# Frost J. Regression Analysis. An Intuitive Guide 2019

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My Approach to Teaching Regression and Statistics
Correlation and an Introduction to Regression
Graph Your Data to Find Correlations
Interpret the Pearson’s Correlation Coefficient
Examples of Positive and Negative Correlations
Graphs for Different Correlations
Pearson’s Correlation Measures Linear Relationships
Hypothesis Test for Correlations
Interpreting our Height and Weight Example
Correlation Does Not Imply Causation
How Strong of a Correlation is Considered Good?
Common Themes with Regression
Regression Takes Correlation to the Next Level
Fundamental Terms and Goals of Regression
Dependent Variables
Independent Variables
Simple versus Multiple Regression
Goals of Regression Analysis
Example of a Regression Analysis
Regression Analyzes a Wide Variety of Relationships
Using Regression to Control Independent Variables
What does controlling for a variable mean?
How do you control the other variables in regression?
An Introduction to Regression Output
Review and Next Steps
Regression Basics and How it Works
Data Considerations for OLS
How OLS Fits the Best Line
Observed and Fitted Values
Residuals: Difference between Observed and Fitted Values
Using the Sum of the Squared Errors (SSE) to Find the Best Line
Implications of Minimizing SSE
Other Types of Sums of Squares
Displaying a Regression Model on a Fitted Line Plot
Importance of Staying Close to Your Data
Review and Next Steps
Interpreting Main Effects and Significance
Regression Notation
Fitting Models is an Iterative Process
Three Types of Effects in Regression Models
Main Effects of Continuous Variables
Graphical Representation of Regression Coefficients
Confidence Intervals for Regression Parameters
Example Regression Model with Two Linear Main Effects
Interpreting P-Values for Continuous Independent Variables
Recoding Continuous Independent Variables
Standardizing the Continuous Variables
Interpreting Standardized Coefficients
Why Obtain Standardized Coefficients?
Main Effects of Categorical Variables
Coding Categorical Variables
Interpreting the Results for Categorical Variables
Example of a Model with a Categorical Variable
Controlling for other Variables
Blurring the Continuous and Categorical Line
The Case for Including It as a Continuous Variable
The Case for Including It as a Continuous Variable
Constant (Y Intercept)
The Definition of the Constant is Correct but Misleading
The Y-Intercept Might Be Outside of the Observed Data
The Constant Absorbs the Bias for the Regression Model
Generally, It Is Essential to Include the Constant in a Regression Model
Interpreting the Constant When You Center All the Continuous Independent Variables
Review and Next Steps
Fitting Curvature
Example Curvature
Graphing the Data for Regression with Polynomial Terms
Graph Curvature with Main Effects Plots
Why You Need to Fit Curves in a Regression Model
Difference between Linear and Nonlinear Models
Linear Regression Equations
Nonlinear Regression Equations
Finding the Best Way to Model Curvature
Curve Fitting using Polynomial Terms in Linear Regression
Curve Fitting using Reciprocal Terms in Linear Regression
Curve Fitting with Log Functions in Linear Regression
Curve Fitting with Nonlinear Regression
Comparing the Curve-Fitting Effectiveness of the Different Models
Closing Thoughts
Another Curve Fitting Example
Linear model
Example of a nonlinear regression model
Comparing the Regression Models and Making a Choice
Review and Next Steps
Interaction Effects
Example with Categorical Independent Variables
How to Interpret Interaction Effects
Overlooking Interaction Effects is Dangerous!
Example with Continuous Independent Variables
Important Considerations for Interaction Effects
Interaction effects versus correlation between independent variables
Combinations of significant and insignificant main effects and interaction effects
When an interaction effect is significant but an underlying main effect is not significant, do you remove the main effect from the model?
The coefficient sign for an interaction term isn’t what I expected.
Different statistical software packages estimate different interaction effects for the same dataset.
The lines in my interaction plot don’t cross even though the interaction effect is statistically significant?
The lines in my interaction plot appear to have different slopes, but the interaction term is not significant.
Review and Next Steps
Goodness-of-Fit
Assessing the Goodness-of-Fit
R-squared
Visual Representation of R-squared
R-squared has Limitations
Are Low R-squared Values Always a Problem?
Are High R-squared Values Always Great?
