1 chapter 4: introduction to predictive modeling: regressions 4.1 introduction 4.2 selecting...
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Chapter 4: Introduction to Predictive Modeling: Regressions
4.1 Introduction
4.2 Selecting Regression Inputs
4.3 Optimizing Regression Complexity
4.4 Interpreting Regression Models
4.5 Transforming Inputs
4.6 Categorical Inputs
4.7 Polynomial Regressions (Self-Study)
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Chapter 4: Introduction to Predictive Modeling: Regressions
4.1 Introduction4.1 Introduction
4.2 Selecting Regression Inputs
4.3 Optimizing Regression Complexity
4.4 Interpreting Regression Models
4.5 Transforming Inputs
4.6 Categorical Inputs
4.7 Polynomial Regressions (Self-Study)
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Model Essentials – Regressions
Predict new cases.
Select useful inputs.
Optimize complexity.
...
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Model Essentials – Regressions
Best modelfrom sequence
Sequentialselection
Predict new cases.
Select useful inputs
Optimize complexity
Select useful inputs.
Optimize complexity.
...
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Linear Regression Prediction Formula
parameterestimate
inputmeasurement
interceptestimate
= w0 + w1 x1 + w2 x2 ^ ^ ^y · · prediction
estimate^
Choose intercept and parameter estimates to minimize:
∑( yi – yi )2
trainingdata
^squared error function
...
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Linear Regression Prediction Formula
parameterestimate
inputmeasurement
interceptestimate
= w0 + w1 x1 + w2 x2 ^ ^ ^y · · prediction
estimate^
Choose intercept and parameter estimates to minimize.
∑( yi – yi )2
trainingdata
^squared error function
...
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Logistic Regression Prediction Formula
= w0 + w1 x1 + w2 x2 ^ ^ ^· · logit scores
...
^log
p
1 – p( )^
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Logit Link Function
= w0 + w1 x1 + w2 x2 ^ ^ ^· ·
...
logitlink function
0 1
5
-5
The logit link function transforms probabilities (between 0 and 1) to logit scores (between −∞ and +∞).
^log
p
1 – p( )^
logit scores
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Logit Link Function
= w0 + w1 x1 + w2 x2 ^ ^ ^· · logit scores
...
logitlink function
0 1
5
-5
The logit link function transforms probabilities (between 0 and 1) to logit scores (between −∞ and +∞).
^log
p
1 – p( )^
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Logit Link Function
= w0 + w1 x1 + w2 x2 ^ ^ ^· ·
...
^log
p
1 – p( )^
1
1 + e-logit( p )p = ^^
^logit( p )
To obtain prediction estimates, the logit equation is solved for p. ^
=
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Logit Link Function
= w0 + w1 x1 + w2 x2 ^ ^ ^· ·
...
^log
p
1 – p( )^
1
1 + e-logit( p )p = ^^
^logit( p )
To obtain prediction estimates, the logit equation is solved for p. ^
=
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Logit Link Function
...
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Simple Prediction Illustration – Regressions Predict dot color for each x1 and x2.
You need intercept and parameter estimates.
...
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Simple Prediction Illustration – Regressions
You need intercept and parameter estimates.
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Simple Prediction Illustration – Regressions
log-likelihood function
Find parameter estimates by maximizing
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Simple Prediction Illustration – Regressions
log-likelihood function
Find parameter estimates by maximizing
...
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x1
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Simple Prediction Illustration – Regressions
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x1
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Using the maximum likelihood estimates, the prediction formula assigns a logit score to each x1 and x2.
...
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4.01 Multiple Choice PollWhat is the logistic regression prediction for the indicated point?
a. 0.243
b. 0.56
c. yellow
d. It depends.
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4.01 Multiple Choice Poll – Correct AnswerWhat is the logistic regression prediction for the indicated point?
a. 0.243
b. 0.56
c. yellow
d. It depends.
0.0 0.50.1 0.2 0.3 0.4 0.6 0.7 0.8 0.9 1.0
x1
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Regressions: Beyond the Prediction Formula
Manage missing values.
Interpret the model.
Account for nonlinearities.
Handle extreme or unusual values.
Use nonnumeric inputs.
...
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Regressions: Beyond the Prediction Formula
Manage missing values.
