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A barn has 4 sections of animals. Within each section are 4 goats. Each goat is given one of four types of food. One of each goat's kidneys is randomly selected to be inspected after 3 days for level of carbohydrates. To measure the level, the goats' kidneys are divided into three pieces, and three measurement techniques are used.

I understand two treatments are applied: $\text{Food}$ (4 fixed levels) and $\text{Measurement Technique}$ (3 fixed levels). I am wondering a couple of things:

  • Are the kidneys "factors"?
    • If so, are they nested or fixed?

My original thinking is that they are fixed...We have a left and right kidney for each animal. Some of my classmates are arguing that the kidneys are nested because each kidney differs for every goat. I don't understand this logic, because EVERY experimental unit differs (even if they are homogeneous, they are different still). So, I'm thinking:

  1. $\text{Kidney}$ is a crossed factor with 2 random levels.
  2. $\frac 1 3$ of $\text{Kidney}$ is a split-plot experimental unit, that is, a factor that is crossed with $\text{Kidney}$ and has 3 fixed levels.

My classmates:

  1. $\text{Kidney}$ is nested (2 levels) with the goat.
  2. $\frac 1 3$ of $\text{Kidney}$ is nested, with the $\text{Kidney}$ (3 levels).

What are your thoughts?

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There are 2 concepts here: Crossed vs. Nested and Fixed vs. Random.

The fixed vs. random applies to single variables, but crossed vs. nested applies to relationships between variables.

When thinking about fixed vs. random, think about future predictions, who or what do we want to make predictions about. The same variable can be considered either fixed or random depending on how the data was collected and what predictions are desired. For example consider that measurements are made on households within 10 cities, is city fixed or random? If those 10 cities were specifically chosen because we are interested in those cities and want to make future predictions about those cities, then cities is fixed. However if the 10 cities were chosen to represent a larger population of possible cities and we want to make future predictions for cities that are not included in our sample, then cities is random.

So are kidneys fixed or random? Well, do you specifically select left and right? or just take one of the 2 at random? Do you care about making specific future predictions based on right or left? or just kidneys in general? (I would lean towards random from your description, but if right vs. left is of interest then they would be fixed).

For crossed vs. nested you need to look at relationships. Nested means that subjects/experimental units at 1 level only occur inside a given unit of the variable that it is nested in. Houses are nested in cities, cities are nested in states (with a couple of exceptions where a city spans the border, but then the 2 parts of the city are often treated as separate cities, each nested in their state). Crossed means that you can potentially take observations within every possible combination of the 2 variables (I cannot take measurements in Denver, Kansas so city is not crossed with state). Sex and treatment are crossed if there are patients of both sexes in each treatment group. For your example kidney is nested in goat because goats don't share kidneys, kidney 1 in goat 1 is different from kidney 1 in goat 2. But kidney is crossed with measurement technique because every kidney is measured by each technique, so every combination of kidney and technique has a measurement.

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