I would like to understand the math behind how survey constructs are measured within a model. For example, if there is a construct C with 3 correlated indicators A, B, and C, what is the equation tying these together?
On one hand, it would make sense that Y would be a reflective construct, in which case each indicator I would have its own equation as follows:
I = Beta(Y) + error
On the other, it would also make sense knowing that indicators are correlated that the construct would be their sum:
Y = A + B + C
Are either of these correct? I am looking to use these within a structural equation model.
Researchers can use lots of different models and hypothesis constructs in lots of different ways. There are lots of alternatives but the two approaches you describe are similar to the two main approaches that social scientists use. Your first is a latent variable measurement models. Here the norm is to describe each observed variable (indicator) as a weighted sum of the latent constructs plus error. The general latent variable model (Bartholomew et al) also includes models like item response theory, which is widely used in education.
Your second approach is like principal component analysis. Here, the components are a weighted sum of the constructs.
There is a lot of debate about which of these is preferred. Historically, statisticians have been weary of latent variable approaches. It is a little spooky that you are letting these hypothesized unmeasured constructs do so much. But this is probably why a lot of social scientists like them, we know the things we are theorizing about are often a bit amorphous. There is probably some debate on this on stackexchange (my lecture on this shows a slide with two monsters fighting). The two are very different procedures, though each takes a covariance matrix and creates variables that in some way account (PCA) or explain (EFA) the associations, so sometimes are used in similar ways.
btw, PCA and methods based on it are used more with 'big data'.
I am having a bit of difficulty understanding whether the three "indicators" are all measure of the same concept, or whether you mean that there three different inter-correlated independent variables.
If A, B, & C are all measures of the same concept, then you would expect them to relatively highly correlated, which would lead you to construct a single index or scale.
Alternatively, if the three are conceptually distinct, then you should use them as separate predictors where
Y = a + b1A + b2B + b3C
In this second case, each of the b's is a partial coefficient that controls for the effects of the other independent variables.
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