Is it possible to do statistic analysis from only knowing my groups's means and sample sizes?

Question

I am writing about a problem my colleagues (in human health field) consider impossible to solve, but I want to believe mathematics have an answer. I am being told it is impossible to do statistical analyses of the experiment described below:

We had 32 individuals that we considered initially identical (mice from same batch) and divided them into 8 groups of 4 individuals. Each group received a different treatment.

Then, at two different timepoints, for each group, we sampled the blood of its 4 individuals, pooled the blood and then ran analyses on the pooled blood. The data we obtained was the number of different cell types (for instance: lymphocytes) in our pooled blood sample.

I would like to know if it possible to determine if our pooled blood samples’ lymphocyte numbers are statistically different.

The next level would be to know if for the same group, the pooled blood sample at the two timepoints differ statistically.

On the Internet I looked up tests like the “Aspin-Welch t-test” that seem promising, but still, it looks like we need to be able to provide SD values, which we lack?

If anyone could help, I would be very grateful.

Have a good day!

Answer 1

You can say that your 8 samples will be different at each time point. But you can't say whether this is systematic, or just due to random variation, so in particular you can't say anything about statistical significance.

You could in principle look at how the samples change between the time points, and run statistical analyses on whether one of your eight groups changes "by exceptionally much". But this will also not be easy, because after all, one group must have the largest change. (You should include this effect in any analysis.)

This is another argument for why it makes sense to consult the statistician before running the study. R. A. Fisher:

To consult the statistician after an experiment is finished is often merely to ask him to conduct a post mortem examination. He can perhaps say what the experiment died of.

In the present case, I don't understand why you would pool the blood from the four individuals, rather than analyze each mouse separately, which would give you much more data to work with and actually allow to address the kinds of questions you pose (even if your sample size is still far too small - better to run a decent power analysis before planning the study). Did you do this to save money, because now you only had to run 1/4 the blood tests? If so, this was an instance of saving money in the wrong place.

Seriously, next time, think about how you will analyze your data before you collect it, spending money and injuring animals.