by Christopher Bingham

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1 a D D E!"! #%$&')( $* '$&$,+.-&$0/!12"!"! &$9 ;:=<)5 '+>1@?A: (BC8 DFE 2HG 2!HG=*!"2! IJ, KML ( ON DFE 2HG 2!!G "!2/ IJPQ :=:%$ &6 N 4R ST$- L < $ by Christopher Bingham X Z [ \ ] ` b 02cde 7C87 gf $h i<:1j$ 7'5 $hk$&c? o 9 4 I E N E G E G E G E E E E G E E E G E G E G E G E E E - E G E E G E G E E G E E E E E G E E G E G E G E E G E G E E E G E G E E E E G E E G E E G E G E E G E E E G E G E E G E E E E E E E E E VVV VWWW WYXXX VVV V!ZZZ [[[ VVV VHZZZ ZC\\\ VVV V!]]] ZZZ Z_^^^ ^``` ZZZ Z_aaa Y&! 8$f lm A5 :)k %n o@9 op4 94 oq rs (t',: ( Y$,:u :)k v8 f $hb :%$,: ;:w,+x y$_8$ - z&'{ f $h i<: lm A5 %l - z&'{ hv O}7$ /6K b ~$ C+x y$ 1 5&$ o@94 :)k %n L()v 5&$ G E $ %+ $ :)) ;: - y '{?F,:=f0 ƒ5&$ E $ %+ $ : ;: - z&'{??yk G E n I E NJ1 -,1s E n,1s-,1 61s -7 2

2 1 [ [ [ y$ Cf 08 5&$ f $h i<:??? 5 m 2c Qf $ ;<:=1 I E N = B 6( x' Y$ h 2 pf $h i<: 0< 2 $ %+ $ : ;: z&'{ >h O} $ 2 K 9 z&'{? 5 5&$ $ %+T$ :)) l.5 '5 k5 $ o@94 :)kk x$hkb! z$ :)) m $ G E,:=f - z&'{??f5 5&$ k $ +T$ :)) mlx5 '5 5&$ o@94 :kk x$hkb! z$ :)) m $ E K o@94 5&$ p,:=f0- y '{? :) ;: ;:=<ui E N. k5 $!"#$%!&(')& #+*K 5, (, l.5 '5 e 7C8,+x- ;: z : $ : l.5 '5 - y&_{k?a: ( ;:=<5&$ f $h i<:1 & ( l ( ;f0k,:=f,+ u'5&h Y$,: 7C( - z&'{ -&$ - z&'{?,:&fk,:=f +m &h A z : 5&$ k $ +T$ :)) ;: -&_5 - y&'{, &K o &(0 $ :%$$f8 < $_ 5&$ l.5&6 z$ f $h i<: 5&$ 7 ;:w! - z&'{?yk -? I E rs (t',: < $_v5&$ k $ +T$ :).,+.- ;: y: :0- z&'{?? - +t( O A ;:=<.- - z&'{ -,1 f) ;:=<0,: /=( $h N m "A m " [[[ [[[?: e 7C & ( < $_v5&$ C+ $ $ %+T$ :)) lx5 $ : & ( +>( O - - x v -761R5&$ _5&$ $ %+ $ :x,+.- ;: z : 5 v $ :%_ ;: - z&'{?k b $ C+. z$ 1 I E 1 4 [ 1-7 N [[[ -7 - "-" [[[ -7 7 " "

3 5&$ $hk$&cu 6!l A5 - z&'{ 5&$ e 7C8v- y&_{ +Tf $6 d 9 9 lx5 $ $ m k,:=f,+ ub &$f0- z&'{ $hk$&c K 5&$ : $h +. 8$ I G G G G N IJ9 G 9 N,2 "! lx5 $ $ # $ &% E "! I 9 G 9 N',2 - :)k C+ : 8$h - 5&$ $h +x 8$K Sx5 $ :0- z&'{ m $ k,:=f,+t1q Am<) $ ;:B A 8$h 5&$,:=f) fm$k K S.5&$ :05&$ $ w &$f1m Av- Y$h o ƒ5&$ _5&$x :)k ) m Y&! 8$f l O5 e 7C87 Q$hk$&C) m $ (:w : (:=f $f l A5 - z&'{ m-&$&',( Y$ 5&$ :)k ) m $ B5&< : 8 5&$ oq94 :)k K ( K KKK KKK KK KKK KKK KK 1j$_ K )"! *' 9+ -, m &$ Y 5&$h Y$ $h +x 8$h m $ :%_ : (:=f $f l Ak5 - z&'{ $hk$&c) &K ' t5 f $h : $hk$&c) m,:=f %l hg7lx ;:)8$k 7C z : m $ (:B : (:=f $fk. : 5&$ 5 $$hg lx = ;:)8$k 7C z : : (:=f $fk 5 6