Interpret the model.
Account for nonlinearities.
Handle extreme or unusual values.
Use nonnumeric inputs.
...
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Missing Values and Regression Modeling
Training Datatargetinputs
Problem 1: Training data cases with missing values on inputs used by a regression model are ignored.
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Consequence: missing values can significantly reduce your amount of training data for regression modeling!
Missing Values and Regression Modeling
Training Datatargetinputs
Problem 1: Training data cases with missing values on inputs used by a regression model are ignored.
...
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Missing Values and Regression Modeling
Consequence: Missing values can significantly reduce your amount of training data for regression modeling!
Training Datatargetinputs
...
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Missing Values and the Prediction Formula
Predict: (x1, x2) = (0.3, ? )
Problem 2: Prediction formulas cannot score cases with missing values.
...
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Missing Values and the Prediction Formula
Predict: (x1, x2) = (0.3, ? )
Problem 2: Prediction formulas cannot score cases with missing values.
...
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Missing Values and the Prediction Formula
...
Problem 2: Prediction formulas cannot score cases with missing values.
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Missing Values and the Prediction Formula
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Problem 2: Prediction formulas cannot score cases with missing values.
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Missing Value Issues
Manage missing values.
Problem 1: Training data cases with missing values on inputs used by a regression model are ignored.
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Problem 2: Prediction formulas cannot score cases with missing values.
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Missing Value Issues
Manage missing values.
Problem 1: Training data cases with missing values on inputs used by a regression model are ignored.
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Problem 2: Prediction formulas cannot score cases with missing values.
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Missing Value Causes
Manage missing values.
Non-applicable measurement
No match on merge
Non-disclosed measurement
...
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Missing Value Remedies
Manage missing values.
xi = f(x1, … ,xp)
Non-applicable measurement
No match on merge
Non-disclosed measurement
...
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Managing Missing Values
This demonstration illustrates how to impute synthetic data values and create missing value indicators.
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Running the Regression Node
This demonstration illustrates using the Regression tool.
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Chapter 4: Introduction to Predictive Modeling: Regressions
4.1 Introduction
4.2 Selecting Regression Inputs4.2 Selecting Regression Inputs
4.3 Optimizing Regression Complexity
4.4 Interpreting Regression Models
4.5 Transforming Inputs
4.6 Categorical Inputs
4.7 Polynomial Regressions (Self-Study)
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Predictionformula
Model Essentials – Regressions
Best modelfrom sequence
Sequentialselection
Predict new cases.
Select useful inputs
Optimize complexity.
Select useful inputs.
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Sequential Selection – Forward
Entry CutoffInput p-value
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Sequential Selection – Forward
Entry CutoffInput p-value
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Sequential Selection – Forward
Entry CutoffInput p-value
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Sequential Selection – Forward
Entry CutoffInput p-value
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Sequential Selection – Forward
Entry CutoffInput p-value
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Sequential Selection – Backward
Stay CutoffInput p-value
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Sequential Selection – Backward
Stay CutoffInput p-value
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Sequential Selection – Backward
Stay CutoffInput p-value
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Sequential Selection – Backward
Stay CutoffInput p-value
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Sequential Selection – Backward
Stay CutoffInput p-value
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Sequential Selection – Backward
Stay CutoffInput p-value
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Sequential Selection – Backward
Stay CutoffInput p-value
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Sequential Selection – Backward
Stay CutoffInput p-value
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Sequential Selection – StepwiseInput p-value Entry Cutoff
Stay Cutoff
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Sequential Selection – StepwiseInput p-value Entry Cutoff
Stay Cutoff
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Sequential Selection – StepwiseInput p-value Entry Cutoff
Stay Cutoff
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Sequential Selection – StepwiseInput p-value Entry Cutoff
Stay Cutoff
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Sequential Selection – StepwiseInput p-value Entry Cutoff
Stay Cutoff
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Sequential Selection – StepwiseInput p-value Entry Cutoff
Stay Cutoff
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Sequential Selection – StepwiseInput p-value Entry Cutoff
Stay Cutoff
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4.02 PollThe three sequential selection methods for building regression models can never lead to the same model for the same set of data.
