The disposed-cases mean as a bound: what the pending cases imply

A cohort of \(n\) disposed cases with durations \(t_i\) and \(m\) pending cases with elapsed ages \(a_j\). The published disposed-cases mean \(\bar t\) is compared with the cohort's true mean, whose pending durations are unknown but at least the ages. The true mean is at least \(L = (\sum t + \sum a)/(n + m)\) (Research.P16Censoring.true_mean_ge), and \(L - \bar t = \frac{m}{n+m}(\bar a - \bar t)\) (lower_bound_sub), so the published mean understates the truth for certain whenever the pending cases are on average older than it (disposed_understates) by at least that amount (bias_lower). No upper bound exists without a cap on the pending durations (no_upper_bound): this is the P1 dark-figure structure again.

Usage

morie_backlog_censoring(disposed, pending_ages)

Arguments

disposed: Durations of the disposed cases (non-negative).
pending_ages: Elapsed ages of the pending cases (non-negative).

Value

A list with n, m, disposed_mean, pending_age (mean age of the pending cases), lower_bound (\(L\)), bias_lower (\(\frac{m}{n+m}(\bar a - \bar t)\)), understates (\(\bar t \le \bar a\)), upper_bound (Inf) and theorems.

Examples

b <- morie_backlog_censoring(disposed = c(30, 45, 60, 90, 120), pending_ages = c(100, 150, 200))
c(disposed = b$disposed_mean, lower = b$lower_bound, bias = b$bias_lower, understates = b$understates)
#>    disposed       lower        bias understates 
#>      69.000      99.375      30.375       1.000