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# modified on June 22, 2020
# (1) rename 'alpha' in powerEQTL.scRNAseq' by 'FWER'
# (3) rename 'delta' in powerEQTL.scRNAseq' by 'slope'
#
# created on Apr 7, 2020
# (1) power for simple linear mixed effects model
#
# We assume the following linear mixed effects model to characterize
# the association between x and y:
# y_{ij} = beta_{0i} + beta_1 * x_i + epsilon_{ij},
# beta_{0i} ~ N(beta_0, sigma^2_{\beta})
# epsilon_{ij} ~ N(0, sigma^2)
#
# slope - slope under alternative hypothesis
# n - number of subjects
# m - number of observations per subject
# sigma.y - standard deviation of the outcome y
# sigma.x - standard deviation of the predictor x
# rho - intra-class correlation (i.e., correlation between y_{ij} and y_{ik})
# rho = sigma^2_{beta} / (sigma^2_{beta}+sigma^2)
# FWER - family-wise type I error rate
# nTests = number of tests
powerLME.default=function(
slope,
n,
m,
sigma.y,
sigma.x,
rho=0.8,
FWER=0.05,
nTests=1)
{
alpha2=FWER/nTests
za2=qnorm(1-alpha2/2)
part0=sigma.x*slope*sqrt(m*(n-1))/(sigma.y*sqrt(1+(m-1)*rho))
part1 = za2-part0
part2 = -za2-part0
power = 1- pnorm(part1) + pnorm(part2)
return(power)
}
powerLME=function(
slope,
n,
m,
sigma.y,
sigma.x,
power = NULL,
rho=0.8,
FWER=0.05,
nTests=1,
n.lower = 2.01,
n.upper = 1e+30)
{
if (is.null(slope) == TRUE &&
is.null(n) == FALSE && is.null(power) == FALSE) {
slope = minSlope.LME(n = n,
m = m,
sigma.y = sigma.y,
sigma.x = sigma.x,
power = power,
rho=rho,
FWER=FWER,
nTests=nTests
)
names(slope) = "slope"
return(slope)
} else if (is.null(slope) == FALSE &&
is.null(n) == TRUE && is.null(power) == FALSE) {
n = ssLME( slope = slope,
m = m,
sigma.y = sigma.y,
sigma.x = sigma.x,
power = power,
rho=rho,
FWER=FWER,
nTests=nTests,
n.lower = n.lower,
n.upper = n.upper
)
names(n)="n"
return(n)
} else if (is.null(slope) == FALSE &&
is.null(n) == FALSE && is.null(power) == TRUE) {
power = powerLME.default(
slope = slope,
n = n,
m = m,
sigma.y = sigma.y,
sigma.x = sigma.x,
rho=rho,
FWER=FWER,
nTests=nTests)
names(power) = "power"
return(power)
} else {
stop("One and only one of the 3 parameters (slope, n, power) can be NULL!\n")
}
}
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