© Joseph J. Nahas 2012 10 Dec 2012

!"#$%$&#'()*%+,-(./(012*3+4*-"%56#3"(777((8-#'9%+%(./(:#3+#-&*(

!"#$%&'!(')*&*#'

;( <(© Joseph J. Nahas 2012

10 Dec 2012

=*/*3*-&*%(•  >7!?(0!@(;ABACAD(E-*5F#&".3(8>E:8(#-G(;ABACAC(HI'$5/#&".3(

8-#'9%+%(./(:#3+#-&*(•  >7!?(0!@(JADAB(83*("K*(4*#-%(*LI#'M(#-G(%IN%*&$.-%(

–  JADABA;(;5O#9(8>E:8(.P*3P+*Q(–  JADABA<(?K*(;5Q#9(8>E:8(4.G*'(#-G(#%%I42$.-%(–  JADABAB(?K*(8>E:8("#N'*(#-G("*%"%(./(K92."K*%*%(#N.I"(4*#-%(–  JADABAD(;5O#9(8>E:8(&#'&I'#$.-%((–  JADABAC(R.-SG*-&*(+-"*3P#'%(/.3("K*(G+T*3*-&*(./("3*#"4*-"(4*#-%(–  JADABAU(8%%*%%+-,("K*(3*%2.-%*(/3.4(#-9(/#&".3(&.4N+-#$.-((–  JADABAJ(?K*("Q.5Q#9(8>E:8((–  JADABAV(H.G*'%(#-G(&#'&I'#$.-%(/.3("K*("Q.5Q#9(8>E:8((

•  >7!?(0!@(JADAD(OK#"(#3*(P#3+#-&*(&.42.-*-"%M(

B(© Joseph J. Nahas 2012

10 Dec 2012

EI"'+-*(•  8-#'9%+%(./(:#3+#-&*(W8>E:8X(Y#%+&%(•  <(O#9(8>E:8(•  <(O#9(8>E:8(01#42'*(•  !+,-+S&#-&*(?*%$-,(I%+-,("K*(F(G+%"3+NI$.-(

D(© Joseph J. Nahas 2012

10 Dec 2012

8-#'9%+%(./(:#3+#-&*(•  7-(#-(8-#'9%+%(./(:#3+#-&*Z("K*(P#3+#$.-(+-("K*(3*%2.-%*(

4*#%I3*4*-"%(+%(2#3$$.-*G(+-".(&.42.-*-"%("K#"(&.33*%2.-G(".(G+T*3*-"(%.I3&*%(./(P#3+#$.-A(

•  ?K*(23+4#39(&.42.-*-"%(#3*[(–  7-2I"(/#&".3%(–  =#-G.4(P#3+#$.-(

C(© Joseph J. Nahas 2012

10 Dec 2012

!I4(./(!LI#3*%(•  ?K*(P#3+#-&*(./(-(4*#%I3*4*-"%(+%(,+P*-(N9(

•  8-#'9%+%(./(:#3+#-&*(&.-&*-"3#"*%(.-("K*(-I4*3#".3(./("K*(P#3+#-&*(&#'&I'#$.-A(–  ?K+%("*34(+%(&#''*G("K*(?."#'(!I4(./(!LI#3*%(.3(!!W?."#'X(

•  ?K*(-I4*3#".3(.3(!!W?."#'X(+%(/#&".3*G(+-".(&.42.-*-"(2#3"%A(!

