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Allan Sims
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1 Rows 1-5 Rows 1-5 Rows 1-5 Rows 1-5 Rows 1-5 ^!"! #%$&')( $" *' +%$-,/.&$ 01243!"!"3 ^^^^ Si$ 6 -+j$ kl&$mn o6&$ ;%$]Ep 819oq -;>m -, $]q$&1rm%$] <=;n s C 7( td j$ q -;>m -, $]q$&1np 8198 um%$] <=;&J :!() N s j C 7( v' j$ ]T j]e ' w$-m ;j$]9$m m $] x=;&y z{ 1,/ k$ <;n6&$ 1,v w$ $ { 1, &$: <;>=)6 ',?2A@B;(C19 D 013FE D 3!FE>G!"3! HI- JLK ( NM D 0'3FE D 3!!E-01"!3O HIPQ ;>;%$ &7 M 5R ST$. K = $ by Christopher Bingham Cmd> data <- read("","stomata") stomata 70 2 ) Data on the number of stomata in 5 rows randomly selected ) on each of 14 randomly selected evergreen needles ) Col. 1: Needle number (1-14) ) Col. 2: stomata per cm. ) Source of the data is unknown Read from file "TP1:Stat5303:Data:stomata.dat" Cmd> makecols(data,needle,stomata) Cmd> needle <- factor(needle) '$}$ ;%$$md k$] ~i$}$ C C ;>m -,s j$7 k$&19$mj!6&$;2i ƒ ˆn$ 8'6 ;%$$-md w$ 2 }-~Š ~T$}$ q -;>m -,Š B j$7 )Œs jc ]Tq}$>$]E l&$ )(Ž1(}$ Experiment Needle 1 Needle 2 Needle Needle 13 Needle 14 2

2 H ( ' -; m $]C <;%$ I 19 o}-~ 2R6 ;%$]9$m ~s B6 <;Š$ 8'6 ;%$$md k$-y Cmd> row <- factor(rep(run(5),14)) Cmd> hconcat(needle,row)[run(10),] # first 10 cases (1,1) 1 1 (2,1) 1 2 (3,1) 1 3 (4,1) 1 4 (5,1) 1 5 (6,1) 2 1 (7,1) 2 2 (8,1) 2 3 (9,1) 2 4 (10,1) 2 5 Nothing in common between different instances of row 2, say, or any other row number!6&$,tm $7 ~T$ 6 -+%$ 6 -+%$ }$ + k ( (j$m 6 s6 s.&$$ ; 2 A;Š$>( <+ k$ ;)Š,im $] ~T ( xm.y$ ~v6y$}$ r6&$ q -;>m -, $]q$&1 T}-~ ~s B6 <; ;%$$md k$ -;>m J!6&$ ;%d k ; ph BM/ <;)9$ ;>m $m9 ;+%$ 6 6 s md q$}$ ;)s,t$ -; <;>= $ 8'6 2 6 $ 8'6 ;%$$md k$j '$}$ 6&-~ &( ~T ( xmn -; Y$ NŠ~Š Bq6 6&$ ;%$]9$m,Tm $7 xj Cmd> anova("stomata=needle+row.needle") Model used is stomata=needle+row.needle DF SS MS CONSTANT e e+06 needle row.needle ERROR1 0 0 undefined <;Ž$ 6&$}$? ; 0,i$ ->(}$-,i$ ;o &$ }-~?2 6&$}$ }$ ;% $q}rmjk}j m%$ ;jr x=; ; <; $E row.needle 81 k;r6&$}$.-() ;%$] <;&=s] row ~Š B6 <; needle J J d-~ d -;sy (r $] up9-, 6> m $ >;Œ <;>md ' 9$ <;)9$q 81 k ;! row.needle " E 0 E 0 """ J$#9-, 6&$.Ž$ ;Ž$ <;%$ row!6y$ m $-=)}$>$] op9$$-m, %#'&( s needle 0'J 6&$ m $-=}$>$ p9$>$-m, row.needle row ;%$ $-m <; needle MT < %#)* & s H.E-0&M+ 01O-HIFE-0&M+ D J 3 4

