1) Answer the following questions as true (T) or false (F) by circling the appropriate letter.

Transcription

1 1) Answer the following questions as true (T) or false (F) by circling the appropriate letter. T F T F T F a) Variance estimates should always be positive, but covariance estimates can be either positive or negative. b) The influence diagnostics called DFFITS and DFBETAS are calculated on deleted observations, while the calculation of Cook s D does not involve a deleted observation. c) The value of R 2 will always increase and the value of the MSE will always decrease when additional variables are added to the model. T F d) The value of R 2 for a linear regression ranges from 0.0 to 1.0. T F e) The confidence interval of a simple linear regression line will be narrowest (i.e. have the smallest variance) at the intercept. T F f) Logarithmic transformation of a Yi variable in regression does not affect homogeneity of variance. T F g) As a result of the assumptions for regression, we expect the regression coefficients, bi, to be independent of each other. T F h) For regression the TYPE I and TYPE II sums of squares are the same. T F i) The sequential sums of squares for the SAS model statement MODEL Y = X1 X2 X3; would be represented in extra sum of squares notation as: SSX1 X2 X3; SSX2 X1 X3; SSX3 X1 X2 T F j) The general linear hypothesis test is used to test for a difference between two models where one model has a subset of variables contained in the other model. 2) Examine the list of diagnostic statistics below. For the questions that follow, choose a diagnostic from this list (SAS names used). a) STUDENT b) RSTUDENT c) COOK's D d) DFFITS e) DFBETAS f) h ii or HATDIAG g) R 2 h) VIF i) Standardized b i j) Sequential b i k) Partial R 2 Type III Description of the statistic : Which of the statistics named in the list above have the property listed below? Choose one statistic and enter the corresponding letter in the blank space to the right. a) Which could be obtained by doing regression on standardized variables (e.g. where Yi and Xij have a mean zero and variance of 1)? Letter from the list above b) Which could be used to examine the observations for unusual values of Xi? 1

2 c) Which could be used to determine which observations strongly influence the fit of the data (e.g. judging by the effect on the predicted Yi values)? d) Which would be the best statistic to use as a diagnostic for multicolinearity? e) Which would be the best statistic to use to diagnose outliers? f) Which would be used to examine changes in the regression coefficients as additional variables are added one by one? 3) Which ONE of the following IS an intrinsically linear regression model? a) a model linearized by taking the logarithm of the dependent variable Yi b) a model linearized by taking the logarithm of the independent variable Xi c) a model linearized by taking the inverse of the independent variable Xi d) a polynomial regression with successively higher powers of Xi 4) Some statistics are diagnostics for observations and some for variables. Which of the statistics below is a diagnostic for a variable? a) STUDENT b) RSTUDENT c) HATDIAG d) VIF 5) Which of the statistics below could be produced by "standardizing" the Yi variable and the Xi variables before doing the regression? a) RSTUDENT b) COOK's D c) HATDIAG d) Standardized b i 6) Which of the diagnostic tools below would NOT be used to examine the observations for potential outliers of Yi values? a) STUDENT b) RSTUDENT c) HATDIAG d) a residual plot 7) Given the model, Yi b0 bx 1 1i bx 2 2i bx 3 3i bx 4 4 i ei, fitted in SAS as MODEL Y = X1 X2 X3 X4;, which statement below best describes the Extra SS for X4? a) The TYPE I SS for X4 will be LARGER than the TYPE II or TYPE III for X4. b) The TYPE I SS for X4 will be SMALLER than the TYPE II or TYPE III for X4. 2

