## Mean Structures

So far we have been concerned with models that focus on relationships among variables and have made the assumption that our variables were centered, meaning we have subtracted the mean from the original variable to simplify our equations and eliminate the intercept. This is appropriate when analyzing covariances or correlations which are invariant when adding or subtracting constants. However, centering can be limiting because there are many important research questions in behavioral medicine that require retaining information about the means. For instance, in randomized clinical trials the focus is often a comparison of two group means. Questions related to health disparities often require testing hypotheses about group means. Longitudinal studies of health outcomes also often focus on changes in mean levels over time. Therefore, if the SEM latent variable framework is to be useful in those situations, we need to consider means.

When we work with mean structures, our equations include intercepts. You may recall when learning about regression that the intercept was associated with a vector of 1's in the data. This is because when we regress a variable, say Y, on the constant 1, the regression coefficient is the mean of Y. Also, if we regress a variable Y on a constant 1 and another predictor, say X, the regression coefficient for the constant is the intercept. These concepts apply in SEM. In a path diagram we indicate the inclusion of means and/or intercepts by specifying a triangle with the number 1 in it. This triangle represents a constant which will be useful in estimating means and intercepts. The arrows going from the triangle to a variable represent either a mean or an intercept, depending on whether there are other predictors also going to the variable.

As I mentioned earlier, mean structures are useful when comparing multiple groups or examining change in a group over time. In a subsequent section, I will introduce latent growth models, which are relevant for studying change over time.

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