How to Use Tracing Rules to Express Variance/Covariance Matrices: Unstandardized Coefficients
Phillip Wood (please let me know if you find errors or have comments/suggestions for this-email:
Phil Wood wood@psysparc.psyc.missouri.edu
Introduction to Unstandardized Path Diagrams
The procedures which are used to translate unstandardized path coefficients into a variance/covariance matrix are really the same as those outlined for standardized coefficients. Figures showing unstandardized paths are not often published, usually because the researcher is interested in showing the relative magnitude of effects associated with variables in different scales.
However, just as unstandardized coefficients can be helpful in comparing results across studies, the unstandardized coefficients from a structural model can also help communicate which effects from some studies are similar to effects found in other studies.
The logic presented on pages 12-14 of Loehlin can easily be extended to show how you can use a variance/covariance matrix and an unstandardized path diagram to solve for unstandardized regression coefficients. After reading this section of Loehlin's book and going through the math outlined here, I believe you should be able to use a path diagram to produce the formulae for unstandardized regression weights.
What does an Unstandardized path model represent?
When you were considering a diagram containing standardized coefficients, you were examining a statement of the structural relationships between variables which produced a particular correlation matrix. (The handout above showed you how to translate a path diagram into a statement of predicted correlations between variables.)
Variance/Covariance matrices - What are they?
When you are looking at a path diagram which shows unstandardized coefficients, the matrix which is being reproduced is a variance/covariance matrix. Variance/covariance matrices summarize the patterns of variability and covariation between variables in the metric of the observed variables. Like correlation matrices, variance/covariance matrices are symmetric. Instead of having 1's on the diagonal, though, the diagonal elements of a variance/covariance matrix contain the variances of the observed variables. Off-diagonal elements, instead of containing correlations, contain covariances.
How to turn a correlation matrix & standard deviations into a variance/covariance matrix.
APA desiderata for reporting results now include some statement about the desirability of including correlations and standard deviations in articles so that researchers can replicate/extend reported analyses.
As such, it's helpful to think about how to take a reported correlation matrix and corresponding standard deviations and convert it to a covariance matrix suitable for statistical analysis.
As with many things that I try to communicate in statistics, there are many ways to accomplish this. Let me for the moment consider one way which is helpful if you are doing this with paper and pencil, and another using matrix algebra:
Suppose that you had the following correlation matrix and you also knew that the standard deviations of the variables A, B and C were 5, 10, and 15, respectively.:
- First, you write in the standard deviations associated with the variables for each column and for each row like this:
- Then you multiply each cell by the value of its respective column like this:
- Now do the same thing, but for the rows. I.e.:
You can check your work here a bit by noting that this matrix is, as promised above, symmetric (the covariance of A with C is the same as the covariance of C with A, for example) and that the diagonal elements of this matrix are the square of the standard deviations associated with that variable (e.g., the variance of C is 225 and the standard deviation for C which we are given is 15).
It's much easier to convert correlations with a matrix algebra program such as IML, particularly if your matrix is large. Myself I'm a fan of having humans manipulate numbers as little as possible, just so long as we are intelligent consumers about the numbers which the computer produces. To convert a correlation matrix into a covariance matrix,
Which gives the same result as we had by doing it all by hand.Unstandardized Path Diagram Example
OK- let's now go through our example. You'll find that this is fairly straightforward to do, given that we used a slightly more detailed way of representing and manipulating the standardized path diagram.
The scenario is that the researcher has published an unstandardized diagram which looks like:
As before, this diagram is a statement about:
the predicted variances and covariances among predictor variables. (You can use these information to calculate the correlation between the two variables, but it takes a small amount of math.)
The covariance between of the variables A and B with the criterion,
The
variance
of the endogenous variable, 
The
variance in the dependent variable
accounted for by the predictors (also known as 
and (again with a bit of math) the proportion of variability in the criterion accounted for the by predictors .
Implied Covariance Matrix
For the purposes of parallelism with the previous handout, what we're interested in doing is filling in the elements of the variance/covariance matrix for the researcher's data.
Variances of the predictor variables.
As you would expect, the slings associated with A and B are the variances of these two predictors. I.e.:
Covariances between predictor variables.
Similarly, the sling between the predictors is the covariance.
(Which is comforting, because that's the value we had for the original correlation matrix.)Covariance of Predictor A with Criterion C.
The covariance of A with C follows the same rules as before.
Specifically, the model says that the covariance of variables A and C in the model is a sum of two components:
This is calculated by taking the sling of 25 associated with A and multiplying that by the path coefficient from A to C. i.e.:
The additional covariation of A with C occurs by virtue of A's being related to B. I.e.:
Covariance of Predictor B with Criterion C.
This follows the same procedure as described above. In pictures:
The covariance of variables B and C in the model is a sum of two components:
This is calculated by taking the sling of 100 (the variance) associated with B and multiplying that by the path coefficient from B to C. i.e.:
The additional covariation o A with C occurs by virtue of A's being related to B. I.e.:
Variance of the Endogenous Variable.
The variance of the endogenous variable proceeds the same as for the correlation above. We again use our little "airplane" to "take off" from C and return to C by use of the path diagrams. I.e.:
Consider, for starters, variance components involving the path from A to C. To do this we "skip out" of C on the path and then return to C by two ways:
We grab the sling which is the variance of A (25) and return the way we came on 1.2 i.e.:
Next, we skip out on the 1.2 again and return to C this time by way of the path from B to C:
Now we will consider the same paths, but this time involving skipping out on the path from B to C.
First off, we consider skipping out on .75, taking the sling of 100 at B and returning the way we came, i.e.,
Now we skip out again and this time take the sling between A and B and return on the path from A to C. I.e.:
Variance in C accounted for.
Notice that the quantity that we have computed so far his one that we have seen before in multiple regression. It is the variability in C accounted for by A and B. (known in notational form as: 
We can, using this quantity, calculate the
proportion
of variability in C accounted for by A and B, (known notationally as:
) by dividing this quantity by the total amount of variability in C. For our example this is: 
which is a comforting value, because it is exactly what we got when we were calculating this proportion using the standardized values.
Now we consider the path involved by skipping out on the path from X (the
error term
) to C. This quantity, is going to be the remaining variability in C not accounted for in the model. I.e., 
We skip out on the number 1, take the sling at X and return the way we came. I.e.
As before, we could just have easily have found the same quantity had the researcher published a model in which the error variance was set to 1 and a path estimated from it to the dependent variable. I.e.:
