Solved – Mean fold change and confidence intervals

confidence interval

I have paired data that come in the form of a pre-treatment ($Pre$) measurement and a post-treatment ($Post$) measurement. So for sake of example you can imagine my data set is the following $n=5$ pairs of measurements:

Subject   Pre-treatment   Post-treatment
  1            10              20
  2            28              43
  3            20              80
  4             0              15
  5            18              65

So the way a fold-change is defined is
$$\text{Fold Change}=\frac{Post-Pre}{Pre}$$

So my first question is how do I calculate a fold change when I have a 0 as a pre-treatment measurement? Is there some sort of valid translation or transformation I can apply such that I am not dividing by 0 and the meaning of the fold change is preserved?

Second question, which assumes we have valid fold change measurements (so for this part you don't have to worry about the problem in the first question), how can I calculate a confidence interval for the average fold change? I know, in my example above, I can calculate the 5 fold changes and take an average to get an average point estimate, but is there a closed form for the interval? Would one way be to assume normality of the fold changes and simply calculate it as point estimate plus or minus the standard error?

Best Answer

There are two approaches that you can take to your first question.

  1. Recognize that for any value where the pre-treatment value was $0$ will have an undefined fold change.
  2. Add an arbitrarily small value to both sides. The smaller the value, the greater the precision for the fold change for all other values. This also comes with the risk that the smaller the value, the larger the fold change will be for any value that was originally $0$. This also deserves a greater explanation in your results than if you just labeled the results as undefined.

Confidence intervals for fold changes would likely follow a $\chi^2$ distribution, based upon the similarity of the fold change and the $\chi^2$ test for equal proportions.

Another approach you may want to consider for this data would be a $\chi^2$ test for equal proportions, with its associated critical values and $P$ value.

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