Find the value of p1 that achieves a target waiting time percentile

Uses a binary search algorithm to find the value of p1 (the proportion parameter) that achieves a specified target waiting time (target_time) at a given percentile (hardcoded as 92nd percentile) of the cumulative waiting list size, given a treatment capacity in the first month (mu_1) and other parameters.

Usage

find_p(
  target_time = 4 + (68/487),
  renege_params,
  mu_1,
  p1_lower = 0.1,
  p1_upper = 0.85,
  tolerance = 0.001,
  referrals,
  percentile = 0.92,
  max_iterations = 15
)

Arguments

target_time

numeric of length 1; the target waiting time, in months, to achieve the specified percentile.

renege_params

numeric; proportion (0 ≤ r_m ≤ 1) of waiting patients reneging in the m-th month of waiting, where r₂ = 0.1 refers to 10% of those in the second month of waiting reneging.

mu_1

numeric of length 1; treatment capacity in the first month.

p1_lower

numeric of length 1; lower bound for the binary search of p1. Must be between 0 and 1 and less than p1_upper. Defaults to 0.1.

p1_upper

numeric of length 1; upper bound for the binary search of p1. Must be between 0 and 1 and more than p1_lower. Defaults to 0.85.

tolerance

numeric of length 1; tolerance for the binary search stopping criterion. The unit is months.

referrals

numeric of length 1; value representing the steady-state number of referrals.

percentile

Numeric value between 0 and 1 indicating the desired percentile (default is 0.92).

max_iterations

numeric of length 1; maximum number of iterations to search for a valid result

Value

A list containing:

p1

The value of p1 found to achieve the target waiting time.

time_p

The interpolated waiting time at the specified percentile.

mu

The total number of treatments required when the solution is found.

wlsize

The total size of the waiting list when the solution is found.

waiting_list

The data frame with historical data formatted and waiting list sizes calculated.

niterations

The number of iterations performed in the binary search.

status

Converged indicates a solution was found, and Not converged indicates no solution was found

Details

The function repeatedly uses the removals table (from initialise_removals()) and calculates waiting list sizes for candidate values of p1, using binary search to home in on the value that achieves the target waiting time at the required percentile. The search stops when the difference between upper and lower bounds is less than tolerance, or when the calculated waiting time is within tolerance of the target.

See also

initialise_removals, calc_wl_sizes, hist_percentile_calc

Examples

find_p(
  target_time = 4 + (68 / 487),
  renege_params = c(0.04, 0.04, 0.03, 0.01, 0.02, 0.02, 0.01),
  mu_1 = 2651.227,
  p1_lower = 0.1,
  p1_upper = 0.85,
  tolerance = 0.001,
  referrals = 12000,
  percentile = 0.92
)
#> $p1
#> [1] 0.8200623
#> 
#> $time_p
#> [1] 4.139275
#> 
#> $mu
#> [1] 10836.18
#> 
#> $wlsize
#> [1] 24630.12
#> 
#> $waiting_list
#>   months_waited_id    r   service     sigma    wlsize
#> 1                0 0.04 2651.2270 2651.2270 8868.7730
#> 2                1 0.04 2174.1712 2174.1712 6339.8509
#> 3                2 0.03 1782.9557 1782.9557 4366.6996
#> 4                3 0.01 1462.1347 1462.1347 2860.8979
#> 5                4 0.02 1199.0415 1199.0415 1604.6385
#> 6                5 0.02  983.2887  983.2887  589.2571
#> 7                6 0.01 4377.9222  583.3645    0.0000
#> 
#> $niterations
#> [1] 13
#> 
#> $status
#> [1] "Converged"
#>