Statistics 423 Midterm Examination Winter 2009

Transcription

1 Sttstcs 43 Mdterm Exmnton Wnter 009 Nme: e-ml: 1. Plese prnt your nme nd e-ml ddress n the bove spces.. Do not turn ths pge untl nstructed to do so. 3. Ths s closed book exmnton. You my hve your hnd clcultor nd you my use the sheet of condensed wsdom you hve brought wth you. You should hve no other wrtten mterl wth you durng the exmnton. 4. If you do not hve enough room for your work n the plce provded, use the bck of nerby pge (but be sure to mrk clerly whch problem the mterl on the bck of ny pge refers to). If you pull the pges prt, sgn ll pges. 5. Ths exmnton conssts of 3 nlytcl questons nd 1 seres of questons concernng R output for dt nlyss. There re 10 pges, ncludng ths cover pge (but you cn gnore p. 10). 6. To receve full credt for problem, you must mke t completely cler how you rrved t your nswer. Problem # Totl ponts possble Totl 60 Grde 1

2 1. Testng the smple dentty regresson model. (10 pts) Consder smple lner regresson problem for whch we wnt to test the null hypothess gnst the generl lterntve : 0 y ( = x + e α = 0, β = 0) : y = α + β x + e = 1,,,n Wrte down n expresson for the F-sttstc tht we would use to test ths hypothess. ow mny degrees of freedom does t hve? Note: You obtn nerly full credt for n expresson referrng just to sums of squres. For full credt wrte these sums of squres more explctly usng expressons for the lest squres estmtors or usng mtrx notton. F = = ( ), 0 (RegSS Reg SS ) / /( n ) ( RSS RSS )/ where 0 /( n ) RSS = ( y ˆ α ˆ βx ) RSS RSS RSS = y x 0

3 . Stndrd errors of predctons. Consder the smple lner regresson model wrtten n the followng form wth men-centered x s. y = α+β ' ( x +ε wth the usul ssumptons bout ndependent nd dentclly dstrbuted (d) errors: The lest squres estmtors re ˆ y β= ˆ ( x x )( y y ( x x ) α= nd ( ) ( ) The stndrd errors of the lest squres estmtors for smple lner regresson re: S ( ˆ ) E se α = nd ˆ S ( ) E se β = n ( x It s esy to show tht these lest squres estmtors re uncorrelted. Use these fcts to ε N σε (0, ). () Wrte n expresson for the stndrd error of yˆ = α+β ˆ ˆ x, whch we wll denote by se( yˆ ). ( se( yˆ) = Vr( y) = Vr( ') + ( x x) Vr( ˆ β) ˆ ˆ ˆ ˆ α ˆ 1 ( x = se + x x se = SE + n ( x ( ( ˆ α') ) ( ) ( ( ˆ β) ) (b) Sketch (smll) pcture showng how se( yˆ ) vres s ccordng to the vryng vlues of j x. (c) In predcton problem we re sked to predct the response y 0 for future observton t some vlue x = x0. We use the predcton yˆ ˆ 0 =α ˆ +β( x0 x ). The unobserved response cn be wrtten y0 =α+β ' ( x 0 +ε 0 nd the predcton error wll be ( y 0 y ˆ0 ). Usng the result n (), gve n expresson for the stndrd error of ( y 0 yˆ 0 ). At wht vlues of x 0 wll the predcton be most ccurte? ( ˆ α ) ( ˆ β ) se y yˆ = Vr ˆ y yˆ = Vr ˆ y + Vr ˆ ˆ α + x x Vr ˆ ˆ β ) ( 0 ) ( 0 ) ( 0) ( ') ( ) ( 1 ( x = SE + se( ') + ( x se( ) = SE 1+ + n ( x j 3

4 3. The expected vlue of the ordnry lest squres estmte of the slope ˆβ 1 n ftted smple lner regresson model when the usul ssumptons ren t true. Suppose the true model s not smple lner regresson, but the multple regresson wth second predctor vrble: y = α + β1x + βz + e. () Derve n expresson for the expected vlue of the lest squres estmtor of slope, ˆβ 1, computed under the (erroneous) ssumpton of smple lner regresson on the x ' s. (Note tht the formul for ˆβ 1 s gven n problem.) (b) Under wht condton s ˆβ 1 stll unbsed? I.e. when s E( ˆ β ) = β? ˆ α ˆ α 1 1 E E ( 1' 1) 1' ( 1' 1) ˆ 1' β1 ˆ = X X X Y = X X X X β 1 ˆ β 1 α α 1 = ( X1' X1) X1' X1 + zβ = + ( X1' X1) X1' z β1 β1 β Or E( ˆ β ) = ( α β β α β β z) ) ( x ( + x + z ) ( + x ( x = = β + β ( β β ) ( x x) ( x ) + ( ( z z) 1 ( x ( x x)( z z) 1 ( x Note: you could nswer ths queston wth mtrx lgebr. If you choose to use mtrx lgebr, I suggest wrtng the desgn mtrx for the multple regresson model s 1 x1 z1 1 x1 z1 1 x z 1 x z X = = ( X1 z), where X1 = nd z = 1 xn z n 1 x n z n 4

5 4. Juul s IGF dt The dt nlyzed below represent reference smple for the dstrbuton of the nsuln-lke growth fctor IGF-1, one observton per subject collected n the course of school physcl observtons. The complete dtset hs 1339 rows wth observtons on the followng vrbles: ge menrche sex gf1 tnner testvol heght weght numerc vector (yers). numerc vector. s menrche occurred (code 1: no, : yes)? numerc vector (1: boy, : grl). numerc vector, nsuln-lke growth fctor (μg/l). numerc vector, codes 1 5: Stges of puberty d modum Tnner. numerc vector, testculr volume (ml). numerc vector (cm). numerc vector, weght (kg). We wll only be concerned wth gf1 s response vrble together wth ge, heght, nd weght on the subset of 1 subjects wth observtons on gf1 nd ge > 5. In Fgure 4.1 you wll fnd sctterplot mtrx nd n Fgure 4. the usul dgnostc plots derved from smple lner regresson of gf1 on ge. () Expln wht these fgures tell you regrdng the usul ssumptons for smple lner regresson of gf1 on ge. 5

