Post on 23-May-2018
transcript
Neil W. Polhemus, CTO, StatPoint Technologies, Inc.
Regression Analysis Using
Statgraphics Centurion
Copyright 2011 by StatPoint Technologies, Inc.
Web site: www.statgraphics.com
Outline
Regression Models
Examples – Single X
Simple regression
Nonlinear models
Calibration
Comparison of regression lines
Examples – Multiple X
Regression model selection (stepwise, all possible)
Logistic regression
Poisson regression
2
Regression Model Setup
3
Dependent variable: Y
Independent variable(s): X1, X2, …, Xk
Error term: e
Model: Y = f (X1, X2, …, Xk) + e
Types of Regression Models (#1)
4
Procedure Dependent variable Independent variables
Simple Regression continuous 1 continuous
Polynomial Regression continuous 1 continuous
Box-Cox Transformations continuous 1 continuous
Calibration Models continuous 1 continuous
Comparison of Regression
Lines
continuous 1 continuous and 1
categorical
Types of Regression Models (#2)
5
Procedure Dependent variable Independent variables
Multiple Regression continuous 2+ continuous
Regression Model Selection continuous 2+ continuous
Nonlinear Regression continuous 1+ continuous
Ridge Regression continuous 2+ continuous
Partial Least Squares continuous 2+ continuous
General Linear Models 1+ continuous 2+ continuous or categorical
variables
Types of Regression Models (#3)
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Procedure Dependent variable Independent variables
Logistic Regression proportions 1+ continuous or categorical
Probit Analysis proportions 1+ continuous or categorical
Poisson Regression counts 1+ continuous or categorical
Negative Binomial
Regression
counts 1+ continuous or categorical
Life Data - Parametric
Models
failure times 1+ continuous or categorical
Example 1: Stability study
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Y: percent of available chlorine
X: number of weeks since production
Lower acceptable limit for Y: 0.40
X-Y Scatterplot with Smooth
8
Simple Regression
9
Analysis Options
10
Tables and Graphs
11
Analysis Window
12
Analysis Summary
13
Lack-of-Fit Test
14
Comparison of Alternative Models
15
Fitted Reciprocal-X Model
16
Plot of Fitted Model
chlorine = 0.368053 + 1.02553/weeks
0 10 20 30 40 50
weeks
0.38
0.4
0.42
0.44
0.46
0.48
0.5
ch
lori
ne
Lower 95% Prediction Limit
17
Outlier Removal
18
Plot of Fitted Model
chlorine = 0.366628 + 1.02548/weeks
0 10 20 30 40 50
weeks
0.38
0.4
0.42
0.44
0.46
0.48
0.5
ch
lori
ne
Example 2: Nonlinear Regression
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Draper and Smith in Applied Regression Analysis suggest fitting
a model of the form
Y = a + (0.49-a)exp[-b(x-8)]
Since the model is nonlinear in the parameters, it requires a
search procedure to find the best solution.
Data Input Dialog Box
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Initial Parameter Estimates
21
Analysis Options
22
Plot of Fitted Model
23
Plot of Fitted Model
0 10 20 30 40 50
weeks
0.38
0.4
0.42
0.44
0.46
0.48
0.5
ch
lori
ne
chlorine = 0.390144+(0.49-0.390144)*exp(-0.101644*(weeks-8))
Example 3: Calibration
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The general calibration problem is that
of determining the likely value of X
given an observed value of Y.
Typically: X = item characteristic, Y =
measured value
Step 1: Build a regression model using
samples with known values of X
(“golden samples”).
Step 2: For another sample with
unknown X, predict X from Y.
Data Input Dialog Box
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Reverse Prediction
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Plot of Fitted Model
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Plot of Fitted Model
measured = -0.0896667 + 1.01433*known
0 2 4 6 8 10
known
0
2
4
6
8
10
12m
easu
red
5.85573 (5.59032,6.1215)
5.85
Example 4: Comparison of
Regression Lines
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Y: amount of scrap produced
X: production line speed
Levels: line number
Data Input Dialog Box
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Analysis Options
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Plot of Fitted Model
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Line1
2
Plot of Fitted Model
100 140 180 220 260 300 340
Speed
140
240
340
440
540
Sc
rap
Significance Tests
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Parallel Slope Model
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Example 5: Multiple Regression
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Stepwise Regression
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Analysis Options
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Selected Variables
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Residual Plot
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All Possible Regressions
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Analysis Options
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Best Adjusted R-Squared Models
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Example 6: Logistic Regression
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Response variable may be in the form of proportions or binary (0/1).
Logistic Model
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)...(exp1
1)(
22110 kk XXXEventP
kk XXXEventP
EventP
...
)(1
)(log 22110
Let P(Event) be the probability an event occurs at specified values of
the independent variables X.
(1)
(2)
Data Input - Proportions
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Analysis Options
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Plot of Fitted Model
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0 20 40 60 80 100
Load
Plot of Fitted Model
with 95.0% confidence limits
0
0.2
0.4
0.6
0.8
1
Fa
ilu
res
/Sp
ec
ime
ns
Statistical Results
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Data Input - Binary
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Analysis Options
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Analysis Summary
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Example 7: Poisson Regression
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Response variable is a count.
Poisson Model
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Values of the response variable are assumed to follow a Poisson
distribution:
kk XXX ...log 22110
The rate parameter is related to the predictor variables through a log-
linear link function:
!Y
eYp
Y
Data Input
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Analysis Options
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Statistical Results
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Plot of Fitted Model
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Thickness=170.0Extraction=75.0Height=55.0
0 10 20 30 40
Years
Plot of Fitted Model
with 95.0% confidence limits
0
1
2
3
4
5In
juri
es
References
Applied Logistic Regression (second edition) – Hosmer and
Lemeshow, Wiley, 2000.
Applied Regression Analysis (third edition) – Draper and
Smith, Wiley, 1998.
Applied Linear Statistical Models (fifth edition) – Kutner et
al., McGraw-Hill, 2004.
Classical and Modern Regression with Applications (second
edition) – Myers, Brookes-Cole, 1990.
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More Information
Go to www.statgraphics.com
Or send e-mail to info@statgraphics.com
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