adherence_garriguet: Estimate a Participant's Probability of Adhering to "N Days...
In vandomed/accelerometry: Functions for Processing Accelerometer Data

Description Usage Arguments Details References Examples

Implements the Bayesian approach described by Garriguet (Statistics Canada 2016).

1	adherence_garriguet(n, x, alpha, beta, n.rec = 7, x.rec = 5)

`n`	Number of monitoring days.
`x`	Number of exercise days.
`alpha`	Parameter in p_d ~ Beta(alpha, beta). Corresponds to `shape1` in `Beta` functions.
`beta`	Parameter in p_d ~ Beta(alpha, beta). Corresponds to `shape2` in `Beta` functions.
`n.rec`	Denominator for recommendation.
`x.rec`	Numerator for recommendation.

The approach aims to estimate a participant's probability of meeting guidelines of the form "at least x minutes per day for at least y days per week" based on observing X active days out of n monitoring days.

The prior assumption for the participant's daily adherence probability is:

p_d ~ Beta(alpha, beta)

where alpha and beta are estimated via maximum likelihood using the observed sample proportions if active days for all study participants. This can be done separately via mles_beta.

Given p_d, the number of active days out of n monitoring days is distributed:

X|p_d ~ Bin(n, p_d)

It can be shown that the posterior for p_d is:

p_d|X ~ Beta(alpha2 = alpha + X, beta2 = beta + n - X)

Garriguet then uses the Beta-binomial distribution, which describes binomial data with success probability randomly drawn from Beta(alpha, beta). The weekly adherence estimator is defined as:

p_w.hat <- P(Y >= 5) with Y ~ Betabin(7, alpha2, beta2)

which can be calculated using mles_beta.

Garriguet, D. (2016). Using a betabinomial distribution to estimate the prevalence of adherence to physical activity guidelines among children and youth. Statistics Canada, Catalogue no. 82-003-X. Health Reports 27(4): 3-9. Available at: https://www150.statcan.gc.ca/n1/pub/82-003-x/2016004/article/14489-eng.pdf.

# Generate data from hypothetical study with 1000 subjects, valid days 
# randomly sampled from 1-7, and p_d's drawn from Beta(0.5, 3).
set.seed(1)
n <- sample(1: 7, size = 1000, replace = TRUE)
p_d <- rbeta(n = 1000, shape1 = 0.5, shape2 = 3)
x <- rbinom(n = 1000, size = n, prob = p_d)

# First step: Estimate (alpha, beta) via maximum likelihood. Have to change 
# 0's to 0.01 and 1's to 0.99 to avoid Inf's
p_d.hat <- x / n
p_d.hat[p_d.hat == 0] <- 0.01
p_d.hat[p_d.hat == 1] <- 0.99
mles <- mles_beta(x = p_d.hat)

# Estimate each subject's weekly adherence probability
p_w.hat <- adherence_garriguet(n = n, x = x, alpha = mles$par[1], beta = mles$par[2])

# Note that the mean p_w.hat differs considerably from the true mean p_w, 
# reflecting bias in the estimator.
mean(p_w.hat)
mean(pbinom(q = 4, size = 7, prob = p_d, lower.tail = FALSE))