3 c) The TYPE I SS for X4 will be the SAME AS the TYPE II or TYPE III for X4. d) Sometimes the TYPE I SS for X4 will be larger and sometimes the TYPE II or TYPE III will be larger. 8) Use of the word partial in association to a statistic in multiple regression implies which of the following (for example partial standardized regression coefficients )? a) The statistics are given for only part of the reduced model b) The statistic is fully adjusted for all other variables c) The statistics is only partly adjusted for some other variables d) The result is an intermediate result 9) Which of the following is not fully adjusted for all variables in the multiple regression? a) TYPE I sum of squares b) TYPE II sum of squares c) regression coefficients (bi) d) squared partial correlations Type II 10) Which of the following will alter homogeneity of variance in a simple linear regression? a) taking a logarithm of the dependent variable (Yi) b) taking a logarithm of the dependent variable (Xi) c) fitting a polynomial with power terms of X d) adding additional independent variables 11) Given two models where the full model is Yi 0 1X1 i 2X 2i 3X3i 4X 4i e and the reduced i model is Yi 0 1X1 i 2X 2i e the test of difference between the models corresponds to which of the i hypotheses tests below? a) H0: b1 = b2 = 0. b) H0: b3 = b4 = 0. c) H0: β1 = β2 = 0. d) H0: β3 = β4 = 0. 3

4 12) Which of the following is a characteristic or likely consequence of strong multicollinearity? Circle all that apply. a) poor estimates of the regression coefficients (βi). b) increased variance estimates for the regression coefficients (bi). c) correlations among combinations of the independent variables (Xi). 13) Which of the following statistics is not used to evaluate multicollinearity. a) correlations among independent variables. b) the hat value (hii). c) the variance inflation factor. d) the sequential parameter estimates. 14) If a 3 factor regression is first fitted in SAS as Y = X1 X2 X3 and then refitted as Y = X1 X2 which of the following is true of the TYPE I sum of squares for X1? a) It will be larger. b) It will be smaller c) It will not change d) It can be larger, smaller or the same 15) Given 4 independent variables available for a multiple regression; if the residuals of the model Y = X3 X1 X4 are plotted on the residuals of the model X2 = X3 X1 X4 what is the graphic called? a) The ordinary residual plot of a 3 factor model (e.g. X1, X3, X4) b) The partial residual plot of the variable X2 c) The scatter plot of Y on the predicted value of Y (i.e. Ŷ ) d) This is a plot of the hat values for X4 16) Given the results below, what is SS(x3, x1 x2)? 4

5 ID Model SSReg SSE SSTotal Model 1 Y = X Model 2 Y = X Model 3 Y = X Model 4 Y = X1 X Model 5 Y = X1 X Model 6 Y = X2 X Model 7 Y = X1 X2 X The remaining questions refer to the SAS output pages given separately. These pages contain output for a multiple regression fitted with PROC REG. Questions requesting a P value may not always have a meaningful P value associated. If no P value is associated with the answer, state what statistic or graphic was used to answer the question. The SAS program is an analysis of attendance at Saturday Australian Football League (AFL) games at the Melbourne Cricket Ground (MCG). The variables used are described in the program comments. The computer output provided contains the SAS program and a full model analysis with selected diagnostics. The program is provided to help you to determine what was done. 17a) Do you believe there is multicolinearity in this multiple regression? Circle one: Yes / No P value or statistic used =. 17b) Disregarding of whether it was statistically significant or not (for this question only), do higher forecast temperatures tend to increase or decrease game attendance? Circle one: Increase / Decrease Statistic used =. 17c) Is there evidence that the size of the membership of the club (Members) has a statistically significant impact on the attendance? Circle one: Yes / No P value or statistic used =. 17d) What is the raw sum of squares of the variable Other (i.e. )? 2 X other 5

6 Requested Value =. 17e) What is the covariance between coefficients of the variables Temp and Top50? Requested Value =. 17f) Does the assumption of normality appear to have been met for this analysis? Circle one: Yes / No P value or statistic used =. 17g) One hypotheses of the investigators was that about 80% of the club members would attend a game and that half of them would bring a guest. The expected coefficient for the slope (Attendence / Member) would then be = 1.2? Does the observed value differ significantly from this hypothesized value (e.g. result in rejecting H : Members ). Circle one: Yes / No P value or statistic used =. 17h) Was the assumption of homogeneity of variance met for this analysis? Circle one: Yes / No P value or statistic used =. 17i) Using statistics provided to evaluate the importance or impact of variables in the regression other than the parameter estimates and the t test of those estimates, which of the variables appears to be the SECOND best variable in the model? Variable name Statistic used =. 17j) What is the standard error of the intercept?. 17k) It was thought by the investigators that the conditions that existed for observation n on 24/7/93 were nearly ideal for game attendance? What is the confidence interval for all games that might be played under the same conditions, that is, have the same values for the four independent variables. P( true mean ) = l) What fraction of the (corrected) total variation is accounted for by the model? 6

