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When you log transform data what is being tested on the original scale of the data? And why can we use our log transformed data to answer our scientific question?

We are looking at data comparing large cell lung cancer vs small cell lung cancer survival times. The original data is time and it’s separated by large or small cell cancer. The data that was log transformed was time.

The scientific question is: does the type of cell cancer you have affect your survival time?

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  • $\begingroup$ Welcome to our site. What is being tested depends on what test you are performing! Ditto for answering a scientific question. Could you supply details? $\endgroup$ – whuber Mar 26 '19 at 17:11
  • $\begingroup$ I was writing an answer when the question got unexpectedly closed. It could be reasonable to close the question if the OP had asked if he should take logs of his data, but he isn't. The question is general enough to be answered with a few examples of situations where taking logs is useful, and that could be a useful answer for the site. $\endgroup$ – Pere Mar 26 '19 at 17:16
  • $\begingroup$ One example: When you need to perform a procedure which works better with normal data and your data isn't normal but their logs are. For example, performing a t-test or an ANOVA with highly skewed data. It might be inaccurate to test if means of each group are equal using t-test or ANOVA because of non-normality, but it might be possible to test if means of logs of data of each group are equal. $\endgroup$ – Pere Mar 26 '19 at 17:18
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    $\begingroup$ More information should be included in the question, or in a new question. $\endgroup$ – Pere Mar 26 '19 at 17:19
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When you log transform data what is being tested on the original scale of the data?

When you log transform data, you change additive relationships to multiplicative ones. If you take log(x) you change:

If x goes up from 1 to 2, or 2 to 3, or 10 to 11 etc. what happens to y?

to

If x goes from 1 to 2, or 2 to 4, or 10 to 20, or 20 to 40 etc. what happens to y?

And why can we use our log transformed data to answer our scientific question?

It's not at all clear that you can do this. It depends on what your question is. If it is very vague (if x goes up, does y go up?) then perhaps you can. If it is a multiplicative question (if x doubles, what happens to y?) then taking the log makes it possible to answer your question. But sometimes taking logs makes it harder or impossible to answer your question.

This is a reason why I often say that you should take logs (or other transformations) for substantive reasons, not statistical ones.

You took log of time. It's not completely clear to me exactly analysis you did - it looks like a t-test but maybe survival analysis - but if you did a t-test then you changed the scale of "survival time". So, how do you want to look at survival time? On the original scale, 1 year, 2 years, 3 years, 4 years are equidistant from each other. On the log scale they are not. On the log scale, 1 year, 2 years, 4 years, 8 years are equidistant.

Are you after "survived 3 years longer" or "survived 1.3 times as long"?

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