Package 'truelies'

Title: 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.
Authors: David Hugh-Jones <[email protected]>
Maintainer: David Hugh-Jones <[email protected]>
License: MIT + file LICENSE
Version: 0.2.0.9000
Built: 2024-07-04 03:47:12 UTC
Source: https://github.com/hughjonesd/truelies

Help Index


Calculate probability that one posterior is larger than another

Description

Given two distributions with density functions ϕ1,ϕ2\phi_1, \phi_2, this calculates:

010l1ϕ1(l1)ϕ2(l2)dl2dl1,\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

dist1

Density of distribution 1, as a one-argument function.

dist2

Density of distribution 2.

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

dist1, dist2

Probability density functions

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

dist

A one-argument function

conf_level

A scalar between 0 and 1

bounds

A length 2 vector of the bounds of the distribution's support

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

dist

A one-argument function returned from update_prior()

l

Lower bound of the density's support

r

Upper bound of the density's support

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

dist

A one argument function

probs

A vector of probabilities

bounds

A length 2 vector of the bounds of the distribution's support

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

heads

A vector of numbers of the good outcome reported

...

Ignored

N

A vector of sample sizes

P

Probability of bad outcome

formula

A two-sided formula of the form heads ~ group. heads is a logical vector specifying whether the "good" outcome was reported. group specifies the sample.

data

A data frame or matrix. Each row represents one individual.

subset

A logical or numeric vector specifying the subset of data to use

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

N

Total number in sample

P

Probability of bad outcome

lambda

Probability of a subject lying

alpha

Significance level to use for the null hypothesis

prior

Prior over lambda. A function which takes a vector of values between 0 and 1, and returns the probability density. The default is the uniform distribution.

nsims

Number of simulations to run

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 λ\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

N1

N of sample 1

N2

N of sample 2

P

Probability of bad outcome

lambda1

Probability of lying in sample 1

lambda2

Probability of lying in sample 2

alpha

Significance level

alternative

"two.sided", "greater" (sample 1 is greater), or "less". Can be abbreviated

prior

Prior over lambda. A function which takes a vector of values between 0 and 1, and returns the probability density. The default is the uniform distribution.

nsims

Number of simulations to run

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

x

The object

...

Unused

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:

ϕ(λR;N,P)=Pr(Rλ;N,P)ϕ(λ)/Pr(Rλ;N,P)ϕ(λ)dλ\phi(\lambda | R; N,P) = Pr(R|\lambda; N,P) \phi(\lambda) / \int Pr(R | \lambda'; N,P) \phi(\lambda') d \lambda'

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

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

Usage

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

Arguments

heads

Number of good outcomes reported

N

Total number in sample

P

Probability of bad outcome

prior

Prior over lambda. A function which takes a vector of values between 0 and 1, and returns the probability density. The default is the uniform distribution.

npoints

How many points to integrate on?

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)