4 G 1 Z rs (t',: ( Y$,: _5&$e 7C87 A Q$hk$&C :)k v8 f $ w ;:%$ - z&'{ &K _$ $,:%_5&$02c Qf $h i<: l O5 f $hb ;: ;:=<> :)k o@9 I E 1MG E 1MG E 1 E 1 E G E 1G E 1 E NJK??? I E N *',l kk 8 $h +x 8$ ( ;:=<5&$ o@9 :)k K rs (t',:t'5&$&_{ A 5 5&$ + I G G G G N I N,2 *',l 5&$ oq9 :)8$k 7'7 z : : (:=f $f1s-,() :% _5&$ =1 :%_~$ %$ : oq94jk o f%$h ;<: - Y$fm :mo f $hb ;: :=<> :kk x : (:&f 5&$ o +. : $hk$&c K ``` ` ZZZ Z'^^^ ^,``` ` ZZZ Z'aaa a [[[ [ VVV V!ZZZ Z\\\ UU \ VVV V!]]] ] ZZZ Z_^^^ ^ XXX X ^^^ ^ ^^^ ^ XXX [[[ [ ZZZ Z VVV V 111 X ``` ` aaa a ZZZ Z ``` ` ZZZ b 2 Qf%$ <:=1F+. _ confound2() k ;:&f 5&$ J C8,+.- ;: z : ;:m$ 7_5 - z&'{ lx5 $ :0&( =( A f $hb ;: ;:=<> :)k ) &K 2c Ml O5 f $hb ;: ;:=<> :)k ~o@94 Cmd> confound2(vector(1,1,1)') WARNING: searching for unrecognized macro confound2 near confound2( component: block1 (1) "(1)" (2) "ab" () "ac" (4) "bc" component: block2 (1) "a" (2) "b" () "c" (4) "abc" 5 : (:=f o@9,4 o 9 4 zk *'_8$ 5&$ -7 + $+- I, N :5=$ <(&+ $ :) K 5 +. {&$h Au m l %$&C8 I +x 7 lm A5 :%$,lxn Cmd> confound2(vector(1,1,0)') # defining contrast is AB component: block1 (1) "(1)" (2) "ab" () "c" (4) "abc" component: block2 (1) "a" (2) "b" () "ac" (4) "bc" counfound() l {, l 5 Qf%$ <: K 7 8

5 R w$ k ;:&f0 5 $ f%$h <: 2 Qf $h i<: confound2() l O5 f $hb ;: ;:=<> :)k oq94s6k 5&$ $ $ z&'{ >hv A} $ 2c8!K. : $hk$&c oq94s : (:=f $fk Cmd> confound2(vector(1,1,1,1)') # 2^(4-1) component: block1 (1) "(1)" (2) "ab" () "ac" (4) "bc" (5) "ad" (6) "bd" (7) "cd" (8) "abcd" component: block2 (1) "a" (2) "b" () "c" (4) "abc" (5) "d" (6) "abd" (7) "acd" (8) "bcd" : (:=f o@9,4 1M,1M ;:0 2 gf $h i<:1( w$ confound2(vector(1,1,1,0,1)') o 9 4 b v 2 Qf $h i<:1 5&$ =)<(&+ $ :) m E -= { +x k7 v,l %$&C8K b v 2 pf $h i<:1 5&$ <(&+ $:)m+t( -&$ - { +.,1 lm A5 $ 7'5,l $&! e ;:=<0 f $hb ;: ;:=< :)k Sx5= u - z&'{ O} $h >h 2 $ 8 < $K b ~$ C+. y$ 1s& ( $ f%6 :=<0 2cp,:=f lx :) 8 ( Y$ - y&'{, hv A} $ 2 y1r5 2c f $h i<:=k / 2 rs ( :%$$f8 ( Y$ %l f $hb ;: ;:=<> :)k ) &K Y$ :%$ 8 A ;:%lt <) ( >h,:=f 5&$ Y$& :=f8 % A $ 7'5 <) ( hv/ ;:)8 %l <) (, h, I E N - G E 1 E 1 E E 1 E G E E 1 E E E E 1 E 1 E G E E 1 E I E N - o@9 I E N - #%$_ k o@9,4,:=f o@96n G o@96n 1MG 1MG 1G 9 z&'{ =(-f_ if $f ?????????,:&f?? :) ;: $ %+T$ :)) e,+?f,:=f?? lm A5 o@9 E K?,:&f?? :) ;:0 $ )G + $:)) e,+?,:=f??ql A5 o@9 G E 9 10

