In brendonmcconnell/Optimal.sample: Optimal Sample Design for Cluster Randomized Trial with Different Costs in Treatment that in Control.

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This vignette will introduce you to the maxpower.opt and mincost.opt() functions which calculate the optimal sample design for cluster randomized trials with different costs in treatment than in control. The maxpower.opt() function maximizes the statistical power - the probability that the null hypothesis of zero effect is correctly rejected - of a cluster RCT given a cost constrain. Meanwhile, the mincost.opt() function solves the dual problem, minimizing costs subject to achieving a given level of power.

We will start by discussing the usage of maxpower.opt() and mincost.opt(), and show how to use these functions with multiple scenarios. Then we will show that the dual problem mincost.opt() provides the same optimal sample design as the primal problem maxpower.opt().

Basic usage

maxpower.opt() and mincost.opt() have 16 arguments, however, one of the most important is .optimal.s. This argument provides the optimal number of clusters and units within clusters for three different cases:

1. when both the optimal number of clusters and units are allowed to be different between treatment and control arms (optimal.s = "CLUST-IND").
2. when the number of clusters are allowed to be different between treatment and control arms, but the number of units is constrained to be the same in both arms (optimal.s = "CONST-IND").
3. when the number of units within cluster are allowed to be different between treatment and control arms, but the number of clusters is constrained to be the same in both arms (optimal.s = "CONST-CLUST").

Here are some examples of maxpower.opt() and mincost.opt() for multiple scenarios in combination with different options in the argument .optimal.s:

• In this example, both fixed costs per cluster and variable cost per unit within cluster are different between treatment and control. There are three different scenarios, each with a different total costs. The syntax (optimal.s = "CLUST-IND") allows both the optimal number of clusters and units per cluster to be different between the treatment and control arms.

 maxpower.opt(delta = 0.25,
              sigma = 1,
              rho = 0.05,
              alpha = 0.05,
              C = c(815052.294, 974856.169, 1095876.675),
              v0 = 150,
              v1 = 2200,
              f0 = 500,
              f1 = 18000,
              optimal.s = "CLUST-IND")

There are two ways to report the results: The first is directly through a matrix as in the following section; and the second is through a .xlsx file using the argument .output and the name of the file (output = "name_of_the_file")

  scenario      1.0000000      2.0000000       3.000000
  delta         0.2500000      0.2500000       0.250000
  sigma         1.0000000      1.0000000       1.000000
  rho           0.0500000      0.0500000       0.050000
  C        815052.2940000 974856.1690000 1095876.675000
  v0          150.0000000    150.0000000     150.000000
  v1         2200.0000000   2200.0000000    2200.000000
  f0          500.0000000    500.0000000     500.000000
  f1        18000.0000000  18000.0000000   18000.000000
  k0           87.9674349    105.2148436     118.276433
  k1           14.6612409     17.5358068      19.712737
  m0            7.9582249      7.9582260       7.958223
  m1           12.4681418     12.4681435      12.468142
  power         0.6740177      0.7524999       0.800811

• In this example, both fixed costs per cluster and variable cost per unit within cluster are different between treatment and control. There are three different scenarios, each with different value of the fixed cost per cluster in the control group (f0). The syntax (optimal.s = "CLUST-IND") allows both the optimal number of clusters and units per cluster to be different between the treatment and control arms. The initial conditions are specified for the optimization algorithm: m0=20, m1=18, k0=15 and k1=18.

   mincost.opt(delta = 0.25,
               sigma = 1,
               rho = 0.05,
               alpha = 0.05,
               beta = 0.80,
               v0 = 150,
               v1 = 2200,
               f0 = c(500, 1500, 5000),
               f1 = 18000,
               optimal.s = "CLUST-IND",
               initial.cond = c(20, 18, 15, 18))

• This is an example with three scenarios, each with a different value of the fixed cost per cluster in the treatment group (f1). The syntax (optimal.s = "CONST-IND") requests that the number of units per cluster is constrained to be the same in treatment as in control.

   maxpower.opt(delta = 0.25,
                sigma = 1,
                rho = 0.27,
                alpha = 0.05,
                C = c(75862.836, 145230.184, 204196.756),
                v0 = 25,
                v1 = 100,
                f0 = 381,
                f1 = c(500, 1981, 3500),
                optimal.s = "CONST-IND")

• This is an example with four scenarios, each with a different value of the variable cost per unit in the treatment group (v1). The syntax (optimal.s = "CONST-CLUST") requests that the number of clusters is constrained to be the same in treatment as in control.

   mincost.opt(delta = 0.25,
               sigma = 1,
               rho = 0.05,
               alpha = 0.05,
               beta = 0.80,
               v0 = 150,
               v1 = c(250, 750, 1500, 2000),
               f0 = 500,
               f1 = 18000,
               optimal.s = "CONST-CLUST")

Dual problem

In this section we will provide some examples to show that the dual problem mincost.opt() provides the same optimal sample design as the primal problem maxpower.opt().

We will start by maximizing the statistical power of a cluster RCT given a cost constrain using the maxpower.opt() function. Then, we will solve the the dual problem, minimizing costs subject to achieving a given level of power, using the mincost.opt() function.

• In this example, both fixed costs per cluster and variable cost per unit within cluster are different between treatment and control. The syntax (optimal.s = "CLUST-IND") allows both the optimal number of clusters and units per cluster to be different between the treatment and control arms.

• Primal problem:

  maxpower.opt(delta = 0.25,
               sigma = 1,
               rho = 0.05,
               alpha = 0.05,
               C = 1177102.663661,
               q = 1,
               v0 = 100,
               v1 = 2150,
               f0 = 500,
               f1 = 18000,
               optimal.s = "CLUST-IND")

The results report for the primal problem:

  scenario       1
  delta          0.250000
  sigma          1.000000
  rho            0.050000
  C        1177102.663661
  v0           100.000000
  v1          2150.000000
  f0           500.000000
  f1         18000.000000
  k0           130.875225
  k1            21.812541
  m0             9.746796
  m1            12.612286
  power          0.850000

1. Dual problem:

 mincost.opt(delta = 0.25,
             sigma = 1,
             rho = 0.05,
             alpha = 0.05,
             beta = 0.85,
             q = 1,
             v0 = 100,
             v1 = 2150,
             f0 = 500,
             f1 = 18000,
             optimal.s = "CLUST-IND")

The results report for the dual problem:

  scenario       1
  delta          0.250000
  sigma          1.000000
  rho            0.050000
  v0           100.000000
  v1          2150.000000
  f0           500.000000
  f1         18000.000000
  k0           130.875241
  k1            21.812541
  m0             9.746794
  m1            12.612286
  cost     1177102.663661

library(Optimal.sample)

brendonmcconnell/Optimal.sample documentation built on May 30, 2022, 2:22 a.m.