s2 =

(yi " y )2i =1

n

#n " 1

U(© Joseph J. Nahas 2012

10 Dec 2012

!+-,'*(F#&".3(012*3+4*-"(•  F.3(#(%+-,'*(/#&".3(*12*3+4*-"(Q+"K(\("3*#"4*-"%(#-G(-+(

%#42'*%(#"(*#&K('*P*'[(!!W?."#'X(](!!?(^(!!0(

(QK*3*(–  !!W?."#'X(+%("K*(?."#'(!I4(./(!LI#3*%(–  !!?(+%("K*(?3*#"4*-"(!I4(./(!LI#3*%(–  !!0(+%("K*(033.3(!I4(./(!LI#3*%(–  \(+%("K*(-I4N*3(./("3*#"4*-"%(–  -+(.%("K*(-I4N*3(./(.N%*3P#$.-%(#"(*#&K("3*#"4*-"('*P*'A(–  8("3*#"4*-"(+%(#(%2*&+S&(&.4N+-#$.-(./(/#&".3('*P*'%(QK.%*(*T*&"(+%(".(

N*(&.42#3*G(".(."K*3("3*#"4*-"%A(

!

(yijj =1

ni

"i =1

k

" # y )2 = ni (y ii =1

k

" # y )2 + (yij # yi )2

j =1

ni

"i =1

k

"

J(© Joseph J. Nahas 2012

10 Dec 2012

EI"'+-*(•  8-#'9%+%(./(:#3+#-&*(W8>E:8X(Y#%+&%(•  <(O#9(8>E:8(•  <(O#9(8>E:8(01#42'*(•  !+,-+S&#-&*(?*%$-,(I%+-,("K*(F(G+%"3+NI$.-(

V(© Joseph J. Nahas 2012

10 Dec 2012

?Q.(F#&".3(H.G*'(•  ?Q.(F#&".3(8-#'9%+%(./(:#3+#-&*(

–  ?Q.(/#&".3%(8(#-G(Y(!  8(K#%('*P*'%(;Z(<Z(A(A(A(#(!  Y(K#%('*P*'%(;Z(<Z(A(A(A(N(

•  H.G*'(*#&K(%#42'*(G#"#(2.+-"Z(_+`\(#%(4#G*(I2(./(/.I3(2#3"%[(–  ?K*(4*#-(P#'I*[(a(–  ?K*(3*%2.-%*(".(/#&".3(8[(b+((–  ?K*(3*%2.-%*(".(/#&".3(Y[(c`((–  8-(*33.3("*34(#%%.&+#"*G(Q+"K("K#"(%#42'*[(d+`\(

!

Yijk = µ + " i + # j + $ ijki = 1,2,!a; j = 1,2,!b;k = 1,2,!r

NIST ESH 1.4.3.7

e(© Joseph J. Nahas 2012

10 Dec 2012

?K*(!#42'*()#"#(!"3I&"I3*(

F#&".3(Y(

+' ,' ('('('' -'.*/0"1'2'

+' 3+++4'3++,4'54'3++1'

3+,+43+,,4'(((43+,1'

('('('' 3+-+4'3+-,4'54'3+-1'

,' 3,++4'3,+,4'54'3,+1'

3,,+43,,,4'54'3,,1'

('('('' 3,-+4'3,-,4'54'3,-1'

('('('' ('('(' ('('('' ('('('' ('('(''

*' 3*++4'3*+,4'54'3*+1'

3*,+43*,,4'54'3*,1'

('('('' 3*-+4'3*-,4'54'3*-1'

;f(© Joseph J. Nahas 2012

10 Dec 2012

?K*(H.G*'(W&.-"AX(•  O*(&#-(G*S-*("K*(G*P+#$.-%(/3.4("K*(4*#-(%.("K#"("K*9(%I4(

".(g*3.A(

•  O*(&#-(*%$4#"*("K*(P#'I*(./(b+(/3.4("K*(#P*3#,*(.P*3("K*(%#42'*(&.-"#+-+-,("K#"(/#&".3[(

!

" ii=1

a

# = 0

$ jj =1

b

# = 0

!

