mare 250 dr. jason turner
DESCRIPTION
Multiple Regression. MARE 250 Dr. Jason Turner. 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). Multiple Regression. - PowerPoint PPT PresentationTRANSCRIPT
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
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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…