Some Additional Notes on ANOVA-Based Gauge
R&R Estimation
We consider the model (2.4), page 21 of V&J and a standard Gauge R&R data
set consisting of m (repeat) measurements of I different parts by each of J
operators. ANOVA calculations produce mean squares M SA, M SB, M SAB,
and M SE. Point estimators for the quantities of most interest in a Gauge
R&R study are partially summarized on the bottom of page 27 in V&J. These
are
σ̂ 2rep eatability = σ̂ 2 = M SE
and
σ̂ 2reproducibility
µ
¶
M SB
(I − 1)
1
= max 0,
+
M SAB − M SE
mI
mI
m
Although it is not presented in V&J, an appropriate estimator for σ2R&R =
σ 2β + σ 2αβ + σ 2 (that is called σ2overall in V&J) is
σ̂ 2R&R =
1
I −1
m−1
M SB +
M SAB +
M SE
mI
mI
m
It is possible to state standard errors for σ̂rep eatability , σ̂ reproducibility and σ̂ R&R .
In fact, it is possible to use these estimates to make an exact confidence interval
for σ rep eatability and to use the Satterthwaite approximation to make at least
rough confidence limits for σ reproducibility and σ R&R . That goes as follows.
Let
ν rep eatability = IJ (m − 1)
Then, confidence limits for σrep eatability are
⎛
⎞
s
s
ν
ν
rep eatability
rep eatability
⎝σ̂ rep eatability
⎠
, σ̂ rep eatability
χ2ν r e p e a t a b i l i t y , upp er
χ2ν r e p e a t a b i l i t y , lower
and a standard error for σ̂ rep eatability is
s
σ̂ rep eatability
1
2ν rep eatability
For estimating σ reproducibility , let
ν̂ reproducibility
=
¡ MSB ¢2
mI
J −1
=
1
m2
µ
+
³
σ̂ 4reproducibility
´2
(I−1)MSAB
mI
(I − 1) (J − 1)
+
¡ MSE ¢2
m
IJ (m − 1)
σ̂ 4reproducibility
¶
M SB 2
(I − 1) M SAB 2
M SE 2
+
+
I 2 (J − 1)
I 2 (J − 1)
IJ (m − 1)
1
Then approximate confidence limits for σ reproducibility are
!
Ã
s
s
ν̂ reproducibility
ν̂ reproducibility
, σ̂ reproducibility
σ̂ reproducibility
χ2ν̂ r e p r o d u c i b i l i t y ,upp er
χ2ν̂ r e p r o d u c i b i l i t y ,lower
and a standard error for σ̂ reproducibility is
s
1
2ν̂ reproducibility
σ̂ reproducibility
For estimating σ R&R , let
ν̂ R&R
=
¡ MSB ¢2
mI
J −1
=
1
m2
µ
+
³
σ̂ 4R&R
´2
(I−1)MSAB
mI
(I − 1) (J − 1)
+
σ̂ 4R&R
³
(m−1)MSE
m
´2
IJ (m − 1)
M SB 2
(I − 1) M SAB 2 (m − 1) M SE 2
+
+
I 2 (J − 1)
I 2 (J − 1)
IJ
Then approximate confidence limits for σ R&R are
Ã
!
s
s
ν̂ R&R
ν̂ R&R
σ̂ R&R
, σ̂ R&R
χ2ν̂ R & R ,upp er
χ2ν̂ R & R ,lower
and a standard error for σ̂ R&R is
σ̂R&R
r
2
1
2ν̂ R&R
¶