ˆ " i =

yijkj =1

b

#k =1

r

#

rb$ y = y i $ y

;;(© Joseph J. Nahas 2012

10 Dec 2012

R.-"3+NI$.-%(".("K*(?."#'(:#3+#$.-(•  Y*&#I%*("K*(G*P+#$.-%(#3*(G*S-*G(#3.I-G("K*(4*#-Z("K*(

P#3+#$.-(+-("K*(%#42'*G(G#"#(/3.4("K*(,'.N#'(4*#-(&#-(N*(4#G*(I2(./("K3**(2#3"%[(–  ?K*(P#3+#$.-(GI*(".(F#&".3(8Z(–  ?K*(P#3+#$.-(GI*(".(F#&".3(YZ(–  ?K*(P#3+#$.-(GI*(".(-.+%*Z(+A*A("K*(*33.3A(

•  O*(&#-(N3*#\("K*(P#3+#$.-(G.Q-(+-".(+"%(&.42.-*-"(2#3"%(N9('..\+-,(#"(#(%I4(./("K*(%LI#3*%[(

(QK*3*((–  !!W8X(+%("K*(Q*+,K"*G(%I4(./(%LI#3*%(/.3(/#&".3(8Z(–  !!WYX(+%("K*(Q*+,K"*G(%I4(./(%LI#3*%(/.3(/#&".3(YZ(#-G(–  !!0(+%("K*(%I4(./(%LI#3*%(/.3("K*(*33.3(!

SS(Total) = SS(A) + SS(B) + SSE

;<(© Joseph J. Nahas 2012

10 Dec 2012

!I4(./(!LI#3*(F.34I'#$.-(•  ?K*(/.I3(&.42.-*-"%(./("K*(!I4(./(!LI#3*%(#3*[(

!

SS( A) = rb ˆ " i =1

a

# = rb (y ii =1

a

# $ y )2

!

SS(B) = ra ˆ " j =1

b

# = ra (y jj =1

b

# $ y )2

!

SSE = (yijki =1

a

"j =1

b

"k =1

r

" # y ij )2

!

SS(Total) = (yijki =1

a

"j =1

b

"k =1

r

" # y )2

where rb is the number of samples in yi

where ra is the number of samples in yi

Note the similarities in the formulations

;B(© Joseph J. Nahas 2012

10 Dec 2012

012#-%+.-(./(#(:#3+#-&*(R#'&I'#$.-(

!

s2 =

(yi " y )2i =1

n

#n " 1

=

yi "

y jj =1

n

#

n

$

%

& & & &

'

(

) ) ) )

2

i =1

n

#

n " 1

=

yi2

i =1

n

# "

yii =1

n

#$

% &

'

( )

2

nn " 1

Correction for the mean

CM Raw Sum of Squares RSS

Corrected Sum of Squares

CSS Degrees of Freedom DF

;D(© Joseph J. Nahas 2012

10 Dec 2012

R#'&I'#$-,(!!(

!

CM =(SumOfAllObservatios)2

rab=

yijki=1

a

"j =1

b

"k=1

r

"#

$ % %

&

' ( (

2

rab

!

SStotal = (EachObservation)2" # CM

!

Ai = yijkj =1

b

"k=1

r

"

!

SS(A) =

Ai2

i=1

a

"rb

# CM where

!

SSE = SStotal " SS(A) " SS(B)

!

SS(B) =

Bj2

j =1

a

"

ra# CM

!

Bj = yijki=1

a

"k=1

r

"where

;C(© Joseph J. Nahas 2012

10 Dec 2012

8>E:8(?#N'*(

!.I3&*( !I4(./(!LI#3*%(W!!X(

)*,3**%(./(F3**G.4(

WG/X(

H*#-(!LI#3*(:#3+#$.-(WH!X(

.*/0"1'2' 66728' 7*9+8' 66728:7*9+8'

.*/0"1';' 667;8' 7-9+8' 667;8:7-9+8'

<11"1' 66<' 7)9*9-9+8' 66<:7)9*9-9+8'

="0*>' 667="0*>8' 7)9+8'

Relative contribution to the total variation

;U(© Joseph J. Nahas 2012

10 Dec 2012

EI"'+-*(•  8-#'9%+%(./(:#3+#-&*(W8>E:8X(Y#%+&%(•  <(O#9(8>E:8(•  <(O#9(8>E:8(01#42'*(•  !+,-+S&#-&*(?*%$-,(I%+-,("K*(F(G+%"3+NI$.-(

;J(© Joseph J. Nahas 2012

10 Dec 2012

8-(01#42'*(

NIST ESM 7.4.3.8

H#"*3+#'%(WYX(

+' ,' ?'h#N%(W8

X( +'@(+'?(A'@(?'

