument. The population value will be different for other correlations, such as the Spearman, percentage bend and winsorised correlations available in bootcorci. See details here. This means that if we use the percentage bend sample correlation to estimate the association between variables, we make inferences about the percentage bend population correlation.

Illustration:

# library(ggplot2)
df <- as_tibble(d)
ggplot(data=df, aes(x=x, y=y)) + theme_bw() +
  geom_point() +
  theme(axis.text = element_text(size = 14),
        axis.title = element_text(size = 16)) +
  coord_cartesian(xlim = c(-3, 3), ylim = c(-3, 3)) +
  scale_x_continuous(breaks = seq(-3, 3, 1)) +
  scale_y_continuous(breaks = seq(-3, 3, 1))

Estimation + bootstrap confidence interval:

res <- corci(d$x, d$y, method = "spearman")

The default outputs are:
- res$estimate = sample correlation
- res$conf.int = confidence interval
- res$p.value = p value
- res$bootsamples = bootstrap samples (can be suppressed by setting saveboot=FALSE)
The method argument can be: - "spearman" = Spearman correlation
- "pearson" = Pearson correlation
- "pbcor" = Percentage bend correlation
- "wincor" = Winsorised correlation

The sample Spearman correlation is r round(res$estimate, digits=2), with a 95% percentile bootstrap confidence interval of [r round(res$conf.int[1], digits=2), r round(res$conf.int[2], digits=2)] and a p value of r res$p.value.

Illustrate bootstrap correlations in the order they were sampled:

n.show <- 500 # show only n.show first bootstrap correlations
df <- tibble(x = 1:n.show, y = res$bootsamples[1:n.show])
ggplot(df, aes(x = x, y = y)) + theme_bw() +
  geom_hline(yintercept = res$estimate, colour = "grey", size = 1) +
  # comment next line to make a scatterplot instead of a lollipop chart:
  geom_point() + 
  scale_x_continuous(breaks = c(1, seq(50, 500, 50))) +
  theme(axis.text = element_text(size = 14),
        axis.title = element_text(size = 16)) +
  labs(x = "Bootstrap samples", y = "Bootstrap correlations")

The sample correlation is marked by a horizontal grey line.

Illustrate bootstrap samples using a density plot:

df <- as_tibble(with(density(res$bootsamples),data.frame(x,y)))
ci <- res$conf.int

ggplot(df, aes(x = x, y = y)) + theme_bw() +
  geom_vline(xintercept = res$estimate, colour = "grey", size = 1) +
  geom_line(size = 2) +
  scale_x_continuous(breaks = seq(-1, 1, 0.1)) +
  theme(axis.text = element_text(size = 14),
        axis.title = element_text(size = 16),
        axis.text.y = element_blank(),
        axis.ticks.y = element_blank()) +
  labs(x = "Bootstrap means", y = "Density") +
    # confidence interval ----------------------
  geom_segment(x = ci[1], xend = ci[2],
               y = 0, yend = 0,
               lineend = "round", size = 3, colour = "orange") +
  annotate(geom = "label", x = ci[1], y = 0.1*max(df$y), size = 7,
             colour = "white", fill = "orange", fontface = "bold",
             label = paste("L = ", round(ci[1], digits = 2))) +
  annotate(geom = "label", x = ci[2], y = 0.1*max(df$y), size = 7,
             colour = "white", fill = "orange", fontface = "bold",
             label = paste("U = ", round(ci[2], digits = 2)))

The horizontal line marks the 95% confidence interval. The boxes report the values of the CI bounds. L stands for lower bound, U for upper bound.

2 independent correlations

Generate data:

set.seed(21) # reproducible example
n1 <- 50
n2 <- 70
d1 <- mkcord(rho=0, n = n1)
d2 <- mkcord(rho=0.4, n = n2)

Illustrate data:

df <- tibble(x = c(d1$x, d2$x),
             y = c(d1$y, d2$y),
             Group = factor(c(rep("Group 1", n1), rep("Group 2", n2)))
             )
ggplot(data=df, aes(x=x, y=y)) + theme_bw() +
  geom_point() +
  theme(axis.text = element_text(size = 14),
        axis.title = element_text(size = 16),
        strip.text.x = element_text(size = 20, colour="white", face="bold"),
        strip.background = element_rect(fill="darkgrey")) +
  coord_cartesian(xlim = c(-3, 3), ylim = c(-3, 3)) +
  scale_x_continuous(breaks = seq(-3, 3, 1)) +
  scale_y_continuous(breaks = seq(-3, 3, 1)) +
  facet_wrap(facets = vars(Group), ncol = 2)

Estimation + bootstrap confidence intervals:

res1 <- corci(d1$x, d1$y, method = "pbcor", saveboot = FALSE)
res2 <- corci(d2$x, d2$y, method = "pbcor", saveboot = FALSE)
res <- twocorci(d1$x, d1$y, d2$x, d2$y, method = "pbcor")

For group 1, the sample percentage bend correlation is r round(res1$estimate, digits=2) [r round(res1$conf.int[1], digits=2), r round(res1$conf.int[2], digits=2)]; for group 2 it is r round(res2$estimate, digits=2) [r round(res2$conf.int[1], digits=2), r round(res2$conf.int[2], digits=2)]. The group difference is r round(res$difference, digits=2) [r round(res$conf.int[1], digits=2), r round(res$conf.int[2], digits=2)]. Given the data and our bootstrap sampling, the confidence interval is compatible with a range of large population percentage bend correlation differences.

