When only a count of people present on certain days is available –
not a record per person – the person-days over the period are the
mean of those counts scaled to the period's length.
Use it with adp() or alos() when the source
is a daily or weekly
count sheet rather than admission and release dates.
Value
A numeric scalar: the estimated person-days over the period,
mean(counts) * t. Despite the function's name this is a total of
person-days, not an average daily population; divide by t (or
pass it to adp()) for the population held on an average day. It has
class c("rmbl_adpcnts", "rmbl_measure", "numeric") and an
attribute "lakner" (the number of counts, their mean and
t), so it prints as a box with its inputs, formula and caveats;
arithmetic, round(), sum(), mean(), [,
c() and as.numeric() give a plain number, and
adp() takes it as days.
Details
Lakner's \(X_t = (\frac{1}{C}\sum N_i) t\) (1976, p.21). The counts need not cover every day: counting on weekdays only is the usual case, and the mean of the days counted stands in for the days not counted. That substitution is an assumption, and it fails if the days counted differ systematically from the days missed – weekday-only counting where weekend admissions are released before Monday, for instance.
Examples
# Lakner (p.21): counts on 255 days summing to 34,935 imply just over
# fifty thousand detention days across the year.
adpcnts(rep(34935 / 255, 255))
#> ╭── Person-days from headcounts ──────────────────────────────────────────────╮
#> │ Person-days 50,005 over the period │
#> │ Headcounts 255 days counted │
#> │ Mean count 137 people │
#> │ Period 365 days │
#> │ Formula days = mean(counts) x t = 137 x 365 │
#> │ │
#> │ Counts on 255 days average 137 people: about 50,005 person-days in 365 │
#> │ days. │
#> │ │
#> │ Lakner (1976, p.21): the mean of the days counted stands in for the │
#> │ days not counted, which fails if they differ systematically (weekday │
#> │ counts where weekend admissions leave before Monday). │
#> │ A total of person-days, not a population: adp(x, t) divides by t. │
#> ╰─────────────────────────────────────────────────────────────────────────────╯
# weekly counts over a 31-day month, then the average daily population
pd <- adpcnts(c(120, 130, 125, 128), t = 31)
pd
#> ╭── Person-days from headcounts ──────────────────────────────────────────────╮
#> │ Person-days 3898 over the period │
#> │ Headcounts 4 days counted │
#> │ Mean count 125.8 people │
#> │ Period 31 days │
#> │ Formula days = mean(counts) x t = 125.8 x 31 │
#> │ │
#> │ Counts on 4 days average 125.8 people: about 3898 person-days in 31 days. │
#> │ │
#> │ Lakner (1976, p.21): the mean of the days counted stands in for the │
#> │ days not counted, which fails if they differ systematically (weekday │
#> │ counts where weekend admissions leave before Monday). │
#> │ A total of person-days, not a population: adp(x, t) divides by t. │
#> ╰─────────────────────────────────────────────────────────────────────────────╯
adp(pd, t = 31)
#> ╭── Average daily population ─────────────────────────────────────────────╮
#> │ Average daily population 125.8 people on an average day │
#> │ Person-days 3898 │
#> │ Period 31 days │
#> │ Formula adp = days / t = 3898 / 31 │
#> │ │
#> │ 3898 person-days over 31 days: 125.8 people held on an average day. │
#> │ │
#> │ Lakner (1976, p.15): a stock -- people held at once, not people who │
#> │ passed through. Count only the days served inside the period. │
#> │ adp = n x alos / t links it to alos() and admits(), exactly only in a │
#> │ steady state. │
#> ╰─────────────────────────────────────────────────────────────────────────╯
# a missing count must be dropped or filled first
try(adpcnts(c(120, NA, 125)))
#> Error : adpcnts: `cnts` must not contain NA