# If I count cells should I consider them as Poisson distributed?

I calculate (with flow cytometry) pecentage of lymphocytes with a specific receptor (Lph*) as a ratio to general number of lymphocytes (Lph). Should I consider them (Lph*) as Poisson distributed?
(My data set is here.)

• What is your goal on generating a probability model for your percentage? Is it possible to generate the necessary descriptives and inference without imposing assumptions about the nature of the data generating process? Feb 12, 2013 at 17:01
• @AdamO: I don't model the process. My goal is to understand whether substances activate the lymphocyte population of interest or not.
– abc
Feb 19, 2013 at 6:49
• Then you should eschew making any assumptions about the probability model for that data unless there is a clear need to do so. Feb 21, 2013 at 21:19
• @AdamO: I would really like to do so. I was advised to use SAS PROC GLIMMIX procedure but DIST option in its MODEL statement requires a type of distribution.
– abc
Jun 9, 2013 at 7:19

Poisson distribution makes sense from the general point of view of flow cytometry measurements: you "sit on your detector", and wait a random time for a/the next lymphocyte to come along. The same is true for lymphocytes expressing the receptor you're interested in. But you then ask for the proportion of two such Poisson distributions.

If you focus on the proportion, you can assume that a true underlying proportion $p$ (possibly depending on the treatment) of lymphocytes exists that does express the receptor. That would be more like a Bernoulli experiment (binomial distribution). You sit on your detector and look at whether the next lymphocyte coming along does express the receptor (it doesn't matter how long you have to wait for it), which happens with probability $p$.
Note that the binomial distribution is related to the beta distribution - you get beta distributions when estimating the true proportion $p$ of a binomial distribution from Bernoulli experiments.

If you look at large enough numbers of cells, you can use approximations (e.g. normal approximation of the binomial if the smaller of $np$ and $n(1-p)$ exceeds 5 or better 10.
Assuming you have 10⁵ lymphocytes per FACS run, that means that for 0.0001 $\leq p \leq$ 0.9999 you should be OK with the normal approximation. As the lowest proportion you report is 0.0015, you are on the safe side even if you add a bit more "safety margin" for the fact that you only have an observed $\hat p$, not the true proportion $p$ (unless your FACS run takes only a very small aliquote of the sample).

See Wikipedia on distributions related to the Poisson distribution and Wikipedia on distributions related to the binomial distribution for relationships and also rules of thumb about the approximations.

The short answer is probably not, since:

• the Poisson distribution is discrete, your data is continuous;
• the Poisson distributions has support on 0,1,2, ..., whereas (I think) your data has a range from 0 to 100.

Without seeing your data and knowing your problem, it's tricky to give you a suggestion. A good starting position would be to look at the statistical analysis section of publications that analyse data similar to your data.

• @stan: Once you calculate the ratio: Lph*/Lph, your data is no longer discrete - or am I missing something? Jun 16, 2011 at 13:59
• @cs I think the point is that there are several analytical options: one is to view the ratio as a continuous variable; another is to use a Poisson model for Lph* with Lph as an offset.
– whuber
Jun 16, 2011 at 17:50
• With the percentages if you think of them as being between 0 to 1 instead of how you've written them then you could use the beta distribution. Otherwise you might be able to approximate it using a gamma distribution which is used in biology and even flow cytometry just realize that gamma is unbounded above. Jun 17, 2011 at 5:06
• @ChrisSimokat: Please note that the paper you linked for gamma distribution in flow cytometry is about a kinetic model (which is fundamentally different from the proportion counts the OP is talking about. Feb 12, 2013 at 17:46
• @csgillespie: "Once you calculate the ratio: Lph*/Lph, your data is no longer discrete" in theory it is as it can only take the values {0, 1, 2, ..., Lph}/Lph. In practice, the variance due to meeting cells that have or have not expressed the receptor is not the only source of variance: cells can stick together, resulting in too low counts. "Something" may flow along that confuses the detector and is counted although it was no cell. Receptor expression is probably a dichotomized continuous fluorescence intensity, implying that false positives and false negatives can occcur. Feb 12, 2013 at 17:56