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.:
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.
As you would expect, the slings associated with A and B are the variances of these two predictors. I.e.:
Similarly, the sling between the predictors is the covariance.
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.:
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.:
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.:
(Which is comforting, because that's the value we had for the original correlation matrix.)