RAM Diagrams for Structural Models

Proper RAM Diagrams

A proper RAM diagram is a representation of data. Data from a study may contain one or more variables taken on at least one variable from either several individuals or several measurement occasions (or both). Ram Diagrams use squares (e.g., to represent variables which are either present or directly calculable by a model and use circles (e.g. ) to denote latent variables which are implied, but not directly calculable by the model (although one may use a structural model to compute estimates of such latent variables). Ram diagrams express relationships between these variables by means of slings and Arrows . Each sling and arrow is associated with some numerical path value. For purposes of convenience some authors (such as Loehlin) assume that a path with no number is assumed to be 1 and omit slings on exogenous variables when they are assumed to be 1. Because we are all beginners at this point, I prefer to be explicit about everything. This will also make it easier when we translate path diagrams into computer programs.

Exogenous Variables

Variables which are thought to be exogenous to the system, either by theoretical assumption or experimenter manipulation are termed exogenous variables. Other names for exogenous variables include predictor variables, independent variables, and upstream variables. Exogenous variables:

Always have a sling attached to them

This sling begins and ends within that variable. This sling has, as its interpretation the variance of the exogenous variable. For example:. or. are examples of valid exogenous variables. The placement of the double headed arrow (sling) does not matter.

May have slings attaching them to other exogenous variables.

For example, in the following diagram, the blue sling is a valid construction.

However, it should be noted that these slings may only connect exogenous variables to other exogenous variables. It is not allowed to have a sling between an exogenous variable and an endogenous variable nor is it allowed to join two endogenous variables with a sling.

May not be pointed at by another variable

Although, of course, exogenous variables may "point to" endogenous variables

Interpretation of slings which join different variables

When the variances of the exogenous variables have a value of 1, the sling joining the two variables is known as a correlation. As in the diagram shown above. When either or both are not equal to 1, the sling joining the two variables is known as a covariance. For example: For example, in the following diagram, . Recall that a covariance can be converted to a correlation by dividing the covariance by the standard deviation of the two variables. Because the slings on A and B represent variances, the correlation can be computed as:

Endogenous Variables

Variables which are explained by other variables in the study/system are known as endogenous variables. Other names for endogenous variables include criterion variables, dependent variables and downstream variables. Endogenous variables:

Have at least one arrow directed at them from either an exogenous variable or some other endogenous variable.

May not have slings joining them with other variables nor may they have slings attached to them.

May point at other endogenous variables

For example, in the following diagram, the variable C is an endogenous variable.

Some Diagrams Taken from Loehlin.

The diagrams below represent RAM versions of the figures in Chapter 1 of Loehlin.

Figure 1.1

 

Figure 1.2

 

Figure 1.3

 

Find the Proper Ram Diagram

Below are some path diagrams (some of which have been taken from previous projects for this course). Which of these are proper and improper?

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

 

Example 1

This is a proper diagram. As a matter of fact, it represents the way that you create a standardized version of a variable in a path diagram. You'll see why this is the case when we cover the tracing rules later.

Example 2

This is also a proper diagram. Notice that it differs from the two variable regression diagram shown in the book (and above) in that there is no sling joining the two predictor variables. This diagram demonstrates a property of structural equation models. They have many strengths in that they allow you to incorporate additional information into you models (in this case, we are assuming that it is known that the covariance between A and B is zero). This may happen, for example, in cases where the predictor variables are manipulated by the researcher, when the researcher has systematically sampled individuals so as to produce a zero correlation in the sample, or when it is well-known in the area that these two variables are unrelated. On the other hand, to the extent that this diagram makes an unwarranted assumption about the data, it may lead to faulty conclusions.

Example 3

This diagram is also a proper RAM diagram. As we will see later, this model has is called a one factor model for the data, and assumes that each variable is determined in part by an unmeasured variable and some component unique to that variable. Error variables of this type are called "uniquenesses" even though, from a path perspective, they are just another error variable.

Example 4

This diagram is not a proper diagram. The variables, e2, e3, and e4 cannot simultaneously have arrows from one variable to another and also have slings. Although this type of model can be fit using some SEM programs, it is not a proper path diagram according to the rules we've outlined here. One "fix" to make this diagram look proper would be to attach error terms on to the error terms of the other variables.

Example 5

This diagram is a proper diagram. The manifest variables B, C, and D are endogenous variables and there is nothing to prevent us from allowing that, in addition to the causes of the central latent variable T and the uniquenesses that other manifest variables cause a particular manifest variable as well. As a matter of fact, when the variables A, B, C, and D are actually the same manifest variable over time, we have a model called the "State/Trait" model originally proposed by Rolf Steyer (as far as I know). These types of modifications to a model are often proposed spontaneously by Tetrad, a computer program for generating structural models from data. To by knowledge, however, such models are not often considered by researchers.

Example 6

This diagram is also a proper diagram. The unmeasured variables T and e4 are both exogenous and so may be joined by a covariance. Note that, as before, we could not put a straight-headed arrow from T to e4 without making a further error variable for e4 (nor, for that matter from e4 to T without making a disturbance for T).