R-squared Is Not Always Straightforward
Some Problems with R-squared
What Is the Predicted R-squared?
Example of an Overfit Model and Predicted R-squared
A Caution about Chasing a High R-squared
Standard Error of the Regression vs. R-squared
Standard Error of the Regression and R-squared in Practice
Example Regression Model: BMI and Body Fat Percentage
I Often Prefer the Standard Error of the Regression
The F-test of Overall Significance
Additional Ways to Interpret the F-test of Overall Significance
Review and Next Steps
The Importance of Graphing Your Data
Statistical Methods for Model Specification
Mallows' Cp
P-values for the independent variables
Stepwise regression and Best subsets regression
Real World Complications
Practical Recommendations
Theory
Simplicity
Residual Plots
Omitted Variable Bias
What Are the Effects of Omitted Variable Bias?
Synonyms for Confounding Variables and Omitted Variable Bias
What Conditions Cause Omitted Variable Bias?
Practical Example of How Confounding Variables Can Produce Bias
How the Omitted Confounding Variable Hid the Relationship
Correlations, Residuals, and OLS Assumptions
Predicting the Direction of Omitted Variable Bias
How to Detect Omitted Variable Bias and Identify Confounding Variables
Obstacles to Correcting Omitted Variable Bias
Recommendations for Addressing Confounding Variables and Omitted Variable Bias
What to Do When Including Confounding Variables is Impossible
Automated Variable Selection Procedures
How Stepwise Regression Works
How Best Subsets Regression Works
Comparing Stepwise to Best Subsets Regression
Using Stepwise and Best Subsets on the Same Dataset
Example of Stepwise Regression
Example of Best Subsets Regression
Using Best Subsets Regression in conjunction with Our Requirements
Assess Your Candidate Regression Models Thoroughly
Stepwise versus Best Subsets
How Accurate is Stepwise Regression?
When stepwise regression is most accurate
The role of the number of candidate variables and authentic variables in stepwise regression accuracy
The role of multicollinearity in stepwise regression accuracy
The role of sample size in stepwise regression accuracy
Closing Thoughts on Choosing the Correct Model
Review and Next Steps
Problematic Methods of Specifying Your Model
Using Data Dredging and Significance
Regression Example that Illustrates the Problems of Data Mining
Using Stepwise Regression on Random Data
Lessons Learned from the Data Mining Example
How Data Mining Causes these Problems
Let Theory Guide You and Avoid Data Mining
Overfitting Regression Models
Graphical Illustration of Overfitting Regression Models
How Overfitting a Model Causes these Problems
Applying These Concepts to Overfitting Regression Models
How to Detect Overfit Models
How to Avoid Overfitting Models
Review and Next Steps
Checking Assumptions and Fixing Problems
Deterministic Component
Stochastic Error
How to Check Residual Plots
How to Fix Problematic Residual Plots
Residual Plots are Easy!
The Seven Classical OLS Assumptions
OLS Assumption 1: The correctly specified regression model is linear in the coefficients and the error term
OLS Assumption 2: The error term has a population mean of zero
OLS Assumption 3: All independent variables are uncorrelated with the error term
OLS Assumption 4: Observations of the error term are uncorrelated with each other
OLS Assumption 5: The error term has a constant variance (no heteroscedasticity)
OLS Assumption 6: No independent variable is a perfect linear function of other explanatory variables
OLS Assumption 7: The error term is normally distributed (optional)
Why You Should Care About the Classical OLS Assumptions
Next Steps
Heteroscedasticity
How to Identify Heteroscedasticity with Residual Plots
What Causes Heteroscedasticity?
Heteroscedasticity in cross-sectional studies
Heteroscedasticity in time-series models
Example of heteroscedasticity
Pure versus impure heteroscedasticity
What Problems Does Heteroscedasticity Cause?
How to Fix Heteroscedasticity
Redefining the variables
Weighted least squares regression
Transform the dependent variable
Multicollinearity
Why is Multicollinearity a Potential Problem?
What Problems Do Multicollinearity Cause?
Do I Have to Fix Multicollinearity?