3 @Bi6&$ $ { &$8,T$ ;)9$r,/ m $ ; > >( 'l (;)) $ 8'6 }-~ ] $ 8'6 ;%$$md k$ 2Lj 6&$}$ ~i$}$ 01O > > 3 01" + <(%$]>2 q6y$ ;n -; } 8 9$,im $7 A~T ( xm.&$ ~v6y$}$ =) ;2 6&$ ;%d w ;rlh pmv,i$ -;) 9 <;>md ' 9$ 6 o6&$ w$ +%$] l Y$&! C 9 6&$ C 7( o}-~ A~Š N6 <; ;%$$md k$ 9J Sv6Y$ ;n&(n6 +%$ -;Š$ { &$Y,T$ ;)o6 v ;E )Š] q -;>m -,Š j$7 k$&1 ;>= $ ;) B k$]š i &$ HI;%$$md k$] M. $ ;) B k$]š i &$ : H }-~Š&2L M ~s B6 <; $ 8d6 &$ $ ;) N $ ;) B k$]š i &$ 5 H q -;>m -, 8$] <; }-~ 2L M ~Š N6 <;Š$ 8'6 &$ : $ ;) Bq P l <;>= ;,T$ >(}$-,T$ ;))s ;Š$ 8'6 &$ 5 $ ;) B 6&$ ;%$]9$m,im $7 A~T ( xm.&$ :-H QMƒ5H A:M z q}ch :5M *'d9$ 6&$}$ }$ ;% 1,/.&7 ;) <; <;>= ~T r,t}$ k$dq9$&j!6 '6 q 8'}$Y ] ( ;%$]9$m m $] x=;&j!6&$ 2 2 -;>m.&$ q -;>m -, + 8. k$] ~Š N6 }$ >(&,T$m9 2 $},T$ -;rh 8 B ;Ž$] ; -;) #% }$ )r -;&mš ;Ž xm $ ;Ž$ <;)9$)+ s& ( >(&,T$ q -;>m -, + 8 B. w$]s }$ ;%,/!6&$ q 1,T$d9$s }$ ;>m 6&$ 8 >ˆ! #"$&%'"ˆ(ˆ *) ;>m M!6&$ + 8 -;Ž$ ]T <;>=d k$.j$)+ k ; H, -!6Y$ + 8 ;Ž$ ] 6&$ =)q ;>m,t$ -;/. H. M ; 5 6

4 J E '$}$ r -;Š$ { 1,/ k$j A;Š$ { Y$8 x,t$ ;)Š~v - m $] x=;%$m9 (>m) 6&$ j ( $ Š]+ 8 E. B <;,T$ >(}$-,T$ ;))?]T6&$ p / ;)9$ ;) ] m)8 w$m ~/6&7 k$ $==&J { O D 3 z ; 3 u,/ 9$8 9.&$ -; Y$m?' 1,T$ p}-, <;>=d k$ ~T$7,Š $m?' -;>J 3O % 1,v k$] p}-, 6&$ ' -; ~T$}$ 8'lE = $-m j$ ;>md <;>=9 B.-&J 1,/ k$] ~i$$ j$ ; 9 $ 8'6 ].rh QM ~v6 '6 ' -;r.&$ ; xm $}$mn q -;>m, 1,/ k$ ].&J L $ 8d6.-2 $ 8'6 ]T. -; )rhi:-m ;n6&$ qt~i$}$ =d <+%$ ; n%~t 1,/ k$]rh 5Mƒ9 -; Y$J 8d6 -; BŠ,/ m $ m $d9$,š <; k ; ]T6&$ p v ;)9$ ;) ]T6&$ 1,/ k$j Cmd> data <- read("","data") data 48 4 ) Col. 1: Lab (1-6) ) Col. 2: Analyst (2 per lab) ) Col. 3: Sample (2 per experimenter) ) Col. 4: Percent fat as measured by experimenter Read from file "TP1:Stat5303:Displays:edp.046.dat" Cmd> makecols(data,lab,analyst,sample,y) Cmd> lab <- factor(lab); exper <- factor(exper) Cmd> sample <- factor(sample) Cmd> anova("y = lab+ analyst.lab+sample.analyst.lab",\ fstat:t) Model used is y = lab+ analyst.lab+sample.analyst.lab DF SS MS F P-value CONSTANT e-21 lab e-06 lab.analyst lab.analyst. sample ERROR z 8'6! -,v -(}$-m p}-, 6&$,T$ -;s 6 w$ +j$7 <J z{ 1,/ k$-y )* & ;C H.. ;Ž ;>(&,v.&$ ]+ <(%$]s -+%$q = $m9 -,ŠE -( $. UUU U ``` ` VVV V fff f ccc c \\\ \ ccc c VVV V'ccc c Cmd> y_ij_dotdot <- tabs(y,lab,analyst,mean:t); y_ij_dotdot (1,1) = Means for 12 analysts (2,1) (3,1) (4,1) (5,1) (6,1) Cmd> y_i_dot <- tabs(y,lab,mean:t); y_i_dot (1) (6) = Means for 6 labs Cmd> a <- 6; b <- 2; c <- 2; n <- 2 Cmd> c*n*sum(vector((y_ij_dotdot- y_i_dot)^2)) (1) = SS_lab.analyst M 7 8

