powerEQTL (version 0.3.1)

powerLME: Power Calculation for Simple Linear Mixed Effects Model

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.

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.

Details

We assume the following simple linear mixed effects model to characterize the association between the predictor x and the outcome y: $$y_{ij} = \beta_{0i} + \beta_1 * x_i + \epsilon_{ij},$$ where $$\beta_{0i} \sim N\left(\beta_0, \sigma^2_{\beta}\right),$$ and $$\epsilon_{ij} \sim N\left(0, \sigma^2\right),$$ \(i=1,\ldots, n\), \(j=1,\ldots, m\), \(n\) 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: $$H_0: \beta_1=0,$$ and $$H_1: \beta_1 = \delta,$$ where \(\delta\neq 0\).

We can derive the power calculation formula is $$power=1- \Phi\left(z_{\alpha^{*}/2}-a\times b\right) +\Phi\left(-z_{\alpha^{*}/2} - a\times b\right),$$ where $$a= \frac{\hat{\sigma}_x }{\sigma_y}$$ and $$ b=\frac{\delta\sqrt{m(n-1)}}{\sqrt{1+(m-1)\rho}} $$ and \(z_{\alpha^{*}/2}\) is the upper \(100\alpha^{*}/2\) percentile of the standard normal distribution, \(\alpha^{*}=\alpha/nTests\), nTests is the number of tests, \(\sigma_y=\sqrt{\sigma^2_{\beta}+\sigma^2}\), \(\hat{\sigma}_x=\sqrt{\sum_{i=1}^n\left(x_i-\bar{x}\right)^2/(n-1)}\), and \(\rho=\sigma^2_{\beta}/\left(\sigma^2_{\beta}+\sigma^2\right)\) is the intra-class correlation.

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

# NOT RUN {
  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)
  


# }