power.3m {odr}

Budget and/or sample size, power, MDES calculation for three-level MRTs detecting main effects

Description

This function can calculate required budget for desired power, power or minimum detectable effect size (MDES) under fixed budget for three-level multisite randomized trials (MRTs). It also can perform conventional power analyses (e.g., required sample size, power, and MDES calculation).

Usage

power.3m(
  cost.model = TRUE,
  expr = NULL,
  constraint = NULL,
  sig.level = 0.05,
  two.tailed = TRUE,
  d = NULL,
  power = NULL,
  m = NULL,
  n = NULL,
  J = NULL,
  K = NULL,
  p = NULL,
  icc2 = NULL,
  icc3 = NULL,
  r12 = NULL,
  r22 = NULL,
  r32m = NULL,
  q = NULL,
  c1 = NULL,
  c2 = NULL,
  c3 = NULL,
  c1t = NULL,
  c2t = NULL,
  omega = NULL,
  dlim = NULL,
  powerlim = NULL,
  Klim = NULL,
  mlim = NULL,
  rounded = TRUE
)

Arguments

cost.model

Logical; power analyses accommodating costs and budget (e.g., required budget for desired power, power/MDES under fixed budget) if TRUE, otherwise conventional power analyses (e.g., required sample size, power, or MDES calculation); default value is TRUE.

expr

Returned objects from function od.3m; default is NULL; if expr is specified, parameter values of icc2, icc3, r12, r22, r32m, c1, c2, c3, c1t, c2t, p, n, and J used or solved in function od.3m will be passed to current function; only the values of p, n, and/or J that specified or solved in function od.3m can be overwritten if constraint is specified.

constraint

Specify the constrained values of p, n, and/or J, in list format to overwrite those from expr; default value is NULL.

sig.level

Significance level or type I error rate, default value is 0.05.

two.tailed

Logical; two-tailed tests if TRUE, otherwise one-tailed tests; default value is TRUE.

d

Effect size.

power

Statistical power.

m

Total budget.

n

The level-1 sample size per level-2 unit.

J

The level-2 sample size per level-3 unit.

K

The level-3 sample size per level-4 unit.

p

The proportion of level-2 units to be assigned to treatment.

icc2

The unconditional intraclass correlation coefficient (ICC) at level 2.

icc3

The unconditional intraclass correlation coefficient (ICC) at level 3.

r12

The proportion of level-1 variance explained by covariates.

r22

The proportion of level-2 variance explained by covariates.

r32m

The proportion of variance of site-specific treatment effect explained by covariates.

q

The number of covariates at level 3.

c1

The cost of sampling one level-1 unit in control condition.

c2

The cost of sampling one level-2 unit in control condition.

c3

The cost of sampling one level-3 unit.

c1t

The cost of sampling one level-1 unit in treatment condition.

c2t

The cost of sampling one level-2 unit in treatment condition.

omega

The standardized variance of site-specific treatment effect.

dlim

The range for solving the root of effect size (d) numerically, default value is c(0, 5).

powerlim

The range for solving the root of power (power) numerically, default value is c(1e-10, 1 - 1e-10).

Klim

The range for searching the root of level-3 sample size (K) numerically, default value is c(4, 1e+10).

mlim

The range for searching the root of budget (m) numerically, default is the costs sampling Klim level-3 units or c(4 * Kcost, 1e+10 * Kcost) with Kcost = ((1 - p) * (c1 * n * J + c2 * J) + p * (c1t * n * J + c2t * J) + c3.

rounded

Logical; round the values of p, n/J/K that are from functions od.4 to two decimal places and integer, respectively if TRUE, otherwise no rounding; default value is TRUE.

Value

Required budget (and/or required level-3 sample size), statistical power, or MDES depending on the specification of parameters. The function also returns the function name, design type, and parameters used in the calculation.

References

Shen, Z., & Kelcey, B. (in press). Optimal sampling ratios in three-level multisite experiments. Journal of Research on Educational Effectiveness.

Examples

# Unconstrained optimal design #---------
  myod1 <- od.3m(icc2 = 0.2, icc3 = 0.1, omega = 0.02,
              r12 = 0.5, r22 = 0.5, r32m = 0.5,
              c1 = 1, c2 = 5,
              c1t = 1, c2t = 200, c3 = 200,
              varlim = c(0, 0.005))
  myod1$out # n = 13.1, J = 15.3, p = 0.23

# ------- Power analyses by default considering costs and budget -------
# Required budget and sample size
  mym.1 <- power.3m(expr = myod1, d = 0.2, q = 1, power = 0.8)
  mym.1$out  # m = 15491, K = 13.6
  # mym.1$par  # parameters and their values used for the function
# Or, equivalently, specify every argument in the function
  mym.1 <- power.3m(d = 0.2, power = 0.8, q = 1,
                 icc2 = 0.2, icc3 = 0.1, omega = 0.02,
                  r12 = 0.5, r22 = 0.5, r32m = 0.5,
                  c1 = 1, c2 = 5,
                  c1t = 1, c2t = 200, c3 = 200,
                  n = 13, J = 15, p = 0.23)
# Required budget and sample size with constrained p
  mym.2 <- power.3m(expr = myod1, d = 0.2, q = 1, power = 0.8,
                 constraint = list(p = 0.5))
  mym.2$out  # m = 21072, K = 10.9
# Required budget and sample size with constrained p and n
  mym.3 <- power.3m(expr = myod1, d = 0.2, q = 1, power = 0.8,
                 constraint = list(p = 0.5, n = 20))
  mym.3$out  # m = 21252, K = 10.4

# Power calculation
  mypower <- power.3m(expr = myod1, q = 1, d = 0.2, m = 15491)
  mypower$out  # power = 0.80
# Power calculation under constrained p (p = 0.5)
  mypower.1 <- power.3m(expr = myod1, q = 1, d = 0.2, m = 15491,
                 constraint = list(p = 0.5))
  mypower.1$out  # power = 0.62

# MDES calculation
  mymdes <- power.3m(expr = myod1, q = 1, power = 0.80, m = 15491)
  mymdes$out  # d = 0.20


# ------- Conventional power analyses with cost.model = FALSE-------
# Required sample size
  myK <- power.3m(cost.model = FALSE, expr = myod1, d = 0.2, q = 1, power = 0.8)
  myK$out  # K = 13.6
  # myK$par  # parameters and their values used for the function
# Or, equivalently, specify every argument in the function
  myK <- power.3m(cost.model = FALSE, d = 0.2, power = 0.8, q = 1,
                  icc2 = 0.2, icc3 = 0.1, omega = 0.02,
                  r12 = 0.5, r22 = 0.5, r32m = 0.5,
                  c1 = 1, c2 = 5,
                  c1t = 1, c2t = 200, c3 = 200,
                  n = 13, J = 15, p = 0.23)

# Power calculation
  mypower1 <- power.3m(cost.model = FALSE, expr = myod1, K = 13.6, d = 0.2, q = 1)
  mypower1$out  # power = 0.80

# MDES calculation
  mymdes1 <- power.3m(cost.model = FALSE, expr = myod1, K = 13.6, power = 0.8, q = 1)
  mymdes1$out  # d = 0.20