1. MARE 250 Dr. Jason Turner Multiple Regression

2. y Linear Regression y = b 0 + b 1 x y = dependent variable b 0 + b 1 = are constants b 0 = y intercept b 1 = slope x = independent variable Urchin density = b 0 + b 1 (salinity)

3. Multiple regression allows us to learn more about the relationship between several independent or predictor variables and a dependent or criterion variable For example, we might be looking for a reliable way to estimate the age of AHI at the dock instead of waiting for laboratory analyses Multiple Regression y = b 0 + b 1 x y = b 0 + b 1 x 1 + b 2 x 2 …b n x n

4. In the social and natural sciences multiple regression procedures are very widely used in research Multiple regression allows the researcher to ask “what is the best predictor of...?” For example, educational researchers might want to learn what are the best predictors of success in high-school Psychologists may want to determine which personality variable best predicts social adjustment Sociologists may want to find out which of the multiple social indicators best predict whether or not a new immigrant group will adapt and be absorbed into society. Multiple Regression

5. The general computational problem that needs to be solved in multiple regression analysis is to fit a straight line to a number of points Multiple Regression In the simplest case - one dependent and one independent variable Can be visualized this in a scatterplot

6. A line in a two dimensional or two-variable space is defined by the equation Y=a+b*X; the animation below shows a two dimensional regression equation plotted with three different confidence intervals (90%, 95% 99%) The Regression Equation In the multivariate case, when there is more than one independent variable, the regression line cannot be visualized in the two dimensional space, but can be computed rather easily

7. The smaller the variability of the residual values around the regression line relative to the overall variability, the better is our prediction Coefficient of determination (r 2 ) - If we have an R- square of 0.4 we have explained 40% of the original variability, and are left with 60% residual variability. Ideally, we would like to explain most if not all of the original variability Therefore - r 2 value is an indicator of how well the model fits the data (e.g., an r 2 close to 1.0 indicates that we have accounted for almost all of the variability with the variables specified in the model Residual Variance and R- square

8. Assumption of Linearity It is assumed that the relationship between variables is linear - always look at bivariate scatterplot of the variables of interest Normality Assumption It is assumed in multiple regression that the residuals (predicted minus observed values) are distributed normally (i.e., follow the normal distribution) Most tests (specifically the F-test) are quite robust with regard to violations of this assumption Review the distributions of the major variables with histograms Assumptions, Assumptions…

9. A data point whose removal causes the regression equation (line) to change considerably Consider removal much like an outlier If no explanation – up to researcher To index Effects of Outliers Outliers may be influential observations

10. For Example… We are interested in predicting values for Y based upon several X’s…Age of AHI based upon FL, BM, OP, PF, GR FL OP PF GR

11. For Example… We are interested in predicting values for Y based upon several X’s…Age of AHI based upon FL, BM, OP, PF, GR We run multiple regression and get the equation: Age = - 12.2 + 0.0370 Fork Len. + 0.093 Body Mass + 0.126 Operculum + 0.463 Pect. Fin + 0.129 Girth

12. So What’s the Problem? It has been said that the simplest explanation is often the best Age = - 12.2 + 0.0370 Fork Len. + 0.093 Body Mass + 0.126 Operculum + 0.463 Pect. Fin + 0.129 Girth Difficult for scientists or fishermen to estimate age based upon 5 measurements

13. Variable Reduction 2 Methods for variable reduction: Stepwise Regression - removes and adds variables to the regression model for the purpose of identifying a useful subset of the predictors Best Subsets Regression - generates regression models using maximum R 2 criterion by first examining all one-predictor regression models and selecting the 2 best, then all two- predictor models and selecting the 2 best, etc.

14. Building Models via Stepwise Regression Stepwise model-building techniques for regression The basic procedures involve: (1) identifying an initial model (2) iteratively "stepping," that is, repeatedly altering the model at the previous step by adding or removing a predictor variable in accordance with the "stepping criteria," (3) terminating the search when stepping is no longer possible given the stepping criteria Stepwise Regression:

15. How does it work? Stepwise Regression Mallows’ CP – can be used to determine the precision of one model over another Lower Mallows’ CP – more precise; better model

16. Building Models via Best Subsets Regression Best Subsets model-building techniques for regression The basic procedures involve: (1) first examining all one-predictor regression models and selecting the 2 best (2) next examining all two-predictor regression models and selecting the 2 best, etc. for all variables (3) from output WE decide the best model based upon high adjusted R 2 and small Mallows' Cp Best Subsets Regression:

17. How does it work? Best Subsets Regression

18. Who Cares? Stepwise and/or Best Subsets analysis allows you (i.e. – computer) to determine which predictor variables (or combination of) best explain (can be used to predict) Y Much more important as number of predictor variables increase Helps to make better sense of complicated multivariate data