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. Basically, you take a sequence over the real space $^5$, and then map each dimension to your variables (so if you get something in the lower half of your $$ dimension for your first 2-level variable, then you choose level 1, etc.). If you expect that your effect size is going to be small relative to noise, then choose a larger sample size.Īnother method that might be sensible is to use a Low-discrepancy sequence, like the Sobol sequence. 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).
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. You can't technically do standard LHC sampling, or orthogonal sampling, because it requires each dimension to have the same number of levels. Depending on your experiment (and the difficulty of taking samples), you should ideally just sample everything.
The total number of sample combinations you have is $2\times 3 \times 2 \times 3 \times 3 = 108$ (or what ever).