There is a general trend to do log-transformations when we face skewed and non-normal distributions. However, it is quite unusual that the authors assess the (real) possibility of having biased the model with that procedure.
On the other hand, perhaps it is not important if it occurs, considering the advantages (not-skewed distribution, normality, parametric tests...)
Is there any simple procedure to check this bias? What do you think about this matter?
Newman MC (1993): Regression analysis of log-transformed data: statistical bias and its correction.
http://www.vims.edu/people/newman_mc/pubs/Newman1993.pdf
Stow CA, Reckhow KH, Qian SS. A bayesian approach to retransformation bias in transformed regression. Ecology. 2006;87(6):1472-7.
https://www.researchgate.net/publication/6917795_A_Bayesian_approach_to_retransformation_bias_in_transformed_regression
Article A Bayesian Approach to Retransformation Bias in Transformed Regression
The bias only matters if you do the inverse transformation of the model. That is, after fitting the log-transformed data like this:
log( Y ) ~ a + b * log( X )
...you apply the reverse transformation (exponent) to achieve the following equation
Y ~ e^a * X ^ b
for the purposes of future predictions.
So, if your question is, for example, whether the correlation between the log-transformed variables is significant, you are fine -- there is no bias. Only if you want to make new predictions for a non-transformed variable you can run into the bias issue, since one omits the reverse transformation of the error. As for dealing with that -- please read about "smearing retransformation". See for example
http://healthcare-economist.com/2010/11/16/duans-smearing-estimator/
https://mywebspace.wisc.edu/jmullahy/web/manningmullahy.pdf
If yo do log-transformation of a covariate, then this transformation give a significant coefficient no need check the the possibility of a bias, because it is helpful transfor for the covariate. Othervise (do not give a significant coefficient) no need in the model, try some other transform or not
The bias only matters if you do the inverse transformation of the model. That is, after fitting the log-transformed data like this:
log( Y ) ~ a + b * log( X )
...you apply the reverse transformation (exponent) to achieve the following equation
Y ~ e^a * X ^ b
for the purposes of future predictions.
So, if your question is, for example, whether the correlation between the log-transformed variables is significant, you are fine -- there is no bias. Only if you want to make new predictions for a non-transformed variable you can run into the bias issue, since one omits the reverse transformation of the error. As for dealing with that -- please read about "smearing retransformation". See for example
http://healthcare-economist.com/2010/11/16/duans-smearing-estimator/
https://mywebspace.wisc.edu/jmullahy/web/manningmullahy.pdf
You could make simulations to estimate bias in your context.
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