normalizing proliferation percentages

Question

I have a control sample that has one cell type (AP). the control sample has 100 cells total of AP and the 95th percentile proliferation score is 0.5, meaning 95 of AP cells are below this and 5 percent of cells are above this. now I also have two treatment samples with the same cell type. the first treatment sample has 50 cells of AP and second treatment sample has 5 AP cells. the first treatment sample has 15 cells that are above the 0.5 threshold and the second treatment sample has 2 cells above the 0.5 threshold. so treatment 1 has 30 percent of cells within it that are above the 0.5 95th percentile threshold of the control sample, and treatment 2 has 40 percent of cells within it that are above the 0.5 95th percentile threshold of the control sample. so relative to control, treatment 1 has (30 -5) 25 percent more proliferating while treatment 2 has (40 -5) 35 percent more proliferation. However, this is obviously skewed bc control has more cells compared to treatment 1 and 2. so how to normalize this? The goal is to see what proportion of the treatment cells are above the 0.5 proliferation score for the control. so if 20 percent of treatment cells are above the 0.5 95th percentile, then that means that (20 -5) 15 percent of cells are proliferating relative to control as if the treatment sample was the same as control then 5 percent of cells would have been above the threshold. Also, the proliferation score is calculated based on the expression of proliferation markers in each cell. I was thinking of dividing the percent of cells above the threshold by the average total cells of all 3 sample, but am not sure if this is correct. any help is appreciated?

Answer 1

For a binary outcome, as in your example of whether a cell has a proliferation score greater than 0.5, you should use a binomial regression; logistic regression is a common choice. For that you provide the actual numbers of cells within each category as the outcome values; the model is fit in a way that takes the number of cells into account. That inherently does the "normalization" you seem to want while recognizing that fewer cells means a less precise estimate.

As you seem to have the value of the proliferation score for each cell, however, I'd recommend some related but potentially more useful steps in analysis.

First, look at all your data with the empirical distribution function plotted for each control/treatment group. As a function of proliferation score, that tells you the fraction of cells having a lower score. That's a useful generalization of the single 95% fraction having a value less than 0.5 in your example with control cells. Such a plot can often show you details missing when you arbitrarily choose a cutoff. The stat_ecdf() function in ggplot makes that pretty simple if you are familiar with the ggplot system (which is good to learn if you aren't yet familiar with it).

Second, for statistical comparison among control/treatment groups, ordinal regression is a good choice that doesn't require strong assumptions about the distribution of the data values. It works with the rank ordering of the values and thus can use all the data, not restricting to whether the value is above or below a cutoff. Although ordinal regression is often just used with only a few outcome levels, as on this example page, it can be used with up to a few thousand different ordered outcome values. Frank Harrell provides resources on this page; the orm() function in his rms package provides an efficient implementation.

That's an extension of a set of "non-parametric" tests like the Kruskal-Wallis test, which could also be used with your data to compare the overall distributions of single-cell values among the groups. The drawback with Kruskal-Wallis is that it only tells you whether there are any differences among the groups, and you have to do further tests to determine which particular groups differ.

In response to comments:

First, if 20 percent of treated cells have a score greater than 0.5, versus only 5 percent of control cells, then just say that directly. If you say "15 percent more cells are in a proliferative state relative to control," there's an ambiguity: someone could take you to mean 15% more than the 5% in the control state (for only $1.15 \times 5 = 5.75$ percent of treated in a proliferative state) rather than 20% versus 5%. (That's how I interpreted the statement at first.)

Avoid using percentage changes. They usually only end up confusing things. See this post.

That said, there's no need for normalization when you report the point estimates of cell fractions in a proliferative state as a function of treatment. 2 out of 10 cells or 200 out of 1000 cells both represent 20% of evaluated cells.

Second, the problem is that the precision of your estimate in the first case of the previous sentence is much lower than in the second, and might not even be statistically different from the control fraction of proliferative cells. To evaluate that (and to convince a skeptical reviewer) you need some statistical test that takes into account the variability of both the control and the treated values.

That's where logistic regression comes into play. Standard logistic regression can be used for any outcome with a yes/no value. In your situation, it would be a model of whether the fraction of cells in a proliferative state (based on a score greater than 0.5) differs as a function of treatment/control group. In your case, it models the (log) odds of a cell's being in a proliferative state as a function of treatment/control group, which can be translated to the probability of being in a proliferative state. In this simple case, those predictions should be the same as what you calculate by hand, but logistic regression allows you to test whether the differences that you observed are greater than you might expect by chance and provides confidence intervals around the individual estimates.