True
False
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4.02 Poll – Correct AnswerThe three sequential selection methods for building regression models can never lead to the same model for the same set of data.
True
False
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Selecting Inputs
This demonstration illustrates using stepwise selection to choose inputs for the model.
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Chapter 4: Introduction to Predictive Modeling: Regressions
4.1 Introduction
4.2 Selecting Regression Inputs
4.3 Optimizing Regression Complexity4.3 Optimizing Regression Complexity
4.4 Interpreting Regression Models
4.5 Transforming Inputs
4.6 Categorical Inputs
4.7 Polynomial Regressions (Self-Study)
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Model Essentials – Regressions
Predict new cases.
Select useful inputs.
Optimize complexity.
Predictionformula
Sequentialselection
...
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Model Fit versus Complexity
1 2 3 4 5 6
Model fit statistic
training
validation
...
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Select Model with Optimal Validation Fit
1 2 3 4 5 6
Model fit statistic
Evaluate eachsequence step.
...
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Optimizing Complexity
This demonstration illustrates tuning a regression model to give optimal performance on the validation data.
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Chapter 4: Introduction to Predictive Modeling: Regressions
4.1 Introduction
4.2 Selecting Regression Inputs
4.3 Optimizing Regression Complexity
4.4 Interpreting Regression Models4.4 Interpreting Regression Models
4.5 Transforming Inputs
4.6 Categorical Inputs
4.7 Polynomial Regressions (Self-Study)
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Beyond the Prediction Formula
Manage missing values.
Interpret the model.
Account for nonlinearities.
Handle extreme or unusual values.
Use nonnumeric inputs.
...
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Beyond the Prediction Formula
Manage missing values
Interpret the model.
Account for nonlinearities.
Handle extreme or unusual values.
Use nonnumeric inputs.
...
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Logistic Regression Prediction Formula
...
= w0 + w1 x1 + w2 x2 ^ ^ ^· ·
^log
p
1 – p( )^
logit scores
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Odds Ratios and Doubling Amounts
Odds ratio: Amount odds change with unit change in input.Doubling amount:
How much does an input have to change to double the odds?
1 odds exp(wi)
odds 20.69wi
Δxi consequence
...
= w0 + w1 x1 + w2 x2 ^ ^ ^· ·
^log
p
1 – p( )^
logit scores
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Interpreting a Regression Model
This demonstration illustrates interpreting a regression model using odds ratios.
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Chapter 4: Introduction to Predictive Modeling: Regressions
4.1 Introduction
4.2 Selecting Regression Inputs
4.3 Optimizing Regression Complexity
4.4 Interpreting Regression Models
4.5 Transforming Inputs4.5 Transforming Inputs
4.6 Categorical Inputs
4.7 Polynomial Regressions (Self-Study)
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Beyond the Prediction Formula
Manage missing values.
Interpret the model.
Account for nonlinearities.
Handle extreme or unusual values.
Use nonnumeric inputs.
...
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Extreme Distributions and Regressions
high leverage pointsskewed inputdistribution
standard regression
true association
standard regression
true association
Original Input Scale
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Extreme Distributions and Regressions
high leverage pointsskewed inputdistribution
standard regression
true association
standard regression
true association
Original Input Scale
more symmetricdistribution
Regularized Scale
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Original Input Scale
Regularizing Input Transformations
more symmetricdistribution
Regularized Scale
standard regression
standard regression
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Original Input Scale
high leverage pointsskewed inputdistribution
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Regularizing Input TransformationsRegularized Scale
standard regression
standard regression
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Original Input ScaleOriginal Input Scale
regularized estimate
regularized estimate
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Regularizing Input TransformationsRegularized Scale
standard regression
standard regression
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Original Input Scale
regularized estimate
regularized estimate
true association
true association
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4.03 Multiple Choice PollWhich statement below is true about transformations of input variables in a regression analysis?
a. They are never a good idea.
b. They help model assumptions match the assumptions of maximum likelihood estimation.
c. They are performed to reduce the bias in model predictions.
d. They typically are done on nominal (categorical) inputs.
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4.03 Multiple Choice Poll – Correct AnswerWhich statement below is true about transformations of input variables in a regression analysis?
a. They are never a good idea.
b. They help model assumptions match the assumptions of maximum likelihood estimation.
c. They are performed to reduce the bias in model predictions.
d. They typically are done on nominal (categorical) inputs.
