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SEMI E89-1104 E © SEMI 1999, 2004 20 R5-2.5 Calculate the Mean Squares for both Load ( MS L ) and Repeat ( MS r ) as the Sum of Squares divi ded by degrees of f r eedom: 0.6830 0.3415 2 L L L SS MS df   ( R5-6) 5.2911…

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SEMI E89-1104
E
© SEMI 1999, 2004 19
RELATED INFORMATION 5
EXAMPLE OF A MEASUREMENT SYSTEM ANALYSIS WITH DAY AND
LOAD ONLY
NOTICE: This related information is not an official part of SEMI E89. It was derived from task force deliberations
during the revision of SEMI E89-0999 in 2001-2003. This related information was approved by full letter ballot
procedures and was approved for publication by the NA RSC on August 16, 2004.
R5-1 Introduction
R5-1.1 The following example illustrates how an MSA
is performed for a hypothetical measurement instrument
where only day and load are considered. Hand
calculations are provided and possible because the data
are balanced (i.e., each load has the same number of
repeated measurements). Rounding of value may cause
results may vary slightly. The data are shown in Table
R5-1.
Table R5-1 Data for MSA with Load and Repeat
Repeat Load 1 Load 2 Load 3
1 593.46 593.43 594.11
2 593.15 594.22 594.03
3 592.87 593.63 593.92
4 593.25 592.73 593.2
5 593.43 592.98 593.16
6 593.21 593.18 593.41
7 593.08 592.74 593.57
8 593.68 592.65 593.46
9 593.06 592.69 592.7
10 592.68 593.38 593.42
11 593.20 593.86 593.5
12 593.00 593.27 593.39
The model for the MSA is
Y
ij
=
+ l
i
+ r
ij
(R5-1)
Y
ij
= measurement on the i
th
load and j
th
repeat,
= true value of the measurand,
l
i
= error term associated with the i
th
load, and
r
ij
= error term associated with the i
th
load, j
th
repeat.
R5-1.2 Load and Repeat are treated as random effects.
Repeat is nested within Load.
R5-2 Analysis
R5-2.1 Calculate the mean over all observations (the
grand mean,
Y

) and the mean for each load (
i
Y
).
Table R5-2 Mean by Load and Grand Mean
N Obs Mean
Load 1 12 593.17
Load 2 12 593.23
Load 3 12 593.49
Grand Mean 36 593.30
R5-2.2 Calculate the Sum of Squares for Load (SS
L
):

3
2
1
12 0.6830
Li
i
SS Y Y


(R5-2)
R5-2.2.1 In general, the formula for SS
L
is

2
1
L
Li
i
SS n Y Y


(R5-3)
where
n = number of repeats per load and,
L = number of loads.
R5-2.3 Calculate the Sum of Squares for Repeatability
(SS
r
):

12 3
2
11
5.2911
riji
ji
SS Y Y



(R5-4)
where:
ij
Y = observation for the i
th
load, j
th
repeat.
R5-2.3.1 In general, the formula for SS
r
is

2
11
nL
riji
ji
SS Y Y



(R5-5)
R5-2.4 Calculate the degrees of freedom for Load (df
L
)
as L – 1. Calculate the degrees of freedom for
repeatability (df
r
) as L(n – 1).
SEMI E89-1104
E
© SEMI 1999, 2004 20
R5-2.5 Calculate the Mean Squares for both Load
(MS
L
) and Repeat (MS
r
) as the Sum of Squares divided
by degrees of freedom:
0.6830
0.3415
2
L
L
L
SS
MS
df
 (R5-6)
5.2911
0.1603
33
r
r
r
SS
MS
df
 (R5-7)
R5-2.6 The variance component for Repeatability
(VC
r
) is MS
r
. The variance component for Load is
0.01510
12
Lr
L
MS VC
VC
 (R5-8)
R5-2.6.1 In general, the formula for VC
L
is
L
r
L
M
SVC
VC
n
(R5-9)
R5-2.7
Reproducibility is
0.1603 0.01510
0.4188
rrL
VC VC


(R5-10)
SEMI E89-1104
E
© SEMI 1999, 2004 21
RELATED INFORMATION 6
EXAMPLE ILLUSTRATING THE DIFFERENCES BETWEEN NESTED
AND CROSSED EFFECTS
NOTICE: This related information is not an official part of SEMI E89. It was derived from task force deliberations
during the revision of SEMI E89-0999 in 2001-2003. This related information was approved by full letter ballot
procedures and was approved for publication by the NA RSC on August 16, 2004.
R6-1 Introduction
R6-1.1 The following example illustrates the
differences between an MSA where all effects are
nested versus the same data when one of the factors is
treated as crossed. It is performed on a hypothetical
measurement instrument. Hand calculations are
provided and possible because the data are balanced.
Rounding of value may cause results may vary slightly.
R6-1.2
The data, shown in Table R6-1, consists of
seven repeated measurements per load and three loads
per sample. There are four samples.
Table R6-1 Data for Example
Sample Repeat Load 1 Load 2 Load 3
1 1 493.72 493.79 492.92
1 2 492.91 493.77 492.78
1 3 493.08 494.07 492.57
1 4 493.78 491.28 493.52
1 5 492.69 492.19 493.79
1 6 492.57 491.82 493.46
1 7 491.21 492.95 493.33
2 1 516.27 514.21 515.20
2 2 516.26 514.54 515.55
2 3 516.35 514.93 515.28
2 4 515.68 515.14 515.47
2 5 515.46 515.21 515.28
2 6 515.67 515.07 514.87
2 7 514.87 515.00 514.77
3 1 541.30 541.84 541.57
3 2 541.38 541.92 541.59
3 3 541.48 542.33 541.71
3 4 541.53 540.61 541.02
3 5 541.31 541.37 540.76
3 6 541.41 541.21 540.85
3 7 540.85 540.80 541.73
4 1 551.00 551.03 552.02
4 2 550.83 551.02 552.23
4 3 551.01 550.98 551.78
4 4 551.60 551.38 550.12
4 5 550.90 551.13 550.58
4 6 551.29 550.74 550.27
4 7 551.91 549.06 550.98
R6-2 Nested Effects
R6-2.1 Assume that each sample represents
measurements made on different days. Load would be
nested in day and repeat in Load and Day. Repeats
cannot be randomized among Loads or Days. A given
Repeat must occur in a specific Load and Day. It is
nested in both Load and Day. Likewise, Loads cannot
be randomized among Days. Load is nested in Day.
R6-2.2
The model for the MSA is
Y
ijk
=
+ d
i
+ l
ij
+ r
ijk
(R6-1)
Y
ijk
= measurement on the i
th
day, j
th
load, and k
th
repeat,
= true value of the measurand,
d
i
= error term associated with the i
th
day,
l
ij
= error term associated with the i
th
day, j
th
load,
and
r
ijk
= error term associated with the i
th
day, j
th
load,
and k
th
repeat.
R6-2.3 Day, Load, and repeatability are treated as
random effects. Load is nested in Day. Repeat is
nested within Day and Load.
R6-2.4 Calculate the mean over all observations (
Y

),
the mean for each day (
i
Y
), and the mean for each
load-by-day combination (
ij
Y
).
Table R6-2 Means for Each Day
Day 1 Day 2 Day 3 Day 4 Grand
Mean
492.96 515.29 541.36 551.04 525.16
Table R6-3 Means for Each Load by Day
Day 1 Day 2 Day 3 Day 4
Load 1 492.85 515.79 541.32 551.22
Load 2 492.84 514.87 541.44 550.76
Load 3 493.20 515.20 541.32 551.14