metadat (version 1.4-0)

dat.bartos2023: Results of 350,757 Coin Flips to Examine Same-Side Bias

Description

Results from 350,757 coin flips by 48 people to examine the presence of same-side bias.

Usage

dat.bartos2023

Format

The data frame contains the following columns:

person	character	person identifier
hsame	numeric	number of flips where the coin landed on heads and on the same side as where it started
hdiff	numeric	number of flips where the coin landed on heads and on the different side as where it started
tsame	numeric	number of flips where the coin landed on tails and on the same side as where it started
tdiff	numeric	number of flips where the coin landed on tails and on the different side as where it started
same	numeric	number of flips where the coin landed on the same side as where it started
flips	numeric	total number of flips

Author

Wolfgang Viechtbauer, , https://www.metafor-project.org

Concepts

physics, human factors, proportions, multivariate models

Details

In a landmark study by Bartoš et al. (2023), 48 people flipped a coin (of various currencies and/or denominations) a total of 350,757 times, recording on each flip whether it landed on heads or tails and whether the coin landed on the same side as where it started or on the different side. The goal of this experiment was to examine the model by Diaconis, Holmes, and Montgomery (2007), according to which flipped coins have a slightly higher than 50% chance (of around 51% according to the D-H-M model) of landing on the same side as where they started.

References

Diaconis, P., Holmes, S., & Montgomery, R. (2007). Dynamical bias in the coin toss. SIAM Review, 49(2), 211--235. https://doi.org/10.1137/s0036144504446436

Examples

### copy data into 'dat' and examine data
dat <- dat.bartos2023
dat

if (FALSE) {
### load metafor package
library(metafor)

### compute proportions and the corresponding sampling variances
dat <- escalc(measure="PR", xi=same, ni=flips, data=dat, slab=person)
dat

### compute confidence intervals for the individual proportions (as in Table 1)
summary(dat, digits=3)[c(1,6:8,13,14)]

### compute a confidence interval based on the column totals
summary(escalc(measure="PR", xi=sum(dat$same), ni=sum(dat$flips)), digits=3)

### this is the same as meta-analyzing the proportions directly using an equal-effects
### model and also computing the sampling variances under the assumption that the true
### proportions are homogeneous
rma(measure="PR", xi=same, ni=flips, vtype="AV", method="EE", data=dat, digits=3)

### fit a random-effects model
res <- rma(yi, vi, data=dat)
res

### profile likelihood confidence interval for tau^2
confint(res, type="PL")

### forest plot
forest(res, refline=0.5, xlim=c(0.38,0.72), digits=c(3,2), efac=c(0,1))

### funnel plot
funnel(res, xlim=c(0.45,0.6), ylim=c(0,0.02))

### fit a random-effects model excluding those with same-side proportions larger than 0.53
res <- rma(yi, vi, data=dat, subset=yi<=0.53)
res
confint(res, type="PL")

### fit a binomial-normal model
res <- rma.glmm(measure="PLO", xi=same, ni=flips, data=dat)
res
predict(res, transf=plogis)

### conduct a meta-analysis for the proportions of heads (to examine heads-tails bias)
dat <- escalc(measure="PR", xi=hdiff+hsame, ni=flips, data=dat)
res <- rma(yi, vi, data=dat)
res
confint(res, type="PL")

### restructure the dataset for a bivariate meta-analysis of same-side and heads proportions
dat <- dat.bartos2023
dat <- dat[rep(1:nrow(dat), each=2),]
rownames(dat) <- NULL
dat$outcome <- c("heads", "same")
dat <- escalc(measure="PR", xi=hsame+hdiff, ni=flips, data=dat, include=outcome=="heads")
dat <- escalc(measure="PR", xi=hsame+tsame, ni=flips, data=dat, include=outcome=="same")
dat

### construct the 2x2 variance-covariance matrix of the proportions within persons
dat$cov <- with(dat, (hsame/flips * (1-hsame/flips) - hsame/flips * tsame/flips -
                      hsame/flips * hdiff/flips - hdiff/flips * tsame/flips) / flips)
V <- lapply(split(dat, dat$person), \(x) matrix(c(x$vi[1], x$cov, x$vi[2]), nrow=2))

### fit bivariate meta-analysis model
res <- rma.mv(yi, V, mods = ~ 0 + outcome, random = ~ outcome | person, struct="UN", data=dat)
res

### create plot with confidence ellipses ('ellipse' package must be installed)
library(ellipse)
plot(NA, xlim=c(0.45,0.62), ylim=c(0.45,0.62), bty="l", xlab="Pr(heads)", ylab="Pr(same)")
abline(h=0.5, lty="dotted")
abline(v=0.5, lty="dotted")
# add confidence ellipses for persons
invisible(tapply(dat, dat$person, \(x) {
   xy <- ellipse(matrix(c(x$vi[1],x$cov,x$vi[2]), nrow=2), centre=x$yi, level=0.95)
   lines(xy[,1],xy[,2], col="gray80")
}))
# add the points
invisible(tapply(dat, dat$person, \(x) points(x$yi[1], x$yi[2], pch=21, bg="gray80", cex=1.5)))
# add the 95% PI ellipsis based on the model
xy <- ellipse(res$G, centre=coef(res), level=0.95)
lines(xy[,1],xy[,2], col="gray30", lwd=3, lty="dotted")
# add the 95% CI ellipsis based on the model
xy <- ellipse(vcov(res), centre=coef(res), level=0.95)
lines(xy[,1],xy[,2], col="gray30", lwd=3)
# add the point for the pooled effects
points(coef(res)[1], coef(res)[2], pch=21, bg="gray40", cex=2)
}