class: center, middle, inverse, title-slide .title[ # <font size="7" color="black">Multiple Linear Regression </font> ] .subtitle[ ## <font size="6" color="black"> Model Building and Analysis of Flight Delay Times </font> ] .author[ ### <font size="5" color="black">Ian Vanwright </font> ] .institute[ ### <font size="6" color="white">West Chester University of Pennsylvania</font><br> ] .date[ ### <font color="black" size="4"> Prepared for<br> </font> <font color="gold" size="6"> STA490: Data Visualization </font> <br><br> <font color="white" size="3"></font> ] --- class: top, center # Table of Contents: .left[ - Introduction - Research Question - Recap of Building the Final Model - Bootstrapping Part 1: Observations - Bootstrapping Part 2: Residuals - Summary of Results/Model Comparisons - Conclusion ] --- class: top, center # Introduction: .left[ - Data Set was taken from 'Applied Analytics Through Case Studies Using SAS and R by Deepti Gupta' - Data set consists of 3593 observations and 11 variables - Variables are as follows: - 1) Carrier (Categorical) - The airline that the flight is taken with - 2) Airport_Distance (Numeric) - The Distance between the airports (in miles) - 3) Number_of_flights (Numeric) - Total number of flights in airport - 4) Weather (Numeric) - Delay due to weather condition ranked 0-10, with 0 being mild and 10 being extreme - 5) Support_Crew_Available (Numeric) - Total number of support crew available - 6) Baggage_loading_time (Numeric) - Time (in minutes) for loading the baggage - 7) Late_Arrival_o (Numeric) - Time (in minutes) for late arriving aircraft of the same flight - 8) Cleaning_o (Numeric) - Time (in minutes) for aircraft cleaning - 9) Fueling_o (Numeric) - Time (in minutes) for aircraft fueling - 10) Security_o (Numeric) - Time (in minutes) for security checking - 11) Arr_Delay (Numeric) - Flight arrival time (in minutes). This will be the dependent variable in the model ] --- class: top, center # Summary of Building the Final Model .left[ - Fit the full model with Arr_Delay as the response and all other variables as predictors - T-tests determined Cleaning_o, Fueling_o, Security_o were all insignificant (p > 0.05) and I decided to remove carrier (this created the reduced model, which analysis was done with) - Diagnostics of the reduced model were checked, and the assumptions of normality and homoscedasticity (constant variances) were violated - A box-cox transformation was applied to Arr_Delay (λ = 0.5) , fixing issues with normality and constant variances - Comparing the goodness-of-fit measures of the full model, the reduced model, and the model with the transformation, the model with the transformation was chosen ] --- class: top, center # Final Model .center[ <div style="text-align: center;"> <img src= "img/MLRInitialFinalModel.PNG" width = 600 height=450;"> </div> ] --- class: top, center # Residual Analysis of the Final Model <div style="text-align: center;"> <img src= "img/MLRResidualAnalysis.PNG" width = 300 height=300;"> <img src= "img/MLRResidualHistogram.PNG" width = 300 height=300;"> <img src= "img/MLRMeanResiduals.PNG" width = 300 height=300;"> </div> --- class: top, center # Bootstrapping (Part 1): Observations .left[ - Bootstrapping: Resampling with replacement - Because our sample size is big enough, bootstrapping is a good method to use - Sampling Distributions of Coefficient Estimates are pictured - Red density curves represent the actual sampling distribution of the coefficients, while the blue density curves represent the sampling distribution created from bootstrapping ] <div style="text-align: center;"> <img src= "img/MLRBootstrap1.PNG" width = 300 height=300;"> <img src= "img/MLRBootstrap2.PNG" width = 300 height=300;"> </div> --- class: top, center # Bootstrapping Part 1 (Continued) - We can check how well the bootstrap did by summarizing the results and building a confidence interval - The estimated parameters and the 95% confidence interval for the bootstrapped sampling distribution are pictured - All parameter estimates in the coefficient matrices match (within rounding) <div style="text-align: center;"> <img src= "img/MLRInitialFinalModel.PNG" width = 300 height=300;"> <img src= "img/MLRBootstrapModel1.PNG" width = 300 height=300;"> </div> --- class: top, center # Bootstrapping (Part 2): Residuals .left[ - Now we will look at the residuals of our sample and resample - Residual sampling distribution values: fitted value + residual - Sampling distributions of coefficient estimates are pictured Red density curves represent the actual sampling distribution of the coefficients, while the blue density curves represent the sampling distribution created from bootstrapping ] <div style="text-align: center;"> <img src= "img/MLRBootstrap3.PNG" width = 300 height=300;"> <img src= "img/MLRBootstrap4.PNG" width = 300 height=300;"> </div> --- class: top, center # Boostrap Part 2 (Continued) - Again, we can check how well the bootstrap did by summarizing the results and building a confidence interval - The estimated parameters and the 95% confidence interval for the bootstrapped residual sampling distribution are pictured - Again, all parameter estimates in the coefficient matrices match (within rounding) <div style="text-align: center;"> <img src= "img/MLRInitialFinalModel.PNG" width = 300 height=300;"> <img src= "img/MLRBootstrapModel2.PNG" width = 300 height=300;"> </div> --- class: top, center # Summary of Results - Essentially no difference in parameter estimation for all 3 models - Will choose parametric (original) model <div style="text-align: center;"> <img src= "img/MLRInitialFinalModel.PNG" width = 400 height=400;"> </div> --- class: middle, center # That's all, folks! # Any Questions, Comments, Thoughts, Concerns?