Analysis of Variance for Attribute Data in Taguchi ‘s Approach Zurnila Marli Kesuma, M.Si
Department of mathematics, Syiah Kuala University, Banda aceh, Indonesia kesumakuyahoo.com
Abstract.
Taguchi has extended the audio concept of the signal-to-noise ratio SN ratio to experiments involving many factors. The formula for signal-to-noise ratio are designed so
that an experimenter can always select the largest factor level setting to optimize the quality characteristic of an experiment. When the quality characteristic is a proportion, such as the
fraction defective, denoted p, which can take values between 0 and 1, we use the omega Ω
transformation as an objective characteristics. The selection of important factor is not only based on the response table, but also uses attribute accumulation analysis - analysis of
variance and the contribution ratio to establish significant factors. The method of conducting the analysis of variance is create a table of data, Calculate the overall total sum of squares
due to class I and class II. Calculate the total degree of freedom. Draw the analysis of variance. Calculate the predicted optimum process and calculate the confidence interval of a
predicted mean. 2000 Mathematics Subject Classification: 30C45, Secondary 30C80
Keywords : omega
Ω transformation, analysis of variance,contribution ratio.
1. Introduction
Analysis variance ANOVA was first introduced by Sir Ronald Fisher. It is a method of partitioning variability into identifiable sources of variation and the associated degrees of
freedom in an experiment. Taguchi recommends that statistical experimental design method be employed to assist in quality improvement, particularly during parameter design and
tolerance design, specifically to reduce the variability. He proposed some steps in analysis varians, that is calculate the percent contribution and pool insignificant factors.
Attribute data analysis such as the fraction defective, denoted p, can take values between 0 and 1. Frequently, p is expressed as a percentage where it can take values between 0 and
100. With fraction defective, the best value for p is zero. Attribute accumulation is used when the experimental data can be ranked or categorized.
Attribute accumulation analysis uses analysis of variance and the contribution ratio to establish significant factors.
2.
Fraction Defective Analysis and Weighting
Fraction defective type of data can be analyzed through fraction defective analysis. A better method of analysis is to use the omega
Ω transformation for improving the additivity of such characteristics. The respon table for fraction defective data is constructed by
determining the totals for each factor level in each category. For each factor, the total of the data in level 1 and level 2 should be the same.
Fraction defective of each class can be formulated as follows :
III I
III II
I I
I
f f
f f
f f
p =
+ +
=
III II
III II
I II
I
f f
f f
f f
p =
+ +
=
III III
III II
I III
III
f f
f f
f f
p =
+ +
=
2.1 Accumulation analysis needs some understanding of the binomial distribution. If the fraction
defective is p, then the corresponding variance is:
2
σ
= p
x 1-p
2.2 This implies that the variance depends on p. However, if we are to compare two distributions
corresponding to classes or categories in an experiment, we can only make a fair comparison if the varians are approximately the same. Since, the sum of squares of different
Classes in accumulation analysis analysis would have different bases, it is important to normalize these basis, by dividing the sum of squares of each class by its variance. This
procedur is frequently called weighting. The weight of each class is:
1 1
1
1 2
1
P x
P
I I
− =
= σ
ω
2.3 For ease of calculation, it may be better to use:
1 1
1
1 2
1
P x
P
I I
− =
= σ
ω
⎟ ⎟
⎠ ⎞
⎜ ⎜
⎝ ⎛
− =
1 1
III I
III I
f f
x f
f
⎟ ⎟
⎠ ⎞
⎜ ⎜
⎝ ⎛
− =
1
III I
III III
I
f f
f x
f f
2 I
III I
III
f f
x f
f −
=
2.4
2 II
III II
III II
f f
x f
f −
= ω
2.5
3. The Analysis of Variance
