# Test for multidimensional uniform density and representing lack thereof?

$Scientist_1$ seeks to maximize pumpkin size. They suspect that there are 6 key variables, that there are likely interactions between variables, and that there are kinks in the causal model. One research design is to choose a "high", "medium," and "low" value for each (naturally non-negative) variable [water, sunlight, worms, etc.], and analyze pumpkin size from each of the 729 = $3^6$ combinations. Unfortunately, $Scientist_1$'s research grant only provides for 200 experiments.

A second research design is to assign each of the 6 variables a random value (from a uniform distribution between 0 and "extremely high," inclusive) for each of the 200 experiments.

$Scientist_2$ hypothesizes that "high" worms, "low" sunlight, and "medium" water maximizes pumpkin size and asks if such a combination was tested.

Uniform Distribution of Experiments Questions

1. Can $Scientist_1$ use a statistic to test whether the distribution of the 200 6-dimensional experiments was uniformly distributed throughout the problem space (made up of combinations of the 6 key variables)?

2. Can $Scientist_1$ quickly describe the "density" of experiments (finding over- and underrepresented "regions") in 6-dimensional terms and mathematically / statistically answer $Scientist_2$'s question?

Research Design Questions

1. Is either research design preferable (for linear regression techniques)? Is there an even better design using 200 plots of land?

2. If there was no experiment with "high" worms, "low" sunlight, and "medium" water, can $Scientist_1$ still use their data to test $Scientist_2$'s hypothesis?

• The first situation is routine: chi-squared tests address this situation, at least provided you roll the dice a few more times (180 rolls will suffice). In the second situation under a uniform distribution there are $20^{20}\approx 10^{26}$ equally possible outcomes. Your options are limited: how many rolls do you think it would take before even a single outcome were to show up twice? – whuber Nov 30 '17 at 21:27
• With many dice you either need a lot of rolls or you need to be much more specific about the kinds of non-uniformity you're looking for -- being sufficiently specific about the likely alternatives (or alternatives of primary interest) can cut the necessary calculation dramatically. [NB: While 180 rolls of 2d6 would be sufficient to apply the usual chi-squared test, that doesn't mean it's enough to have very good power against all the alternatives it is able to pick up. Again, being more specific about alternatives of interest will improve things.] – Glen_b Nov 30 '17 at 23:37
• @whuber and glen_b, thanks for comments. to former, if there were 200 perfume molecules in a room with $10^{26}$ molecules, that is an extremely low density but can't we use x, y, z spatial dimensions and test if those 200 molecules are NOT uniformly distributed? can't we also represent the density in various ways? to latter, unfortunately, i'm not familiar enough with non-uniformity varieties to specify one, but my intuitions are spatial. thanks. – jtd Dec 1 '17 at 16:10
• That's a helpful comment. Please incorporate its ideas into your question, because as it's currently stated, it's not spatial at all. Faces on a die are just faces--the fact that the objects used to identify them are numbers (with their attendant order, distances, and interpretations as magnitudes) is incidental, leading to purely non-spatial methods to assess uniformity of distribution. – whuber Dec 1 '17 at 19:15