Package ‘odr’

The costs of sampling each additional unit in multilevel experimental studies vary across levels of hierarchy and treatment conditions due to the hierarchical sampling and the delivery of treatment. This package is a tool to optimize the designs of multilevel experimental studies such that the variances of treatment effects are minimized under a fixed budget and cost structure, or the budget is minimized to achieve same level design precision or statistical power. The optimal sample allocation or optimal design parameters include

1. Function od

Given cost structure (i.e., the costs of sampling each unit at different levels and treatment conditions), this function solves the optimal sample allocation with and without constraints.

To solve the optimal sample allocation of a two-level cluster-randomized trial, we need the following information

icc: intraclass correlation coefficient
r12: the proportion of level-one outcome variance explained by covariates
r22: the proportion of level-two outcome variance explained by covariates
c1: the cost of sampling each additional level-one unit in the control condition
c2: the cost of sampling each additional level-two unit in the control condition
c1t: the cost of sampling each additional level-one unit in the experimental condition
c2t: the cost of sampling each additional level-two unit in the experimental condition
m: a total fixed budget used to plot the variance curves, default value is the cost of sampling 60 level-two units across treatment conditions.

1.1 Examples

library(odr)

 # unconstrained optimal design
myod1 <- od.2(icc = 0.2, r12 = 0.5, r22 = 0.5, c1 = 1, c2 = 5, c1t = 1, c2t = 50, 
              varlim = c(0.01, 0.02))

## The optimal level-1 sample size per level-2 unit (n) is 8.878572.
## The optimal proportion of level-2 units in treatment (p) is 0.326828.

 # The function by default prints messages of output and plots the variance curves; one can turn off message and specify one or no plot.
 # myod1$out # output; 
 # myod1$par # parameters used in the calculation.

 # constrained optimal design with n = 20
myod2 <- od.2(icc = 0.2, r12 = 0.5, r22 = 0.5, c1 = 1, c2 = 5, c1t = 1, c2t = 50,
              plot.by = list(p = "p"), n = 20, varlim = c(0.005, 0.030))

## The constrained level-1 sample size per level-2 unit (n) is 20.
## The optimal proportion of level-2 units in treatment (p) is 0.3740667.

 # myod2$out # output
 # myod2$par # parameters used in the calculation.

 # constrained optimal design with p = 0.5
myod3 <- od.2(icc = 0.2, r12 = 0.5, r22 = 0.5, c1 = 1, c2 = 5, c1t = 1, c2t = 50, 
             p = 0.5, varlim = c(0.005, 0.020))

## The optimal level-1 sample size per level-2 unit (n) is 10.48809.
## The constrained proportion of level-2 units in treatment (p) is 0.5.

 # myod3$out # output; 
 # myod3$par # parameters used in the calculation.

 # constrained n and p, no calculation performed
myod4 <- od.2(icc = 0.2, r12 = 0.5, r22 = 0.5, c1 = 1, c2 = 5, c1t = 1, c2t = 50,
              plots = FALSE, n = 20, p = 0.5, varlim = c(0.005, 0.025))

## ===============================
## Both p and n are constrained, there is no calculation from other parameters.
## ===============================
## The constrained level-1 sample size per level-2 unit (n) is 20.
## The constrained proportion of level-2 units in treatment (p) is 0.5.

1.2 Examples for other types of trials

Please see examples in corresponding functions by uncommenting below lines.

# ?od.1 
# ?od.3
# ?od.4
# ?od.2m
# ?od.3m
# ?od.4m

2. Function power

This function by default can perform power analyses accommodating cost structures (i.e., cost.model = TRUE), one of ‘power’, ‘m’, and ‘d’ must be NULL. For example, if ‘power’ is NULL, the function calculates statistical power under a fixed budget and cost structure; if ‘d’ is NULL, the function calculates minimum detectable effect size (i.e., d) under a fixed budget and desired power level; if ‘m’ is NULL, the function calculate required budget (and required sample size) to achieve desired power level to detect a treatment effect.

