8.2 Maximum likelihood estimation and interpretation of coefficients

Logistic regression uses maximum likelihood estimation to find the best-fitting coefficients for predicting binary outcomes. This method maximizes the probability of observing the given data under the assumed model, iteratively adjusting coefficients until convergence.

Interpreting logistic regression coefficients is crucial for understanding predictor impacts. Coefficients represent changes in log-odds, while exponentiated coefficients give odds ratios. Visualizations and marginal effects help illustrate non-linear relationships between predictors and outcome probabilities.

Maximum Likelihood Estimation in Logistic Regression

Principles of Maximum Likelihood Estimation

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• Maximum likelihood estimation (MLE) estimates logistic regression parameters by maximizing the likelihood function
• Likelihood function represents probability of observing given data under assumed model and parameter values
• MLE finds coefficients making observed data most probable or likely
• Process iteratively adjusts coefficients to maximize log-likelihood function using numerical optimization algorithms (Newton-Raphson, gradient descent)
• Provides asymptotically unbiased, efficient, and normally distributed estimates of logistic regression coefficients
• Based on assumption that observed data follows Bernoulli distribution
• Convergence criteria determine when maximum likelihood estimates sufficiently approximated in iterative process

Constructing and Optimizing the Likelihood Function

• Likelihood function constructed as product of Bernoulli probabilities for each observation in dataset
• Log-likelihood function used for computational convenience, transforming product into sum of log probabilities
• Initial coefficient values chosen, often using ordinary least squares estimates as starting points
• Algorithm iteratively updates coefficient estimates until convergence achieved
• Convergence based on predefined criteria (changes in log-likelihood or parameter estimates)
• Standard errors of coefficient estimates derived from observed information matrix (negative of Hessian of log-likelihood function)
• Statistical software packages provide built-in functions to perform MLE for logistic regression, automating process for users

Interpreting Logistic Regression Coefficients

Understanding Coefficient Meaning and Interpretation

• Coefficients represent change in log-odds of outcome for one-unit increase in corresponding predictor variable, holding other variables constant
• Exponential of coefficient (e^β) gives odds ratio, indicating multiplicative change in odds for one-unit increase in predictor
• Positive coefficients associated with increased probability of outcome, negative coefficients with decreased probability
• Magnitude of coefficient reflects strength of relationship between predictor and outcome
• Larger absolute values indicate stronger associations
• Standardized coefficients compare relative importance of predictors measured on different scales

Statistical Significance and Confidence Intervals

• Statistical significance of coefficients assessed using Wald tests or likelihood ratio tests
• P-values indicate probability of observing such coefficients under null hypothesis
• Confidence intervals provide range of plausible values for true population parameter
• Narrower intervals indicate more precise estimates
• Typical confidence levels used (95%, 99%)
• Interpretation example: "We are 95% confident that the true population coefficient lies between X and Y"

Predictor Variable Impact on Probability

Non-linear Relationships and Marginal Effects

• Impact of predictor variables on outcome probability non-linear in logistic regression due to logistic function
• Marginal effects calculate change in probability for unit change in predictor, holding other variables constant at specific values
• Average marginal effects provide summary measure of predictor impact across all observations in dataset
• Predicted probabilities computed for specific combinations of predictor values to illustrate model's predictions
• Interaction terms assess how impact of one predictor on outcome probability depends on values of another predictor

Visualizing and Interpreting Predictor Effects

• Graphical methods visualize relationship between predictors and outcome probability (probability plots, effect plots)
• Odds and log-odds crucial for understanding how predictor variables influence binary outcome
• Probability plots show predicted probabilities across range of predictor values
• Effect plots display marginal effects or average marginal effects for different predictors
• Example: Plot showing how predicted probability of event changes as continuous predictor increases, holding other variables constant

Estimating Logistic Regression Coefficients

Numerical Optimization Techniques

• Numerical optimization algorithms (Newton-Raphson, gradient descent) employed to maximize log-likelihood function
• Newton-Raphson method uses first and second derivatives of log-likelihood function to update parameter estimates
• Gradient descent uses first derivative (gradient) to iteratively move towards maximum
• Algorithms balance computational efficiency and convergence speed
• Choice of algorithm may depend on dataset size, number of predictors, and computational resources available

Practical Considerations in Coefficient Estimation

• Software packages automate MLE process for logistic regression (R, Python, SAS, SPSS)
• Users typically specify model formula, dataset, and optional settings (convergence criteria, maximum iterations)
• Importance of checking convergence and model diagnostics after estimation
• Potential issues: separation, multicollinearity, influential observations
• Strategies for addressing estimation problems (regularization techniques, variable selection)
• Cross-validation or bootstrapping to assess model stability and generalizability