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. 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. 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: When is too much – too much

9. 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

10. 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 How does it work… 1. Simplest model with the highest R 2 wins! 2. Use Mallows’ Cp to break the tie Who decides – YOU!

11. 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 How does it work… You should also look a

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

13. Who Cares? 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