adherence_dodd: 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 developed by Dodd and used in the landmark Troiano et al. paper (MSSE 2008).

1	adherence_dodd(n, x, n.rec = 7, x.rec = 5, posterior = NULL)

`n`	Number of monitoring days.
`x`	Number of active days.
`n.rec`	Denominator for recommendation.
`x.rec`	Numerator for recommendation.
`posterior`	Can be `NULL` for original Dodd method or `"mean"` or `"median"` for modified version described above.

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. We illustrate here with the "5+ active days per week" guideline that motivated the approach.

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

p_d ~ Uni(0, 1)

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(X + 1, n - X + 1)

Under a somewhat questionable independence assumption, the weekly adherence probability is p_w = P(Y >= 5) with Y ~ Bin(7, p_d). Dodd estimates p_w as:

p_w.hat = P(p_d >= 5/7 | X)

which can be calculated using pbeta.

In my view, the quantity P(p_d >= 5/7 | X) is not a good estimator for p_w. Consider what would happen in a really long protocol. The Beta posterior for p_d would be very tightly centered around the true p_d, and p_w.hat = P(p_d >= 5/7 | X) would be very close to either 0 or 1 – not very close to what we're trying to estimate, p_w.

A solution is to define p_d.hat as the posterior mean, median, or mode, and map that estimate to p_w, i.e. p_w.hat = P(Y >= 5) with Y ~ Bin(7, p_d.hat). So there is an option for that.

Dodd, K. (2008). Estimation of the population prevalence of adherence to physical activity recommendations based on NHANES accelerometry measurements. Technical Report. Available at: https://epi.grants.cancer.gov/nhanes_pam/bayesian_adherence_estimation.pdf. Accessed Nov. 13, 2018.

Troiano, R.P., Berrigan, D., Dodd, K.W., Masse, L.C. and McDowell, M. (2008). Physical activity in the United States measured by accelerometer. Medicine \& Science in Sports \& Exercise 40(1): 181–188.

# 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)

# Estimate p_w's using Dodd's method
p_w.hat <- adherence_dodd(n = n, x = x)

# 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))