MARE 250Dr. Jason Turner

Multiple Regression

y

Linear Regression

y = b0 + b1xy = dependent variable

b0 + b1 = are constants

b0 = y intercept

b1 = slope

x = independent variable

Urchin density = b0 + b1(salinity)

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 = b0 + b1x

y = b0 + b1x1 + b2x2 …bnxn

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

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

SL

Age

77.575.072.570.067.565.062.560.0

13

12

11

10

9

8

7

6

5

4

Scatterplot of Age vs SL

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

SL

Age

786858

14

13

12

11

10

9

8

7

6

5

BM

363024

OP

16128

PF

642

Matrix Plot of Age vs SL, BM, OP, PF

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

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

S B O PVars 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 R2 wins!2. Use Mallows’ Cp to break the tieWho decides – YOU!

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:

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…Stepwise Regression: Age versus SL, BM, OP, PF Alpha-to-Enter: 0.15 Alpha-to-Remove: 0.15Response is Age on 4 predictors, with N = 84

Step 1 2 3Constant -0.8013 -1.1103 -5.4795

BM 0.355 0.326 0.267T-Value 16.91 13.17 6.91P-Value 0.000 0.000 0.000

OP 0.096 0.101T-Value 2.11 2.26P-Value 0.038 0.027

SL 0.087T-Value 1.96P-Value 0.053

S 0.962 0.943 0.926R-Sq 77.71 78.87 79.84R-Sq(adj) 77.44 78.35 79.08Mallows 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

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

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

However…

SL

Age

786858

14

13

12

11

10

9

8

7

6

5

BM

363024

OP

16128

PF

642

Matrix Plot of Age vs SL, BM, OP, PF

There are 3-dimensional graphical techniques to encompass multivariate datasets

Don’t Despair Grasshopper…

Cool! When do we learn…

“First learn stand, then learn fly. Nature rule, Daniel-san, not mine.”

Miyagi Says…

All in good time Daniel-san…