Why is a non-linear transformation a parameter? This in reference to my answer at What model should I use to prove statistical significance?.
I test the correlation between x and log10(y), specifically, in R:
Satisfaction = c(55, 34, 24, 12, 17, 10, 14)
Tenure = c(0.5, 2, 4.5, 8, 13, 17.5, 20)

cor.test(Satisfaction, log10(Tenure))

Whuber in comments points out that the model has 3 parameters: In addition to the "familiar" slope and intercept, the third is the choice of non-linear transformation.
I'm puzzled by the non-linear transformation being counted as a parameter and I would appreciate some explanation.
I understand that choosing a transformation, especially after having seen the data, should be penalized as a form of multiple hypothesis testing. But I don't see it as a parameter and in fact I'm not estimating it, I'm just imposing it. I'm also puzzled because I have never seen in literature transformations being considered a parameter. Usually, you transform to make the data better behaved or more interpretable, but I haven't seen this done at a cost.
 A: I think I disagree that the transformation is a third parameter in cor(x, log10(y)).
I agree that choosing a transformation after having seen the data is a form of estimation and therefore should impact on the interpretation of the results (correlation coefficient and p-value in this case).
However, if I receive a new dataset as a replicate experiment, I wouldn't decide on a different transformation. I would choose the logarithmic transformation even before seeing the new data. This is in contrast with the slope and intercept that would have to be re-estimated on the new dataset.
Perhaps more importantly, even if I didn't log-transform I would still have considered in my mind various options and decided for the identity transformation (i.e. leave the data as it is). Effectively this is what I've done for the variable Satisfaction. Would that be a fourth parameter that I estimated from the data?
I guess this is what Andrew Gelman calls Garden of Forking Paths. As a very rough guideline when I see a p-value, including those from my own analyses, I mentally multiply it by, say, 10x to account for all the things that have been or couldn't have been done on the data before the results have been finalized.
