$Scientist_1$ 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
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)?
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
Is either research design preferable (for linear regression techniques)? Is there an even better design using 200 plots of land?
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?