6 c c c Sx5= v5 =&$ : lx5&$ : & (0k 8 $ i+. 8$ 4 ( ;:=<> :)k G E 1MG E 1G E 1MG E 1 E 1 E 1 E 1 E ' I G G G G N I G=9 G&9 G&9 G= N! I G=9 G&9 9 9 N,/! :) C+ ;: 8$f lm A5 - z&'{ $hk$&c) Y 4 : (:&f $f l A5 - y '{ &K *'_8$,n o@9 o@94 o m,: $ C+. z$ h 5&$ +.&B,:) aaa a XXX X ZZZ Z XXX X WWW W WWW W ]]] ] XXX X n Sx5 $ :0&(>: (:=f0,: %lt :)k ) =1 5&$6 0< $ :%$B A} $f0 f(wc m Y : (:=f $fk _$ $ k ;:&f m 5 $ 8$ %+ $:,+.- : y : confound2() ~$ 7'5 - z&'{k Cmd> confound2(matrix(vector(1,1,1, 1,1,0),)') # ABC and AB component: block1 (1) "(1)" (2) "ab" component: block2 (1) "c" (2) "abc" component: block (1) "ac" (2) "bc" component: block4 (1) "a" (2) "b" 5,( =1 lm A5 u : (:=f $f +. : $hk$&c1 k5 $ '5&6 $ h oq9,:=f o@94 f $hb ;: ;:=<> :))G k ) :%_ < fm :%$K rs(>',: ( Y$ k ;:&f f%$ k ;: i:&<t:, ) choosedef2() 5 $ -&$h u ;: u$b : Y$ : Y$,K component: generators (1) "BC" (2) "AB" component: aberration component: basis (1,1) (2,1) <_ Y$h f%$hb : ;:=<> :)k ) A B C o@9,:=f 94 5 m-&$_k8$ ;:w$ o@9 9,4 o@9 O4 op4jk PR ;: $hk$&c) m,:=f o@94 $ :%_. : (:=f $fk 2!G lx ;:)8$k 7C z : m $ : (:=f $fk 11 12

7 4,+T,:w$: generators (),(). :) ;: k5 $ f $hb ;: ;:=<> :)k ) m choosedef2() z$_k $ &K 4 +.&:%$:) 5= m5&$ '+T$ i:g +x 7 z : m basis lm A5u6 ;(&+>: k $h & :=f_ ;:=<8 e 7C8 &K E : u6 ;(&+t: + $,: k5= e 7C8u ;:05&$ : (:=f $f $h$&' K rj(t_,: ( w$,,: = <(7+ $:~ basis choosedef2() K Cmd> basis <- choosedef2(,2,all:t)$basis Cmd> confound2(basis) component: block1 (1) "(1)" (2) "abc" component: block2 (1) "ab" (2) "c" component: block (1) "a" (2) "bc" component: block4 (1) "b" (2) "ac" :Y$:) aberration $h ; (m5 l +x : $ B$&' >h~f_ k$ $ :)~)f%$ $ : (:=f $fk rj (t',:08$6 " +. : $hk$&c) %l hg l. ;:)8$k 7C z : m,:=f " 5 $$ G lx ;:)8$k 7C y : $ : (:=f%$fk #%$_, k l Ak5 2 ;: - y '{ >hv/hn 2 Cmd> stuff <- choosedef2(4,2,all:t) Cmd> stuff component: generators (1) "ABC" (2) "ABD" component: aberration (1) component: basis (1,1) (2,1) m Y : (:=f o@9,4 oq9! o ;9 O4s 04sK 5 5&$ :%$ : (:=f $f%lthg l. ;:)8$k 7C y:=k o@1soq9!1soq4s1soq!19,4s19!1sop4j!194j :=f0oq94s $ (:w : (:=f $fk 1 14

