Answer all three questions. All questions carry equal marks.

Transcription

1 UNIVERSITY OF DUBLIN TRINITY COLLEGE Faculty of Engineering, Mathematics and Science School of Computer Science and Statistics Postgraduate Diploma in Statistics Trinity Term 2 Introduction to Regression Thursday 5 May RDS main Time (9:3 2:3 hours) Prof. Haslett, Mr Mullins Answer all three questions. All questions carry equal marks. Non-programmable calculators are permitted for this examination please indicate the make and model of your calculator on each answer book used. Page of 9

2 Q. Write short notes on EIGHT of the following as they arise in regression. You should illustrate your notes by referring to examples. You may draw on other questions in this exam paper or on examples discussed in class. However, additional credit will be given for the use of other examples. a) Lessons from my project b) Multiplicative and Additive models c) s d) Over-fitting e) Linear regression does not necessarily mean straight lines f) Confidence and Prediction Intervals g) Influential cases h) Binary (indicator) variables i) The NoConstant option when using multiple indicator variables. j) Sampling Distributions and Standard Errors in Regression Q2. Data are available on the Price (in Singapore $) and Carat (the carat weight) of diamonds. Of central interest is the importance or otherwise of Clarity (VS2 or VVS) on the average price. Some descriptive summaries, T-tests and regression analyses are below. For regression, indicator variables have been created to correspond to the Clarity grades but only one is used in the regression models. The analyses are conducted both on Price and on Log price It is observed (see Two Sample tests, below) that usual tests to compare the mean prices for the two groups conclude that the observed price difference is not statistically significantly different from zero. This surprised the investigators. However the regression analyses suggest otherwise. Page 2 of 9

3 (a) Consider the regression of Price on Carat (left column) i. Interpret the coefficients of the fitted regression model. Illustrate your answer by showing how to use these to compute rough 95% ( 2S) prediction intervals for the Price of diamonds, whose Carat weights are.3 and., for each type of Clarity. (ie four intervals). Discuss the intervals. ii. iii. Explain the use and interpretation of standard errors for coefficients in regression, illustrating your answer by discussing those for Carat and for Clarity. Explain the use and interpretation of S and R-Sq values in regression, illustrating by reference to the analysis of variance table. iv. The Analysis of Variance table and the tables of coefficients report p- values. What hypotheses are being tested? (b) Consider the regression of Log Price on Carat. i. Interpret the coefficients of the fitted model. Illustrate your answer by computing and commenting on rough 95% ( 2S) prediction intervals for the Log Price, and hence for the Price, of diamonds whose Carat weights are.3 and., for each type of Clarity. Contrast with (a)i above. ii. iii. iv. Interpret the standard errors for the coefficients of Carat and of Clarity contrasting with (a) above. Interpret the S and R-Sq value, contrasting with (a) above. The Plots suggest fewer problems with regression analysis in the log scale. Discuss. (c) In respect of the importance or otherwise of Clarity, the conclusions on from the regressions analyses contrast with the Two Sample T-tests, involve change both in the estimated price difference and in its Standard Error. Discuss. (d) These regression analyses provide examples of using regression to control for the external variation that can often arise in observational studies. Discuss. Page 3 of 9

4 Descriptive Statistics: Carat, Price Sing $, logcarat, logprice Total Variable Clarity Count Mean StDev Carat VS VVS Price VS VVS logprice VS VVS Two-Sample T-Test and CI: Price, Clarity Two-sample T for Price Sing $ Clarity N Mean StDev SE Mean VS VVS Difference = mu (VS2) - mu (VVS) Estimate for difference: 29 95% CI for difference: (-5, 596) T-Test of difference = (vs not =): T-Value =.44 P-Value =.66 DF = 3 Both use Pooled StDev = Two-Sample T-Test and CI: logprice, Clarity Two-sample T for logprice Clarity N Mean StDev SE Mean VS VVS Difference = mu (VS2) - mu (VVS) Estimate for difference: % CI for difference: (-.963,.56) T-Test of difference = (vs not =): T-Value =.47 P-Value =.64 DF = 3 Both use Pooled StDev =.326 Page 4 of 9

