Power maximizing sample design for cluster Randomized Control Trials with only an endline measurement subject to a costs constraint.
maxpower.opt.RdThis function delivers the number of clusters and the number of units (individuals, firms, etc) within cluster that maximizes power subject to a cost constraint in a cluster Randomized Control Trial (RCT). The function assumes that the outcome variable is continuous and has only been measured at endline. This function assumes that the cost function includes a fixed costs per cluster (that can be different for treatment and control clusters) as well as a cost per unit within a cluster (that can be different for treatment and control clusters): \(Costs = k0*(f0 + (v0*m0)) + k1*(f1 + (v1*m1))\). The function 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, (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, and (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.
Usage
maxpower.opt(
delta,
sigma,
rho,
alpha,
C,
q = 1,
v0,
v1,
f0,
f1,
optimal.s = c("CLUST-IND", "CONST-IND", "CONST-CLUST"),
initial.cond = NULL,
seed = 210613,
lb = NULL,
ub = NULL,
temp = NULL,
output = NULL
)Arguments
- delta
Vector. Size of the effect on the outcome variable (effect size measured in same units as the outcome variable).
- sigma
Vector. Standard deviation of the outcome variable.
- rho
Vector. Intra-cluster correlation.
- alpha
Vector. Significance level for the null hypothesis of no effect.
- C
Vector. Maximum level of costs of implementing the RCT. It includes data collection costs (baseline and endline) and the costs of implementing the intervention under study.
- q
Where (K - q) are the degrees of freedom to test the null hypothesis of null effect. Default is 1.
- v0
Vector. Variable costs per unit in the control clusters. It includes the cost of data collection (baseline and endline) and the cost of implementing the intervention under study.
- v1
Vector. Variable costs per unit in the treatment clusters. It includes the cost of data collection (baseline and endline) and the cost of implementing the intervention under study.
- f0
Vector. Fixed costs per control cluster. It includes the total fixed cost: baseline and endline.
- f1
Vector. Fixed costs per treatment cluster. It includes the total fixed cost: baseline and endline.
- optimal.s
Indicates whether the sample design should constrain the number of units per treatment and control clusters to be the same ("CONST-IND") or whether the sample design should constrain the treatment and control clusters to be the same "CONST-CLUST" or whether the solution should be fully unconstrained ("CLUST-IND").
- initial.cond
Vector. Initial values of the number of sample units per cluster (m0, m1) and the number of clusters (k0, k1) - keep the order- that the optimization routine will use. Default is NULL, in which case, the function will compute these initial conditions.
- seed
Integer. Seed for the random number generator that the optimization routine GenSA will use. Default is 210613.
- lb
Vector. Minimum possible value for the optimal number of clusters and optimal number of units. Default is 1 for each parameter.
- ub
Vector. Maximum possible value for the optimal number of clusters and optimal number of units. Default is 1000 for each parameter.
- temp
Numeric. Temperature parameter for the GenSA optimization function. Default is NULL, in which case, the default value in GenSA function will be used.
- output
Indicates the name of the xlsx file where you want to save the results. Default is NULL, in which case, the results will be presented in a matrix.
Value
maxpower.opt returns a matrix of size (14 x number of Scenarios). Each scenario is a combination of of specify parameters, and fixed and variable costs per unit. For each scenario the matrix provides the following components:
- scenario
This is a vector of the number of the scenario displayed.
- delta
This is a vector of the size of the effect on the outcome variable.
- sigma
This is a vector of the standard deviation of the outcome variable.
- rho
This is a vector of the intra-cluster correlation.
- C
This is a vector of the maximum level of total costs of implementing the RCT. It includes data collection costs (baseline and endline) and the costs of implementing the intervention under study.
- v0
This is a vector of the variable cost per control unit.
- v1
This is a vector of the variable costs per treatment unit.
- f0
This is a vector of the fixed costs per control cluster.
- f1
This is a vector of the fixed costs per treatment cluster.
- k0
This is a vector of the optimum number of control clusters that maximize power.
- k1
This is a vector of the optimum number of treatment clusters that maximize power.