8 '$ $ $ 5&$ $ %+ $ :)) n Cmd> confound2(stuff$basis) component: block1 (1) "(1)" (2) "ab" () "acd" (4) "bcd" component: block2 (1) "c" (2) "abc" () "ad" (4) "bd" component: block (1) "ac" (2) "bc" () "d" (4) "abd" component: block4 (1) "a" (2) "b" () "cd" (4) "abcd" I E N 7 - = hf f f hf -7hf -f -f -7hf k 2 c ;:=<05&,l z :=< A {&$h &K Cmd> timeit(stuff1 <- choosedef2(7,,all:t);;) Elapsed time is 10.8 seconds 5 =:#, %$k y:=<-,(u A, m :%_ $ - z$ lx O K Cmd> print(stuff1,format:"4.0f") stuff1: component: generators (1) "ABDE" (2) "ACDF" () "BCDG" component: aberration (1) component: basis (1,1) (2,1) (,1) K Sx5 _5&$ $hk$&c) m $ : (:=f $f ;: 5&$ 2 c f $ ;<: -&$h if $h oq4s 19,4s b 1 :=f o b b b o b b b b o b b K 9,4 %.(~-= choosedef2() K o < $ :%$k A} $f f(w' >h 5&$h w$ 5 $$,n oq4s 94s oq94 ; moq9 oq4s o@94 ;94 ; 9! oq4s 94s oq94 ; moq9,:=f 15 16

9 5 ; v :k %+> l.5 _5 =, * (G l. ;:)8$k 7C z : m $ aberration : (:=f $f1s-,() :% +x ;: $hk$&c) =1R%lThG l.,1r5 $$hg l.,1 B Y$hG lx 1M G l.,1j Y$ %$ :G l. : $7G 7C z : =1,:=f5&$ $ $ y >h (:w :G (:=f $f / lx = ;:)8$k 7C z : &K #%$_, $_ '5 Av8 2 c n Cmd> timeit(stuff2 <- choosedef2(8,,all:t);;) Elapsed time is 87.8 seconds Cmd> print(stuff2,format:".0f") stuff2: component: generators (1) "ABDEF" (2) "ACDEG" () "BCDEH" component: aberration (1) component: basis (1,1) (2,1) (,1) { i+th H" Y$& :=f > : PR 7! ;:)8h 5 E H"K PM(B'5 < $ < $_) z :=< $t5 :m :Y$ l.,:)) 8 lx A K rs (t',: %$ + $ - f_ $&C ;:=<k5= choosedef2() :Y_t+ { $ :t$=5= ( ) Y$ Y$ '5K S> A 5 tries:m i: $ Cf all:t 1 choosedef2() k i:&f 5 $ - $ x m :G f,+ A Y$6 z$&c8$f Y$_) >h~f $hb : :=<> :G k ) &K?, ',: -&$ +>(B'5 /=( '{&$1s,:=f A5& (=<)5 O =:, <( k,: $=$fu8 B ;:=f5&$ -&$h A~hJ8$ : f $h &K Cmd> timeit(stuff <- choosedef2(7,,tries:1000);;) Elapsed time is 2.5 seconds Cmd> print(abberation:stuff$aberration,format:"4.0f") aberration (1) Cmd> timeit(stuff4 <- choosedef2(8,,tries:1000);;) Elapsed time is 2.4 seconds Cmd> print(abberation:stuff4$aberration,format:"4.0f") abberation: (1) S $ < _v5&$ C+T$ -&$kk y : k8$): m lm A5 5&$ $5,( %$ Y$ '5&$h m-,() -&_5 l $$ +t(w'5 e 8$K 17 18

