Statistics Profile Analysis Gary W. Oehlert School of Statistics 313B Ford Hall

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1 Statitic Profile Aalyi Gary W. Oehlert School of Statitic 313B Ford Hall Let me add a few more thig about imultaeou iferece before goig o to profile aalyi. Advatage of the Boferroi approach. Eay to compute ad udertad. If we do reject H, we have iformatio about how it wa violated (which variable differ). Oly require uivariate aumptio, veru multivariate aumptio for T. Ca ue upooled whe variace differ. No imple mall ample fix for T. Will give horter cofidece iterval for a prechoe et of liear combiatio (icludig the coordiate themelve). Advatage of T. T i ivariat uder oigular liear traformatio. The Boferroi approach i ot. T ca have greater power tha the Boferroi approach, particularly whe correlatio are trog. The area (or volume) of a T cofidece regio ca be much le tha that of the correpodig Boferroi regio. The T cofidece regio ca be ued for liear combiatio uggeted by the data. Coider a ituatio where we have variable, all expreed i the ame uit, o that compario amogt them make ee. We have. The profile i a coect-the-dot plot of the mea profile plot u i t compoet I may cae, each compoet correpod to a differet treatmet, ad the variable may be repoe to the treatmet. Drug cocetratio i blood 1, 2, 3, 4, ad 5 hour after admiitratio. Cor yield uig five differet varietie. IQ meaured uig four differet techique. 1

2 I will ue the treatmet/repoe termiology, eve though there are other poibilitie. What might we like to kow about profile? Doe? I the profile flat? profile plot u i t compoet Doe? I the profile liear? profile plot u i t compoet Doe? I the profile quadratic? 2

3 profile plot 6 5 u i t compoet Wome track record from Table 1.9 of text. Cmd> readdata("") Read from file "/HOME/faculty/gary/clae/5401/JW5data/T1-9b.dat" Colum 1 aved a REAL vector x100 Colum 2 aved a REAL vector x200 Colum 3 aved a REAL vector x400 Colum 4 aved a REAL vector x800 Colum 5 aved a REAL vector x1500 Colum 6 aved a REAL vector x3000 Colum 7 aved a REAL vector xm Colum 8 aved a factor coutry Cmd> x800 <- x800*60 Cmd> x1500 <- x1500*60 Cmd> x3000 <- x3000*60 Cmd> X <- hcocat(x100,x200,x400, x800,x1500,x3000) Cmd> xbar <- tab(x,mea:t) Cmd> plot(1,xbar,lie:t, ylab:"ecod",title:"record time") 3

4 record time 500 e c o d Cmd> xbarl <- tab(log(x),mea:t) Cmd> plot(1,xbarl,lie:t, ylab:"ecod",title:"log record time") log record time e c o d I each cae, we ca fid a matrix ad chage the quetio to doe. Ofte there are may choice for. For cocretee, aume. For a flat profile? 4

5 The firt three of thee have full rak ( ). alo ha rak ( look at pair of ucceive mea:. A ozero differece idicate a chage poit i the mea. compare the firt compoet to ucceive compoet. ), but it i ot itelf of full rak. Thi might be helpful if the firt treatmet repreeted a cotrol or a tadard, ad we were itereted i which treatmet differed from cotrol. compare the th compoet to average of the precedig compoet 5.

6 Thi could alo be ued to look for a chage poit. look at all pairwie compario betwee treatmet. There i reducacy i thi et of compario. What if we were itereted i liearity of the mea? Liearity mea that the icremet from treatmet to i the ame a the icremet from to. Thu we might coider lookig at the differece of thee icremet: Let be the matrix that compute i the ame a. To tet liearity of a -vector, ue ucceive differece from a -vector. Note that Namely,. If we are itereted i the quadratic ature of the profile, we wat to kow if the chage i icremet from to i the ame a the chage i icremet from to. I differece: Note,. For a cotat,. For a liear,, but. For a quadratic,, ad, but. Cmd> C1 %*% xbar (1,1) (2,1) (3,1) (4,1) (5,1) Cmd> C5 %*% xbar (1,1) (2,1) (3,1) (4,1) Cmd> C6 %*% xbar (1,1) (2,1) above 6

