class: center, middle, inverse, title-slide .title[ # Week 2 Assignment: Presentation ] .subtitle[ ## MLR of CO2 Emissions for Vehicles ] .author[ ### Alice Xiang ] .date[ ### 2024-02-18 ] --- ## Table of Contents - Introduction to the Dataset - Research Question - The Full Model + Discussion - The Edited Model + Discussion - The Transformed Model + Discussion - Model Selection - Conclusion --- class: inverse center middle ## Introduction to the Dataset --- ## Introduction I chose [this dataset](https://www.kaggle.com/datasets/bhuviranga/co2-emissions) on CO2 emissions of different cars to do multiple linear regression. --- ## Variables The following are the variables included in the dataset (6 continuous, 2 categorical): - Engine.Size - Cylinders - Fuel.Type - Fuel.Consumption.City - Fuel.Consumption.Hwy - Fuel.Consumption.Combined - Fuel.Consumption.mpg - CO2.Emissions --- class: inverse center middle ## Research Question: How do different predictor variables relate to the CO2 emissions of the vehicle? --- class: inverse center middle ## Full Model + Discussion --- ## The Full Model Using R, we create the full model. ```r full.model = lm(CO2.Emissions ~ ., data = emissions) ``` --- ## Summary of the Full Model <div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-97d749de279cb80fc5e9" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-97d749de279cb80fc5e9">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["(Intercept)","Engine.Size","Cylinders4","Cylinders5","Cylinders6","Cylinders8","Cylinders10","Cylinders12","Cylinders16","Fuel.TypeE","Fuel.TypeN","Fuel.TypeX","Fuel.TypeZ","Fuel.Consumption.City","Fuel.Consumption.Hwy","Fuel.Consumption.Combined","Fuel.Consumption.mpg"],[93.86700592931325,0.8218282259850136,-0.9584579335314239,-3.55010448014747,-0.0901626353848913,1.308276132454805,6.303207027193385,8.605655485194246,28.89624790824825,-137.7060177396539,-111.3336334409743,-30.52596134597003,-31.1123288437751,6.096225527958519,5.48267524056471,8.194797893051557,-0.9647167053907203],[1.642289915788377,0.1440849931055685,0.5227130642987137,1.103369636917398,0.5888234796809556,0.7376997733144125,1.094563313100982,0.9572174584848991,3.067556709696526,0.5364754699883367,4.925878253242038,0.3843916506436318,0.3887747869664007,0.7424400201668332,0.6119734545924317,1.346873652311458,0.0253005770175994],[57.15617262634937,5.703773920319897,-1.833621539223087,-3.217511486056276,-0.1531233697299986,1.773453347527612,5.758650003841183,8.990282624823235,9.419955568191259,-256.6865130714135,-22.60178342160578,-79.41369510720862,-80.02661151599817,8.211068049091212,8.959008269755945,6.084310788163335,-38.13022543792781],[0,1.217199897848648e-08,0.06675050754875948,0.00129867210619025,0.8783051799495434,0.07619491435061164,8.819850037038467e-09,3.089637050615245e-19,5.917493871500079e-21,0,2.043217625385835e-109,0,0,2.567098223922559e-16,4.092108154177914e-19,1.22846934786612e-09,1.667283597557993e-290]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>Estimate<\/th>\n <th>Std. Error<\/th>\n <th>t value<\/th>\n <th>Pr(>|t|)<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"columnDefs":[{"className":"dt-right","targets":[1,2,3,4]},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"Estimate","targets":1},{"name":"Std. Error","targets":2},{"name":"t value","targets":3},{"name":"Pr(>|t|)","targets":4}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> --- ## Residual Analysis <img src="xaringan_files/figure-html/unnamed-chunk-4-1.png" width="100%" /> --- ## VIF ```r vif(full.model) ``` ``` ## GVIF Df GVIF^(1/(2*Df)) ## Engine.Size 11.668643 1 3.415940 ## Cylinders 14.403671 7 1.209896 ## Fuel.Type 2.475681 4 1.119984 ## Fuel.Consumption.City 2069.965111 1 45.496869 ## Fuel.Consumption.Hwy 