Optimal.sample
Optimal.sample.RmdThis 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:
- when both the optimal number of clusters and units are allowed to be
different between treatment and control arms
(
optimal.s = "CLUST-IND"). - 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"). - 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- 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