How Rao's Orthogonal arrays helps in developing Taguchis Design of Experiments.

1. Rao-Taguchi-
Rao-Taguchi-Mahalanobis
Life and Research
Alex Thumba
alexthumba@gmail.com

2. Prof. Calyampudi Radhakrishna Rao
C.R. Rao, Sc.D. (Cantab), F.R.S
Member, National Academy of Sciences, USA.
National Medal of Science Laureate,USA
Eberly Professor Emeritus of Statistics, PSU, USA
Distinguished Professor Emeritus and Advisor CR Rao AIMSCS
Padma Vibhushan Awardee, India

3. C R Rao was born in Hadagali, Karnataka, India. His Parents were C
Daraiswamy Naidu, a police inspector, and A Laxmikanthamma. He was the
eighth of his parents. His name Radhakrishna comes from the god Krishna (who
was the eighth of his parents children and, for that reason, the custom was to
name the eighth child after Krishna). In 1931 CR's father retired and the family
settled down in Visakhapatnam, on the coast of Andhra Pradesh. CR studied
there for ten years, first at high school, then mathematics, physics, and chemistry
at the intermediate Mrs A V N College before attending Andhra University. At
the intermediate college he won the Chandrasekara Iyer Scholarship in each of
his two years. He has had the unique distinction of knocking off the most coveted
prizes throughout his school career.

4. He graduated M.A. with First Class Honours in Mathematics from Andhra University in
1940. He applied for a research scholarship from Andhra University but his application
was rejected on the grounds that it had been received after the deadline. At this stage,
encouraged by his family, he decided to sit the competitive Indian Civil Service
examinations but, being only twenty years old, he had to wait eighteen months before
being allowed to take the examinations. He applied for job as a mathematician in an
army survey unit to fill out the time before taking the Civil Service examinations. He
was called to Calcutta for an interview but failed to get the job. However, this was a
turning point for CR, for he stayed in the South Indian Hotel before his interview and
there he met a young man who was being trained in statistics at the Indian Statistical
Institute. CR had taken a course on probability while studying for his Master's degree at
Andhra University but he had never heard of the Indian Statistical Institute. The young
man took CR to visit the Institute, at that time located in the Physics Department of
Presidency College. It seemed to provide both a job and a chance to test whether he
would like research so CR applied for the one-year training course in statistics.

5. The family were in some financial difficulties by this time since CR's father had died in
the previous year. However, one of his brothers and his mother managed to finance him
through the year at the Institute. The training course was rather a disappointment, taught
by people with little understanding of statistical theory. However, there was the head of
the Institute P C Mahalanobis, as well as other top researchers working at the Institute
such as K Raghavan Nair, Samarendra Nath Roy and Raj Chandra Bose. CR began
undertaking research with Nair and they published a joint paper Confounded designs for
asymmetrical factorial experiments(1941). In the following year he published six
experiments(1941)
papers, four of them joint publications with Nair, for example A general class of quasi-
quasi-
factorial designs leading to confounded designs for factorial experiments and A note on
partially balanced incomplete block designs.
designs.
CR was appointed as a Technical Apprentice at the Indian Statistical Institute beginning
in November 1943 and, a few months later, in June 1944, he also worked as a part-time
lecturer at Calcutta University. By the end of 1946 he had over thirty papers in print

6. Way to success
M.A. degree in mathematics with a first class and first rank at the
Andhra University in Waltair, , Andhra Pradesh-(1941)
M.A. degree in statistics from Calcutta University in Kolkata, West
Bengal with a first class, first rank and a record of marks unbeaten
till now, and a gold medal- (1943)
Ph.D. in 1948 from Cambridge University with R.A. Fisher, the
father of modern statistics, as his thesis advisor.
Sc.D (Doctor of science) degree in 1965 from the same university
31 Honorary Doctoral Degrees from universities in 18 countries
spanning six continents.

