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I am designing an experiment with 5 variables. Some of the variables have 2 and others have 3 levels. The method that I am going to use is the Latin Hypercube, but I do not not what the sample size should be for a valid experiment?

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The total number of sample combinations you have is $2\times 3 \times 2 \times 3 \times 3 = 108$ (or what ever). Depending on your experiment (and the difficulty of taking samples), you should ideally just sample everything. If not, there a a few other options.

You can't technically do standard LHC sampling, or orthogonal sampling, because it requires each dimension to have the same number of levels. However, you can do LHC if you use $6n$ (lowest common multiple of 3 and 2) levels, and then map that to your 2- and 3-level spaces.

The number of samples you choose is up to you, but more samples will give you more reliable results, and will also help avoid correlation between variables (you should check this when you decide what your samples are, before you actually take them). If you expect that your effect size is going to be small relative to noise, then choose a larger sample size.

Another method that might be sensible is to use a Low-discrepancy sequence, like the Sobol sequence. Basically, you take a sequence over the real space $[0,1]^5$, and then map each dimension to your variables (so if you get something in the lower half of your $[0,1]$ dimension for your first 2-level variable, then you choose level 1, etc.). This has the advantage over LHC that you can decide to add more samples later, while retaining relatively even sample coverage, and low correlations between variables. Also, you're not restricted to sample sizes of $6n$. I successfully used this method with sample sizes as low as 25.

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