STAT 5031 Statistical Methods for Quality Improvement



STAT 5031 Statistical Methods for Quality Improvement

A Taguchi experiment (Table 13.3 of Montgomery)

The original data set

Outer E -1 -1 -1 -1 1 1 1 1

array F -1 -1 1 1 -1 -1 1 1

A B C D G -1 1 -1 1 -1 1 -1 1 YBAR S SN

-1 -1 -1 -1 15.6 9.5 16.9 19.9 19.6 19.6 20.0 19.1 17.525 3.613 24.025

-1 0 0 0 15.0 16.2 19.4 19.2 19.7 19.8 24.2 21.9 19.425 2.908 25.522

-1 1 1 1 16.3 16.7 19.1 15.6 22.6 18.2 23.3 20.4 19.025 2.883 25.335

0 -1 0 1 18.3 17.4 18.9 18.6 21.0 18.9 23.2 24.7 20.125 2.598 25.904

0 0 1 -1 19.7 18.6 19.4 25.1 25.6 21.4 27.5 25.3 22.825 3.428 26.908

0 1 -1 0 16.2 16.3 20.0 19.8 14.7 19.6 22.5 24.7 19.225 3.380 25.326

1 -1 1 0 16.4 19.1 18.4 23.6 16.8 18.6 24.3 21.6 19.850 2.985 25.711

1 0 -1 1 14.2 15.6 15.1 16.8 17.8 19.6 23.2 24.2 18.313 3.729 24.852

1 1 0 -1 16.1 19.9 19.3 17.3 23.1 22.7 22.6 28.6 21.200 3.948 26.152

The SN is Taguchi’s signal to noise ratio for a setting like this where the objective is to maximize the Y value. A one-way main effects anova of SN gives:

DF SS MS

CONSTANT 1 5864.2 5864.2

a 2 1.7763 0.88813

b 2 0.47689 0.23845

c 2 2.8447 1.4223

d 2 0.16486 0.08243

ERROR1 0 0 undefined

(note that this is a saturated main effects design; there are no degrees of freedom left to do tests or to check interactions.) A table of means is

-1 0 1

A 24.961 26.046 25.572

B 25.213 25.761 25.604

C 24.734 25.859 25.985

D 25.695 25.520 25.364

We want a high SN. So set A =0, B=0, C=1, D=-1

This is rather unsatisfying. We do seem to have a higher SN ratio from this setting than from some other settings, but what is the origin of the difference?

To explore this, investigate the full data set as a ‘combined array’ of 34-2x23.

A sample MacAnova analysis that pulls the high points is

DF SS MS F P-value

CONSTANT 1 28010 28010 6658.21989 < 1e-08

a 2 51.4 25.7 6.10925 0.0055305

b 2 12.817 6.4085 1.52338 0.23291

c 2 68.717 34.358 8.16745 0.0013135

d 2 24.077 12.038 2.86170 0.071436

e 1 276.52 276.52 65.73162 < 1e-08

f 1 159.91 159.91 38.01191 5.9348e-07

e.f 1 3.8735 3.8735 0.92077 0.34425

g 1 0.91125 0.91125 0.21662 0.64469

e.g 1 2.6068 2.6068 0.61967 0.43679

f.g 1 1.4168 1.4168 0.33679 0.56563

a.e 2 2.5386 1.2693 0.30173 0.74156

a.f 2 2.9286 1.4643 0.34808 0.7086

a.g 2 26.881 13.44 3.19496 0.053905

b.e 2 9.9836 4.9918 1.18662 0.31794

b.f 2 1.5436 0.77181 0.18347 0.83322

b.g 2 0.16583 0.082917 0.01971 0.98049

c.e 2 6.3369 3.1685 0.75319 0.47879

c.f 2 4.4353 2.2176 0.52716 0.59517

c.g 2 4.3225 2.1613 0.51376 0.60295

d.e 2 21.005 10.503 2.49661 0.097798

d.f 2 15.087 7.5435 1.79318 0.18227

d.g 2 7.6758 3.8379 0.91232 0.41148

ERROR1 33 138.82 4.2068

This shows hugely significant E and F main effects, and smaller A and C main effects. There is no perceptible interaction between any design and any noise variable. This is quite unsettling. A robust design reduces the variability of the measured variable either by reducing the inherent random variability, or by finding interactions between the design and the noise variables that can be capitalized on to set the design variables to nullify the effects of the noise variables. But as far as we can tell, this data set is additive, so there are no interactions that can be used to cancel the large E and F main effects.

And furthermore, if we look at the problem, the real objective is to make the pull force as large as possible. Following up the anova fit by listing the effects of these four significant factors gives:

-1 0 1

A -1.0653 1.0014 0.063889

C -1.3694 0.52639 0.84306

E -1.9597 1.9597

F -1.4903 1.4903

Noted that E and F, being noise variables, must be harder to control than are the design variables A and C. But given that their effects are so much larger than those of A and C, you have to ask ‘How much is higher pull force worth, and how much would it cost to control the conditioning time and temperature at the high level?’ Perhaps the smart solution is to ‘just do it….’

Checking the residuals to check the other source of improvement, we find that D seems to have a large effect. The spread decreases as D goes from low to medium to high. This gives a pointer for setting D high to get another smaller layer of improvement.

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