truelies: Bayesian Methods to Estimate the Proportion of Liars in Coin Flip Experiments

Description

Implements Bayesian methods, described in Hugh-Jones (2019) <doi:10.1007/s40881-019-00069-x>, for estimating the proportion of liars in coin flip-style experiments, where subjects report a random outcome and are paid for reporting a "good" outcome.

Usage

To estimate the proportion of liars in an experiment, use update_prior() followed by dist_mean():

posterior <- update_prior(heads = 33, N = 50, P = 0.5, prior = dunif)
dist_mean(posterior)

To get confidence intervals for an estimate, use dist_hdr():

dist_hdr(posterior, conf_level = 0.95)

To test whether two different samples have the same proportion of liars, use difference_dist() followed by dist_hdr():

p2 <- update_prior(heads = 42, N = 49, P = 0.5, prior = dunif)
dd <- difference_dist(posterior, p2)
dist_hdr(dd, 0.95, bounds = c(-1, 1))

To test power for detecting a given proportion of liars, use power_calc():

power_calc(N = 100, P = 0.5, lambda = 0.2)

To test power for detecting differences between groups, use power_calc_difference():

power_calc_difference(N1 = 100, P = 5/6, lambda1 = 0.1, lambda2 = 0.25)

To compare different samples by empirical Bayes estimation, use empirical_bayes():

heads <- c(Baseline = 30, Treatment1 = 38, Treatment2 = 45)
N <- c(50, 52, 57)
result <- empirical_bayes(heads, N, P = 0.5)

Testing the package

To run tests on the package:

source(system.file("test-statistics.R", package = "truelies"))

You will need dplyr, purrr, tidyr and ggplot2 installed.

This will take some time and will produce data frames of test results for different parameter values, along with some plots.

Author(s)

David Hugh-Jones

References

Hugh-Jones, David (2019). True Lies: Comment on Garbarino, Slonim and Villeval (2018). Journal of the Economic Science Association. https://link.springer.com/article/10.1007/s40881-019-00069-x.

See Also

Useful links:

• https://github.com/hughjonesd/truelies

• Report bugs at https://github.com/hughjonesd/truelies/issues

Calculate probability that one posterior is larger than another

Description

Given two distributions with density functions \phi_1, \phi_2, this calculates:

\int_0^1 \int_0^{l_1}\phi_1(l_1) \phi_2(l_2) d l_2 d l_1,

the probability that the value of the first distribution is greater.

Usage

compare_dists(dist1, dist2)

Arguments

Value

A probability scalar.

Examples

d1 <- update_prior(30, 50, P = 0.5, prior = stats::dunif)
d2 <- update_prior(25, 40, P = 0.5, prior = stats::dunif)
compare_dists(d1, d2)

Find density of the difference of two distributions

Description

Given two probability density functions dist1 and dist2, difference_dist returns the density of “dist1 - dist2'.

Usage

difference_dist(dist1, dist2)

Arguments

Details

At the moment this only works when dist1 and dist2 are defined on [0, 1].

Value

A probability density function defined on [-1, 1].

Examples

d1 <- update_prior(30, 50, P = 0.5, prior = stats::dunif)
d2 <- update_prior(32, 40, P = 0.5, prior = stats::dunif)
dd <- difference_dist(d1, d2)
dist_hdr(dd, 0.95)

Compute highest density region for a density function

Description

This is a wrapper for hdrcde::hdr. The highest density region is the interval that covers conf_level of the data and has the highest average density. See:

Usage

dist_hdr(dist, conf_level, bounds = attr(dist, "limits"))

Arguments

Details

Rob J Hyndman (1996) “Computing and graphing highest density regions”. American Statistician, 50, 120-126.

