# How to present Confidence Interval for Log-Transformed Means & Mean Difference?

After trying to read on this topic, I still have some clarifications remaining.

Context:

Comparing between 2 arms (categorical), measuring microbiological plate-counted bacteria concentration (colony forming units/cm2). Results are positive skewed, log transformation gives me a normal distribution which I am interested in as it loses less data points than a non-parametric test (EDIT #1: I read often that non-parametric tests is inefficient, and 'wastes data' compared to parametric tests).

Problem:

1. I am expressing the results as back-transformed mean ($$\exp(\log(\text{mean}))$$). Should I present the standard deviation as $$\exp(\log(\text{sd}))$$?

2. If I am interested in mean difference (since I'm using a student t test), is it a $$\exp(\log(\text{mean}_A)) - \exp(\log(\text{mean}_B))$$, or $$\exp(\log(\text{mean}_A/\text{mean}_B))$$?

3. How best to express confidence interval for this t-test comparison of log-transformed data? I read differing opinions - some suggest to back-transform the CI generated by the t-test(log data), accepting that the CI will not contain the back-transformed mean. Others say that back-transformed CI has no meaning and hence should not be presented.

Example:

NON-TRANSFORMED, NATIVE VALUES

mean(GrpA) = 2.11 (SD 3.13) cfu/cm2, median(GrpA) = 1.01 (0.54 to 2.44) cfu/cm2

vs

mean(GrpB) = 1.80 (SD 3.24) cfu/cm2, median(GrpB) = 0.61 (0.29 to 1.48) cfu/cm2

Comparing GrpA vs GrpB will require non-parametric test (Mann-Whitney) since they are not distributed normally.

SO, USE TRANSFORMED LOG VALUES

exp(mean(log(GrpA))) = 1.107

exp(sd(log(GrpA))) = 3.094

vs

exp(mean(log(GrpB))) = 0.738

exp(sd(log(GrpB))) = 3.585

Comparing log(GrpA) vs log(GrpB) can be done with t-test since log(GrpA) and log(GrpB) are normally distributed

T-test of log(GrpA) vs log(GrpB):

95% CI -0.08288 to 0.89421

exp(-0.08288)=0.92047, exp(0.89421)=2.44534

Can I then express this statement on a scientific journal (stating in methods that data was log transformed to allow for parametric statistical testing): Mean Grp A was 1.107 (sd=3.094). Mean Grp B was 0.738 (sd=3.585). Mean difference was 1.107-0.738=0.369, 95% CI 0.92047 to 2.44534?

EDIT #1

Thank you for your replies! (Will read more on the suggested generalised linear model with logarithmic link) I included above a specific example, given my untrained eyes and mind thought maybe this will express my dilemma and help my understanding of your pointers.

• "as it loses less data points than a non-parametric test": what does this mean? Feb 25, 2020 at 18:11
• I would just use a generalised linear model with logarithmic link here. Switch to 1970s technology from older technique. Feb 25, 2020 at 18:12
• Do you have access to the raw individual values, or only to the arithmetic (regular) mean and SD? Feb 27, 2020 at 15:50
– ltw
Feb 28, 2020 at 17:35