Illustrate bootstrap differences using a density plot:

df <- as_tibble(with(density(res$bootsamples),data.frame(x,y)))
ci <- res$conf.int

ggplot(df, aes(x = x, y = y)) + theme_bw() +
  geom_vline(xintercept = res$estimate, colour = "grey", size = 1) +
  geom_line(size = 2) +
  scale_x_continuous(breaks = seq(-2, 2, 0.2)) +
  coord_cartesian(xlim = c(-1.4, 0)) + 
  theme(axis.text = element_text(size = 14),
        axis.title = element_text(size = 16),
        axis.text.y = element_blank(),
        axis.ticks.y = element_blank()) +
  labs(x = "Bootstrap means", y = "Density") +
    # confidence interval ----------------------
  geom_segment(x = ci[1], xend = ci[2],
               y = 0, yend = 0,
               lineend = "round", size = 3, colour = "orange") +
  annotate(geom = "label", x = ci[1], y = 0.1*max(df$y), size = 7,
             colour = "white", fill = "orange", fontface = "bold",
             label = paste("L = ", round(ci[1], digits = 2))) +
  annotate(geom = "label", x = ci[2], y = 0.1*max(df$y), size = 7,
             colour = "white", fill = "orange", fontface = "bold",
             label = paste("U = ", round(ci[2], digits = 2)))

2 dependent correlations: overlapping case

Here we deal with 3 dependent measurements. We correlate x1 with y and x2 with y and compare the two correlations.

Generate data:

set.seed(21) # reproducible example
n <- 50
d <- mkcord3(rho.x1y=0, rho.x2y=0.4, rho.x1x2=0.2, n = n)

Illustrate data:

df <- tibble(x = c(d$x1, d$x2),
             y = c(d$y, d$y),
             Group = factor(c(rep("Correlation(x1, y)", n), rep("Correlation(x2, y)", n)))
             )
ggplot(data=df, aes(x=x, y=y)) + theme_bw() +
  geom_point() +
  theme(axis.text = element_text(size = 14),
        axis.title = element_text(size = 16),
        strip.text.x = element_text(size = 20, colour="white", face="bold"),
        strip.background = element_rect(fill="darkgrey")) +
  coord_cartesian(xlim = c(-3, 3), ylim = c(-3, 3)) +
  scale_x_continuous(breaks = seq(-3, 3, 1)) +
  scale_y_continuous(breaks = seq(-3, 3, 1)) +
  facet_wrap(facets = vars(Group), ncol = 2)

Estimation + bootstrap confidence intervals:

res1 <- corci(d$x1, d$y, method = "pbcor", saveboot = FALSE)
res2 <- corci(d$x2, d$y, method = "pbcor", saveboot = FALSE)
res <- twocorci.ov(d$x1, d$x2, d$y, method = "pbcor")

For group 1, the sample percentage bend correlation is r round(res1$estimate, digits=2) [r round(res1$conf.int[1], digits=2), r round(res1$conf.int[2], digits=2)]; for group 2 it is r round(res2$estimate, digits=2) [r round(res2$conf.int[1], digits=2), r round(res2$conf.int[2], digits=2)]. The group difference is r round(res$difference, digits=2) [r round(res$conf.int[1], digits=2), r round(res$conf.int[2], digits=2)]. Given the data and our bootstrap sampling, the confidence interval is compatible with a range of large population percentage bend correlation differences.

Illustrate bootstrap differences using a density plot:

df <- as_tibble(with(density(res$bootsamples),data.frame(x,y)))
ci <- res$conf.int

ggplot(df, aes(x = x, y = y)) + theme_bw() +
  geom_vline(xintercept = res$estimate, colour = "grey", size = 1) +
  geom_line(size = 2) +
  scale_x_continuous(breaks = seq(-2, 2, 0.2)) +
  # coord_cartesian(xlim = c(-1.4, 0)) + 
  theme(axis.text = element_text(size = 14),
        axis.title = element_text(size = 16),
        axis.text.y = element_blank(),
        axis.ticks.y = element_blank()) +
  labs(x = "Bootstrap means", y = "Density") +
    # confidence interval ----------------------
  geom_segment(x = ci[1], xend = ci[2],
               y = 0, yend = 0,
               lineend = "round", size = 3, colour = "orange") +
  annotate(geom = "label", x = ci[1], y = 0.1*max(df$y), size = 7,
             colour = "white", fill = "orange", fontface = "bold",
             label = paste("L = ", round(ci[1], digits = 2))) +
  annotate(geom = "label", x = ci[2], y = 0.1*max(df$y), size = 7,
             colour = "white", fill = "orange", fontface = "bold",
             label = paste("U = ", round(ci[2], digits = 2)))

2 dependent correlations: non-overlapping case

Here we deal with 4 dependent measurements. We correlate x1 with y1 and x2 with y2 and compare the two correlations.