This function also can conduct conventional power analysis or power analysis without accommodating cost structures by specifying cost.model = FALSE, the conventional power analyses include statistical power calculation, minimum detectable effect size calculation, and required sample size calculation.

2.1 Examples of power analyses accommodating cost structures (cost.model = TRUE)

Required budget for desired power

Required budget calculation

mym <- power.2(expr = myod1, d = 0.3, q = 1, power = 0.8)
# mym$out  # m =1702, J = 59

Effects on required budget to maintain same level power when designs depart from the optimal one

figure <- par(mfrow = c(1, 2))
budget <- NULL
nrange <- c(2:50)
for (n in nrange)
  budget <- c(budget, power.2(expr = myod1, constraint = list (n = n), d = 0.3, q = 1, power = 0.8)$out$m)
plot(nrange, budget, type = "l", lty = 1, xlim = c(0, 50), ylim = c(1500, 3500),
     xlab = "Level-1 sample size: n", ylab = "Budget", main = "", col = "black")
 abline(v = 9, lty = 2, col = "Blue")
 
budget <- NULL
prange <- seq(0.05, 0.95, by = 0.005)
for (p in prange)
  budget <- c(budget, power.2(expr = myod1, constraint = list (p = p), d = 0.3, q = 1, power = 0.8)$out$m)
plot(prange, budget, type = "l", lty = 1, xlim = c(0, 1), ylim = c(1500, 7000),
     xlab = "Porportion groups in treatment: p", ylab = "Budget", main = "", col = "black")
 abline(v = 0.33, lty = 2, col = "Blue")

par(figure)

Statistical power under a fixed budget

Power calculation

mypower <- power.2(expr = myod1, q = 1, d = 0.3, m = 1702)
# mypower$out  # power = 0.80

Effects on power under same budget when designs depart from the optimal one

figure <- par(mfrow = c (1, 2))
pwr <- NULL
nrange <- c(2:50)
for (n in nrange)
  pwr <- c(pwr, power.2(expr = myod1, constraint = list (n = n), d = 0.3, q = 1, m = 1702)$out)
plot(nrange, pwr, type = "l", lty = 1, xlim = c(0, 50), ylim = c(0.4, 0.9),
     xlab = "Level-1 sample size: n", ylab = "Power", main = "", col = "black")
 abline(v = 9, lty = 2, col = "Blue")
 
pwr <- NULL
prange <- seq(0.05, 0.95, by = 0.005)
for (p in prange)
  pwr <- c(pwr, power.2(expr = myod1, constraint = list (p = p), d = 0.3, q = 1, m = 1702)$out)
plot(prange, pwr, type = "l", lty = 1, xlim = c(0, 1), ylim = c(0.1, 0.9),
     xlab = "Porportion groups in treatment: p", ylab = "Power",  main = "", col = "black")
 abline(v = 0.33, lty = 2, col = "Blue")

 par(figure)

Minimum detectable effect size under a fixed budget

minimum detectable effect size calculation

mymdes <- power.2(expr = myod1, q = 1, power = 0.80, m = 1702)
# above experssion takes parameters and outputs from od.2 function. Equivalently, each parameter can be explicitly specified.
# mym <- power.2(icc = 0.2, r12 = 0.5, r22 = 0.5, c1 = 1, c2 = 5, c1t = 1, c2t = 50,
#                     n = 9, p = 0.33, d = 0.3, q = 1, power = 0.8)
# mymdes$out  # d = 0.30

Effects on minimum detectable effect size under same budget when designs depart from the optimal one

figure <- par(mfrow = c (1, 2))
MDES <- NULL
nrange <- c(2:50)
for (n in nrange)
  MDES <- c(MDES, power.2(expr = myod1, constraint = list (n = n), power = 0.8, q = 1, m = 1702)$out)
plot(nrange, MDES, type = "l", lty = 1, xlim = c(0, 50), ylim = c(0.3, 0.8),
     xlab = "Level-1 sample size: n", ylab = "MDES", main = "", col = "black")
 abline(v = 9, lty = 2, col = "Blue")
 
