Show pageBacklinksCite current pageExport to PDFBack to top This page is read only. You can view the source, but not change it. Ask your administrator if you think this is wrong. Linear [[modeling]] is a fundamental statistical technique used to describe the relationship between one or more independent variables (predictors) and a dependent variable (outcome). It assumes a linear relationship between these variables. Types of Linear Models Simple Linear Regression: Describes the relationship between a single independent variable ( 𝑋 X) and a dependent variable ( 𝑌 Y). Equation: 𝑌 = 𝛽 0 + 𝛽 1 𝑋 + 𝜖 Y=β 0 +β 1 X+ϵ 𝑌 Y: Dependent variable. 𝑋 X: Independent variable. 𝛽 0 β 0 : Intercept (the value of 𝑌 Y when 𝑋 = 0 X=0). 𝛽 1 β 1 : Slope (the change in 𝑌 Y for a one-unit change in 𝑋 X). 𝜖 ϵ: Error term (captures random variation not explained by 𝑋 X). Multiple Linear Regression: Extends simple linear regression to include multiple independent variables. Equation: 𝑌 = 𝛽 0 + 𝛽 1 𝑋 1 + 𝛽 2 𝑋 2 + … + 𝛽 𝑝 𝑋 𝑝 + 𝜖 Y=β 0 +β 1 X 1 +β 2 X 2 +…+β p X p +ϵ 𝑋 1 , 𝑋 2 , … , 𝑋 𝑝 X 1 ,X 2 ,…,X p : Predictors. Generalized Linear Models (GLM): A flexible extension of linear models for non-normal dependent variables (e.g., binary, count). Includes logistic regression and Poisson regression. Hierarchical Linear Models (HLM): Used for data with a nested structure (e.g., students within schools). Assumptions of Linear Models For valid results, linear modeling relies on the following assumptions: Linearity: The relationship between predictors and the outcome is linear. Independence: Observations are independent of each other. Homoscedasticity: The variance of the errors is constant across all levels of the independent variables. Normality of Residuals: The residuals (differences between observed and predicted values) are normally distributed. No Multicollinearity: Independent variables are not highly correlated. Steps in Linear Modeling Formulate the Model: Define the dependent and independent variables based on the research question. Fit the Model: Use statistical software (e.g., R, Python, SPSS) to estimate the coefficients ( 𝛽 β). Evaluate Model Fit: R-squared ( 𝑅 2 R 2 ): Measures the proportion of variance in 𝑌 Y explained by 𝑋 X. Adjusted 𝑅 2 R 2 : Adjusts for the number of predictors in the model. Residual Analysis: Check for patterns in residuals to ensure assumptions are met. Interpret Coefficients: Each 𝛽 β represents the change in 𝑌 Y associated with a one-unit change in the corresponding 𝑋 X, holding other variables constant. Validate the Model: Use cross-validation or a separate test dataset to assess the model's predictive accuracy. Applications of Linear Modeling Medicine: Predicting patient outcomes based on clinical factors (e.g., blood pressure, cholesterol levels). Analyzing treatment effects in clinical trials. Social Sciences: Studying relationships between demographic variables and outcomes (e.g., income, education level). Business: Forecasting sales based on advertising spend and market trends. Engineering: Modeling physical systems and processes. linear_modeling.txt Last modified: 2024/12/18 07:50by 127.0.0.1