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# INTROTOLINEARMODELS BSTA651

Penn

GPA 3.68

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This 5 page Class Notes was uploaded by Mason Crist on Monday September 28, 2015. The Class Notes belongs to BSTA651 at University of Pennsylvania taught by Staff in Fall. Since its upload, it has received 36 views. For similar materials see /class/215362/bsta651-university-of-pennsylvania in Epidemiology at University of Pennsylvania.

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Date Created: 09/28/15

Statistical Methods amp Data Analysis I BSTA630 Part 11 Fall Semester 20062007 Knashawn H Morales ScD kmorales ccebmedupennedu SLR Least Squares Estimation Modi ed mm 71112005 some notes WeirTnAg Hwang Instructor Slide 1 Scallerplol Weight and Height Weight kg HE EhHEm Estimating B1 002 I Methods of estimation MLE Moment I Least square LS estimates in SLR Basic Idea gt Find 30 and 31 st Observed data K as close to u 31X regression line as possible gt Find 30 and 31 st 5175 as small as possible I closeness Slide 3 Square 53 small Absolute value l e l small Slide 2 Least Square LS Estimates Find 30 and 31 st Minimize 211z 7 30 312 Solve for 66191 0 and 66130 0 9 67 2 gillz 30 31 0 5 Zyzi oi iz h0 11 11 then all divided by n 5 5017515 Slide 4 Estimate 72 Varei Least Square LS Estimates 3 Since 39 Minimize 27111 u 131 5512 51 N N002 and Va39r5 T2 a 11 7 2 0 ml 2 u 1 1gtgt7 1 A n A n I If 3 1 known then i Z1y17 02 17 1230 0 1 11 11 11 51 K 7 g 7 3le gt 211111 7 17 113111 9113 0 N N002 1111 11 n 11 we estimate 02 by i 2x1y17n517751237 20 1 7quot 11 A 11 211 30 319512 i 5111 7 13151111 0 11 gt The sample variance with a known 0 mean Slide 5 Slide 7 2 Estimate 0 Varei When 30131 are unknown replace 30131 by ao l 1 n A A Least Square LS Estimates 211111 7 1 0 7 1312 Since SW 221Xz 7 7 Y 271 my 7 nail n 7 2 n 2 72 I But need to reduce degrees of frfedqm df because we estimate u and 31 And SW 2171 z X 21 1 m 62 72 111 7 90 7 My n 7 We have 1 1 RSS A s 7 7 A 2 7 191 Si 1129 1quotquot 112 30 1 15 Q1 30 1 1 tted value of y 7 Residual zyz 7 11 observed yz 7 tted Value of y1 7 RSSResidual Sum of Square 27191 7 1 7 919602 27191 7 131 Slide 6 Slide 8 Adjustment for the df Estimate normal variance with 1952y We N NWJTZ 1 When mean M is known A 1 g2 2ng 7 M2 MLE unbiased 2 When mean M is unknown A2 L1 9 e 502 MLE unbiased 11 Loose df because we estimate M by a Slide 9 Inference for 3031 I Inferences based on ha I Model YL O1Xz51 EZNN0U2 i12 I LS estimates of 130131 A 7 Sxy 131 7 SW 30 1 19 I Properties of LS estimates 1 Mean Unbiasedness M51 191 and Elo u 2 Standard Errors for 3150 Slide 10 Means of Bl and Bo A 5 271X17XXY17Y SW 7 BLUE 7 XV 271 EXz 7 XYz 7Xz 7 XW 2211097 XV BLOC XV 2109 XW of le 222109 7 XV since XXX 7 X 0 0 ELIMXZAW 221X17W 7 1 E 0 u Exercise Slide 11 Mean of 6392 2 22191 7w Rss 2 7L 7 2 Since 7 A2 w M X72172 Exercise E M72 7 7L72T2 U2 Hence E72 72 More references Weisberg Appendix A Montgomery Appendix C Linear models course BSTA 651 Slide 12 Properties of LS estimates Standard Errors of 6130 I Means 131 7451 Var y Vadw 3 m m E T T VWZLAXZ 7 XYz 2 1 n XI 7 X2Va39rYz so 21 I Standard Errors SW 72 A 53x 02 SW mel gm A 1 2 Va39r 0 02 Slide 13 Slide 15 Standard Errors of 6130 A A When will we estimate Bl precisely VWWO Va 1919 VW17 izVaTlt51gt 250041151 72 U2 7 i2 7 0 Exercise A 71 Saw I Precision gt Small Va39r 1 1 22 2 7 a n SW A A I Important factors Also 31130 are correlated 7L SW A A 0292 Caz3111 7 S Exercise Slide 14 Slide 16 Sampling Distributions for 6150 31 so 2221Xz7 mg 7 Y Sm 1ltXz7 Xv 221Xr m 1ltXz7 Xv gt Weighted sum of data K Since 31 is a linear combination of K and K are Normal rv gt al also follows Normal distribution 2 91 N wag 522 m 90 N N o02s Slide 17 Sampling Distribution for 6392 7 A 2 w Exercise 7 N X3172 Slide 18

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