Testing for Multicollinearity with Variance Inflation Factors (VIFs)
Multicollinearity Example: Predicting Bone Density in the Femur
Center the Independent Variables to Reduce Structural Multicollinearity
Regression with Centered Variables
Comparing Regression Models to Reveal Multicollinearity Effects
How to Deal with Multicollinearity
Next Steps
Unusual Observations
Observations in Regression
Unusual Observations
Outliers (Unusual Y-values)
High Leverage Observations (Unusual X-values)
Influential Points
Managing Unusual Observations and Influential Points
Next Steps
Using Data Transformations to Fix Problems
Determining which Variables to Transform
Determining which Transformation to Use
Box-Cox Transformation
Johnson Transformation
How to Interpret the Results for Transformed Data
Use data transformation as a last resort!
Cheat Sheet for Detecting and Solving Problems
Using Regression to Make Predictions
Explanatory versus Predictive Models
The Regression Approach for Predictions
Example Scenario for Regression Predictions
Finding a Good Regression Model for Predictions
Assess the Residual Plots
Interpret the Regression Output
Other Considerations for Valid Predictions
Using our Regression Model to Make Predictions
Interpreting the Regression Prediction Results
Next Steps: Don’t Focus On Only the Fitted Values
The Illusion of Predictability
Studying How Experts Perceive Prediction Uncertainty
Use a Regression Model to Make a Decision
The Difference between Perception and Reality
Low R-squared Values Should Have Warned of Low Precision
Graph the Model to Highlight the Variability
Graphs Are One Way to Pierce the Illusion of Predictability
Display Prediction Intervals on Fitted Line Plots to Assess Precision
Different Example of Using Prediction Intervals
Tips, Common Questions, and Concerns
Five Tips to Avoid Common Problems
Tip 1: Conduct A Lot of Research Before Starting
Tip 2: Use a Simple Model When Possible
Tip 3: Correlation Does Not Imply Causation . . . Even in Regression
Tip 4: Include Graphs, Confidence, and Prediction Intervals in the Results
Tip 5: Check Your Residual Plots!
Differences Between a Top Analyst and a Less Rigorous Analyst
Identifying the Most Important Variables
Do Not Associate Regular Regression Coefficients with the Importance of Independent Variables
Do Not Link P-values to Importance
Do Assess These Statistics to Identify Variables that might be Important
Standardized coefficients
Change in R-squared for the last variable added to the model
Example of Identifying the Most Important Independent Variables in a Regression Model
Cautions for Using Statistics to Pinpoint Important Variables
Non-Statistical Issues that Help Find Important Variables
Comparing Regression Lines with Hypothesis Tests
Hypothesis Tests for Comparing Regression Constants
Interpreting the Results
Hypothesis Tests for Comparing Regression Coefficients
Interpreting the Results
How High Does R-squared Need to Be?
How High Does R-squared Need to be is the Wrong Question
Define Your Objectives for the Regression Model
R-squared and Understanding the Relationships between the Variables
R-squared and Predicting the Dependent Variable
Using Prediction intervals to Assess Precision
R-squared Is Overrated!
Five Reasons Why R-squared can be Too High
High R-squared Values can be a Problem
Reason 1: R-squared is a biased estimate
Reason 3: Data mining and chance correlations
Reason 4: Trends in Panel (Time Series) Data
Reason 5: Form of a Variable
Interpreting Models that have Significant Variables but a Low R-squared
Comparing Regression Models with Low and High R-squared Values
Regression Model Similarities
Regression Model Differences
Using Prediction Intervals to See the Variability
Key Points about Low R-squared Values
Choosing the Correct Type of Regression
Continuous Dependent Variables
Linear regression
Nonlinear regression
Categorical Dependent Variables
Binary Logistic Regression
Ordinal Logistic Regression
Nominal Logistic Regression
Count Dependent Variables
Poisson regression
Alternatives to Poisson regression for count data
Examples of Other Types of Regression
Using Log-Log Plots to Determine Whether Size Matters
Does the Mass of Mammals Affect Their Metabolism?
Example: Log-Log Plot of Mammal Mass and Basal Metabolic Rate
Example: Log-Log Plot of Basal Metabolic Rate and Longevity
Binary Logistic Regression: Statistical Analysis of the Republican Establishment Split
How Does the Freedom Caucus Fit In?
Data for these Analyses
Graphing the House Republican Data
Binary Logistic Regression Model of Freedom Caucus Membership
Graphing the Results
References

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\$14.00 \$3.00