Multiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Re-grades

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1 STAT 101 Dr. Kar Lock Morga 11/20/12 Exam 2 Grades Multple Regresso SECTIONS 9.2, 10.1, 10.2 Multple explaatory varables (10.1) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (10.2) Trasformatos (ot book) Exam 2 Re-grades Re-grade requests due wrtg by class o Tuesday, 11/27/12 Partal credt wll ot be altered oly submt a re-grade request f you thk you have etrely the correct aswer but got pots off Grades may go up or dow If pots were added up correctly, just brg me your exam (o eed for a offcal re-grade) More tha 2 varables! Today we ll fally lear a way to hadle more tha 2 varables! Multple Regresso Multple regresso exteds smple lear regresso to clude multple explaatory varables: y xxx kk Grade o Fal We ll use your curret grades to predct fal exam scores, based o a model from last year s studets Respose: fal exam score Explaatory: hw average, clcker average, exam 1, exam 2 y hw clcker exam1 exam

2 Grade o Fal What varable s the most sgfcat predctor of fal exam score? a) Homework average b) Clcker average c) Exam 1 d) Exam 2 Exam 1 has the lowest p-value Iferece for Coeffcets The p-value for explaatory varable x s assocated wth the hypotheses H : 0 0 H : 0 a For tervals ad p-values of coeffcets multple regresso, use a t-dstrbuto wth degrees of freedom k 1, where k s the umber of explaatory varables cluded the model Grade o Fal Grade o Fal Estmate your score o the fal exam. What type of terval do you wat for ths estmate? a) Cofdece terval b) Predcto terval A cofdece terval s for a average, a predcto terval s for a dvdual. Estmate your score o the fal exam. (hw average s out of 10, clcker average s out of 2) For a HW average of 9, a clcker average of 1.7, ad exams scores of 80 o each exam (these were the averages for each category): (9) (1.7) + 0.4(80) (80) = Grade o Fal Grade o Fal Is the clcker coeffcet really egatve?!? Gve a 95% cofdece terval for the clcker coeffcet (okay to use t* = 2) = = (-12.56, 7.16) Is your score o exam 2 really ot a sgfcat predctor of your fal exam score?!? 2

3 Coeffcets The coeffcet (ad sgfcace) for each explaatory varable deped o the other varables the model! I predctg fal exam scores, f you kow someoe s score o Exam 1, t does t provde much addtoal formato to kow ther score o Exam 2 (both of these explaatory varables are hghly correlated) If you take Exam 1 out of the model Now Exam 2 s sgfcat! Model wth Exam 1: Grade o Fal Multple Regresso The coeffcet for each explaatory varable s the predcted chage y for oe ut chage x, gve the other explaatory varables the model! The p-value for each coeffcet dcates whether t s a sgfcat predctor of y, gve the other explaatory varables the model! If explaatory varables are assocated wth each other, coeffcets ad p-values wll chage depedg o what else s cluded the model If you clude Project 1 the model Model wthout Project 1: Grade o Fal Grades Evaluatg a Model How do we evaluate the success of a model? How we determe the overall sgfcace of a model? How do we choose betwee two competg models? 3

4 Oe way to evaluate a model s to partto varablty Total Explaed by the Model Error A good model explas a lot of the varablty Y Exam Scores Wthout kowg the explaatory varables, we ca say that a perso s fal exam score wll probably be betwee 60 ad 98 (the rage of Y) Kowg hw average, clcker average, exam 1 ad 2 grades, ad project 1 grades, we ca gve a arrower predcto terval for fal exam score We say the some of the varablty y s explaed by the explaatory varables How do we quatfy ths? How do we quatfy varablty Y? a) Stadard devato of Y b) Sum of squared devatos from the mea of Y c) (a) or (b) d) Noe of the above Total Y 2 Y 1 Sums of Squares Explaed by the model Error varablty ˆ 2 Y Y Y ˆ 2 Y 1 1 SST SSM SSE Total Sum of Squares: SST y 2 y 1 Model Sum of Squares: SSM yˆ 2 y 1 Y R 2 2 SSM " Y explaed by the model" R SST "Total varablty Y" Explaed by the Model Total Error Sum of Squares: SSE y ˆ 2 y 1 If SSM s much hgher tha SSE, tha the model explas a lot of the varablty Y R 2 s the proporto of the varablty Y that s explaed by the model 4

