It's a natural question to wonder where the intercept "went to" when we compare unstandardized models in path diagrams with the more familiar output from a regression analysis. Increasingly, models with intercepts are being used in structural equation presentations, particularly when the researcher is interested in describing processes of growth, learning or systematic change in data gathered over time. As before, there does exist an equivalent path diagram for models with an intercept. In order to express this, however, we are going to have to introduce a small "wild card" to our diagram to convey information about the intercept.
Before presenting that, however, it's helpful to consider a little bit of matrix algebra. Recall that when a given body of data is transformed to deviation scores about the mean, a matrix of variances and covariances can be calculated as x'x/N (where N is the sample size- some folks also use N-1, but for large samples there's usually no difference here). Deviation score matrices, as mentioned before, are useful in producing unstandardized regression weights. Such regression lines, however, go through the origin. The act of putting the data in deviation score form eliminates information about the intercept from the data. (This is why, for example, Pedhazur's Multiple Regression in Behavioral Research 3rd Ed. [page 19 Formula 2.8] needs to include a formula for calculating the intercept which uses the means and the unstandardized coefficients.)
There are times, however, we are interested in including information about the intercept. Sometimes we might be interested in actuarial prediction where the real values are of concrete interest to us. For example, we might be interested in knowing the exact predicted grade point average associated with someone's high school core gpa and ACT composite. At other times, the intercept itself can be informative. One example I've encountered in studies of growth and change considers as a predictor the variable time. In these biological studies of embryological development, when the researcher wishes to evaluate the reasonableness of the model, it is interesting to inspect the intercept to make sure that it's close to zero at time zero. Intercept values of weight at time zero which are equal to a few ounces, for example, indicate that the researcher hasn't described the growth pattern well.
I also have a few ulterior motives in presenting data with an intercept. Many of the recent discussions of structural equation modeling have focussed on the need to "include the means in the model." I share this enthusiasm. Including information about mean level can help us understand a great deal about patterns of change and growth, and assist in developing models of how individuals differ from each other. This point also allows us to incorporate results/findings which have often been conducted with analysis of variance simultaneously with patterns of covariation. It's a very exciting area. By the same token, however, I've encountered some opinions in the literature which suggest that putting means in the model (no matter how you do so) is somehow just generically better and the results of such analyses must almost mystically be better than any model which doesn't have mean information in it. Sometimes I find these arguments a bit fuzzy.
To that end, I would like to present a RAM diagram which corresponds to a two-variable prediction model which an intercept. We will make use of a new "wild card" element in our RAM notation to accomplish this, but other than that, the same rules will apply as were described for the unstandardized and standardized models.