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notes7 UPenn STAT 112
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  • Title: notes7
  • Type: Notes
  • School: UPenn
  • Course: STAT 112
  • Term: Fall

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112: Stat Lecture 7 Notes Homework 2: Due next Thursday The Multiple Linear Regression model (Chapter 4.1) Inferences from multiple regression analysis (Chapter 4.2) Interpretation of Regression Coefficients Gas mileage regression from Car89.JMP Response GP1000MHwy Parameter Estimates Term Intercept Weight(lb) Cargo Seating Horsepower Estimate 19.100521 0.0040877 0.0533 0.0268912 0.0426999 Std Error 2.098478 0.001203 0.013787 0.428283 0.01567 t Ratio Prob>|t| 9.10 <.0001 3.40 0.0010 3.87 0.0002 0.06 0.9501 2.73 0.0075 Interpretation of coefficient bweight = 0.004 : The mean of GP1000Mwy is estimated to increase 0.004 for a one pound increase in weight holding fixed cargo, seating and horsepower. E (Y | X 1 = x1 + 1, X 2 = x2 , , X K = xK ) E (Y | X 1 = x1 , , X K = xK ) = ( 0 + 1 ( x1 + 1) + + K xK ) ( 0 + 1 x1 + + K xK ) = 1 Partial Slopes vs. Marginal Slopes Multiple Linear Regression Model: E (Y | X 1 , , X K ) = 0 + 1 X 1 + + K X K The coefficient k is a partial slope. It indicates the change in the mean of y that is associated with a one unit increase in xk while holding all other variables x1 ,..., xk 1 , xk +1, ..., xK fixed. A marginal slope is obtained when we perform a simple regression with only one X, ignoring all other variables. Consequently the other variables are not held fixed. Partial vs. Marginal Slopes Example How much gas will it use to carry an additional 200 pound passenger for 1000 miles? Simple Regression Parameter Estimates Term Intercept Weight(lb) Estimate 18.278693 0.0067879 Std Error 1.872435 0.000655 t Ratio 9.76 10.36 Prob>|t| <.0001 <.0001 Multiple Regression Parameter Estimates Term Intercept Weight(lb) Cargo Seating Horsepower Estimate 19.100521 0.0040877 0.0533 0.0268912 0.0426999 Std Error 2.098478 0.001203 0.013787 0.428283 0.01567 t Ratio 9.10 3.40 3.87 0.06 2.73 Prob>|t| <.0001 0.0010 0.0002 0.9501 0.0075 Marginal slope suggest 0.0068*200=1.36 gallons Partial slope suggests 0.0041*200=0.82 gallons The partial slope better answers the question because when we add the additional passenger, we are planning to keep cargo, seating and horsepower constant. Partial Slopes vs. Marginal Slopes: Another Example In order to evaluate the benefits of a proposed irrigation scheme in a certain region, suppose that the relation of yield Y to rainfall R is investigated over several years. Data is in rainfall.JMP. Bivariate Fit of Yield By Total Spring Rainfall 90 80 70 Yield 60 50 40 30 7 8 9 10 11 12 13 Total Spring Rainfall Linear Fit Linear Fit Yield = 76.666667 - 1.6666667 Total Spring Rainfall Summary of Fit RSquare RSquare Adj Root Mean Square Error Mean of Response Observations (or Sum Wgts) 0.027778 -0.13426 13.94433 60 8 Parameter Estimates Term Intercept Total Spring Rainfall Estimate 76.666667 -1.666667 Std Error 40.5546 4.025382 t Ratio 1.89 -0.41 Prob>|t| 0.1076 0.6932 Bivariate Fit of Average Spring Temperature By Total Spring Rainfall 57.5 Average Spring Temperature 55 52.5 50 47.5 45 42.5 7 8 9 10 11 12 13 Total Spring Rainfall Higher rainfall is associated with lower temperature. Multiple Linear Regression Response Yield Parameter Estimates Term Intercept Total Spring Rainfall Average Spring Temperature Estimate -144.7619 5.7142857 2.952381 Std Error 55.8499 2.680238 0.692034 t Ratio -2.59 2.13 4.27 Prob>|t| 0.0487 0.0862 0.0080 Rainfall is estimated to be beneficial once temperature is held fixed. Multiple regression provides a better picture of the benefits of an irrigation scheme because temperature would be held fixed in an irrigation scheme. Inferences about Regression Coefficients Confidence intervals: (1 )100% confidence interval for k : bk t / 2 sb Degrees of freedom for t equals n-(K+1). Standard error of bk , sb , found on JMP output. Hypothesis Test: H 0 : k = k* k k Decision rule for test: Reject H0 if t > t / 2 or t < t / 2 bk k* t= where sb p-value for testing H 0 : k = 0 is printed in JMP output under Prob>|t|. k H a : k k* Inference Examples Find a 95% confidence interval for horsepower ? Is seating of any help in predicting gas mileage once horsepower, weight and cargo have been taken into account? Carry out a test at the 0.05 significance level. 