5 R 2 R 2 For smple lear regresso, R 2 s just the squared correlato betwee X ad Y 2 R R 0.09 For multple regresso, R 2 s the squared correlato betwee the actual values ad the predcted values Fal Exam Grade Is the model sgfcat? If we wat to test whether the model s sgfcat (whether the model helps to predct y), we ca test the hypotheses: H :... 0 H a : At least oe 0 k We do ths wth ANOVA! ANOVA for Regresso ANOVA for Regresso Source Model Error Total df k -k-1-1 Sum of Squares SSM SSE SST Mea Square MSM = SSM/k MSE = SSE/(-k-1) F MSM MSE p-value Use F k,-k-1 Source Model Error Total df Sum of Squares Mea Square F p-value 0 k: umber of explaatory varables : sample sze 5

6 Fal Exam Grade Smple Lear Regresso For smple lear regresso, the followg tests wll all gve equvalet p-values: t-test for o-zero correlato t-test for o-zero slope ANOVA for regresso Mea Square Error (MSE) Fal Exam Grade Mea square error (MSE) measures the average varablty the errors (resduals) The square root of MSE gves the stadard devato of the resduals (gvg a typcal dstace of pots from the le) Ths umber s also gve the R output as the resdual stadard error, ad s kow as s the textbook y Smple Lear Model x 0 1 ~ 0, N Resdual stadard error = MSE = s e estmates the stadard devato of the resduals (the spread of the ormal dstrbutos aroud the predcted values) Resdual Stadard Error Use the fact that the resdual stadard error s ad your predcted fal exam score to compute a approxmate 95% predcto terval for your fal exam score yˆ NOTE: Ths calculato oly takes to accout errors aroud the le, ot ucertaty the le tself, so your true predcto terval wll be slghtly wder 6

7 Revstg Codtos For smple lear regresso, we leared that the followg should hold for fereces to be vald: Learty Costat varablty of the resduals Normalty of the resduals How do we assess the frst two codtos multple regresso, whe we ca o loger vsualze wth a scatterplot? Resdual Plot A resdual plot s a scatterplot of the resduals agast the predcted resposes Should have: 1) No obvous patter or tred (learty) 2) Costat varablty Resdual Plots Fal Exam Score Obvous patter ot costat Are the codtos satsfed? (a) Yes (b) No Trasformatos If the codtos are ot satsfed, there are some commo trasformatos you ca apply to the respose varable You ca take ay fucto of y ad use t as the respose, but the most commo are log(y) (atural logarthm - l) y (square root) y 2 (squared) e y (expoetal)) Orgal Respose, y: Logged Respose, log(y): log(y) 7

8 y y 2 Orgal Respose, y: Orgal Respose, y: Square root of Respose, y: Squared respose, y 2 : Orgal Respose, y: Expoetated Respose, e y : e y Over-fttg It s possble to over-ft a model: to clude too may explaatory varables The fewer the coeffcets beg estmated, the better they wll be estmated Usually, a good model has prued out explaatory varables that are ot helpg R 2 Addg more explaatory varables wll oly make R 2 crease or stay the same Addg aother explaatory varable ca ot make the model expla less, because the other varables are all stll the model Is the best model always the oe wth the hghest proporto of varablty explaed, ad so the hghest R 2? (a) Yes (b) No Adjusted R 2 Adjusted R 2 s lke R 2, but takes to accout the umber of explaatory varables As the umber of explaatory varables creases, adjusted R 2 gets smaller tha R 2 Oe way to choose a model s to choose the model wth the hghest adjusted R 2 8

9 Fal Exam Grade To Come How do we decde whch explaatory varables to clude the model? How do we use categorcal explaatory varables? What s the coeffcet of oe explaatory varable depeds o the value of aother explaatory varable? Read 9.2, 10.1, 10.2 To Do Do Homework 8 (due Thursday, 11/29) Do Project 2 (poster due Moday, 12/3, paper due 12/6) 9

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