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How we could take a student
analysis one step further This student proposes a very interesting set of hypotheses. Rather than focusing on demographic causes (e.g., age, gender, race), she posits a causal chain among opinion variables. [Actually, the first one is a behavioral variable.] If we draw her hypotheses as a path model, it would look something like this:
Or we could also draw it like this, which will help us in further steps, as you’ll see:
The student tests the correlations between each variable using crosstabs and “gamma” ordinal correlations. If we draw in the correlations, the path model would look like this:
Already, we can see very clearly one of the things she reported. There is a strong connection between the two religious variables, and there are strong connections among the non-religious variables, but there seems to be a break between the two realms. Now, one feature of this path model is that the (non-adjacent) steps along the path are “mediated.” Let’s take a simple example. Suppose we have a causal chain among three variables, A, B, and C that looks like this:
This says, A causes (or leads to) B, B causes C, and any influence of A on C passes through B. This means that B “mediates” the influence of A on C. But we could posit that A also has a direct, “unmediated” influence on C, as does B. Then, our picture would look like this:
If we apply the same logic to the student’s model, and draw in all the direct or “unmediated” paths among the variables, the model might look something like this:
Testing the hypotheses in such a model requires multivariate statistics, and you might not have learned how to do this yet. But even if you haven’t learned the statistics yet, the concept is not too difficult. Consider the third variable in the chain, Trust People. The student has hypothesized a direct effect of Trust Church on Trust People, and a direct effect of Church Attendance on Trust Church. However, we might also ask whether Church Attendance also has a direct effect on Trust People, not just a mediated effect that runs through Trust Church. That direct arrow is now drawn in this picture, as are all similar arrows. Thus, we have hypothesized both direct (unmediated) and indirect (mediated) effects. Again, without being too technical about it, this means that we want to know about the causal effects of each variable on each later variable along the chain, controlling for the effects of all other prior variables. Thus, for instance, for the fourth variable, Trust Neighbors, we would measure the effects of the first three variables, Church Attendance, Trust Church, and Trust People, controlling the effect of each on each other. As you have learned in a statistics class, or will learn, we can test such hypotheses easily with multiple regression. Thus, for each step along the path model, we test the effects of all the (independent) variables that precede each (dependent) variable; and we calculate a set of multiple regression equations for each step along the way. We call this a set of “recursive” regression equations – and we call the picture a “recursive” model – because the arrows only point one way along the causal path. Here are the recursive multiple regression equations for the last path model we drew:
Now, we could draw the same path model as before, but this time put the (standardized) regression coefficients in. The model would now look something like this:
This is a pretty good model, and can tell us a lot, but it is still a little hard to look at because there are so many arrows and numbers in it. We can make one more refinement that will make it graphically easier to look at. Suppose we take out the coefficients that are statistically insignificant (they’re usually small numbers, anyway). The new path model would then look something like this:
This version leaves the insignificant paths in as light dotted lines. If we think this still looks too busy, we could take them out entirely, like this:
[If we want to be
fully accurate, we should recalculate the models, excluding variables that
have no significant effect, but we can leave that issue for a statistics
class.] What conclusions can we draw from these results? Let’s just focus on the simplest ones:
Thus, the student did a very good job of testing a set of related hypotheses, using simple cross-tabulations, and her conclusions were sound as far as they went. This exercise indicates how you can take these hypotheses one step further and learn a little more about the relationships among the variables. Further steps are also possible. If you continue learning data analyses methods, you’ll learn a number of them. |
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