568.001039 1 23.832772 ## Fuel.Consumption.Combined 4651.987253 1 68.205478 ## Fuel.Consumption.mpg 10.261228 1 3.203315 ``` --- ## Issues We See - nonconstant variance - residuals not normal (Q-Q plot) - multicollinearity between all Fuel Consumption variables --- class: inverse center middle ## Edited Model + Discussion --- ## Edited Model: Removing Predictors due to Multicollinearity Of the Fuel Consumption variables, we keep only Fuel.Consumption.mpg and create the following model. ```r emissions.edit <- emissions %>% dplyr::select(-c(Fuel.Consumption.City, Fuel.Consumption.Hwy, Fuel.Consumption.Combined)) full.model.edit = lm(CO2.Emissions ~., data=emissions.edit) ``` --- ## Summary of the Edited Model ```r DT::datatable(summary(full.model.edit)$coef, fillContainer = FALSE, options = list(pageLength = 5)) ``` <div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-b43f6e1586b345c297ce" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-b43f6e1586b345c297ce">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["(Intercept)","Engine.Size","Cylinders4","Cylinders5","Cylinders6","Cylinders8","Cylinders10","Cylinders12","Cylinders16","Fuel.TypeE","Fuel.TypeN","Fuel.TypeX","Fuel.TypeZ","Fuel.Consumption.mpg"],[425.4363233700449,7.499682382390572,-8.933532628831339,-13.77736112544499,-5.947815206359456,11.74102576318419,34.31568642366131,44.38107646366814,145.2516283914759,-77.75131314802286,-100.3179414254768,-25.91604270676288,-29.99438795280482,-6.053155605219331],[2.777403590573778,0.4322623788979119,1.601375202509239,3.383367009818704,1.805902309020907,2.25772613769068,3.333705902701893,2.867431038446597,9.265172051136846,1.465818494809694,15.1136250091658,1.177351206276188,1.193015469041779,0.04164968550129563],[153.177710583342,17.34983831235007,-5.578663023402098,-4.072085908936981,-3.293542057423992,5.200376417306984,10.29355540806681,15.47764388004642,15.67716471856055,-53.04293363969134,-6.637583065918204,-22.01215964158394,-25.14165887295332,-145.3349654952624],[0,4.020980491342882e-66,2.50966758382251e-08,4.708168817950154e-05,0.00099399945392509,2.042365119491379e-07,1.106433811483896e-24,3.353997080133979e-53,1.634740220682596e-54,0,3.414845398701041e-11,4.68227082710005e-104,6.736636070332254e-134,0]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>Estimate<\/th>\n <th>Std. Error<\/th>\n <th>t value<\/th>\n <th>Pr(>|t|)<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"columnDefs":[{"className":"dt-right","targets":[1,2,3,4]},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"Estimate","targets":1},{"name":"Std. 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Error<\/th>\n <th>t value<\/th>\n <th>Pr(>|t|)<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"columnDefs":[{"className":"dt-right","targets":[1,2,3,4]},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"Estimate","targets":1},{"name":"Std. 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Error<\/th>\n <th>t value<\/th>\n <th>Pr(>|t|)<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"columnDefs":[{"className":"dt-right","targets":[1,2,3,4]},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"Estimate","targets":1},{"name":"Std. Error","targets":2},{"name":"t value","targets":3},{"name":"Pr(>|t|)","targets":4}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]}},"evals":[],"jsHooks":[]}</script> --- class: inverse center middle ## Conclusions --- ## Conclusions - Log transformed model chosen as best model due to residual analysis and goodness of fit - Still shows violations to assumptions - variation in residuals - assumption of normality - Includes outliers Further analysis can be done through bootstrapping to eliminate some of these issues --- class: inverse center middle ## Thank you