7 Requested Value =. The investigators were concerned about the individual observation #6 (ObsID f ). They calculated a Bonferroni adjusted t value of 3.52 for =0.05 and 41 observations and a mean value of h ii (p/n = 0.122). Answer the next 4 questions below about this observation. 17m) First place a confidence interval on this single observation. P( true mean ) = n) Does this observation appear to be an outlier given the Bonferroni calculation above? Circle one: Yes / No Statistic(s) used =. 17o) Does this observation appear to be influential? Circle one: Yes / No Statistic(s) used =. 17p) Does this observation appear to have an unusual combination of independent variables? Circle one: Yes / No Statistic(s) used =. 17q) Which observation appears most likely to be an outlier judging from the residual plot? Give an observation number of ObsID value? Indicate if you believe it is an outlier (yes of no). Circle one: Yes / No Observation number =. 7

8 dm'log;clear;output;clear'; options ps=512 ls=111 nocenter nodate nonumber FORMCHAR=" =-/\<>*"; ODS HTML style=minimal body='afl Crowd Attendance.HTML'; ODS listing; ods graphics off; TITLE1 'Exam 1 Example - AFL Crowd Attendance at the MCG'; *****************************************************************; *** AFL Crowd Attendance at the MCG ***; *** Data set was assembled by Rowan Todd and Mark McNaughton, *** *** two stat students in Dr Margaret Mackisack's QUT class. ***; *********************************************************************; *** They collected data involving Saturday Australian Football ***; *** League (AFL) matches at the Melbourne Cricket Ground (MCG). ***; *** They looked only at matches during the normal home and away ***; *** season (i.e. not including finals). They used statistics from ***; *** all such games in 1993 and 1994 (nineteen relevant matches in ***; *** 1993 and twenty-two in 1994). The response variable measured ***; *** was attendance at the MCG, and after consideration, they came ***; *** up with the following covariates: ***; 8

9 *********************************************************************; *** Variable Description ***; *** MCG Attendance at the MCG in 1000's. ***; *** Temp Temperature. The forecast max temp on the day of the***; *** match, in degrees C, from The Weekend Australian. ***; *** Other Attendance at other matches in 1000's. The sum of ***; *** the attendances at other AFL matches in Melbourne ***; *** and Geelong on the same day as the match. ***; *** Members Membership. The sum of the memberships of the two ***; *** clubs whose teams were playing the match in 1000's. ***; *** Top50 Number of players from the top fifty. The number of ***; *** top 50 players from the AFL playing in the match. ***; *** Date Date of the match in the format dd/mm/yy. ***; *** Home Abbreviation for home team. ***; *** Away Abbreviation for away team. ***; **********************************************************************; DATA Attendance; INPUT MCG 1-7 Other 9-16 Temp Members Top Date $ Home $ Away $ 53-58; obs + 1; 9

10 ObsID = byte(obs+96); * lower case *; if obs gt 26 then ObsID = byte(obs+64-26); * upper case *; * ; CARDS; RUN; Data was here ; *proc print data=attendance; run; PROC REG DATA=Attendance lineprinter; id ObsID; TITLE2 'PROC REG'; MODEL MCG = Other Temp Members Top50 / CLB ss1 ss2 XPX I influence vif collin scorr1 pcorr2 covb R stb; OUTPUT out=next1 p=predicted r=residual lclm=lclmean Uclm=Uclmean lcl=lclindiv ucl=uclindiv; Test1:Test Members = 1.2; RUN; proc print data=next1 noobs; TITLE3 'OUTPUT STATEMENT results'; var obs ObsID MCG Other Temp Members Top50 predicted lclmean Uclmean lclindiv uclindiv; run; 10