95% Confidence interval for horsepower : bhorsepower t.025,n (4+1) * sbhorsepower = 0.0427 t.025,104 *0.01567 = 0.0427 1.98*0.01567 = (0.0117, 0.0737) Is seating of any help in predicting gas mileage once horsepower, and weight cargo have been taken into account? H 0 : seating = 0 H a : seating 0 t= bseating 0 sbseating = 0.0269 0 = 0.0628 0.4283 We reject H 0 if | t |> t.025,n ( K +1) = 1.98 Thus, we do not reject H 0 ; there is no Evidence that seating is of any help in Predicting as mileage once horsepower, weight and cargo have been taken into account. Checking Assumptions: Multiple Linear Regression Model E (Y | x1 , , xK )(= y| x1 ,..., xK ) = 0 + 1 x1 + + K xK yi = 0 + 1 x1i + 2 x2i + + K xKi + ei The expected value of the disturbances is zero for each ( x1 , , xK ) , E (ei | xi1 , , xiK ) = 0 2 ei e ,i.e., The variance of each is equal to Var (ei | xi1 , , xiK ) = e2 The ei are normally distributed. The ei are independent. Plots for Checking Assumptions We can construct residual plots of each explanatory variable Xk vs. the residuals. We save the residuals by clicking the red triangle next to Response after fitting the model and clicking Save Columns and then residuals. We then plot Xk vs. the residuals using Fit Y by X (where Y=the residuals). We can plot a horizontal line at 0 by using Fit Y by X (it is a property of multiple linear regression that the least squares line for the regression of the residuals on any Xk is a horizontal line. A useful summary of the residual plots for each explanatory variable is the Residual by Predicted plot that is automatically plotted after using Fit Model. The residual ,by, predicted plot is a plot of the predicted values E (Y | X i1 X iK ) , , vs. the residuals Checking Assumptions Linearity: Check that in residual by predicted plot, the mean of the residuals for each range of the predicted values is about zero. Check that in each residual plot, the mean of the residuals for each range of the explanatory variable is about zero. Constant Variance: Check that in the residual by predicted plot that for each range of the predicted values, the spread of the residuals is about the same. Normality: Plot histogram of the residuals. Check that the histogram is bell shaped. 10 GP1000MHwy Residual 5 Bivariate Fit of Residual GP1000MHwy By Horsepower 10 Residual GP1000MHwy 0 -5 -10 25 30 35 40 45 50 55 GP1000MHwy Predicted 5 0 Residual by predicted plot does not suggest and suggests approximately constant variance 10 Residual GP1000MHwy 5 -5 -10 50 100 150 Horsepower 200 250 0 Plot of horsepower vs. residuals suggests linearity is okay. -5 -10 1500 2000 2500 3000 3500 4000 Weight(lb) Plot of weight vs. residuals suggests linearity is okay. One potential concern is that highest weight cars all have negative residuals. Bivariate Fit of Residual GP1000MHwy By Seating 10 Residual GP1000MHwy Bivariate Fit of Residual GP1000MHwy By Horsepower 10 Residual GP1000MHwy 2 3 4 5 Seating 6 7 8 5 5 0 0 -5 -5 -10 -10 50 100 150 Horsepower 200 250 Plot of residuals vs. seating suggests linearity is not perfect for seating. Residuals for small and high seating seem to have a mean that is smaller than 0. Plot of residuals vs. horsepower suggest linearity is okay. Highest 4 horsepower cars all have negative residuals but next 5 highest horsepower cars all have positive residuals. Coefficient of Determination The coefficient of determination R 2 for multiple regression is defined as for simple linear regression: