Hej,
Given the following study setup: having mice and CRTL (C) or TREATMENT (T) administered to those. All groups here are independent, and my readout (called GU) will be log-normally distributed. I want to log-transform my readout (GU) and carry out a statistical analysis, e.g. independent t-test. Two major points I'm concerned about:
1. Hypothesis testing: What I was initially interested in is the mean difference of GU comparing C and T, using a t-test I could test these mean differences. Once I carry out the t-test on the log-transformed data, is my t-test then a measurement of geometric mean differences between C and T. Would this be still an interesting readout to my question?
Or would it be more reasonable to display log (original values T/original values C)? Is my p-value a meaningful readout (assuming it is meaningful in general)?
2. Data plotting:
I´m uncertain about how I should plot my results because I would say that GU on my non-transformed data gives "interpretable" measurements. In case I want to display my non-transformed data, would geometric mean and 95%CI of GM be reasonable summary statistics, plotting individual data points? Or is a simple scatter plot sufficient?
Or is it more reasonable to plot the transformed data? What would be reasonable summary statistics then (mean, SD?)?
According to this paper:
Article Methods for Comparing the Means of Two Independent Log-Normal Samples
If variance is the same, then the null hypothesis Ho is equivalent to Ho*. However, if variance is unequal, then the null hypothesis Ho is not equivalent to H0*.
As I often observe similar variation in Crtl and Treatment for log-transformed values would my hypothesis test still be reasonable to answer my initial study question.
ad 1)
if, after log-transformation, the variance is similar in both groups (more concrete: can be assumed being identical in both groups), then a test of a mean difference of the transformed variables is equivalent to a test of a mean difference of the measured variables. The mean µ of a log-normal distributed variable is µ = exp(m+s²/2) = exp(m)*exp(s²/2), where m and s² are the mean and the variance on the log-scale. If you now assume that you have two variables with µ1 and µ2 and that s² is the same for both, then the factor exp(s²/2) = v is the same for both, too, and you have
µ1 = exp(m1)*v
µ2 = exp(m2)*v
The hypothesis µ1 = µ2 is thus exp(m1)*v = exp(m2)*v which we can divide on both sides by v and exponentiate both sides to get m1 = m2.
So testing µ1 = µ2 (measured scale) is equvalent to testing m1 = m2 (log-transformed scale). This is provided under the assumption that s² (on the log-transformed scale) is the same in both groups. If s² is positively correlated with m, then the tests are still equivalent, as any µ1 < µ2 still naps to m1 < m2, and any µ1 > µ2 still maps to m1 = m2. Only when s² and m are negatively correlated, then testing m1 = m2 is not equivalent to testing µ1 = µ2. But this is a case I have never seen in practice and is is difficult to find a reason that would explain this (if some expert reads this and knows a possible practical reason or scenario under which this might happen: please raise your hand and let us know!).
Just to round up: under the assumption that s² is constant, a t-test on m1 = m2 {mean(log(Y1)) = mean(log(Y2))} is not only equivalent to testing µ1 = µ2 {mean(Y1)=mean(Y2)} but also to testing geomean(Y1) = geomean(Y2) or mode(Y1)=model(Y2), and geomean(log(Y1))=geomean(log(Y2)) and mode(log(Y1)) = model(log(Y2)).
ad 2)
In any case I strongly suggest plotting the individual values, that is, the data. I would not plot summary statistics of the data. These summary statistics are not relevant to the reader (to your story), since your aim is to analyse the difference in means (or a (log-)ratio). It's often convenient to show expression values or concentrations on a log-scaled axis, especially when the dynamic range of the values is large.
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