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Transforming Inputs
This demonstration illustrates using the Transform Variables tool to apply standard transformations to a set of inputs.
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Chapter 4: Introduction to Predictive Modeling: Regressions
4.1 Introduction
4.2 Selecting Regression Inputs
4.3 Optimizing Regression Complexity
4.4 Interpreting Regression Models
4.5 Transforming Inputs
4.6 Categorical Inputs4.6 Categorical Inputs
4.7 Polynomial Regressions (Self-Study)
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Beyond the Prediction Formula
Manage missing values.
Interpret the model.
Account for nonlinearities.
Handle extreme or unusual values.
Use nonnumeric inputs.
...
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Beyond the Prediction Formula
Manage missing values.
Interpret the model.
Account for nonlinearities.
Handle extreme or unusual values.
Use nonnumeric inputs.
...
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Nonnumeric Input Coding
Level DI
1 0 0 0 0 0 0 0
DA DB DC DD DE DF DG DH
0
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 00 0 1 0 0 0 0 0
0 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 0 0 0 0
00000001
ABCDEFGHI
...
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DI
000000001
DI
000000001
Coding Redundancy
Level
1 0 0 0 0 0 0 0
DA DB DC DD DE DF DG DH
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 00 0 1 0 0 0 0 0
0 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 0 0 0 0
ABCDEFGHI
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DI
000000001
Coding Consolidation
Level
1 0 0 0 0 0 0 0
DA DB DC DD DE DF DG DH
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 00 0 1 0 0 0 0 0
0 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 0 0 0 0
ABCDEFGHI
...
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DI
000000001
Coding Consolidation
Level
1 0 0 0 0 0 0 0
DABCD DB DC DD DEF DF DGH DH
1 0 0 1 0 0 0 0
1 1 0 0 0 0 0 01 0 1 0 0 0 0 0
0 0 0 0 1 0 0 00 0 0 0 1 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 1 10 0 0 0 0 0 0 0
ABCDEFGHI
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Recoding Categorical Inputs
This demonstration illustrates using the Replacement tool to facilitate the process of combining input levels.
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Chapter 4: Introduction to Predictive Modeling: Regressions
4.1 Introduction
4.2 Selecting Regression Inputs
4.3 Optimizing Regression Complexity
4.4 Interpreting Regression Models
4.5 Transforming Inputs
4.6 Categorical Inputs
4.7 Polynomial Regressions (Self-Study)4.7 Polynomial Regressions (Self-Study)
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Beyond the Prediction Formula
Manage missing values.
Interpret the model.
Account for nonlinearities.
Handle extreme or unusual values.
Use nonnumeric inputs.
...
95
Beyond the Prediction Formula
Manage missing values.
Interpret the model.
Account for nonlinearities.
Handle extreme or unusual values.
Use nonnumeric inputs.
...
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Standard Logistic Regression
= w0 + w1 x1 + w2 x2 ^
^ ^ ^log p
1 – p( )^ ·
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x1
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Polynomial Logistic Regression
= w0 + w1 x1 + w2 x2 ^
^ ^ ^log p
1 – p( )^ · ·
quadratic terms
+ w3 x1 + w4 x2 2 2^ ^
+ w5 x1 x2
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x1
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1.0
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0.40 0.50 0.60 0.700.30
0.60
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...
^
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Adding Polynomial Regression Terms Selectively
This demonstration illustrates how to add polynomial regression terms selectively.
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Adding Polynomial Regression Terms Autonomously (Self-Study)
This demonstration illustrates how to add polynomial regression terms autonomously.
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Exercises
This exercise reinforces the concepts discussed previously.
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Regression Tools ReviewReplace missing values for interval (means) and categorical data (mode). Create a unique replacement indicator.
Create linear and logistic regression models. Select inputs with a sequential selection method and appropriate fit statistic. Interpret models with odds ratios.
Regularize distributions of inputs. Typical transformations control for input skewness via a log transformation.
continued...
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Regression Tools Review
Consolidate levels of a nonnumeric input using the Replacement Editor window.
Add polynomial terms to a regression either by hand or by an autonomous exhaustive search.