- m0
This is a vector of the optimum number of sample units per control cluster that maximize power.
- m1
This is a vector of the optimum number of sample units per treatment cluster that maximize power.
- power
This is the vector of the power of the RCT with the optimum number of clusters and units provided by this function.
References
McConnell and Vera-Hernández (2022). More Powerfull Cluster Randomized Control Trials. Mimeo
Author
Nancy A. Daza-Báez, n.baez@ucl.ac.uk
Brendon McConnell, B.I.Mcconnell@soton.ac.uk>
Marcos Vera-Hernández, m.vera@ucl.ac.uk
Examples
## 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. The results will be saved in the "myresults.xlsx" file.
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",
output = "myresults")
## If you wish, you can specify initial conditions for the optimization algorithm: m0=20, m1=18, k0=15 and k1=18.
maxpower.opt(delta = 0.25,
sigma = 1,
rho = 0.05,
alpha = 0.05,
C = c(815065.655, 877717.857, 995811.458),
v0 = 150,
v1 = 2200,
f0 = c(500, 1500, 5000),
f1 = 18000,
optimal.s = "CLUST-IND",
initial.cond = c(20, 18, 15, 18))
#> 1 2 3
#> scenario 1.000000 2.0000000 3.0000000
#> delta 0.250000 0.2500000 0.2500000
#> sigma 1.000000 1.0000000 1.0000000
#> rho 0.050000 0.0500000 0.0500000
#> C 815065.655000 877717.8570000 995811.4580000
#> v0 150.000000 150.0000000 150.0000000
#> v1 2200.000000 2200.0000000 2200.0000000
#> f0 500.000000 1500.0000000 5000.0000000
#> f1 18000.000000 18000.0000000 18000.0000000
#> k0 87.968898 52.6143724 30.4356543
#> k1 14.661482 15.1884612 16.0409986
#> m0 7.958223 13.7840489 25.1661136
#> m1 12.468141 12.4681412 12.4681407
#> power 0.674025 0.6677639 0.6536381
## 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")
#> 1 2 3
#> scenario 1.0000000 2.0000000 3.0000000
#> delta 0.2500000 0.2500000 0.2500000
#> sigma 1.0000000 1.0000000 1.0000000
#> rho 0.2700000 0.2700000 0.2700000
#> C 75862.8360000 145230.1840000 204196.7560000
#> v0 25.0000000 25.0000000 25.0000000
#> v1 100.0000000 100.0000000 100.0000000
#> f0 381.0000000 381.0000000 381.0000000
#> f1 500.0000000 1981.0000000 3500.0000000
#> k0 64.1971019 81.6868247 92.6482906
#> k1 46.2134146 37.2016788 34.2375765
#> m0 4.5447816 7.0129476 8.5481776
#> m1 4.5447816 7.0129476 8.5481776
#> power 0.4971883 0.5342804 0.5467766
## This is an example with three 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.
maxpower.opt(delta = 0.25,
sigma = 1,
rho = 0.05,
alpha = 0.05,
C = c(144412.242, 251543.646, 384610.811),
v0 = 150,
v1 = c(250, 750, 1500),
f0 = 500,
f1 = 18000,
optimal.s = "CONST-CLUST")
#> 1 2 3
#> scenario 1.0000000 2.0000000 3.0000000
#> delta 0.2500000 0.2500000 0.2500000
#> sigma 1.0000000 1.0000000 1.0000000
#> rho 0.0500000 0.0500000 0.0500000
#> C 144412.2420000 251543.6460000 384610.8110000
#> v0 150.0000000 150.0000000 150.0000000
#> v1 250.0000000 750.0000000 1500.0000000
#> f0 500.0000000 500.0000000 500.0000000
#> f1 18000.0000000 18000.0000000 18000.0000000
#> k0 4.7719101 7.1633417 9.6463857
#> k1 4.7719101 7.1633417 9.6463857
#> m0 34.2296240 34.2296138 34.2296186
#> m1 26.5141636 15.3079510 10.8243554
#> power 0.1888505 0.2731188 0.3375474