powerLME: Power Calculation for Simple Linear Mixed Effects Model

In powerEQTL: Power and Sample Size Calculation for Bulk Tissue and Single-Cell eQTL Analysis

Description

Power calculation for simple linear mixed effects model. This function can be used to calculate one of the 3 parameters (power, sample size, and minimum detectable slope) by setting the corresponding parameter as NULL and providing values for the other 2 parameters.

Usage

powerLME(
  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)

Arguments

slope	numeric. Slope under alternative hypothesis.
n	integer. Total number of subjects.
m	integer. Number of observations per subject.
sigma.y	numeric. Standard deviation of the outcome y.
sigma.x	numeric. Standard deviation of the predictor x.
power	numeric. Desired power.
rho	numeric. Intra-class correlation (i.e., correlation between y_{ij} and y_{ik} for the j-th and k-th observations of the i-th subject).
FWER	numeric. Family-wise Type I error rate.
nTests	integer. Number of tests (e.g., number of genes in differential expression analysis based on scRNAseq to compare gene expression between diseased subjects and healthy subjects).
n.lower	numeric. Lower bound of the total number of subjects. Only used when calculating toal number of subjects.
n.upper	numeric. Upper bound of the total number of subjects. Only used when calculating total number of subjects.

Details

We assume the following simple linear mixed effects model to characterize the association between the predictor x and the outcome y:

y_{ij} = β_{0i} + β_1 * x_i + ε_{ij},

where

β_{0i} \sim N≤ft(β_0, σ^2_{β}\right),

and

ε_{ij} \sim N≤ft(0, σ^2\right),

i=1,…, n, j=1,…, m, n is the number of subjects, m is the number of observations per subject, y_{ij} is the outcome value for the j-th observation of the i-th subject, x_i is the predictor value for the i-th subject. For example, x_i is the binary variable indicating if the i-th subject is a diseased subject or not.

We would like to test the following hypotheses:

H_0: β_1=0,

and

H_1: β_1 = δ,

where δ\neq 0.

We can derive the power calculation formula is

power=1- Φ≤ft(z_{α^{*}/2}-a\times b\right) +Φ≤ft(-z_{α^{*}/2} - a\times b\right),

where

a= \frac{\hat{σ}_x }{σ_y}

and

b=\frac{δ√{m(n-1)}}{√{1+(m-1)ρ}}

and z_{α^{*}/2} is the upper 100α^{*}/2 percentile of the standard normal distribution, α^{*}=α/nTests, nTests is the number of tests, σ_y=√{σ^2_{β}+σ^2}, \hat{σ}_x=√{∑_{i=1}^n≤ft(x_i-\bar{x}\right)^2/(n-1)}, and ρ=σ^2_{β}/≤ft(σ^2_{β}+σ^2\right) is the intra-class correlation.

Value

power if the input parameter power = NULL.

sample size (total number of subjects) if the input parameter n = NULL;

minimum detectable slope if the input parameter slope = NULL.

Author(s)

Xianjun Dong <[e-mail]>, Xiaoqi Li<[e-mail]>, Tzuu-Wang Chang <[e-mail]>, Scott T. Weiss <[e-mail]>, Weiliang Qiu <[e-mail]>

References

Dong X, Li X, Chang T-W, Scherzer CR, Weiss ST, and Qiu W. powerEQTL: An R package and shiny application for sample size and power calculation of bulk tissue and single-cell eQTL analysis. Bioinformatics, 2021;, btab385

Examples

  n = 102
  m = 227868
  
  # calculate power
  power = powerLME(
    slope = 0.6, 
    n = n, 
    m = m,
    sigma.y = 0.29, 
    sigma.x = 0.308,
    power = NULL,
    rho = 0.8, 
    FWER = 0.05,
    nTests = 1e+6)

  print(power)
  
  # calculate sample size (total number of subjects)
  n = powerLME(
    slope = 0.6, 
    n = NULL, 
    m = m,
    sigma.y = 0.29, 
    sigma.x = 0.308,
    power = 0.9562555,
    rho = 0.8, 
    FWER = 0.05,
    nTests = 1e+6)

  print(n)
  
  # calculate slope
  slope = powerLME(
    slope = NULL, 
    n = n, 
    m = m,
    sigma.y = 0.29, 
    sigma.x = 0.308,
    power = 0.9562555,
    rho = 0.8, 
    FWER = 0.05,
    nTests = 1e+6)

  print(slope)
  

powerEQTL documentation built on July 22, 2021, 9:08 a.m.