Value

A length 2 vector of region endpoints

Examples

d1 <- update_prior(33, 50, P = 0.5, prior = stats::dunif)
dist_hdr(d1, 0.95)

Find mean of a probability density function

Description

Find mean of a probability density function

Usage

dist_mean(dist, l = attr(dist, "limits")[1], r = attr(dist,
  "limits")[2])

Arguments

Value

A scalar

Examples

d1 <- update_prior(10, 40, P = 5/6, prior = stats::dunif)
dist_mean(d1)

Find quantiles given a probability density function

Description

Find quantiles given a probability density function

Usage

dist_quantile(dist, probs, bounds = attr(dist, "limits"))

Arguments

Value

A vector of quantiles

Examples

d1 <- update_prior(33, 50, P = 0.5, prior = stats::dunif)
dist_quantile(d1, c(0.025, 0.975))

Estimate proportions of liars in multiple samples using empirical Bayes

Description

This function creates a prior by fitting a Beta distribution to the heads/N vector, using MASS::fitdistr(). The prior is then updated using data from each individual sample to give the posterior distributions.

Usage

empirical_bayes(heads, ...)

## Default S3 method:
empirical_bayes(heads, N, P, ...)

## S3 method for class 'formula'
empirical_bayes(formula, data, P, subset, ...)

Arguments

Details

The formula interface allows calling the function directly on experimental data.

Value

A list with two components:

• prior, the calculated empirical prior (of class densityFunction).

• posterior, a list of posterior distributions (objects of class densityFunction). If heads was named, the list will have the same names.

Examples

heads <- c(Baseline = 30, Treatment1 = 38, Treatment2 = 45)
N <- c(50, 52, 57)
res <- empirical_bayes(heads, N, P = 0.5)

compare_dists(res$posteriors$Baseline, res$posteriors$Treatment1)
plot(res$prior, ylim = c(0, 4), col = "grey", lty = 2)
plot(res$posteriors$Baseline, add = TRUE, col = "blue")
plot(res$posteriors$Treatment1, add = TRUE, col = "orange")
plot(res$posteriors$Treatment2, add = TRUE, col = "red")


# starting from raw data:
raw_data <- data.frame(
        report = sample(c("heads", "tails"),
          size = 300,
          replace = TRUE,
          prob = c(.8, .2)
        ),
        group = rep(LETTERS[1:10], each = 30)
    )
empirical_bayes(I(report == "heads") ~ group, data = raw_data, P = 0.5)

Calculate power to detect non-zero lying

Description

This uses simulations to estimate the power to detect a given level of lying in a sample of size N by this package's methods.

Usage

power_calc(N, P, lambda, alpha = 0.05, prior = stats::dunif,
  nsims = 200)

Arguments

Value

Estimated power, a scalar between 0 and 1.

Examples

power_calc(N = 50, P = 0.5, lambda = 0.2)

Estimate power to detect differences in lying between two samples

Description

Using simulations, estimate power to detect differences in lying using compare_dists(), given values for \lambda, the probability of lying, in each sample.

Usage

power_calc_difference(N1, N2 = N1, P, lambda1, lambda2, alpha = 0.05,
  alternative = c("two.sided", "greater", "less"),
  prior = stats::dunif, nsims = 200)

Arguments

Value

Estimated power, a scalar between 0 and 1.

Examples

power_calc_difference(N1 = 100, P = 0.5, lambda = 0, lambda2 = 0.25)

Print/plot an object of class densityFunction.

Description

Print/plot an object of class densityFunction.

Usage

## S3 method for class 'densityFunction'
print(x, ...)

## S3 method for class 'densityFunction'
plot(x, ...)

Arguments

Examples

d1 <- update_prior(33, 50, P = 0.5, prior = stats::dunif)
d1
plot(d1)

# show the actual R code (techies only)
unclass(d1)

Calculate posterior distribution of the proportion of liars

Description

update_prior uses the equation for the posterior:

\phi(\lambda | R; N,P) = Pr(R|\lambda; N,P) \phi(\lambda) / \int Pr(R | \lambda'; N,P) \phi(\lambda') d \lambda'

where \phi is the prior and Pr(R | \lambda; N, P) is the probability of R reports of heads given that people lie with probability \lambda:

Pr(R | \lambda; N, P) = binom(N, (1-P) + \lambda P)

Usage

update_prior(heads, N, P, prior = stats::dunif, npoints = 1000)

Arguments

Value

The probability density of the posterior distribution, as a one-argument function.

Examples

posterior <- update_prior(heads = 30, N = 50, P = 0.5, prior = stats::dunif)
plot(posterior)