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Logit Models
An Introduction


Prof. David Bernstein
James Madison University

Computer Science Department
bernstdh@jmu.edu

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Motivation
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  • Review:
    • Regression analysis can be used to determine the dependence of one variable (called the endogenous or dependent variable) on other variables (called the exogenous or independent or explanatory variables)
    • Linear regression models are linear in parameters, assume the dependent variable is continuous, and the errors are identically and normally distributed
  • Our Interest:
    • Situations in which the dependent variable is not continuous (e.g., 0-1)
A 0-1 Example
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The Data

images/logit_scatterplot.png
A 0-1 Example (cont.)
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Using Linear Regression

images/logit_linear-regression.png
Implications of the 0-1 Example for Linear Regression
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  • Improper Probabilities:
    • The predicted probabilities can be outside of \([0, 1]\)
  • Heteroscedasticity:
    • The variance of the dependent variable is different for different values of the independent variable
  • Non-Normality:
    • The error terms are not normally distributed
Toward an Alternative Approach
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  • Model the Probabilities:
    • \(\frac{p_i}{1-p_i} = e^{\alpha + \beta X_i + \epsilon_i}\)
  • Interpretation of the Results:
    • \(e^{\beta}\) is the effect of the independent variable on the odds ratio (i.e., the probability of the event divided by the probability of the non-event)
The 0-1 Example Revisited
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Using Logistic Regression (i.e., A Binary Logit Model)

images/logit_logistic-regression.png
There's Always More to Learn
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