power.mmrm.ar1 {longpower}

Linear mixed model sample size calculations.

Description

This function performs the sample size calculation for a mixed model of repeated measures with AR(1) correlation structure. See Lu, Luo, & Chen (2008) for parameter definitions and other details.

Usage

power.mmrm.ar1(
  N = NULL,
  rho = NULL,
  ra = NULL,
  sigmaa = NULL,
  rb = NULL,
  sigmab = NULL,
  lambda = 1,
  times = 1:length(ra),
  delta = NULL,
  sig.level = 0.05,
  power = NULL,
  alternative = c("two.sided", "one.sided"),
  tol = .Machine$double.eps^2
)

Arguments

N

total sample size

rho

AR(1) correlation parameter

ra

retention in group a

sigmaa

standard deviation of observation of interest in group a

rb

retention in group a (assumed same as ra if left blank)

sigmab

standard deviation of observation of interest in group b. If NULL, sigmab is assumed same as sigmaa. If not NULL, sigmaa and sigmab are averaged.

lambda

allocation ratio

times

observation times

delta

effect size

sig.level

type one error

power

power

alternative

one- or two-sided test

tol

numerical tolerance used in root finding.

Details

See Lu, Luo, & Chen (2008).

Value

The number of subject required per arm to attain the specified power given sig.level and the other parameter estimates.

Author(s)

Michael C. Donohue

References

Lu, K., Luo, X., Chen, P.-Y. (2008) Sample size estimation for repeated measures analysis in randomized clinical trials with missing data. International Journal of Biostatistics, 4, (1)

See Also

power.mmrm, lmmpower, diggle.linear.power

Examples

# reproduce Table 2 from Lu, Luo, & Chen (2008)
tab <- c()
for(J in c(2,4))
for(aJ in (1:4)/10)
for(p1J in c(0, c(1, 3, 5, 7, 9)/10)){
  rJ <- 1-aJ
  r <- seq(1, rJ, length = J)
  # p1J = p^(J-1)
  tab <- c(tab, power.mmrm.ar1(rho = p1J^(1/(J-1)), ra = r, sigmaa = 1, 
    lambda = 1, times = 1:J,
    delta = 1, sig.level = 0.05, power = 0.80)$phi1)
}
matrix(tab, ncol = 6, byrow = TRUE)

# approximate simulation results from Table 5 from Lu, Luo, & Chen (2008)
ra <- c(100, 76, 63, 52)/100
rb <- c(100, 87, 81, 78)/100

power.mmrm.ar1(rho=0.6, ra=ra, sigmaa=1, rb = rb, 
               lambda = sqrt(1.25/1.75), power = 0.904, delta = 0.9)
power.mmrm.ar1(rho=0.6, ra=ra, sigmaa=1, rb = rb, 
               lambda = 1.25/1.75, power = 0.910, delta = 0.9)
power.mmrm.ar1(rho=0.6, ra=ra, sigmaa=1, rb = rb, 
               lambda = 1, power = 0.903, delta = 0.9)
power.mmrm.ar1(rho=0.6, ra=ra, sigmaa=1, rb = rb,
               lambda = 2, power = 0.904, delta = 0.9)

power.mmrm.ar1(N=81, ra=ra, sigmaa=1, rb = rb, 
               lambda = sqrt(1.25/1.75), power = 0.904, delta = 0.9)
power.mmrm.ar1(N=87, rho=0.6, ra=ra, sigmaa=1, rb = rb,
               lambda = 1.25/1.75, power = 0.910)
power.mmrm.ar1(N=80, rho=0.6, ra=ra, sigmaa=1, rb = rb, 
               lambda = 1, delta = 0.9)
power.mmrm.ar1(N=84, rho=0.6, ra=ra, sigmaa=1, rb = rb,
               lambda = 2, power = 0.904, delta = 0.9, sig.level = NULL)

# Extracting paramaters from gls objects with AR1 correlation

# Create time index:
Orthodont$t.index <- as.numeric(factor(Orthodont$age, levels = c(8, 10, 12, 14)))
with(Orthodont, table(t.index, age))

fmOrth.corAR1 <- gls( distance ~ Sex * I(age - 11), 
  Orthodont,
  correlation = corAR1(form = ~ t.index | Subject),
  weights = varIdent(form = ~ 1 | age) )
summary(fmOrth.corAR1)$tTable

C <- corMatrix(fmOrth.corAR1$modelStruct$corStruct)[[1]]
sigmaa <- fmOrth.corAR1$sigma *
          coef(fmOrth.corAR1$modelStruct$varStruct, unconstrained = FALSE)['14']
ra <- seq(1,0.80,length=nrow(C))
power.mmrm(N=100, Ra = C, ra = ra, sigmaa = sigmaa, power = 0.80)
power.mmrm.ar1(N=100, rho = C[1,2], ra = ra, sigmaa = sigmaa, power = 0.80)