MDES <- NULL
prange <- seq(0.05, 0.95, by = 0.005)
for (p in prange)
  MDES <- c(MDES, power.2(expr = myod1, constraint = list (p = p), power = 0.8, q = 1, m = 1702)$out)
plot(prange, MDES, type = "l", lty = 1, xlim = c(0, 1), ylim = c(0.3, 0.8),
     xlab = "Porportion groups in treatment: p", ylab = "MDES", main = "", col = "black")
 abline(v = 0.33, lty = 2, col = "Blue")

 par(figure)

2.2 Examples of conventional power analyses (cost.model = FALSE)

# Required level-2 sample size calculation
myJ <- power.2(cost.model = FALSE, expr = myod1, d = 0.3, q = 1, power = 0.8)
# above experssion takes parameters and outputs from od.2 function. Equivalently, each parameter can be explicitly specified.
# myJ <- power.2(icc = 0.2, r12 = 0.5, r22 = 0.5, 
#                     cost.model = FALSE, n = 9, p = 0.33, d = 0.3, q = 1, power = 0.8)
myJ$out  # J = 59

## $J
## [1] 58.99295

# Power calculation
mypower1 <- power.2(cost.model = FALSE, expr = myod1, J = 59, d = 0.3, q = 1)
mypower1$out  # power = 0.80

## $power
## [1] 0.8000486

# Minimum detectable effect size calculation
mymdes1 <- power.2(cost.model = FALSE, expr = myod1, J = 59, power = 0.8, q = 1)
mymdes1$out  # d = 0.30

## $d
## [1] 0.2999819

2.3 Examples of conventional power curves

figure <- par(mfrow = c (1, 2))
pwr <- NULL
mrange <- c(300:3000)
for (m in mrange)
  pwr <- c(pwr, power.2(expr = myod1, d = 0.3, q = 1, m = m)$out)
plot(mrange, pwr, type = "l", lty = 1, xlim = c(300, 3000), ylim = c(0, 1),
     xlab = "Budget", ylab = "Power", main = "", col = "black")
 abline(v = 1702, lty = 2, col = "Blue")
 
pwr <- NULL
Jrange <- c(4:100)
for (J in Jrange)
  pwr <- c(pwr, power.2(expr = myod1, cost.model = FALSE, d = 0.3, q = 1, J = J)$out)
plot(Jrange, pwr, type = "l", lty = 1, xlim = c(4, 100), ylim = c(0, 1),
     xlab = "Level-2 sample size: J", ylab = "Power", main = "", col = "black")
 abline(v = 59, lty = 2, col = "Blue")

par(figure)

2.4 Examples for other types of trials

Please see examples in corresponding functions by uncommenting below lines.

# ?power.1
# ?power.3
# ?power.4
# ?power.2m
# ?power.3m
# ?power.4m

3. Function re

Calculate the relative efficiency (RE) of two designs, this function uses the returns from od function

3.1 Examples

Based on above examples in od functions, calculate the relative efficiency

# relative efficiency (RE) of a constrained design comparing with the optimal design
myre <- re(od = myod1, subod= myod2)

## The relative efficiency (RE) of the two two-level CRTs is 0.8790305.

myre$re # get the output (i.e., RE = 0.88)

## [1] 0.8790305

# relative efficiency (RE) of a constrained design comparing with the unconstrained optimal one
myre <- re(od = myod1, subod= myod3)

## The relative efficiency (RE) of the two two-level CRTs is 0.8975086.

# relative efficiency (RE) of a constrained design comparing with the unconstrained optimal one
myre <- re(od = myod1, subod= myod4)

## The relative efficiency (RE) of the two two-level CRTs is 0.8266527.

3.2 Examples for other types of trials

For additional examples, please see example sections in corresponding od functions by uncommenting below lines.

# ?od.1
# ?od.2
# ?od.3
# ?od.4
# ?od.2m
# ?od.3m
# ?od.4m