2.4 Regression analysis

Regression analysis is a powerful statistical tool used to model relationships between variables. It helps us understand how changes in one or more independent variables affect a dependent variable, allowing for predictions and insights across various fields.

This section covers different types of regression, key assumptions, parameter estimation, and model evaluation. We'll explore linear and nonlinear models, simple and multiple regression, and techniques for handling categorical predictors and complex relationships.

Types of regression analysis

• Regression analysis is a statistical modeling technique used to examine the relationship between a dependent variable and one or more independent variables
• It helps to understand how changes in the independent variables are associated with changes in the dependent variable
• Regression models can be used for prediction, forecasting, and inferring causal relationships in various fields such as economics, social sciences, and engineering

Linear vs nonlinear regression

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• Linear regression assumes a linear relationship between the dependent variable and independent variables
  • The model is represented by the equation $y = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_px_p + \epsilon$, where $y$ is the dependent variable, $x_i$ are the independent variables, $\beta_i$ are the coefficients, and $\epsilon$ is the error term
• Nonlinear regression models the relationship between variables using nonlinear functions (exponential, logarithmic, polynomial)
  • These models are more flexible and can capture complex relationships that linear models cannot
• The choice between linear and nonlinear regression depends on the nature of the relationship between the variables and the underlying assumptions

Simple vs multiple regression

• Simple regression involves only one independent variable and one dependent variable
  • It examines the relationship between two variables and can be represented by the equation $y = \beta_0 + \beta_1x + \epsilon$
• Multiple regression involves two or more independent variables and one dependent variable
  • It allows for the analysis of the combined effect of multiple predictors on the dependent variable
• Multiple regression can help identify the relative importance of each predictor and control for confounding variables

Logistic regression for classification

• Logistic regression is a type of regression analysis used for binary classification problems
  • The dependent variable is categorical and takes on two values (0 or 1, yes or no)
• The logistic regression model estimates the probability of an event occurring based on the values of the independent variables
  • The model equation is $\log(\frac{p}{1-p}) = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_px_p$, where $p$ is the probability of the event occurring
• Logistic regression is widely used in medical research (predicting disease outcomes), marketing (customer churn prediction), and social sciences (voting behavior analysis)

Assumptions of regression models

• Regression models rely on certain assumptions to ensure the validity and reliability of the results
• Violating these assumptions can lead to biased estimates, incorrect standard errors, and misleading conclusions
• It is essential to check and address any violations of assumptions to obtain accurate and meaningful insights from the regression analysis

Independence of observations

• The observations in the dataset should be independent of each other
  • Each observation should not be influenced by or related to other observations
• Violation of independence can occur due to clustering, repeated measures, or temporal dependencies
• Techniques such as mixed-effects models or time series analysis can be used to handle dependent observations

Linearity between variables

• The relationship between the dependent variable and independent variables should be linear
  • A scatterplot of the variables can help visualize the linearity assumption
• If the relationship is nonlinear, transformations (logarithmic, polynomial) or nonlinear regression models may be more appropriate
• Residual plots can also be used to assess linearity by checking for patterns or trends in the residuals

Homoscedasticity of residuals

• Homoscedasticity assumes that the variance of the residuals is constant across all levels of the independent variables
  • The spread of the residuals should be consistent and not vary systematically with the predicted values
• Heteroscedasticity (non-constant variance) can be detected using residual plots or statistical tests (Breusch-Pagan, White's test)
• Remedies for heteroscedasticity include weighted least squares, robust standard errors, or transforming the dependent variable

Normality of residuals

• The residuals should follow a normal distribution with a mean of zero
  • This assumption is required for valid hypothesis testing and confidence intervals
• Normality can be assessed using histograms, Q-Q plots, or statistical tests (Shapiro-Wilk, Kolmogorov-Smirnov)
• Non-normality can be addressed by transforming the dependent variable, using robust regression methods, or considering alternative error distributions

Estimating regression parameters

• Regression parameters, including the intercept and coefficients, need to be estimated from the data
• The goal is to find the parameter values that minimize the difference between the observed and predicted values of the dependent variable
• Two common methods for estimating regression parameters are ordinary least squares (OLS) and maximum likelihood estimation (MLE)

Ordinary least squares (OLS)

• OLS is a widely used method for estimating regression parameters
  • It minimizes the sum of squared residuals, which are the differences between the observed and predicted values of the dependent variable
• The OLS estimates are obtained by solving the normal equations, which are derived from the least squares criterion
  • The estimates are unbiased and have the lowest variance among all linear unbiased estimators (Gauss-Markov theorem)
• OLS assumes that the errors are uncorrelated, have constant variance, and follow a normal distribution

Maximum likelihood estimation (MLE)

• MLE is a general approach for estimating parameters by maximizing the likelihood function
  • The likelihood function measures the probability of observing the data given the parameter values
• MLE finds the parameter values that make the observed data most likely to occur
  • It involves solving the likelihood equations, which are obtained by setting the partial derivatives of the log-likelihood function to zero
• MLE is more flexible than OLS and can be used for a wider range of models, including logistic regression and generalized linear models

