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10 Dec 2012
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s2 =
(yi " y )2i =1
n
#n " 1
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10 Dec 2012
!+-,'*(F#&".3(012*3+4*-"(• F.3(#(%+-,'*(/#&".3(*12*3+4*-"(Q+"K(\("3*#"4*-"%(#-G(-+(
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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
"
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10 Dec 2012
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10 Dec 2012
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!
Yijk = µ + " i + # j + $ ijki = 1,2,!a; j = 1,2,!b;k = 1,2,!r
NIST ESH 1.4.3.7
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10 Dec 2012
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F#&".3(Y(
+' ,' ('('('' -'.*/0"1'2'
+' 3+++4'3++,4'54'3++1'
3+,+43+,,4'(((43+,1'
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,' 3,++4'3,+,4'54'3,+1'
3,,+43,,,4'54'3,,1'
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*' 3*++4'3*+,4'54'3*+1'
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('('('' 3*-+4'3*-,4'54'3*-1'
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10 Dec 2012
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!
" 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
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SS(Total) = SS(A) + SS(B) + SSE
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10 Dec 2012
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!
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 =
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n
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=
yi "
y jj =1
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#
n
$
%
& & & &
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(
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2
i =1
n
#
n " 1
=
yi2
i =1
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# "
yii =1
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#$
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( )
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
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10 Dec 2012
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<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
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10 Dec 2012
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10 Dec 2012
8-(01#42'*(
NIST ESM 7.4.3.8
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+' ,' ?'h#N%(W8
X( +'@(+'?(A'@(?'
?(+',(B'?(?'
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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
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<;(© Joseph J. Nahas 2012
10 Dec 2012
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<<(© Joseph J. Nahas 2012
10 Dec 2012
8>E:8(?#N'*(
!.I3&*( !I4(./(!LI#3*%(W!!X(
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.*/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'
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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