Hi Rayhanul,

I gather from the log10-functions surrounding the sample statistics for the E.coli count that you intend to compare means on log10-transformed counts. Is this correct?

Your outcome distribution is actually a count variable so you could also consider a distribution intended for counts, e.g. the negative-binomial. However, log-transformation may be a reasonable and practical approximation, if your data consists of large counts. But here lies the problem: why do you log10-transform the means and SD? Transforming the count variable does not lead to simple expressions for the mean and variance on the new variables on the log10 scale (such as taking logs of the original variable sample statistics). As an illustration consider generating the following data in R yielding 10 000 cases for a variable with a skewed distribution that - after the log10-transformation - follows the normal distribution with population mean of 6.21 and standard deviation 1.

set.seed(12)

y = 10^(rnorm(10000, 6.21, 1))

The sample mean and SD for the log10-transformed variable are approximately as expected:

mean(log10(y)) = 6.2087 and

sd(log10(y)) = 1.0017,

but note that the latter is not the same as log10(sd(y)) = 8.0494.

I usually use base e for logarithms, but quick calculation with Mathematica reveals that the mean of the log10-normal distribution is equal to 10^m * exp( 1/2 * v * log(10)^2 ) and the variance to 100^m * exp( v * log(10)^2 ) * ( exp(v * log(10)^2) - 1 ), where m and v are the mean and variance parameters on the log10-scale. The implication for sample size computation is that you should also transform the test statistics appropriately. These two equations could be used to get expressions for m and v that you need for the sample size calculation, but - although entertaining - it is not necessary to follow this path.

As useful practical alternative for these equations you could use pilot data for computing the log10-transformed statistics or simulate some data based on the unlogged summary statistics from your post. Then log10-transform the E.coli-count variable in the pilot/simulated data and then compute the means (mean1 and mean2) and variance (var) to be used in power analysis. You can use these values in the equation or you can use the R-package "pwr" that has a simple function for conducting power analysis for the t-test. So in the commands below replace "mean1", "mean2" and "var" with the values that you obtain from the data and submit the following three commands to R:

require(pwr)

dif = (mean1 - mean2) / sqrt(var)

pwr.t.test(n=NULL, d = dif, sig.level = 0.05, power = 0.8, type="two.sample", alternative="two.sided")

Depending on your setting the t-test approach might be more useful than the normal approximation.

Finally, did you round up the 1-log reduction in -log10(1.61/6.21) = 0.586?

BTW: the article's figure is somewhat misleading in that it implies that population variance is alternatively called SD (standard deviation), but variance is well-known to be the square of SD.

Regards,

Timo Törmäkangas

Gerontology Research Center

University of Jyväskylä

Finland

The equation seems improperly printed which may be the problem. I would look at the references in a paper such as the one attached. This by the way seems awfully simple for a modern RCT. Good luck, D. Booth

Hi, the approach is correct in principle. However you may need to consider whether your study is going to be a superiority (clinical vs. statistics), a non-inferiority, or an equivalence trial. You may want to consider the attrition rate as well. I recommend you the paper by Zhong B., 2009 (How to Calculate Sample Size in Randomized Controlled Trial?). A user-friendly, yet a basic, software for this purpose is IBM Sample Size calculator. It includes prospective studies with dichotomous outcomes, controlled. I think this software is now discontinued and it is part of IBM SPSS Statistics (or its plug-ins) nowadays. Besides that, NCSS is the one I recommend and I used myself. I am confident additional software can handle this computations, including G-Power and R. Good luck with your project. P.S. please invest your time in proper sample computation because this is an essential step towards valid results and clinically-informative effect sizes. Michele.