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WGCNA (version 1.49)

empiricalBayesLM: Empirical Bayes-moderated adjustment for unwanted covariates

Description

This functions removes variation in high-dimensional data due to unwanted covariates while preserving variation due to retained covariates. To prevent numerical instability, it uses Empirical bayes-moderated linear regression, optionally in a robust (outlier-resistant) form.

Usage

empiricalBayesLM(
  data,
  removedCovariates,
  retainedCovariates = NULL,
  weights = NULL,
  weightType = c("apriori", "empirical"),
  stopOnSmallWeights = TRUE,
  tol = 1e-4, maxIterations = 1000,
  scaleMeanToSamples = NULL,
  robustPriors = FALSE,
  automaticWeights = c("none", "bicov"),
  aw.maxPOutliers = 0.1)

Arguments

data
A 2-dimensional matrix or data frame of numeric data to be adjusted. Variables (for example, genes or methylation profiles) should be in columns and observations (samples) should be in rows.
retainedCovariates
A vector or two-dimensional object (matrix or data frame) giving the covariates whose effect on the data is to be retained. May be NULL if there are no such "retained" covariates.
removedCovariates
A vector or two-dimensional object (matrix or data frame) giving the covariates whose effect on the data is to be removed. At least one such covariate must be given.
weights
Optional 2-dimensional matrix or data frame of the same dimensions as data giving weights for each entry in data.
weightType
One of (unique abbreviations of) "apriori" or "empirical". Determines whether a standard ("apriori") or a modified ("empirical") weighted regression is used. The "apriori" choice is suitable
stopOnSmallWeights
Logical: should presence of small "apriori" weights trigger an error? Because standard weighted regression assumes that all weights are non-zero (otherwise estimates of standard errors will be biased), this function will by default complain a
tol
Convergence criterion used in the numerical equation solver. When the relative change in coefficients falls below this threshold, the system will be considered to have converged.
maxIterations
Maximum number of iterations to use.
scaleMeanToSamples
Optional specification of samples (given as a vector of indices) to whose means the resulting adjusted data should be scaled (more precisely, shifted). If not given, the mean of all samples will be used.
robustPriors
Logical: should robust priors be used? This essentially means replacing mean by median and covariance by biweight mid-covariance.
automaticWeights
One of (unique abrreviations of) "none" or "bicov", instructing the function to calculate weights from the given data. Value "none" will result in trivial weights; value "bicov" will result
aw.maxPOutliers
If automaticWeights above is "bicov", this argument gets passed to the function bicovWeights and determines the maximum proportion of outliers in calculating the weights. See

Value

  • A list with the following components:
  • adjustedDataA matrix of the same dimensions as the input data, giving the adjusted data. If input data has non-NULL dimnames, these are copied.
  • residualsA matrix of the same dimensions as the input data, giving the residuals, that is, adjusted data with zero means.
  • coefficientsA matrix of regression coefficients. Rows correspond to the design matrix variables (mean, retained and removed covariates) and columns correspond to the variables (columns) in data.
  • coefficiens.scaledA matrix of regression coefficients corresponding to columns in data scaled to mean 0 and variance 1.
  • sigmaSqEstimated error variances (one for each column of input data.
  • sigmaSq.scaledEstimated error variances corresponding to columns in data scaled to mean 0 and variance 1.
  • fittedValuesFitted values calculated from the means and coefficients corresponding to the removed covariates, i.e., roughly the values that are subtracted out of the data.
  • adjustedData.OLSA matrix of the same dimensions as the input data, giving the data adjusted by ordinary least squares. This component should only be used for diagnostic purposes, not as input for further downstream analyses, as the OLS adjustment is inferior to EB adjustment.
  • residuals.OLSA matrix of the same dimensions as the input data, giving the residuals obtained from ordinary least squares regression, that is, OLS-adjusted data with zero means.
  • coefficients.OLSA matrix of ordinary least squares regression coefficients. Rows correspond to the design matrix variables (mean, retained and removed covariates) and columns correspond to the variables (columns) in data.
  • coefficiens.OLS.scaledA matrix of ordinary least squares regression coefficients corresponding to columns in data scaled to mean 0 and variance 1. These coefficients are used to calculate priors for the EB step.
  • sigmaSq.OLSEstimated OLS error variances (one for each column of input data.
  • sigmaSq.OLS.scaledEstimated OLS error variances corresponding to columns in data scaled to mean 0 and variance 1. These are used to calculate variance priors for the EB step.
  • fittedValues.OLSOLS fitted values calculated from the means and coefficients corresponding to the removed covariates.
  • weightsA matrix of weights used in the regression models. The matrix has the same dimension as the input data.
  • dataColumnValidLogical vector with one element per column of input data, indicating whether the column was adjusted. Columns with zero variance or too many missing data cannot be adjusted.
  • dataColumnWithZeroVarianceLogical vector with one element per column of input data, indicating whether the column had zero variance.
  • coefficientValidLogical matrix of the dimension (number of covariates +1) times (number of variables in data), indicating whether the corresponding regression coefficient is valid. Invalid regression coefficients may be returned as missing values or as zeroes.

Details

This function uses Empirical Bayes-moderated (EB) linear regression to remove variation in data due to the variables in removedCovariates while retaining variation due to variables in retainedCovariates, if any are given. The EB step uses simple normal priors on the regression coefficients and inverse gamma priors on the variances. The procedure starts with multivariate ordinary linear regression of individual columns in data on retainedCovariates and removedCovariates. To make the coefficients comparable, columns of data are scaled to (weighted if weights are given) mean 0 and variance 1. The resulting regression coefficients are used to determine the parameters of the normal prior (mean, covariance, and inverse gamma or median and biweight mid-covariance if robust priors are used), and the variances are used to determine the parameters of the inverse gamma prior. The EB step then essentially shrinks the coefficients toward their means, with the amount of shrinkage determined by the prior covariance.

Using appropriate weights can make the data adjustment robust to outliers. This can be achieved automatically by using the argument automaticWeights = "bicov". When bicov weights are used, we also recommend setting the argument maxPOutliers to a maximum proportion of samples that could be outliers. This is especially important if some of the design variables are binary and can be expected to have a strong effect on some of the columns in data, since standard biweight midcorrelation (and its weights) do not work well on bimodal data.

The automatic bicov weights are determined from data only. It is implicitly assumed that there are no outliers in the retained and removed covariates. Outliers in the covariates are more difficult to work with since, even if the regression is made robust to them, they can influence the adjusted values for the sample in which they appear. Unless the the covariate outliers can be attributed to a relevant vriation in experimental conditions, samples with covariate outliers are best removed entirely before calling this function.

See Also

bicovWeights for suitable weights that make the adjustment robust to outliers.