![]() ![]() ![]() ![]() A simple way to think about parameters is to see them as the equations that you ask the program to solve in order to estimate things for you. See? Putting these 15 observations to use involves specifying parameters. Counting them up by hand yields the same result. So there are a total of 15 observations available in this system for any kind of SEM to put to use. If you prefer the alternative formula you’d get: So here, with a 5 x 5 matrix, we’ve got unique observations. In either case, the fool-proof way to figure out the number of observations is to apply the good old lower-triangular formula to find the unique elements in a k x k matrix (it has two forms. When you have only a couple of variables, you can pretty easily count the number of observations by hand, but in larger systems (e.g., 17 variables with a few latent variables thrown in just to piss us off to high hell and back), you’ll lose track quickly. All the covariance elements shown above are the unique elements in that matrix. The number of observations is simply the number of unique elements in the matrix. More importantly, it’s precisely this fact that leads us to the next key aspect of figuring out observations in the SEM context. Why? Simple - the upper triangular portion is an exact copy of the lower triangular portion. This is pretty standard for k x k matrices. Honestly, it makes no difference for illustrative purposes): You’ll notice that half of the matrix is empty. Here’s what they’d look like in a k x k (5×5) covariance matrix (Yes, I realize I swapped the traditional row x column labeling convention around. In my case, the matrix contains 5 variables, so k = 5. If you have k variables, then you can put those variables into a k x k matrix. The easiest way to think about observations in the SEM context is to look at all of the variables you have in your system, and then stick them into a matrix (often a covariance or correlation matrix). This is where two parts of the system come into play – observations and parameters. When you hand the program a specification in this fashion, you’re telling the program to go ahead and solve the equations that are indicated by your model. Now go check the data and see if I’m right (or more accurately, go see how right I am).” I think this layout best describes the ways in which these variables are related to each other. When you specify and analyze a structural model, you’re basically saying “ Hey, program, here’s a layout of a group of variables. To do this, SEM methods primarily involve analyzing variances and covariances among your variables. What SEM consists of is describing relationships among variables (frequently with the goal of figuring out what sorts of things explain variance in one or more endogenous (outcome) variables). Where do a model’s degrees of freedom come from, exactly?The easiest way to understand this is to examine the basic idea behind SEM. So now that we have the data, let’s learn a bit about model df. Here’s what my data look like (yes this simple input matrix below is the entire data set. I named them X1, X2, X3, Y1, and Y2, respectively. I’m assuming that I’m going to be working with a structural model with 3 exogenous (predictor) variables and 2 endogenous (outcome) variables. I set up a mock 5-variable data file containing a correlation matrix. I decided to build my example for this article for use in Mplus. However, I hope that this article at least provides you a better idea of the rudiments lying under the hood of SEM. Honestly, degrees of freedom can be kind of a moving target sometimes – especially in more complex cases. It’s not intended to be a complete solution to all your df woes. If you are facing a similar frustration now, this article might help. Lacking an understanding of how degrees of freedom function in the context of SEM drove me mad during my introductory days. This was certainly a sticking point for me during my introduction to SEM, and can continue to be from time to time (in fact, part of why I’m writing this article is to solidify my own understanding - learning by teaching, as they say). Sometimes it isn’t entirely clear where those degrees of freedom come from or why they have the values that they do. For many students, df is one of the more puzzling aspects of SEM. As you sally forth into the land of structural equation modeling (SEM), you’ll come across terms like identification, and ideas like degrees of freedom ( df) for a chi-square goodness of fit test. ![]()
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