Statistics: Unlocking the Power of Data Lock 5
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1 STAT 0 Dr. Kar Lock Morga Exam 2 Grades: I- Class Multple Regresso SECTIONS 9.2, 0., 0.2 Multple explaatory varables (0.) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (0.2) Exam 2 Re- grades Re- grade requests due wrtg by class o Moday, 4/5/4 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 your exam to your TA (o eed for a ofxcal re- grade) More tha 2 varables! Today we ll Xally lear a way to hadle more tha 2 varables! Multple Regresso Multple regresso exteds smple lear regresso to clude multple explaatory varables: y = β + β x + β x + + β x + ε k k Grade o Fal We ll use your curret grades to predct Xal exam scores, based o a model from prevous 0 studets Respose: Xal exam score Explaatory: hw average, clcker average, exam, exam 2 y = β + β hw + β clcker + β exam + β exam2 + ε
2 Grade o Fal What varable s the most sgxcat predctor of Xal exam score? a) Homework average b) Clcker average c) Exam d) Exam 2 Iferece for CoefOcets The p- value for explaatory varable x s assocated wth the hypotheses H : 0 0 β = H : β 0 a For tervals ad p- values of coefxcets multple regresso, use a t- dstrbuto wth degrees of freedom k, where k s the umber of explaatory varables cluded the model Grade o Fal Grade o Fal Estmate your score o the Xal exam. What type of terval do you wat for ths estmate? a) CoXdece terval b) Predcto terval Estmate your score o the Xal exam. (for ths data hw average was out of 0, clcker average was out of 2) Grade o Fal Grade o Fal Is the clcker coefxcet really egatve?!? Is your score o exam 2 really ot a sgxcat predctor of your Xal exam score?!? 2
3 CoefOcets The coef2cet (ad sg2cace) for each explaatory varable deped o the other varables the model! Grade o Fal If you take Exam out of the model Now Exam 2 s sgxcat! Model wth Exam : Multple Regresso The coefxcet 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 coefxcet dcates whether t s a sgxcat predctor of y, gve the other explaatory varables the model! If explaatory varables are assocated wth each other, coefxcets ad p- values wll chage depedg o what else s cluded the model If you clude Project the model Model wthout Project : Grade o Fal Grades Evaluatg a Model How do we evaluate the success of a model? How we determe the overall sgxcace of a model? How do we choose betwee two competg models? 3
4 Varablty Oe way to evaluate a model s to partto varablty Total Varablty = + Varablty Explaed by the Model Error Varablty A good model explas a lot of the varablty Y Exam Scores Wthout kowg the explaatory varables, we ca say that a perso s Xal exam score wll probably be betwee 60 ad 98 (the rage of Y) Kowg hw average, clcker average, exam ad 2 grades, ad project grades, we ca gve a arrower predcto terval for Xal exam score We say the some of the varablty y s explaed by the explaatory varables How do we quatfy ths? Varablty 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 Varablty ( Y ) 2 Y = Sums of Squares Varablty Explaed by the model = + = + Error varablty ( ˆ ) 2 Y Y ( Y ˆ ) 2 Y = = SST = SSM + SSE Total Sum of Squares: SST = ( y ) 2 y = Model Sum of Squares: SSM = ( yˆ ) 2 y = Varablty Y R 2 2 SSM "Varablty Y explaed by the model" R = = SST "Total varablty Y" Varablty Explaed by the Model Total Varablty Error Sum of Squares: SSE = ( y ˆ ) 2 y = 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 sgocat? If we wat to test whether the model s sgxcat (whether the model helps to predct y), we ca test the hypotheses: H : β = β =... = β = 0 H 0 2 a : At least oe β 0 k We do ths wth ANOVA! ANOVA for Regresso ANOVA for Regresso Source Model Error Total df k -k- - Sum of Squares SSM SSE SST Mea Square MSM = SSM/k MSE = SSE/(-k-) F MSM MSE p-value Use F k,-k- Source Model Error Total df Sum of Squares Mea Square F 20.7 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 = β + β x + ε 0 ε ~ 0, ( σ ) N ε Resdual Stadard Error Use the fact that the resdual stadard error s ad your predcted Xal exam score to compute a approxmate 95% predcto terval for your Xal exam score Resdual stadard error = MSE = s e estmates the stadard devato of the resduals (the spread of the ormal dstrbutos aroud the predcted values) 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 Xrst 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: ) No obvous patter 2) Costat varablty Resdual Plots Fal Exam Score Obvous patter Varablty ot costat Are the codtos satsxed? (a) Yes (b) No Codtos What f the codtos for ferece are t met??? Opto (best opto): Take STAT 20 ad lear more about modelg! Opto 2: Try a trasformato Trasformatos If the codtos are ot satsxed, 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)) 7
8 Orgal Respose, y: log(y) Orgal Respose, y: y Logged Respose, log(y): Square root of Respose, y: Orgal Respose, y: y 2 Orgal Respose, y: e y Squared respose, y 2 : Expoetated Respose, e y : Trasformatos Iterpretato becomes a bt more complcated f you trasform the respose t should oly be doe f t clearly helps the codtos to be met If you trasform the respose, be careful whe terpretg coefxcets ad predctos The slope wll ow have dfferet meag, ad predctos ad coxdece/predcto tervals wll be for the trasformed respose Trasformatos You do NOT eed to kow whch trasformato would be approprate for gve data o the Xal, but they may help f codtos are ot met for Project 2 or for future data you may wat to aalyze 8
9 To Come How do we decde whch explaatory varables to clude the model? How do we use categorcal explaatory varables? What f the coefxcet of oe explaatory varable depeds o the value of aother explaatory varable? Project 2 Project doe your lab groups oe project per group 0 page (max) paper: due Wedesday, 4/23 Choose oe quattatve varable ad aswer questos about t ad t s relatoshp wth other varables Use multple regresso ad aythg else we ve leared the course Project 2 Detals here Project 2 Data Data o college studets: Sleep data from a 2- week sleep dary Geder Class year Early rser, ght owl, or ether? Early classes? Mssed classes Score o a test of cogtve sklls GPA Alcohol cosumpto Depresso, axety, stress, happess To Do Read 9.2, 0., 0.2 Do HW 8 (due Wedesday, 4/6) Do Project 2 (due Wedesday, 4/23) 9
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