Regression analysis in survey research is a powerful tool for understanding relationships between variables. It allows researchers to predict outcomes and examine the impact of multiple factors simultaneously, providing valuable insights into complex social phenomena.

When working with survey data, regression techniques must be adapted to account for sampling design and weights. This ensures accurate estimates and valid statistical inferences, reflecting the true population characteristics rather than just the sample.

Linear and Logistic Regression Models

Fundamentals of Linear Regression

• Linear regression models the relationship between a dependent variable and one or more independent variables using a linear equation
• Dependent variable represents the outcome or response being predicted
• Independent variables act as predictors or explanatory factors in the model
• Linear equation takes the form $Y = β₀ + β₁X₁ + β₂X₂ + ... + βₙXₙ + ε$
  • Y: dependent variable
  • X: independent variables
  • β: coefficients
  • ε: error term
• Coefficient of determination (R-squared) measures the proportion of variance in the dependent variable explained by the independent variables
  • Ranges from 0 to 1, with higher values indicating better model fit
• Residuals represent the differences between observed and predicted values
  • Used to assess model assumptions and identify outliers

Logistic Regression for Binary Outcomes

• Logistic regression predicts the probability of a binary outcome based on one or more independent variables
• Used when the dependent variable is categorical with two possible outcomes (yes/no, success/failure)
• Employs a logistic function to model the relationship between variables
• Logistic function: $P(Y=1) = \frac{1}{1 + e^{-(β₀ + β₁X₁ + β₂X₂ + ... + βₙXₙ)}}$
• Interprets results using odds ratios and predicted probabilities
• Assesses model fit using measures like pseudo R-squared and likelihood ratio tests

Multiple Regression and Model Considerations

Advanced Regression Techniques

• Multiple regression extends simple linear regression to include two or more independent variables
• Allows for simultaneous examination of multiple predictors' effects on the dependent variable
• Equation: $Y = β₀ + β₁X₁ + β₂X₂ + ... + βₙXₙ + ε$
• Interaction effects occur when the relationship between an independent variable and the dependent variable changes based on the value of another independent variable
  • Modeled by including product terms in the regression equation
• Dummy variables represent categorical variables in regression models
  • Created by assigning binary codes (0 or 1) to different categories
  • Allows inclusion of non-numeric variables in regression analysis

Addressing Regression Assumptions and Issues

• Multicollinearity occurs when independent variables are highly correlated with each other
  • Can lead to unreliable coefficient estimates and inflated standard errors
  • Detected using variance inflation factor (VIF) or correlation matrices
• Heteroscedasticity refers to unequal variance of residuals across the range of predicted values
  • Violates the assumption of constant variance in regression models
  • Addressed through robust standard errors or weighted least squares
• Other considerations include:
  • Normality of residuals
  • Linearity of relationships
  • Independence of observations

Regression with Complex Survey Data

Incorporating Survey Design in Regression Analysis

• Weighted least squares regression accounts for unequal sampling probabilities in survey data
  • Assigns weights to observations based on their representation in the population
  • Improves the accuracy of parameter estimates and standard errors
• Survey weights in regression adjust for:
  • Unequal selection probabilities
  • Non-response
  • Post-stratification
• Incorporating weights modifies the estimation procedure:
  • $\hat{\beta} = (X'WX)^{-1}X'WY$
    • W: diagonal matrix of survey weights
• Complex survey design effects impact standard errors and confidence intervals
  • Clustering and stratification in survey designs affect the precision of estimates

Adjusting for Complex Survey Designs

• Design-based approach accounts for survey design features in variance estimation
  • Uses techniques like Taylor series linearization or replication methods
• Specialized software packages (SUDAAN, Stata's svy commands) facilitate regression analysis with complex survey data
• Effective degrees of freedom may be reduced due to design effects
  • Affects hypothesis testing and confidence interval construction
• Goodness-of-fit measures require modification for weighted regression models
  • Pseudo R-squared and F-tests adapted for complex survey data