5 Frequency Frequency Percent Percent Price Sing $ logprice Regressions of Price and of Log Price on Carat Weight, using an indicator for the Clarity variable Scatterplot of Price Sing $ vs Carat Scatterplot of logprice vs Carat Note the shown lines in both scatterplots are smooths, and are not fitted regression lines Clarity VS2 VVS Clarity VS2 VVS Carat Carat Regression Analysis: Price versus Carat, Clarity_VS2 Price = Carat - 89 Clarity_VS2 Predictor Coef SECoef T P Constant Carat Clarity_VS S = R-Sq = 87.6% R-Sq(adj) = 87.4% Analysis of Variance Source DF SS MS F P Regression Res Error Total Regression Analysis: logprice versus Carat, Clarity_VS2 logprice = Carat -.8 Clarity_VS2 Predictor Coef SECoef T P Constant Carat Clarity_VS S =.7985 R-Sq = 94.% R-Sq(adj) = 94.% Analysis of Variance Source DF SS MS F P Regression Error Total Plots for Price Sing $ Plots for logprice Normal Probability Plot 6 Versus Fits Normal Probability Plot.2 Versus Fits Fitted Value Fitted Value 4. 3 Histogram 6 Versus Order 2 Histogram.2 Versus Order Observation Order Observation Order 9 Page 5 of 9

6 Q3 The Math Marks data being marks in each of 5 exams for 88 students are summarised overleaf.several regression analyses are reported; details are below. (a) Below (Math Marks Summary statistics) are two tables summarising the data, a scatter plot matrix and two fitted line plots. Discuss in the light of these: i ii the relationships between the Statistics and Analysis marks and the Statistics and Algebra marks; the direct connection between the tables and the variances of the residuals in the fitted line plots; and hence, by drawing only on these tables (ie without conducting any further regressions) iii the details (coefficients, R-sq, S) of simple linear regression analyses of Statistics on Vectors were it to be carried out. (b) Consider the multiple regression analysis (A) reported below. i ii Explain the analysis and interpret the various reported summaries, contrasting with the analysis in part (a). Discuss the unusual observations and the diagnostic plots, Prepare a rough sketch of the implied relationship between Statistics and Analysis when Algebra marks are 4% and when they are 7%. (c) A law of diminishing returns applies to the addition of explanatory variables, especially when the explanatory variables are correlated, and can lead to misinterpretation of fitted coefficients. Illustrate by reference to the analysis reported in (C) below. Page 6 of 9

7 Stat Stat Descriptive Stats: Stat, Anal, Alg, Vect, Mech Variable Mean StDev Stat Anal Alg Vect Mech Math Marks Summary statistics Correlations: Stat, Anal, Alg, Vect, Mech Stat Anal Alg Vect Anal.67 Alg Vect Mech Cell Contents: Pearson correlation Matrix Plot of Stat, Anal, Alg, Vect, Mech Stat Anal Alg 2 8 Vect Mech Fitted Line Plot Stat = Anal Fitted Line Plot Stat = Alg Regression 95% PI Regression 95% PI 8 6 S R-Sq 36.9% R-Sq(adj) 36.% 8 6 S R-Sq 44.2% R-Sq(adj) 43.5% Anal Alg Page 7 of 9

8 Frequency Percent (A) Analysis: Stat versus Anal, Alg (A) Analysis: Unusual Observations Stat = Anal Alg Predictor Coef SE Coef T P Constant Anal Alg S = R-Sq = 47.9% Analysis of Variance Obs Anal Stat Fit SE Fit StResid X R X R denotes an observation with a large standardized residual. X denotes an observation whose X value gives it large leverage. Source DF SS MS F P Regres Res Total (A) Diagnostic plots Plots for Stat Normal Probability Plot 4 Versus Fits Fitted Value Histogram 4 Versus Order Observation Order 7 8 Page 8 of 9

9 (B) Reg Analysis: Stat versus Alg, Mech Stat = Alg +.36 Mech Predictor Coef SE Coef T P Constant Alg Mech S = 3.3 R-Sq = 44.3% (C) Reg Analysis: Stat versus Alg, Vect, Anal, Mech Stat = Alg +.33 Anal +.26 Vect Mech Predictor Coef SE Coef T P VIF Constant Alg Anal Vect Mech S = R-Sq = 47.9% UNIVERSITY OF DUBLIN 22 Page 9 of 9