You know those “duh” phenomena or experiences that you come across every day? You know, the ones that you would think would have a simple explanation for why they occur but have never actually asked yourself? Those make the best TWSBs.
Take today’s topic, for example: why a three-legged stool is stable, while a four-legged stool (or any higher-legged stool, though I’m pretty sure octuple-legged stools are quite rare) often wobbles.
This article here talks about the fundamentals of why this is. It uses the analogy of a cane: suppose you’re holding a cane unrestrained in the air. You can twirl it in any direction possible, in all three dimensions. Now suppose you set one end of the cane on the ground. You’ve now constrained its motion to two dimensions (you can’t lift or rotate it). Next? Take a pair of canes, connect the tops, and place the other two ends on the ground in a little triangle of cane, cane, ground. The tops are still movable, but only along an arc. The motion is now constrained to one dimension. If you do the same thing with three canes? Can’t move the top at all. Now motion is constrained in all three dimensions, meaning the canes cannot be moved at all. As the article puts it, “each time you add a cane, you remove one dimension in which the top can move freely – that is, each new cane removes one ‘degree of freedom.”
At this point I had to stop and have a little stats freak out, ‘cause this means that stool-leggedness is almost perfectly analogous to model identification in structural equation modeling.
So let’s check that out, shall we?
A structural equation model is made up of variables and parameters (paths between variables, either dependent or independent, and paths between variables and errors). Parameters are, in other words, covariances between the variables. In this example, parameters are analogous to the dimensions in which the stool’s legs can rotate (so # of parameters = 3).
The number of covariances in any given model is called the number of “known values.” In the case of the stool, the number of legs the stool has is analogous to the number of known parameters in a model.
A just-identified (or saturated) structural equation model is one for which the number of parameters is equal to the number of known values. Such a model has zero degrees of freedom (since df = number of known values minus the number of parameters). A three-legged stool is like such a model, since df: 3 parameters – 3 known values = 0). A just-identified model has only one unique solution. The stool, analogously, has only one “solution” in the sense that there are exactly three legs used to stabilize the stool on the plane that is the ground.
Give the stool any more legs, though, and it becomes like an over-identified model, or one for which there are more known values (four in the case of a four-legged stool) than parameters (still only three dimensions in which the stool’s legs can rotate). An over-identified model, unlike a just-identified model, does not have a single unique solution, owing to the non-zero number of degrees of freedom. As the article puts it, “…now you have too many constraints. This means that there are multiple ways that the stool can “solve” the problem of which legs to use for support.”
Statistics analogies FTW!
Haha, sorry, this was a longer blog than originally planned. It got me excited.