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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?

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    $\begingroup$ If you don't want to make parametric assumptions you can always bootstrap, no? $\endgroup$
    – amoeba
    Jan 18, 2017 at 18:19
  • $\begingroup$ Re question 1: Are the zeros true zeros, or are they sampling error about small quantities? In the latter case (just for illustrative purposes, let's imagine that the data are observed via a Poisson process), then you can use the data for each observation to estimate the lambda for the Poission distribution that generated it, and then take the ratio of the lambdas and propagate error accordingly (in this example, this is equivalent to doing inference on the odds ratio of the binomial probability p, where pre-treatment ~ binomial(p, pre+post). ) $\endgroup$ Jan 19, 2017 at 6:42
  • $\begingroup$ Without further information on the data, I don't think it's possible to answer. What are your data? The pre/post-treatment measurements seem to be always integers: are they counts, or do you just truncate to the nearest integer some continous measure? For example, they could be time durations truncated to the nearest minute. If one doesn't know what the measurements are, it's not possible to hypothesize which distribution they come from. If both populations are Poisson-distributed, for example, then by all means your fold change is not normally distributed. [1/2] $\endgroup$
    – DeltaIV
    Jan 19, 2017 at 8:03
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    $\begingroup$ Why are you interested in the fold change and not simply the change? $\endgroup$
    – IWS
    Jan 19, 2017 at 8:19
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    $\begingroup$ You might want to present a summary of results as typical fold changes, but it might be foolish to try to perform statistical analysis on fold changes when some cases have 0 as the Pre values. Dividing by zero, or by values close to 0, is something to be avoided. Have you considered including the Pre values as covariates in models having Post values as outcomes? $\endgroup$
    – EdM
    Jan 19, 2017 at 18:49

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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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