7 (3,1) Cmd> C1 %*% xbarl (1,1) (2,1) (3,1) (4,1) (5,1) Cmd> C5 %*% xbarl (1,1) (2,1) (3,1) (4,1) Cmd> C6 %*% xbarl (1,1) (2,1) (3,1) How do we tet? We ca ue T or Boferroi t-tet. Note: we had,, ad above, all decribig equality of compoet mea. Which hould we ue for tetig? It doe ot matter for T. It doe matter for Boferroi t, which will be more eitive if deviatio from the ull match the patter exemplified by. Cmd> T2 <- <- <- <- <- <- <- <- 1-cumF(@T2,@p,@-@p) tructure(t2:@t2,df:vector(@p,@-@p),pval:@pv)") Cmd> T2(X%*%C1 ) compoet: T2 (1,1) compoet: df (1) 5 50 compoet: pval (1,1) 0 7

8 The origial time are t cotat over ditace, but we kew that ayway. Cmd> T2(Xl%*%C1 ) compoet: T2 (1,1) e+05 compoet: df (1) 5 50 compoet: pval (1,1) 0 The logarithmic time are ot cotat over ditace. Cmd> T2(Xl%*%C5 ) compoet: T2 (1,1) compoet: df (1) 4 51 compoet: pval (1,1) 0 The lope are ot cotat uig logarithmic data, I had high hope here. log record time e c o d Cmd> T2(Xl%*%C6 ) compoet: T2 (1,1) compoet: df (1) 3 52 compoet: pval (1,1) 0 8

9 Not quadratic either. Cmd> tab(xl%*%c5,tddev:t) (1) Cmd> tab(xl%*%c5,mea:t)/ tab(xl%*%c5,tddev:t)*qrt(55) (1) Boferroi make clear that each idividual chage i lope tatitically differet from 0. There are two-ample aalog of thee profile procedure. Suppoe ad are the mea of the two populatio. Are the profile parallel? Ie, doe? If ad are parallel, we ca tet for equal by tetig if. For parallel profile, thi i more powerful tha tetig. For equal profile, we ca tet for flate via. Let be the pooled etimate of variace from two idepedet ample of ize ad. Tet for parallel profile compared with We ca ue,, or ay equivalet form for. Tet of equality give parallel profile compared with Tet of flate give equal profile. Let compared with There i ome muddle over multiple tetig here. We may wat to ru each tet at a maller error rate o that the accumulated error rate doe ot get too large. Alo ote that thi approach i ot uique. To tet for equal, flat profile, we could do a above, or we could tet for equality of mea, ad the tet for flate of the commo mea if equality i ot rejected. We do t have to go through the parallel tage. Split time data ito et of 25 ad 30. 9

10 Cmd> X1 <- X[ru(25),] Cmd> X2 <- X[ru(26,55),] Cmd> S1 <- tab(x1,covar:t) Cmd> S2 <- tab(x2,covar:t) Cmd> Sp <- (24*S1+29*S2)/53 Cmd> xb1 <- tab(x1,mea:t) Cmd> xb2 <- tab(x2,mea:t) Tet for parallel profile Cmd> (xb1-xb2) %*%C1 %*% olve((1/25+1/30)*c1%*%sp%*%c1 ) %*%C1%*%(xb1-xb2) (1,1) Cmd> /5/53*49 (1) Cmd> 1-cumF(.68,5,49) (1) Parallel look OK. Tet for equal, give parallel. Cmd> um(xb1-xb2)ˆ2/ ((1/25+1/30)*um(vector(Sp))) (1) Cmd> 1-cumF(.08,1,53) (1) Equal look OK. We could have goe traight to equal with a two-ample T. Tet for flat. Cmd> xb <- (25*xb1+30*xb2)/55 Cmd> 55*xb %*%C1 %*% olve(c1%*%sp%*%c1 )%*%C1%*%xb (1,1) Cmd> 26905/54/5*50 10

11 (1) Cmd> 1-cumF(4982,5,50) (1) 0 Not flat, but we did t expect it to be. 11