R 2 = SSR = 1 SSE SST SST Represents percentage of variation in y that is explained by the multiple regression line. R2 is between 0 and 1. The closer to 1, the better the fit of the regression equation to the data. Assessing Quality of Prediction (Chapter 3.5.3) R squared measures is a measure of a fit of the regression to the sample data. It is not generally considered an adequate measure of the regression s ability to predict the responses for new observations. One method of assessing the ability of the regression to predict the responses for new observations is data splitting. We split the data into a two groups a training sample and a holdout sample (also called a validation sample). We fit the regression model to the training sample and then assess the quality of predictions of the regression model to the holdout sample. College Data in collegeclass.JMP Training Sample: 40 observations. Holdout Sample: Last 10 observations. Mean Squared Deviation: Mean squared prediction error over the holdout sample (y y ) over the n2 (=10 here) observations n2 2 i =1 i i in the holdout sample. n2

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04-css.ppt
Path: UPenn >> CIT >> 597 Fall, 2009
Description: CSS Applications to HTML and XHTML Apr 10, 2009 The problem with HTML HTML was originally intended to describe the content of a document Page authors didnt have to describe the layout-the browser would take care of that This is a good enginee...
0readme.txt
Path: UPenn >> LDC >> 2006 Fall, 2009
Description: README File for the FRENCH GIGAWORD TEXT CORPUS = First Edition = INTRODUCTION - French Gigaword is a comprehensive archive of newswire text data that has been acquired over several year...
README.pdf
Path: UPenn >> LDC >> 2008 Fall, 2008
Description: THE WEST POINT BRAZILIAN PORTUGUESE SPEECH CORPUS The Center For Technology Enhanced Language Learning United States Military Academy Department Of Foreign Languages 745 Brewerton Road West Point, NY 10996 Email: john.morgan@usma.edu Phone: 845-938-6...
transpec_iraqi.doc
Path: UPenn >> LDC >> 2006 Fall, 2009
Description: Appen Iraqi Transcription Conventions Summary 04 November, 2004 The following LDC conventions were used in the Iraqi Arabic Appen database. Background Noise intermittent noise continuous noise -noise -end noise Speaker Noise lipsmack breath cough lau...
CALLHOME_Mandarin_Chinese_Transcripts_XML_version.doc
Path: UPenn >> LCD >> 2008 Fall, 2009
Description: CALLHOME Mandarin Chinese Transcripts XML version The XML edition of the CallHome Mandarin Chinese Transcripts corpus contains the same 120 transcripts of telephone conversions in the LDCs original release (LDC96T16). The current version is marked u...
0readme.txt
Path: UPenn >> LDC >> 2003 Fall, 2008
Description: README File for the GIGAWORD ARABIC TEXT CORPUS = INTRODUCTION - The Gigaword Arabic Corpus is a comprehensive archive of newswire text data that has been acquired from Arabic news sources by the Linguistic Data Consortium (LDC), at the Uni...
SRO_SPEC.TXT
Path: UPenn >> LDC >> 93 Fall, 2009
Description: File sro-specs.doc. originally drawn from a memo by C. Hemphill of TI (4/18/90). amended 07/91. revised by L. Shriberg (11/10/91). revised by Patti Price (12/09/91), revised 01/21/92. minor revisions made by J. Garofolo on 02/21/92. Please note: in ...
sp_lex.txt
Path: UPenn >> LDC >> 96 Fall, 2009
Description: Garrett, Susan, T. Morton, and C. McLemore. 1997. LDC Spanish Lexicon. Philadelphia: Linguistic Data Consortium, University of Pennsylvania. - Description of the LDC Spanish lexicon - CONTENTS 1. Sum...
PAPER.TXT
Path: UPenn >> LDC >> 96 Fall, 2009
Description: THE CTIMIT CELLULAR BANDWIDTH SPEECH CORPUS E. Bryan George(*), Kathy L. Brown Signal Processing Center of Technology Lockheed-Martin Sanders, Inc. Nashua, NH 03061 (*) E. Bryan George is now with the DSP Research and Development Center, Texas Instru...

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