The following examples demonstrates how to perform constrained optimal sample allocation (COSA) for a two-level cluster randomized trial (CRT) and for the corresponding cluster-level regression discontinuity (CRD) design under three primary constraints. Note: NULL arguments are provided for clarity, otherwise they do not have to be explicit.

Primary Constraint on the Total Cost

# CRT('rhots = 0')
# cost constrained - optimize p and n2
crt <- cosa.crd2r2(rhots = 0,
                   constrain = "cost", cost = 12500, 
                   cn1 = c(5, 2), cn2 = c(50, 20),
                   es = .20,  power = .80, rho2 = .20,
                   g2 = 5, r21 = .20, r22 = .30,
                   p = NULL, n1 = 24,  n2 = NULL)

## 
## Results are equivalent to corresponding random assignment designs 
## Solution converged with LBFGS 
##      [n1]  n2     p [cost] mdes 95%lcl 95%ucl power
## [1,]   24 116 0.388  12478 0.22  0.066  0.374 0.722

# CRD
# cost constrained - optimize n2
crd <- cosa.crd2r2(constrain = "cost", cost = 12500,
                   cn1 = c(5, 2), cn2 = c(50, 20),
                   es = .20,  power = .80, rho2 = .20,
                   g2 = 5, r21 = .20, r22 = .30,
                   p = .387, n1 = 24,  n2 = NULL)

## Solution converged with LBFGS 
##      [n1]  n2   [p] [cost]  mdes 95%lcl 95%ucl power
## [1,]   24 116 0.388  12478 0.356  0.106  0.605 0.351

# example plots
par(mfrow = c(1, 2), mai = c(1, .9, .5, .2))
# compare minimum detectable effect size and 95% CI
plot(crt, ypar = "mdes", xpar = "n2",
     ylim = c(0, 1), xlim = c(10, 200), 
     ylab = "MDES (with Power = .80)", xlab = "Number of Clusters",
     main = "CRT", locate = TRUE)
plot(crd, ypar = "mdes", xpar = "n2",
     ylim = c(0, 1), xlim = c(10, 200),
     ylab = "MDES (with Power = .80)", xlab = "Number of Clusters",
     main = "CRD", locate = TRUE)

# compare statistical power 
plot(crt, ypar = "power", xpar = "n2",
     ylim = c(0, 1), xlim = c(10, 200), 
     ylab = "Power (for ES = .20)", xlab = "Number of Clusters",
     main = "CRT", locate = TRUE)
plot(crd, ypar = "power", xpar = "n2",
     ylim = c(0, 1), xlim = c(10, 200), 
     ylab = "Power (for ES = .20)", xlab = "Number of Clusters",
     main = "CRD", locate = TRUE)

As seen from the MDES and power plots, a little less than 150 clusters are needed to obtain the benchmark power of 80% for the CRT (more than 200 clusters for the CRD). Precise number of clusters can be found via placing the primary constraint on either the effect size or power.

Primary Constraint on the Effect Size

# CRT('rhots = 0')
# cost constrained - optimize p and n2
cosa.crd2r2(rhots = 0,
            constrain = "es", es = .20,
            cn1 = c(5, 2), cn2 = c(50, 20),
            power = .80, rho2 = .20,
            g2 = 5, r21 = .20, r22 = .30,
            p = NULL, n1 = 24,  n2 = NULL)

## 
## Results are equivalent to corresponding random assignment designs 
## Solution converged with SLSQP 
##      [n1]  n2     p  cost [mdes] 95%lcl 95%ucl power
## [1,]   24 140 0.386 15028    0.2   0.06   0.34   0.8

# CRD
# cost constrained - optimize n2
cosa.crd2r2(constrain = "es", es = .20,
            power = .80, rho2 = .20,
            cn1 = c(5, 2), cn2 = c(50, 20),
            g2 = 5, r21 = .20, r22 = .30,
            p = .386, n1 = 24,  n2 = NULL)

## Solution converged with SLSQP 
##      [n1]  n2   [p]  cost [mdes] 95%lcl 95%ucl power
## [1,]   24 363 0.386 38964    0.2   0.06   0.34   0.8

Primary Constraint on the Statistical Power

# CRT('rhots = 0')
# cost constrained - optimize p and n2
cosa.crd2r2(rhots = 0, 
            constrain = "power", power = .80,
            cn1 = c(5, 2), cn2 = c(50, 20),
            es = .20, rho2 = .20,
            g2 = 5, r21 = .20, r22 = .30,
            p = NULL, n1 = 24,  n2 = NULL)

## 
## Results are equivalent to corresponding random assignment designs 
## Solution converged with SLSQP 
##      [n1]  n2     p  cost mdes 95%lcl 95%ucl [power]
## [1,]   24 140 0.386 15028  0.2   0.06   0.34     0.8

# the cost reduction for using an unbalanced design might be trivial
# fix p = .50 to obtain the cost under the balanced design for comparison
cosa.crd2r2(rhots = 0, 
            constrain = "power", power = .80,
            cn1 = c(5, 2), cn2 = c(50, 20),
            es = .20, rho2 = .20,
            g2 = 5, r21 = .20, r22 = .30,
            p = .50, n1 = 24,  n2 = NULL)

## 
## Results are equivalent to corresponding random assignment designs 
## Solution converged with SLSQP 
##      [n1]  n2   [p]  cost mdes 95%lcl 95%ucl [power]
## [1,]   24 133 0.496 15776  0.2   0.06   0.34     0.8

# CRD
# cost constrained - optimize n2
cosa.crd2r2(constrain = "power", power = .80,
            cn1 = c(5, 2), cn2 = c(50, 20),
            es = .20, rho2 = .20,
            g2 = 5, r21 = .20, r22 = .30,
            p = .386, n1 = 24,  n2 = NULL)

## Solution converged with SLSQP 
##      [n1]  n2   [p]  cost mdes 95%lcl 95%ucl [power]
## [1,]   24 363 0.386 38964  0.2   0.06   0.34     0.8

Constrained Optimal Sample Allocation Tutorial

Metin Bulus

2018-04-04

Primary Constraint on the Total Cost

Primary Constraint on the Effect Size

Primary Constraint on the Statistical Power