11 proc univariate data=next1 normal plot; var residual; ods exclude basicmeasures extremeobs quantiles testsforlocation; TITLE3 'Univariate analysis of residuals'; run; proc plot data=next1; plot residual*predicted=obsid / vref=0; TITLE3 'Residual plot'; options ps=56 ls=111; run; options ps=512 ls=111; 11

12 Exam 1 Example - AFL Crowd Attendance at the MCG PROC REG The REG Procedure Model: MODEL1 Dependent Variable: MCG Number of Observations Read 41 Number of Observations Used 41 Model Crossproducts X'X X'Y Y'Y Variable Intercept Other Temp Members Top50 MCG Intercept Other Temp Members Top MCG X'X Inverse, Parameter Estimates, and SSE Variable Intercept Other Temp Members Top50 MCG Intercept Other E Temp Members E Top MCG Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var

13 Parameter Estimates Parameter Standard Standardized Variable DF Estimate Error t Value Pr > t Type I SS Type II SS Estimate Intercept Other Temp Members Top Parameter Estimates Squared Squared Semi-partial Partial Variance Variable DF Corr Type I Corr Type II Inflation 95% Confidence Limits Intercept Other Temp Members Top Covariance of Estimates Variable Intercept Other Temp Members Top50 Intercept Other Temp Members Top Collinearity Diagnostics Condition Proportion of Variation- Number Eigenvalue Index Intercept Other Temp Members Top Test Test1 Results for Dependent Variable MCG Source DF Mean Square F Value Pr > F Numerator Denominator

14 Exam 1 Example - AFL Crowd Attendance at the MCG PROC REG The REG Procedure Model: MODEL1 Dependent Variable: MCG OutputStatistics Obs Dependent Predicted Std Error Std Error Student Cook's Obs ID Variable Value Mean Predict Residual Residual Residual D RStudent 1 a * b * c d e * f ***** g h ** i j ** k * l m n ** o p * q r **** s t ** u * v *** w ** x ** y ** z A * B C *** D E

15 32 F G * H ** I *** J * K L M N * O * OutputStatistics Obs Hat Diag Cov DFBETAS-- Obs ID H Ratio DFFITS Intercept Other Temp Members Top50 1 a b c d e f g h i j k l m n o p q r s t u v w x y z A

16 28 B C D E F G H I J K L M N O Sum of Residuals 0 Sum of Squared Residuals Predicted Residual SS (PRESS) Exam 1 Example - AFL Crowd Attendance at the MCG PROC REG OUTPUT STATEMENT results Obs obs ID MCG Other Temp Members Top50 predicted lclmean Uclmean lclindiv uclindiv 1 a b c d e f g h i j k l m n o p

17 17 q r s t u v w x y z A B C D E F G H I J K L M N O Exam 1 Example - AFL Crowd Attendance at the MCG PROC REG Univariate analysis of residuals The UNIVARIATE Procedure Variable: residual (Residual) Moments N 41 Sum Weights 41 Mean 0 Sum Observations 0 Std Deviation Variance Skewness Kurtosis Uncorrected SS Corrected SS Coeff Variation. Std Error Mean

18 Tests for Normality Test --Statistic p Value Shapiro-Wilk W Pr < W Kolmogorov-Smirnov D Pr > D > Cramer-von Mises W-Sq Pr > W-Sq > Anderson-Darling A-Sq Pr > A-Sq > Stem Leaf Boxplot Normal Probability Plot * * +* * * ** * ***** *** *-----* **** *** ****** * +** * Multiply Stem.Leaf by 10**

19 Exam 1 Example - AFL Crowd Attendance at the MCG PROC REG Residual plot Plot of residual*predicted. Symbol is value of ObsID r C I n w 20 + H x 10 + O b N G D i d R F z q e L K M s c E s i B l d u g o a u l k p J e A a h t y j v f Predicted Value of MCG NOTE: 1 obs hidden. 19