Generate data:

set.seed(21) # reproducible example
n <- 50
d <- mkcord4(rho.x1y1=0, rho.x2y2=0.4, 
             rho.x1x2=0.1, rho.y1y2=0.1,
             rho.x1y2=0.1, rho.x2y1=0.1, n = n)

Illustrate data:

df <- tibble(x = c(d$x1, d$x2),
             y = c(d$y1, d$y2),
             Group = factor(c(rep("Correlation(x1, y1)", n), rep("Correlation(x2, y2)", n)))
             )
ggplot(data=df, aes(x=x, y=y)) + theme_bw() +
  geom_point() +
  theme(axis.text = element_text(size = 14),
        axis.title = element_text(size = 16),
        strip.text.x = element_text(size = 20, colour="white", face="bold"),
        strip.background = element_rect(fill="darkgrey")) +
  coord_cartesian(xlim = c(-3, 3), ylim = c(-3, 3)) +
  scale_x_continuous(breaks = seq(-3, 3, 1)) +
  scale_y_continuous(breaks = seq(-3, 3, 1)) +
  facet_wrap(facets = vars(Group), ncol = 2)

Estimation + bootstrap confidence intervals:

res1 <- corci(d$x1, d$y1, method = "pbcor", saveboot = FALSE)
res2 <- corci(d$x2, d$y2, method = "pbcor", saveboot = FALSE)
res <- twocorci.nov(d$x1, d$y1, d$x2, d$y2, method = "pbcor")

For group 1, the sample percentage bend correlation is r round(res1$estimate, digits=2) [r round(res1$conf.int[1], digits=2), r round(res1$conf.int[2], digits=2)]; for group 2 it is r round(res2$estimate, digits=2) [r round(res2$conf.int[1], digits=2), r round(res2$conf.int[2], digits=2)]. The group difference is r round(res$difference, digits=2) [r round(res$conf.int[1], digits=2), r round(res$conf.int[2], digits=2)]. Given the data and our bootstrap sampling, the confidence interval is compatible with a range of large population percentage bend correlation differences.

Illustrate bootstrap differences using a density plot:

df <- as_tibble(with(density(res$bootsamples),data.frame(x,y)))
ci <- res$conf.int

ggplot(df, aes(x = x, y = y)) + theme_bw() +
  geom_vline(xintercept = res$estimate, colour = "grey", size = 1) +
  geom_line(size = 2) +
  scale_x_continuous(breaks = seq(-2, 2, 0.2)) +
  # coord_cartesian(xlim = c(-1.4, 0)) + 
  theme(axis.text = element_text(size = 14),
        axis.title = element_text(size = 16),
        axis.text.y = element_blank(),
        axis.ticks.y = element_blank()) +
  labs(x = "Bootstrap means", y = "Density") +
    # confidence interval ----------------------
  geom_segment(x = ci[1], xend = ci[2],
               y = 0, yend = 0,
               lineend = "round", size = 3, colour = "orange") +
  annotate(geom = "label", x = ci[1], y = 0.1*max(df$y), size = 7,
             colour = "white", fill = "orange", fontface = "bold",
             label = paste("L = ", round(ci[1], digits = 2))) +
  annotate(geom = "label", x = ci[2], y = 0.1*max(df$y), size = 7,
             colour = "white", fill = "orange", fontface = "bold",
             label = paste("U = ", round(ci[2], digits = 2)))

References

Pernet, C.R., Wilcox, R.R., & Rousselet, G.A. (2013) Robust Correlation Analyses: False Positive and Power Validation Using a New Open Source Matlab Toolbox. Front. Psychol., 3.

Rousselet, G.A., Pernet, C.R., & Wilcox, R.R. (2019) The percentile bootstrap: a primer with step-by-step instructions in R (preprint). PsyArXiv.

Rousselet, G.A., Pernet, C.R., & Wilcox, R.R. (2019) A practical introduction to the bootstrap: a versatile method to make inferences by using data-driven simulations (Preprint). PsyArXiv.

Wilcox, R.R. (2009) Comparing Pearson Correlations: Dealing with Heteroscedasticity and Nonnormality. Communications in Statistics - Simulation and Computation, 38, 2220–2234.

Wilcox, R.R. (2016) Comparing dependent robust correlations. Br J Math Stat Psychol, 69, 215–224.

Wilcox, R.R. (2017) Introduction to Robust Estimation and Hypothesis Testing, 4th edition. edn. Academic Press.

GRousselet/bootcorci documentation built on March 6, 2021, 7:13 a.m.