5 !6&$ l&$7 k$d9 ; A* ' ( $ %# z P4 E-0 >;1, >;Ž, >;.8 :-H QM H.E-0&M >;1 >;Ž 5H A:M.H E-0&M >;1 z ' j$ Cmd> vector(a-1,a*(b-1),a*b*(c-1),a*b*c*(n-1)) (1) d ( $ # z P4 &3 >O :H LM D &3, >O, 57H :-M 013 &3, z C} 3!O #}-, 6?$],v 9$]?]T6&$ Œr }$ rhppl & E PL )* & M 0 ;.872, rhppl ) & E PL & )* M 0 ;Ž72 $d ]J ems() ' -; -,/ ()9$ 6Y$]Ž$ h, ( -Yy Cmd> ems("y = lab+analyst.lab+sample.analyst.lab",\ vector("lab","analyst","sample")) EMS(CONSTANT) = V(ERROR1) + 2V(lab.exper.sample) + 4V(lab.exper) + 8V(lab) + 48Q(CONSTANT) EMS(lab) = V(ERROR1) + 2V(lab.exper.sample) + 4V(lab.exper) + 8V(lab) EMS(lab.exper) = V(ERROR1) + 2V(lab.exper.sample) + 4V(lab.exper) EMS(lab.exper.sample) = V(ERROR1) + 2V(lab.exper.sample) EMS(ERROR1) = V(ERROR1) s.&$]}$ 2 -;>m 6&$ + 8 -;Ž$ T q -;>m -, $]q$&1 -;>m V -;>m o ;)8 E.-() k ; p}-, ;%$ r,t}$ Q C { $mn q E,i$}$8YJ ; C { $mn6&$}$ -;>m Q(CONSTANT) J Cmd> sigmasqa_hat <- (MS[2] - MS[3])/(n*b*c); sigmasqa_hat (1) Cmd> sigmasqb_hat <- (MS[3] - MS[4])/(n*b); sigmasqb_hat (1) Cmd> sigmasqc_hat <- (MS[4] - MS[5])/n; sigmasqc_hat (1) Cmd> sigmasq_hat <- MS[5]; sigmasq_hat ERROR Cmd> vcomp <- varcomp("y=exper + lab + lab.exper + lab.exper.sample",vector("lab","sample")) Cmd> vcomp Estimate SE DF lab exper.lab exper.lab.sample ERROR

6 ^ ( ' -; (j$ 6? () -() 9 -,/ -()9$ } {,/ 9$ ;Ž xm $ ;Ž$ <;)9$)+ ( <;>= H >(&,Š ;>= ;%,/ Bq ] $]q$&1)mpj Cmd> df <- vcomp[1,3]; df DF lab Cmd> estimate <- vcomp[1,1]; estimate Estimate lab Cmd> eps <-.025; chisqpts <- invchi(vector(1-eps/2,eps/2),df) Cmd> vector(df*estimate/chisqpts) # 95% confidence interval (1) ^^^ ^ ^^^!( &]j$ 6&$ %~T $ { &$8 x,t$ ;)9$s }$ j$7 k$&19$m j q6> ;%$ ;j$ { &$8 k$ ;Ž$mH 3 &$ ;n6&$.m ;>m 6&$ d6&$ $ { &$8 k$ ;Ž$mH 3 &$ MpJ z{ &$8 k$ ;Ž$ r p '}]j$m ~s B6. -;>m 1,v k$ ;%$]9$m ~s B6 <; -,/. ; w ;?]. -;>ms$ { &$8 k$ ;Ž$J Cmd> exper <- analyst # experience factor Cmd> anova("y=exper + lab + lab.exper + lab.exper.sample", \ fstat:t) Model used is y=exper + lab + lab.exper + lab.exper.sample DF SS MS F P-value CONSTANT e-21 exper lab e-06 exper.lab exper.lab. sample ERROR !6&$,Tm $7 y=exper+lab+lab.exper+lab.exper.sample Y$& q k$ r}6 exper ;>m 9$ '9]>Ž$m -;&m ;jr;j$ $-m j }6> lab lab.exper <Š q -;>m -, ;)9$q 81 k ;n9$, J sample < ;j$ $-m ~? }6 x; lab.exper J 11 12

7 !6&$,/ 6&$-,v ' A,i-m $7 ƒ ~v6y$}$ 2 -;&m }$ q -;>m -, + 8. k$] ~Š B6 Y$},T$ -;s -;>m + 8 -;Ž$] ;>m J Cmd> ems("y=exper + lab + lab.exper + lab.exper.sample",\ vector("lab","sample")) # exper not a random factor EMS(CONSTANT) = V(ERROR1) + 2V(exper.lab.sample) + 8V(lab) + 48Q(CONSTANT) EMS(exper) = V(ERROR1) + 2V(exper.lab.sample) + 4V(exper.lab) + 24Q(exper) EMS(lab) = V(ERROR1) + 2V(exper.lab.sample) + 8V(lab) EMS(exper.lab) = V(ERROR1) + 2V(exper.lab.sample) + 4V(exper.lab) EMS(exper.lab.sample) = V(ERROR1) + 2V(exper.lab.sample) EMS(ERROR1) = V(ERROR1) *'9$ 6 2.Y$&' (Ž$ exper Š q { $-m '92 Q(exper) ;&m ;j V(exper) <? > >q EMS(exper) J Cmd> varcomp("y=exper + lab + lab.exper + lab.exper.sample",\ vector("lab","sample")) Estimate SE DF lab exper.lab exper.lab.sample ERROR !6Y$}$ ;% <;j$ exper J 13

P a g e 3 6 of R e p o r t P B 4 / 0 9

P a g e 3 6 of R e p o r t P B 4 / 0 9 p r o t e c t h um a n h e a l t h a n d p r o p e r t y fr om t h e d a n g e rs i n h e r e n t i n m i n i n g o p e r a t i o n s s u c h a s a q u a r r y. J