Chapter 9 Outline
  • 1. Modeling multivariate reality
  • 2. The population regression function
  • 3. From two-variable to multiple regression
  • 4. Interpreting multiple regression
  • 5. Which effect is “biggest”?
  • 6. Statistical and Substantive Signicance
  • 7. What happens when we fail to control for 
   Z?
      Crossing the fourth causal
           hurdle
  • In any observational study, how do we 
   “control for” the effects of other variables?
  • Multiple regression, this chapter's topic, is by 
   far the most common method in the social 
   sciences.
               The population regression function
       We can generalize the population regression model from Chapter 8:
         bivariate population regression model:
       to include more than one cause of Y , which we have been calling Z:
            multiple population regression model:
        The interpretation of the slope coefficients here is similar to interpreting
        bivariate coefficients, with one big difference. In both, the coefficient in
        front of X (     in the two-variable model,       in the multiple regression
        model) represents the “rise-over-run” effect of X on Y . In the multiple
        regression case,      represents the effect of X on Y while holding constant
        the effects of Z.
               Remember…
   Recall from Chapter 8 that the formula for a two-variable 
    regression line is the following (in a sample):
   And recall that, in order to understand the nature of the effect that X has on Y , the 
   estimated coefficient      tells us, on average, how many units of change in Y we should 
   expect given a one-unit increase in X. The formula for      in the two-variable model, as 
   we learned in Chapter 8, is:
                           From a line to a plane
       Given that our goal is to “control for” the effects of some third variable,
       Z, how exactly is that accomplished in regression equations? If a
       scatterplot in two dimensions (X and Y ) leaves the formula for a line,
       then adding a third dimension leaves the formula for a plane. And the
       formula for that plane is:
     That might seem deceptively simple. A formula representing a plane simply adds the 
     additional term        to the formula for a line.