I am trying to understand randomization in experiment design, and am very confused, because there appear to be several types of randomization possible.
For example, for a Categorical Factor with Non-Repeated-Measures Units, I identify three possible types of randomization, based on whether or not the Units are pre-existing objects or are constructed on-demand for the experiment, and also whether or not the Factor Level is an intrinsic quality of the Unit or if it can be assigned to the Unit at-will:
(1) Pre-Existing / Intrinsic :: e.g. Male is intrinsic level of factor Sex, in Pre-Existing units of people
(2) Pre-Existing / Extrinsic :: e.g. Blue is extrinsic level of factor House Color, in Pre-Existing units of houses
(3) Constructed / Intrinsic :: e.g. Cotton is intrinsic level of factor Material, in Constructed units of T-shirts
In Experiment Units of Type (1), randomization occurs when the units are sampled from their larger population (picking one male from the pop. of all males).
In Experiment Units of Type (2), randomization occurs when the treatment factor is chosen for application to the already-sampled units (different paints could be applied to one house).
In Experiment Units of Type (3), randomization occurs in the time-order that the units are constructed (cotton T-shirts can be woven before or after T-shirts of other materials).
In a second example, for a Categorical Factor with Fully-Repeated-Measures Units, there are two more types of randomization, depending on whether the factor can be applied at any time, or if the levels must follow the flow of time:
(4) Unit<->Factor Assignment Independent of Time :: e.g. The order that a research subject tries three different weight loss diets, since orderings 1-2-3, 2-3-1, 1-3-2, etc. are all acceptable
(5) Unit<->Factor Assignment Dependent on Time :: e.g. Blood sugar measured at times 0, 1, and 2 hours after a meal, because the only acceptable order is 0-1-2
In Experiment Units of Type (4), there is no randomization in the Unit<->Factor assignment, since the Unit must be measured in every factor level, so it is the time-order of testing which is randomized.
In Experiment Units of Type (5), again there is no randomization in the Unit<->Factor because it is repeated measures, but now there is no longer time-order randomization either, because the time order for testing the factor levels happens in only one way, i.e. with the flow of time.
In the final example, with a Numerical Factor (i.e. covariate), there seems to be no randomization of any sort, since a covariate is simply a single numerical value associated with every Experiment Unit, meaning there is no Unit<->Covariate assignment and no time-ordering of the covariate either:
(6) No assignment or time-ordering of the Unit<->Covariate relationship
So now I am wondering if there are any other types of randomization that I am missing, and also of course how are the different types of randomization dealt with either in the experiment design or in the statistical analysis?
Apologies for the long post, any advice on this topic is appreciated.
