1 Example 1.0: Indirect Effects and the Test of Mediation

2 Example: Community Response to Wildfire in California Shrublands

3 One of the many findings: vegetation recovery was a function of the age of the stand that burned. r = cover is in proportions. age in years.

4 Indirect Effects as Causal Tests: Step 1 B. How do we interpret the observation that plant cover the year after the fires is a function of the age of the stand that burned? A. Let’s start with a simple regression. In this case, we regress the amount of vegetation cover that has developed in the first year following a fire, as a function of how old the stand of shrubs was that burned in the fire. age of stand that burned post-fire vegetation cover e1 C. We might hypothesize that older stands would have more fuel and would burn hotter than young stands, resulting in less post-fire vegetation cover. D. Since we have estimates of fire severity, we can test to see if fire severity can explain the relationship between stand age and cover. This is referred to as the test of mediation. It is a characteristic feature of SEM. Model A

5 Indirect Effects as Causal Tests: Step 2 The test of mediation E. Graphically, our test can be represented as follows: F. To conclude that fire severity mediates effects of age on cover, the paths from age to severity and from severity to cover should be significant, and the coefficients for these paths should add up to the net relationship between age and cover. This test will be conducted using the raw (unstandardized) covariances. However, for simplicity in presenting the results, we will assume the variables are standardized and present the standardized correlations. age of stand that burned fire severity e1 post-fire vegetation cover e2 Model B

6 Indirect Effects as Causal Tests: Step 2 (cont.) The test of mediation age of stand that burned fire severity e1 post-fire vegetation cover e2 cover firesev age cover firesev age G. What do the correlations among variables look like? H. Based on the second rule of path coefficients, if model B is correct, the path coefficients in this case should be the simple covariances/correlations. expected path coefficients if this model is correct I. The model-implied correlation between age and cover = 0.453* = , while the observed correlation is , yielding a standardized residual of

7 Indirect Effects as Causal Tests: Step 3 Model Evaluation age of stand that burned fire severity e1 post-fire vegetation cover e2 Model B J. Lets assume without showing the test results explicitly that the two path coefficients are judged to be statistically significant. This means we can conclude that fire severity does mediate the effect of age on cover, at least in part. K. Since the indirect effect of age on cover (= ) does not exactly explain the observed net effect of , we need to rely on the model chi-square test or some other measure of overall model fit in order to judge Model B. The alternative model is Model C. Which is better? age of stand that burned fire severity e1 post-fire vegetation cover e2 Model C mechanisms other than fire severity whereby age affects cover

8 Indirect Effects as Causal Tests: Step 3 (cont.) Model Evaluation L. If we obtain the model chi-square for model B, we get the following: model chi-square = model degrees of freedom = 1 p-value for chi-square= Since the “single-degree-of-freedom chi-square test value is 3.84, we conclude that model B is adequate and does not require an additional path. M. If we estimate Model C, our chi-square drops to 0 since our model is saturated and the degrees of freedom = 0. As stated above, this drop in chi-square of is not a significant improvement using conventional hypothesis testing logic. Still, we might conclude as scientists that Model C is a better model because it allows for other mechanisms whereby older stands have lower recovery, such as a decline in the seed bank over time. A second data set with a larger sample size might show evidence for such a additional mechanism to operate.

9 Indirect Effects as Causal Tests: Step 3 (cont.) Model Evaluation N. If we assume Model C, we get the following estimates. age of stand that burned fire severity e1 post-fire vegetation cover e2 Model C O. Note that our estimate for the path from severity to cover is different from before. An understanding of the rules of path coefficients (see “SEM Essentials”) shows why this is the case. In any event, here we judge that the coefficient is not reliably different from zero, indicating that the mechanisms it represents are not consistently reliable enough that they must be considered in a parsimonious model. P. More will be said about model testing and selection in a separate tutorial.