What is the sample size calculation for a discrete outcome (count variable)?
For example, I want to design a trial with a placebo and a treatment. Alpha is 0.05 and power 0.8. The outcome is the number of days of treatment : 14 days for placebo and 10 days (expected) for treatment (i.e., reduction of 4 days of treatment). Which formula should I use to calculate the sample size for this trial?
I usually use the book "Sample Sizes for Clinical, Laboratory and Epidemiology Studies (Machin et al., 2018)" for sample sizes calculations but I do not find any example for this type of data. There are formulas for continuous data, but in this case I don't know if I can approximate days as a continuous variable (to me it is rather a discrete variable).
For a trial comparing the average number of treatment days between placebo and treatment groups, you can use the two-sample t-test formula for comparing means. You'll need estimates for the standard deviations of the outcome variable in both groups.
As a side note if the outcome follows a Poisson or negative binomial distribution, I heard that using methods for comparing proportions like chi-square, Fisher's exact, or score tests. I'm not a statistician but I hope it helps!
Hi,
As the outcome is the number of days when the counts are measured and compared, then, refer to the chapter on survival time outcomes. Use Proportional hazards, K-M curves with Log-Rank test, etc. Refer to section 7.2 in Machon's book.
Bhogaraju is right: your variable is not a count variable but a time-to-event variable. Such variables should be analyzed with Weibull, log-normal, or Gamma, logistic of log-logistic distributions, or to model the hazard ratios under the assumption of hazards being proportional.
These sources may be helpful to find out how the sample sized can be calulated:
Article Sample Size Estimation When Time-to-Event is the Primary Endpoint
Article Sample size calculation for two‐arm trials with time‐to‐even...
https://search.r-project.org/CRAN/refmans/gsDesign/html/nSurvival.html
Bhogaraju Anand and Jochen Wilhelm, I do not see an event mentioned in the original question. How did you infer that the outcome is time to some event, and what is the event? Thanks for clarifying. ;-)
Like Bruce Weaver I am a little confused (Bruce may be a little confused, I am a lot confused). The bit I don't understand is "The outcome is the number of days of treatment : 14 days for placebo and 10 days for treatment (reduction of 4 days of treatment)." What does this mean? Do you mean that you stop the study at these two cutoffs so censure at different places, or do you mean you'd like to design a study where those will be some statistic related to when something happens, or is it the number of days within those windows where some event happens ( like I ate applies on 10 of those 14 days). More information is needed, at least for me, to give a useful response.
Bruce Weaver and Daniel Wright , Pauline may explain this more concretely. As she wrote that "number of days" is the outcome, I suspect that the animals are kept either under treatment conditions or under control conditions until something happens (e.g. they develop some disease, show some behaviour, die, ...), that is, "the event". This assumption might be wrong, but I don't have any other idea in which circumstances "number of days" would be entitled as the outcome of an experiment. Did this clarify some of your confusion or did I make it worse?
Jochen Wilhelm , my confusion is that, I think, at least one interpretation is that it is "the outcome is the number of days" that something happens in the 14 and 10 days windows (so counted fractions with different denominators for the two groups, like on how many of those days did the subject do something). I am confused though, as the full phrase is "the outcome is the number of days of treatment" but then I think it is worth knowing what treatment is given to the placebo group. So for me I would want more clarification, but if the meaning is clear to you, that's great and I hope that your answer is useful to the questioner.
Daniel Wright , imagine you have mice with a genetic defect so that their offspring quickly develops cancer and days. Without any treatment - under control or placebo condition - these mice die on average after 10 days. IImagine further that the genetic defet is thought to act via exaggerated oxidative stress resulting from energy metabolism. The treatment is some diet impacting the energy metabolism and so reducing the oxidative stress. Under these treatment conditions the mice survive on average 14 days.
Jochen Wilhelm , that sounds a perfectly plausible study, I am not saying it is not. I am also not claiming it is not what the questioner may mean, though things like the "days of treatment" being the outcome variable for the placebo group makes me wonder. The only difference between us is that I am not claiming this is the only interpretation so think some input from the questioner would have been useful for me, but I guess you do not need the clarification, so I'll stop.
Daniel Wright , of course I believe that this is not the only possible interpretation. It's just the only kind of plausible interpretation that in comes into my mind. Of course, it would be best if Pauline would clarify this.
I should think that Pauline could clear this up. Is she still following her question.
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