Psychology 202b Advanced Psychological Statistics, II February 22, 2011

Overview Sequential regression: –Conceptual discussion –Efficient execution Stepwise regression

Introducing sequential regression The regression models we have worked with so far all performed inference about the direct effects of the predictors. Order of specification doesn’t matter. Sequential regression considers the combination of direct and indirect effects. Order of specification matters. Illustration using an example from Cohen & Cohen.

Direct effects

A conceptual model of causal flow

Direct vs. total effects Conventional regression performs inference only on the direct effects. Sequential regression will have exactly the same parameter estimates, but inference will be with respect to the total effect. On the previous slide, inference about years since PhD would include both the direct effect, and the effect that causally flows through number of publications.

How does sequential regression work? Identify the variables that should have causal precedence. Estimate the regression including only them. Add the next set of variables, observe the change in the model sum of squares, and test for change using the current error term. Continue that process.

How does sequential regression work? In the current example, we cannot assess exactly the causal model we pictured. If “Male” is entered at the same stage as “Years since PhD,” inference about Male will include an indirect effect through “Publications.” If “Male” is entered with publications, inference about “Years” would include a highly implausible indirect effect.

The actual model

How does sequential regression work? Note that structural equation modeling allows greater control over causal paths. Illustration in SAS: –Manually –More efficiently Using proc reg Similarity to Type I sum of squares

What if I do something stupid? It is easy to specify ludicrous models with sequential regression. Illustration in SAS.

Things that are easier in proc reg CIs for regression parameters. The condition number (sort of). Example in SAS with supporting explanatory work in R.

Stepwise procedures The idea of stepwise procedures is to let the software automatically select an optimal model. Forward selection starts with the intercept and adds predictors according to flexible criteria. Backward selection starts with all predictors and selectively removes some.

Stepwise procedures (cont.) The “stepwise” selection method in SAS can remove variables that have already been added. All possible subsets selects the best of all possible regression models according to some criterion.

Selection criteria Maximize increase in F for adding a variable. Minimize reduction in F for removing a variable. Find the subset of variables from among all possible subsets meeting entry criteria that maximizes R 2. Same criterion with adjusted R 2.

Using stepwise procedures Illustration in SAS. Cross-validating. An example of one danger of stepwise approaches.

Reasons not to use stepwise procedures Atheoretic –Can result in nonsensical models. –Cannot test particular models that make sense. Where is the assumption checking? Danger of capitalizing on chance. Difficult to cross-validate.

Next Time No class on Thursday.