1. **Problem Statement:** We analyze a dataset of 52 cystic fibrosis patients with variables including PEmax (response), age, gender, height, weight, BMP, FEV1, FRC, RV, and TLC (explanatory variables). The goal is to identify variables associated with PEmax using multiple regression and evaluate model performance. 2. **Multiple Regression Model:** The general multiple linear regression model is: $$\text{PEmax} = \beta_0 + \beta_1 \text{age} + \beta_2 \text{gender} + \beta_3 \text{height} + \beta_4 \text{weight} + \beta_5 \text{BMP} + \beta_6 \text{FEV1} + \beta_7 \text{FRC} + \beta_8 \text{RV} + \beta_9 \text{TLC} + \epsilon$$ where $\epsilon$ is the error term. 3. **Exploratory Data Analysis (EDA):** - Calculate summary statistics (mean, median, sd) for all variables. - Visualize distributions and relationships using scatterplots and boxplots. - Check for outliers and missing data. 4. **Fit Full Model:** - Use R's lm() function: lm(PEmax ~ age + gender + height + weight + BMP + FEV1 + FRC + RV + TLC, data=PEmax) - Examine coefficients, standard errors, t-values, and p-values. 5. **Model Diagnostics:** - Check assumptions: linearity, normality of residuals, homoscedasticity, independence. - Use residual plots, Q-Q plots, and tests like Shapiro-Wilk. - Check multicollinearity via Variance Inflation Factor (VIF). 6. **Prediction Performance:** - Calculate $R^2$ and adjusted $R^2$. - Use cross-validation or split data to assess prediction error (e.g., RMSE). 7. **Stepwise Model Selection:** - Use stepwise regression (both directions) with AIC criterion. - Identify the optimal subset of explanatory variables. 8. **Compare Models:** - Compare full and stepwise models by $R^2$, adjusted $R^2$, AIC, and prediction error. - Interpret which variables remain significant and how model fit improves or changes. 9. **Conclusions:** - Summarize key variables associated with PEmax. - Discuss model assumptions and prediction accuracy. - Provide recommendations for clinical interpretation and further research. This approach ensures a thorough statistical analysis addressing the research questions.