Multiple Regression MARE 250 Dr. Jason Turner

Linear Regression y = b0 + b1x Urchin density = b0 + b1(salinity) y = dependent variable b0 + b1 = are constants b0 = y intercept b1 = slope x = independent variable y Urchin density = b0 + b1(salinity)

Multiple Regression y = b0 + b1x y = b0 + b1x1 + b2x2 …bnxn 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 y = b0 + b1x y = b0 + b1x1 + b2x2 …bnxn

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

The Regression Equation 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%)       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

Residual Variance and R-square 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 (r2) - 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 - r2 value is an indicator of how well the model fits the data (e.g., an r2 close to 1.0 indicates that we have accounted for almost all of the variability with the variables specified in the model

Assumptions, Assumptions… 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

Effects of Outliers Outliers may be influential observations 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

When is too much – too much Stepwise Regression: When is too much – too much Building Models via Stepwise Regression Stepwise model-building techniques for regression The basic procedures involve: identifying an initial model 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," terminating the search when stepping is no longer possible given the stepping criteria

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

How does it work… Response is Age 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

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

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