Evaluating regression models

• Evaluating the performance and goodness-of-fit of regression models is crucial for assessing their validity and usefulness
• Several metrics and techniques can be used to evaluate regression models, including the coefficient of determination, adjusted R^2, residual analysis, and hypothesis testing

Coefficient of determination (R^2)

• R^2 measures the proportion of variance in the dependent variable that is explained by the independent variables
  • It ranges from 0 to 1, with higher values indicating a better fit
• R^2 is calculated as the ratio of the explained sum of squares (ESS) to the total sum of squares (TSS)
  • $R^2 = \frac{ESS}{TSS} = 1 - \frac{RSS}{TSS}$, where RSS is the residual sum of squares
• R^2 has limitations, such as increasing with the addition of more variables and not accounting for model complexity

Adjusted R^2 for model comparison

• Adjusted R^2 is a modified version of R^2 that adjusts for the number of independent variables in the model
  • It penalizes the addition of unnecessary variables and helps compare models with different numbers of predictors
• Adjusted R^2 is calculated as $1 - \frac{(1-R^2)(n-1)}{n-p-1}$, where $n$ is the sample size and $p$ is the number of independent variables
• A higher adjusted R^2 indicates a better balance between model fit and complexity

Residual analysis and diagnostics

• Residual analysis involves examining the residuals (observed minus predicted values) to assess model assumptions and identify potential issues
  • Residual plots can reveal patterns, outliers, or heteroscedasticity
• Diagnostic plots, such as residual vs. fitted plots, Q-Q plots, and scale-location plots, help visualize the residuals and detect violations of assumptions
• Influential observations and leverage points can be identified using measures like Cook's distance and hat values

Hypothesis testing for coefficients

• Hypothesis testing is used to assess the statistical significance of individual regression coefficients
  • It determines whether the estimated coefficients are significantly different from zero
• The null hypothesis ($H_0$) states that the coefficient is equal to zero, while the alternative hypothesis ($H_1$) states that it is not
• The test statistic (t-statistic) is calculated as the ratio of the estimated coefficient to its standard error
  • $t = \frac{\hat{\beta}_i}{SE(\hat{\beta}_i)}$, where $\hat{\beta}_i$ is the estimated coefficient and $SE(\hat{\beta}_i)$ is its standard error
• The p-value associated with the test statistic determines the significance of the coefficient at a chosen level (e.g., 0.05)

Interpreting regression coefficients

• Interpreting the coefficients in a regression model is essential for understanding the relationship between the independent variables and the dependent variable
• The interpretation depends on the type of variables (continuous or categorical) and the scale of the coefficients (standardized or unstandardized)

Slope and intercept meanings

• In a simple linear regression model, $y = \beta_0 + \beta_1x + \epsilon$, the intercept ($\beta_0$) represents the expected value of the dependent variable when the independent variable is zero
  • The slope ($\beta_1$) represents the change in the dependent variable for a one-unit increase in the independent variable, holding other variables constant
• In multiple regression, the interpretation of the intercept and slopes is similar, but the slopes represent the effect of each independent variable while controlling for the others

Standardized vs unstandardized coefficients

• Unstandardized coefficients are in the original units of the variables and can be directly interpreted based on the scale of the predictors
  • They indicate the change in the dependent variable for a one-unit change in the independent variable
• Standardized coefficients (beta coefficients) are obtained by standardizing the variables to have a mean of zero and a standard deviation of one
  • They allow for comparing the relative importance of predictors measured on different scales
• Standardized coefficients indicate the change in the dependent variable (in standard deviations) for a one-standard-deviation change in the independent variable

Confidence intervals for coefficients

• Confidence intervals provide a range of plausible values for the population coefficients based on the sample estimates
  • They indicate the precision and uncertainty associated with the estimated coefficients
• A 95% confidence interval for a coefficient is calculated as $\hat{\beta}_i \pm t_{1-\alpha/2, n-p-1} \times SE(\hat{\beta}_i)$, where $t_{1-\alpha/2, n-p-1}$ is the critical value from the t-distribution with $n-p-1$ degrees of freedom
• Wider confidence intervals suggest greater uncertainty in the estimates, while narrower intervals indicate more precise estimates

Handling categorical predictors

• Categorical predictors are variables that take on a limited number of distinct values or categories
• To include categorical predictors in a regression model, they need to be properly encoded and interpreted

Dummy variable encoding

• Dummy variable encoding is a method for representing categorical variables as binary (0 or 1) variables
  • Each category of the categorical variable is assigned a separate dummy variable
• For a categorical variable with $k$ categories, $k-1$ dummy variables are created, with one category serving as the reference level
  • The reference level is typically the most common or meaningful category and is omitted from the encoding
• Dummy variables allow the model to estimate the effect of each category compared to the reference level

