**Regression Models Short Notes – AI Generated**

2.1 Introduction to Simple Linear Regression:

- The Regression Equation: In simple linear regression, the relationship between a dependent variable (y) and an independent variable (x) is modeled using a straight line equation: y = β0 + β1x + ε, where β0 is the intercept, β1 is the slope, and ε is the random error term.
- Fitted Value and Residuals: The fitted value (ŷ) is the predicted value of y based on the estimated regression line. The residual (ε) is the difference between the observed value of y and the fitted value (y – ŷ).
- Least Squares: The method of least squares is used to estimate the values of β0 and β1 that minimize the sum of squared residuals, providing the best-fitting regression line.

Introduction to Multiple Linear Regression:

- Assessing the Model: In multiple linear regression, where there are multiple independent variables, various diagnostic measures are used to assess the model’s fit, such as R-squared, adjusted R-squared, and residual analysis.
- Cross-Validation: Cross-validation techniques, like k-fold cross-validation, are used to evaluate the model’s predictive performance and guard against overfitting.
- Model Selection and Stepwise Regression: Stepwise regression methods, like forward selection, backward elimination, and bidirectional elimination, are used to select the most relevant independent variables for the model.
- Prediction Using Regression: Once the regression model is built, it can be used to make predictions for new observations based on their independent variable values.

2.2 Logistic Regression:

- Logistic Response Function and Logit: The logistic response function is a sigmoid curve that maps the linear combination of predictors to a probability value between 0 and 1. The logit is the natural logarithm of the odds ratio, which is used to model the log-odds of the outcome.
- Logistic Regression and GLM: Logistic regression is a type of Generalized Linear Model (GLM) used for binary classification problems, where the dependent variable is categorical (0 or 1).
- Generalized Linear Model: GLMs extend the linear model to allow for non-normal response distributions and link functions that relate the linear predictor to the mean of the response variable.
- Predicted Values from Logistic Regression: The predicted values from logistic regression are probabilities of the binary outcome, based on the values of the independent variables.
- Interpreting Coefficients and Odds Ratios: The coefficients in logistic regression represent the change in the log-odds of the outcome associated with a one-unit change in the predictor variable. Odds ratios are used to interpret the strength and direction of the association.
- Linear and Logistic Regression: Similarities and Differences: While linear regression models a continuous response variable and logistic regression models a binary outcome, both involve estimating coefficients and making predictions based on independent variables. However, the underlying assumptions and interpretations differ.
- Assessing the Models: Various measures, such as deviance, Akaike Information Criterion (AIC), and area under the ROC curve (AUC), are used to assess the goodness-of-fit and predictive performance of logistic regression models.