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, researchers might want to learn what abiotic variables (temp, sal, DO, turb) are the best predictors of plankton abundance/diversity in Hilo Bay Or Which morphometric measurements are the best predictors of fish age 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 This can be visualized in a scatterplot

6. A line in a two dimensional or two-variable space is defined by the equation Y=a+b*X 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. Two Methods: Best Subset & Stepwise Analysis Best Subsets: Best subsets regression provides information on the fit of several different models, thereby allowing you to select a model based on four distinct statistics Stepwise: Stepwise regression produces a single model based on a single statistic. How To – Multiple Regression

8. For data sets with a small number of predictors, best subset regression is preferable to stepwise regression because it provides information on more models. For data sets with a large number of predictors (> 32 in Minitab), stepwise regression is preferable. Stepwise…Subsets

9. S B O P Vars R-Sq R-Sq(adj) C-p S L M P F 1 77.7 77.4 8.0 0.96215 X 1 60.3 59.8 76.6 1.2839 X 2 78.9 78.3 5.4 0.94256 X X 2 78.6 78.0 6.6 0.94962 X X 3 79.8 79.1 3.6 0.92641 X X X 3 79.1 78.3 6.5 0.94353 X X X 4 80.0 79.0 5.0 0.92897 X X X X Response is Age Best Subsets 1. Simplest model with the highest R 2 wins! 2. Use Mallows’ Cp to break the tie Who decides – YOU!

10. 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:

11. For Example… We are interested in predicting values for Y based upon several X’s…Age of AHI based upon SL, BM, OP, PF We run multiple regression and get the equation: Age = - 2.64 + 0.0382 SL + 0.209 BM + 0.136 OP + 0.467 PF We then run a STEPWISE regression to determine the best subset of these variables

12. How does it work… Stepwise Regression: Age versus SL, BM, OP, PF Alpha-to-Enter: 0.15 Alpha-to-Remove: 0.15 Response is Age on 4 predictors, with N = 84 Step 1 2 3 Constant -0.8013 -1.1103 -5.4795 BM 0.355 0.326 0.267 T-Value 16.91 13.17 6.91 P-Value 0.000 0.000 0.000 OP 0.096 0.101 T-Value 2.11 2.26 P-Value 0.038 0.027 SL 0.087 T-Value 1.96 P-Value 0.053 S 0.962 0.943 0.926 R-Sq 77.71 78.87 79.84 R-Sq(adj) 77.44 78.35 79.08 Mallows C-p 8.0 5.4 3.6 Step 1 – BM variable is added Step 2 – OP variable is added Step 3 – SL variable is added

13. Who Cares? Best Subsets & Stepwise 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

14. At this point we are still limited to 2-dimensional graphs; although our statistics have become 3- dimensional… However…

15. There are 3-dimensional graphical techniques to encompass multivariate datasets Don’t Despair Grasshopper… Cool! When do we learn…

16. “First learn stand, then learn fly. Nature rule, Daniel-san, not mine.” Miyagi Says… All in good time Daniel-san…