Interpreting categorical coefficients

• The coefficients for dummy variables represent the difference in the dependent variable between each category and the reference level, holding other variables constant
  • A positive coefficient indicates that the category is associated with a higher value of the dependent variable compared to the reference level
• The interpretation of categorical coefficients depends on the choice of the reference level
  • Changing the reference level will change the interpretation of the coefficients
• It is important to consider the practical and theoretical significance of the categories when interpreting the coefficients

Polynomial and interaction terms

• Polynomial and interaction terms allow for modeling nonlinear relationships and the combined effects of multiple predictors in a regression model

Modeling nonlinear relationships

• Polynomial terms are used to capture curvilinear relationships between the dependent variable and independent variables
  • They are created by raising the independent variable to a power (e.g., $x^2$, $x^3$)
• Polynomial regression models can be represented as $y = \beta_0 + \beta_1x + \beta_2x^2 + ... + \beta_px^p + \epsilon$
  • The coefficients for the polynomial terms indicate the change in the dependent variable for a one-unit change in the corresponding power of the independent variable
• The degree of the polynomial should be chosen based on the observed pattern in the data and the theoretical justification

Interpreting interaction effects

• Interaction terms are created by multiplying two or more independent variables
  • They allow for the effect of one variable to depend on the level of another variable
• An interaction term between variables $x_1$ and $x_2$ can be represented as $x_1 \times x_2$ in the regression model
  • The coefficient for the interaction term represents the change in the effect of $x_1$ on the dependent variable for a one-unit change in $x_2$ (and vice versa)
• Interpreting interaction effects requires considering the main effects of the individual variables and the joint effect of the interaction term
  • The significance and magnitude of the interaction effect can be assessed using hypothesis tests and confidence intervals
• Plotting the predicted values of the dependent variable at different levels of the interacting variables can help visualize the interaction effect

Regression model selection

• Regression model selection involves choosing the best subset of predictors from a larger set of potential variables
• The goal is to find a parsimonious model that balances model fit, complexity, and interpretability

Forward, backward, stepwise selection

• Forward selection starts with an empty model and iteratively adds the most significant predictor until a stopping criterion is met
  • It can be useful when there are many potential predictors and a simple model is desired
• Backward selection starts with a full model containing all predictors and iteratively removes the least significant predictor until a stopping criterion is met
  • It can be useful when there are few potential predictors and a comprehensive model is desired
• Stepwise selection combines forward and backward selection, allowing for the addition and removal of predictors at each step
  • It can be useful when there are moderate to many potential predictors and the best subset is unknown

Criteria for model comparison

• Model selection criteria are used to compare and evaluate different regression models
  • They balance model fit and complexity by penalizing models with too many parameters
• Common criteria include Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and adjusted R^2
  • AIC and BIC are based on the likelihood function and the number of parameters, with lower values indicating better models
  • Adjusted R^2 accounts for the number of predictors and favors models with higher explanatory power relative to their complexity

Bias-variance tradeoff considerations

• The bias-variance tradeoff is a fundamental concept in model selection
  • Bias refers to the error introduced by approximating a complex relationship with a simpler model
  • Variance refers to the sensitivity of the model to the specific training data
• Models with high complexity (many predictors) tend to have low bias but high variance, leading to overfitting
  • Overfitting occurs when the model fits the noise in the training data, resulting in poor generalization to new data
• Models with low complexity (few predictors) tend to have high bias but low variance, leading to underfitting
  • Underfitting occurs when the model is too simple to capture the underlying patterns in the data
• The goal is to find the right balance between bias and variance to achieve good performance on both the training and test data

Regularization techniques

• Regularization techniques are used to control the complexity of regression models and prevent overfitting
• They add a penalty term to the objective function, discouraging large coefficient values and promoting simpler models

Ridge regression (L2 regularization)

• Ridge regression adds a L2 penalty term to the ordinary least squares objective function
  • The L2 penalty is the sum of squared coefficients multiplied by a tuning parameter ($\lambda$)
• The ridge regression objective function is $\sum_{i=1}^n (y_i - \beta_0 - \sum_{j=1}^p \beta_j x_{ij})^2 + \lambda \sum_{j=1}^p \beta_j^2$
  • The tuning parameter $\lambda$ controls the strength of the regularization, with higher values leading to smaller coefficients
• Ridge regression shrinks the coefficients towards zero but does not perform variable selection (coefficients are not exactly zero)

Lasso regression (L1 regularization)

• Lasso (Least Absolute Shrinkage and Selection Operator) regression adds a L1 penalty term to the ordinary least squares objective function
  • The L1 penalty is the sum of absolute values of coefficients multiplied by a tuning parameter ($\lambda$)
• The lasso regression objective function is $\sum_{i=1}^n (y_i - \beta_0 - \sum_{j=1}^p \beta_j x_{ij})^2 + \lambda \sum_{j=1}^p |\beta_j|$
  • The tuning parameter $\lambda$ controls the strength of the regularization and the sparsity of the model
• Lasso regression can perform variable selection by shrinking some coefficients exactly to zero, effectively removing the corresponding predictors from the model

Elastic net for feature selection

• Elastic net is a combination of ridge and lasso regression, incorporating both L1 and L2 penalties