?(+',(B'?(?'

?(C'?(,'?(D'

,',(E'?(+',(D'

+(A',(,',(?'

,(E',(?',(C'

;V(© Joseph J. Nahas 2012

10 Dec 2012

!I44#$.-%(

1 2 3 Total (Ai)1 12.3 9.2 10.3 31.82 8.4 6.4 7.5 22.3

20.7 15.6 17.8 54.1Total(Bj)

Lab (A) Sum

Materials (B) Sum

r 3a 2b 3

r a b 18

Summations in each box on previous slide

Summation over rows

Summation over

Columns

Total Sum

NIST ESH 7.4.3.8

;e(© Joseph J. Nahas 2012

10 Dec 2012

!I44#$.-%(

CM 162.6SS(Total) 7.93SS(A) 5.0139SS(B) 2.1811SSE 0.7344

ANOVA Table

Calculations based on Slide 14

Source SS df MSA 5.0139 1 5.0139B 2.1811 2 1.0906

Error 0.7344 12 0.0612Total 7.93 17

NIST ESH 7.4.3.8 <f(© Joseph J. Nahas 2012

10 Dec 2012

EI"'+-*(•  8-#'9%+%(./(:#3+#-&*(W8>E:8X(Y#%+&%(•  <(O#9(8>E:8(•  <(O#9(8>E:8(01#42'*(•  !+,-+S&#-&*(?*%$-,(I%+-,("K*(F(G+%"3+NI$.-(

<;(© Joseph J. Nahas 2012

10 Dec 2012

!+,-+S&#-&*(./("K*(=*'#$P*(R.-"3+NI$.-((•  83*("K*(P#3+.I%(/#&".3(&.-"3+NI$.-%(%+,-+S&#-"(3*'#$P*(".("K*(

-.+%*(+-("K*(4*#%I3*4*-"A(•  h..\(#"(P#3+#-&*(./(/#&".3(3*'#$P*(".(P#3+#-&*(./("K*(-.+%*A(

<<(© Joseph J. Nahas 2012

10 Dec 2012

8>E:8(?#N'*(

!.I3&*( !I4(./(!LI#3*%(W!!X(

)*,3**%(./(

F3**G.4(WG/X(

H*#-(!LI#3*(:#3+#$.-(WH!X(

F(!"#$%$&(

F(=*/*3*-&*(

.*/0"1'2' 66728' 7*9+8' 66728:7*9+8' F6728:F6<' .G(GC4'*9+4')9*9-9+'

.*/0"1';' 667;8' 7-9+8' 667;8:7-9+8' F67;8:F6<' .G(GC4'-9+4')9*9-9+'

<11"1' 66<' 7)9*9-9+8' 66<:7)9*9-9+8'

="0*>' 667="0*>8'

7)9+8'

Variance of Factor relative to

Variance of Error

F Distribution Comparing

Factor Variance to

Error Variance

<B(© Joseph J. Nahas 2012

10 Dec 2012

012#-G*G(8>E:8(?#N'*(

In Excel: Fref = FINV(Alpha, a– 1, N – a – b – 1) = FINV(0.05, 2 – 1, 18 – 2 – 3 – 1) = FINV(0.05, 1, 12) = 4.75

> >

Source SS df MS Fstat FrefA 5.0139 1 5.0139 81.92 4.75 SignificantB 2.1811 2 1.0906 17.82 3.89 Significant

Error 0.7344 12 0.0612Total 7.93 17

Alpha = 0.05