# sample size for continuous skewed data

I have an outcome which takes values from 0 - 100. I have added a figure of the distribution (below). However, 30% or more will have a value of 0 (so zero inflated). Then the next lowest value it can take is 9 (numbers in between not possible), with all other values being possible. We do expect that the mode will be around 8 – 15 and 5% will have a value of 100.

I am trying to calculate the sample size required (varying levels of alpha, power, MCD and other parameters that one requires for sample size) for a study having this index as primary outcome. The Sample size will also have to account for the fact that data is likely to be cluster randomised, therefore need to take a note of that.

In my search, I have come across a few things that allow modelling of zero inflated and skewed data:

But I have not been able to find a way to calculate the sample size based on those distributions that would account for clustering within the data.

I found the following which seems to be non-parametric approach:

https://www.jstor.org/stable/41242503?seq=3#metadata_info_tab_contents (Power and Sample Size Estimation for the Clustered Wilcoxon Test)

However, when modelling, the wilcoxon test doesn't allow for adjustment for other variables?

So my questions are:

Is it possible to accurate calculate the sample size for a data set such as above without the use of simulation? If so, can anyone point me in the right direction?

If I have to simulate data, am I right to assume that essentially, I will be fixing my sample size and calculating the power under different sample sizes? • To compute power you need a model and an alternative. If 35% of your values are 0 or 100 it's not continuous but mixed. If it was really continuous apart from the endpoints I might be inclined to divide by 100 and consider a 0-1 inflated beta model - but that huge spike at 20 has me concerned: are there really no ties there? Oct 14, 2021 at 0:23
• Sorry @Glen_b, not sure i understand what you mean by ties in this context? Oct 14, 2021 at 7:09
• Your data already have many 0s, for example, and every zero is tied with every other (in the ordinary English sense of coming in this same place in an ordered list). I am asking if there's some values near 20 that have this property of being equal to other values, and if so how many such, as a proportion of the total count. Oct 14, 2021 at 21:22
• @Glen_b, it is really hard to say, as the data doesn't belong to be, and the histogram I have shared is from published paper. But having checked the way the index is scored, it can take values 9, 12, 15, and 17. Most common being 9 then 12 then 15 then 17. The index looks at participants having an event. Events are graded and higher grades weigh more, then participant must have 4 lowest grade event to get score of 17, while next grade scores a grade of 20. As the score increases the likelyhood of ties reduce. Oct 15, 2021 at 20:34
• Didn't you say these were continuous? What you're describing is discrete. How do 0's arise? What's the smallest value larger than zero?. How does the gap between possible values change from 3 to 2 then go back to 3 again? Oct 15, 2021 at 22:33

Proving non-existence is always hard. But yes, I would assume there is no simple recipe to determine your sample size for as complicated a setup as you have.

So you will indeed need to simulate. Any first of all, you will need to decide what parameter you are interested in. The mean? The amount of zero inflation? Something else? Essentially, you will need to figure out which test statistic you are interested in.

Note that this is not only a question of simulating the power under different sample sizes. You will need to make assumptions as follows:

• Distribution of predictors (if any)
• Relationship between predictors and observables, including the amount of zero inflation, and the non-zero distribution
• An effect size (in terms of the parameter you are looking at, see above) that you want to detect