I271b Quantitative Methods Regression Part II

Data points and Regression http://www.math.csusb.edu/faculty/stanton/m262/regress/

Power Transformations Helps to correct for skew and non-symmetric distributions. Yq : reduces negative skew Log Y or –(Y-q ): reduces positive skew

Interpreting a Power Transformation Power transformations are not a magic wand– they only help you achieve symmetry and/or normality, but often symmetry is all you can hope to achieve. And, you still need to interpret the outcome! We must reconvert our transformed Y back to normal units by using the inverse transformation: Y* = Yq  Y* 1/q Y* = Loge Y  eY* So, when you calculate the predicted value from the regression equation, take the inverse transformation of the result: Y2 = 2.5 + 3.2 X = 5.7  5.71/2 = 2.39 Thus, when X =1, Y = 2.39 in its original units.

Multivariate Regression Same basic idea, but allows us to look at more than 2 variables at one time. Each variable has its own independent effect on the slope, given the effect of the other variables in the same model. Control Variables Should one or more variables be ‘controlled’ so that we can examine effect of our main IV? Alternate Predictor Variables Perhaps we have more than one IV that might have a linear relationship with Y? Nested Models We may want to examine more than one model and see which one is a better fit. X1 + X2  Y X1 + X3  Y

Nested Models Model 1 Model 2 Model 3 Control Variable 1 Explanatory Variable 1 Model 3 Explanatory Variable 2

Issue with Multiple Independent Variables: Multicollinearity Occurs when an IV is very highly correlated with one or more other IV’s Caused by many things (including variables computed by other variables in same equation, using different operationalizations of same concept, etc) Consequences For OLS regression, it does not violate assumptions, but Standard Errors will be much, much larger than normal when there is multicollinearity (confidence intervals become wider, t-statistics become smaller) We often use VIF (variance inflation factors) scores to detect multicollinearity Generally, VIF of 5-10 is problematic, higher values considered problematic Solving the problem Typically, regressing each IV on the other IV’s is a way to find the problem variable(s). Once we find IV’s that are collinear, we should eliminate one of them from the analysis.