6 Fgure 4.1 Fgure 4. 6

7 (b) Below you wll fnd R output for smple lner regresson of sqrt(gf1) on ge. (The fct tht ths nlyss uses sqrt trnsformton does not men tht you hve to thnk sqrt s the best choce of trnsformton.) The result of ths lner regresson computton s sved n the R object Tft1b. > Tft1b <- lm(sqrt(gf1) ~ ge, subset=(ge>5) ) > summry(tft1b) Cll: lm(formul = sqrt(gf1) ~ ge, subset = (ge > 5)) Resduls: Mn 1Q Medn 3Q Mx Coeffcents: Estmte Std. Error t vlue Pr(> t ) (Intercept) <e-16 *** ge <e-16 *** --- Sgnf. codes: 0 *** ** 0.01 * Resdul stndrd error: on 10 degrees of freedom (9 observtons deleted due to mssngness) Multple R-squred: 0.446, Adjusted R-squred: F-sttstc: 96.6 on 1 nd 10 DF, p-vlue: <.e-16 > pr(mfrow=c(1,),pty="s") > plot(ftted(tft1b),rstudent(tft1b)) > blne(h=0,lty=) > qq.plot(rstudent(tft1b)) 7

8 eght nd weght re unvlble on mny cses so tht we only hve 36 cses for multple regresson on ge, heght, nd weght. We wll next rerun the smple lner regresson of sqrt(gf1) on ge for ths subset of cses. > # The followng lne computes n ndctor of cses tht re not > # mssng on ny of the vrbles of nterest. > nm <-!s.n(gf1) &!s.n(ge) &!s.n(heght) &!s.n(weght) > Tft1 <- lm(sqrt(gf1) ~ ge, subset=(ge>5 & nm) ) > summry(tft1) Cll: lm(formul = sqrt(gf1) ~ ge, subset = (ge > 5 & nm)) Resduls: Mn 1Q Medn 3Q Mx Coeffcents: Estmte Std. Error t vlue Pr(> t ) (Intercept) e-13 *** ge Sgnf. codes: 0 *** ** 0.01 * Resdul stndrd error:.043 on 34 degrees of freedom Multple R-squred: , Adjusted R-squred: F-sttstc: 3.97 on 1 nd 34 DF, p-vlue: () Comprng ths ftted model Tft1 on 36 cses wth the ftted model Tft1b bove on 1 cses, comment on nythng you fnd surprsngly dfferent n the regresson summry sttstcs for these two dtsets. Less sgnfcnt ge effect n smller nlyss (Tft1) s not surprsng, but no reson to expect such bg decrese n R-squred. () Compute n pproxmte 95% confdence ntervl for the slope usng the output for Tft1 nd usng the output for Tft1b. Do you thnk the estmtes of the slope bsed on these two smples re consstent? Why or why not? [Note: t.05,34 =.03 nd t.05,10 = Approxmte confdence ntervl mens you cn do the clcultons to just couple of dgts wthout clcultor.] 8

9 (c) Fnlly, we present the multple regresson of sqrt(gf1) on ge, heght, nd weght. > Tft <- lm(sqrt(gf1) ~ ge + weght + heght, subset=(ge>5 & nm)) > summry(tft) Cll: lm(formul = sqrt(gf1) ~ ge + weght + heght, subset = (ge > 5 & nm)) Resduls: Mn 1Q Medn 3Q Mx Coeffcents: Estmte Std. Error t vlue Pr(> t ) (Intercept) * ge weght heght Sgnf. codes: 0 *** ** 0.01 * Resdul stndrd error:.087 on 3 degrees of freedom Multple R-squred: , Adjusted R-squred: F-sttstc: 1.44 on 3 nd 3 DF, p-vlue: () Compre ths ftted model, Tft, wth the ftted smple lner regresson model, Tft1, n terms of the usul summry sttstcs, S E, R, nd F. Is nythng surprsng n the dfferences n these three sttstcs between the two models? Surprsng to dd vrbles nd yet see the resdul stndrd error ncrese. The extr vrbles dd not even compenste for the degrees of freedom. Clerly they do not help. R-squred must go up n the bgger model, but you see t went up only slghtly (nd Adusted R- squred decresed). () Wrtng ths multple regresson model s y = α+β ge +β wweght +βhheght, nd usng ether the summry of the regressons Tft1 nd Tft or the nov output below, compute test of the null hypothess 0 : β w =β h = 0. [Note: F.975,,3 = 4.15.] > nov(tft) Anlyss of Vrnce Tble Response: sqrt(gf1) Df Sum Sq Men Sq F vlue Pr(>F) ge weght heght Resduls Sgnf. codes: 0 *** ** 0.01 *

10 F (Reg SS Reg ) / SS 0 = RSS /3 ( ) /.458 / 1.9 = = = = / It s fne tht you remember the reltonshp between n F-sttstc nd the dfference n R-squred vlues, F ( n k 1) ( R R ) = q R1 (I forget ths one myself!) But I hghly recommend tht you commt to memory the more fundmentl defnton n terms of sums of squres. 10

11 You re done! You need not look t ths pge, but for completeness I fgured I should show you the Component+Resdul plots for the ftted multple regresson model. > pr(pty="s") > cr.plots(tft,sk=f,pch=19) 11