10 '$ $ B : A, >h 2 f $h i<:1 E'D - z&'{ >hv O} $!K Cmd> data <- read("","exmpl15.6") exmpl ) A data set from Oehlert (2000) \emph{a First Course in Design ) and Analysis of Experiments}, New York: W. H. Freeman. ) ) Table 15.6, p. 98 ) Data for a 2^7 in standard order. Factors are size of image, ) shape ofimage, color of image, orientation of image, duration ) of image vertical location of image, and horizontal location ) of image. ) ABCD, ACEG, BCE, BCFG, ACF, CDEF, ABG, BDEG, ADE, ADFG, BDF, ) EFG, CDG, ABEF, and ABCDEFG are confounded with blocks. ) Columns are block, A, B, C, D, E, F, G, and response Read from file "TP1:Stat50:Data:OeCh15.dat" Cmd> makecols(data,block,a,b,c,d,e,f,g,y,factors:run(8)) Column 1 saved as factor block with 16 levels Column 2 saved as factor a with 2 levels Column saved as factor b with 2 levels Column 4 saved as factor c with 2 levels Column 5 saved as factor d with 2 levels Column 6 saved as factor e with 2 levels Column 7 saved as factor f with 2 levels Column 8 saved as factor g with 2 levels Column 9 saved as vector y!( lm A5 :% :GB : (:&f $f f%$h ;<: &( ',: y={ r 8$h $hk$&c, &K ',+T$ 1 5&,lT$ %$ =$ : (:=f $f0,:=f +x -&$ < $ -&$&',( Y$ h A =)<%$ - z&'{ f_ k$ $ :w$h &K Cmd> yts <- yates(y) Cmd> chplot(rankits(yts),yts,xlab:"normal scores",\ title:"normal score plot of Yates effects") Cmd> chplot(halfnorm(abs(yts)),abs(yts),\ xlab:"half ormal scores",\ title:"normal score plot of Yates effects") Normal score plot of Yates effects 16 y 0.1 t s Normal scores Half normal scores plot of Yates effects Half normal scores k$&c) E'D 1 E "!!1/2,:=f E 2!* &$ 8 -&$ () y$ &K?A: - ;: k :%_ z :0k5 $h Y$ =$ b! 4s9'o k$&c E_D "!" E "!"!""- E "! E!E " E "!" E - o@ b /2 " E " E " E "- 9! b E 2!* E!EE!E!E!EE - o@94j b *'_8$ 5 vk5 $ v5 $$ $ p C+T :=<k5 $ f $hb ;: :&< :kk m,:=f0 $ $h Y$ : f_ k$g $ :w$h m-&$_%lt$$ : - y '{ =1:%_v $ %+ $:) $hk$&c) &K 19 20

11 '$ $,:mo*. o :B! (=f :=< +. ;: $hk $ C) ;( 2!G lx =,1HG l.,:=f /HG l. :)8$k 76G y: &K Cmd> anova("y=block + (a+b+c+d+e+f+g)^4") Model used is y=block + (a+b+c+d+e+f+g)^4 WARNING: summaries are sequential DF SS MS CONSTANT block a b c d e f g a.b a.c a.d a.e a.f a.g b.c b.d b.e 1 5e-05 5e-05 b.f b.g c.d c.e c.f c.g d.e d.f d.g e.f e.g f.g a.b.c a.b.d a.b.e a.b.f a.b.g 0 0 undefined a.c.d a.c.e a.c.f 0 0 undefined a.c.g a.d.e 0 0 undefined a.d.f 1.125e e-06 a.d.g a.e.f a.e.g a.f.g b.c.d b.c.e 0 0 undefined b.c.f b.c.g b.d.e b.d.f 0 0 undefined b.d.g b.e.f b.e.g b.f.g c.d.e c.d.f c.d.g 0 0 undefined c.e.f c.e.g c.f.g d.e.f d.e.g d.f.g e.f.g 0 0 undefined a.b.c.d 0 0 undefined a.b.c.e a.b.c.f a.b.c.g a.b.d.e a.b.d.f a.b.d.g a.b.e.f 0 0 undefined a.b.e.g a.b.f.g a.c.d.e a.c.d.f a.c.d.g a.c.e.f a.c.e.g 0 0 undefined a.c.f.g a.d.e.f a.d.e.g a.d.f.g 0 0 undefined 22

12 a.e.f.g b.c.d.e b.c.d.f b.c.d.g b.c.e.f b.c.e.g b.c.f.g 0 0 undefined b.d.e.f b.d.e.g 0 0 undefined b.d.f.g b.e.f.g c.d.e.f 0 0 undefined c.d.e.g c.d.f.g c.e.f.g d.e.f.g ERROR &$ $k $+ : ) >ht 5&$ &6 z$f HG=1 G,:&f *!G lx ;:)8$k 7C z :!K *'_8$ 5&$ 6 y,l ;:=<=n 9C e 5&$ A =)<%$h ~$hk$&c +T$,: /=( $ e K $Y+t a.b.g 1 a.c.f 1 a.d.e 1 b.c.e 1 b.c.e, b.d.f, c.d.g, e.f.g, a.b.c.d, a.b.e.f, a.c.e.g, a.d.f.g, 5&, w$ " b.c.f.g, f $<$=$h hve $$f,+ b.d.e.g, K c.d.e.f 5&$h Y$ $ 5&$ E : (:=f $fm$hk$&c) &K 2

choosedef2(7,4,all:t) K

choosedef2(7,4,all:t) K i!"! a ` a c a ``` `aaa ``` aaa ``` `!ccc j'$k$ 1 C l ; B-?hm 4noqsr $h t=;2 4nXu ED4+* J D98 B v-,/. = $-r