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After publication of this article , the authors noticed two errors:
A corrected version of Table 3 is included with this Erratum.
In the “Calculation of partial correlation coefficients” section in Methods, the expression of the Pearson’s covariance matrix is incorrect. The denominator should be n−1 instead of n. In the corrected version, the authors clarify the computational implementation used. Also, the authors now follow the convention where random variables are expressed in uppercase letters.
A corrected version of this follows below (references included in the revised portion are referring to the original article):
To examine associations of the content of inhibitory codon pairs with various gene expression variables controlling for covariates, we first attempted to use multiple linear regression models with exclusion of outliers and logarithmic transformation of skewed variables. However, we found that the models failed to satisfy the assumption of residual homogeneity (see below). We therefore chose to use non-parametric methods throughout the study.
We computed Spearman’s and Kendall’s partial correlation coefficients as described previously . Briefly, we let X be a p-dimensional random vector (X=[X1 X2Xp]T) and cij be the covariance between two random variables Xi and Xj (1≤i,j≤p). We denote the covariance matrix of X as CX, the inverse covariance matrix as DX, and the (i,j) element of DX as dij. We then let XS be a vector that contains all elements of X except Xi and Xj. The partial correlation of Xi and Xj given the vector XS is
The Spearman’s and Kendall’s covariance matrices were constructed as implemented in the cov() function in the R base package .
We computed P values by previously described methods as implemented in the pcor() function in the R ppcor package  as well as by permutation tests. To obtain permutation P values, we randomly permuted the predictor variables and computed correlation coefficients. We repeated the procedure for 10,000 times and computed a permutation P value as (B + 1)/(N + 1), where N is the number of permutations. B represents the number of events where the permutation correlation coefficient exceeds the empirically observed value.
The online version of the original article can be found under doi:10.1186/s12864-017-3749-8