7. C.R.Rao With Prof. P.C. Mahalanobis

8. C. R. Rao with R.A. Fisher

9. C.R.Rao With S. N. Bose of Bose Einstein

10. Positions Held
worked in India at the Indian Statistical Institute for 40 years before he
took mandatory retirement at the age of 60
Director of the Indian Statistical Institute
President of the International Statistical Institute
President of the Institute of Mathematical Statistics (USA)
President of the International Biometric Society
Professor at the University of Pittsburgh and Eberly
Professor and Director of the Center for Multivariate Analysis at
Pennsylvania State University
C.R. Rao Advanced Institute for Mathematics, Statistics and Computer
Science as an honour ,for his outstanding achievements

11. Areas of research contributions
Estimation theory
Statistical inference and linear models
Multivariate analysis
Combinatorial design
Biometry
Statistical genetics
Generalized matrix inverses
Functional equations

12. Awards and medals
Guy Medal in Gold (2011) , Silver (1965) of the Royal Statistical Society[
India Science Award 2010
International Mahalanobis Prize (2003) of the International Statistical Institute
Srinivasa Ramanujan Medal (2003) of the Indian National Science Academy
President George W. Bush, on June 12, 2002, honored him with the National Medal of
Science, the highest award in U.S. in the scientific field
Padma Vibhushan (2001) by the Government of India
Mahalanobis Centenary Gold Medal (1993?) of the Indian Science Congress
Wilks Memorial Award (1989) of the American Statistical Association
Megnadh Saha Medal (1969) of the Indian National Science Academy
S. S. Bhatnagar Award (1963) of Council of Scientific and Industrial Research
JC Bose Gold Medal of the Bose Institute

13. Technical Terms Bearing Rao's Name
Widely quoted in statistical and engineering literature
Cramer – Rao Inequality
Rao-Black wellization
Rao’s Score Test
Rao’ s Orthogonal Array
Generalized Inverse of matrix /Khatri-Rao Product
Quoted in recent research papers
Quatum Cramer-Rao Bound
Rao’s Quadratic Entropy
Fisher–Rao Metric/Rao Distance/Rao Measure
Fisher–Rao Theorem/Cramer-Rao Functional
Rao-Rubin Theorem/ Rao’ s Damage Model
Lau, Rao and Shanbhag Theorems/Lau-Rao Theorem
Kagan, Linnik, Rao Theorem
Rao’ s Paradox in Sample Surveys/Multivariate Analysis
Quoted in specialized literature

14. Dr. Gen'ichi Taguchi (1924-2012)
(1924-
Taguchi was born and raised in the textile town of Tokamachi, in Niigata prefecture. He
initially studied textile engineering at Kiryu Technical College with the intention of
entering the family kimono business.However, with the escalation of World War II in
1942, he was drafted into the Astronomical Department of the Navigation Institute of
the Imperial Japanese Navy.
After the war, in 1948 he joined the Ministry of Public Health and Welfare, where he
came under the influence of eminent statistician Matosaburo Masuyama, who
kindled his interest in the design of experiments. He also worked at the Institute of
Statistical Mathematics during this time,and supported experimental work on the
production of penicillin at Morinaga Pharmaceuticals, a Morinaga Seika company.

15. In 1950, he joined the Electrical Communications Laboratory (ECL) of the Nippon
Telegraph and Telephone Corporation just as statistical quality control was beginning to
become popular in Japan, under the influence of W. Edwards Deming and the Japanese
Union of Scientists and Engineers. ECL was engaged in a rivalry with Bell Labs to
develop cross bar and telephone switching systems, and Taguchi spent his twelve years
there in developing methods for enhancing quality and reliability. Even at this point, he
was beginning to consult widely in Japanese industry, with Toyota being an early adopter
of his ideas.
During the 1950s, he collaborated widely and from 1954 he was visiting professor at
1950s,
the Indian Statistical Institute, He worked under C. R. Rao, Ronald Fisher and Walter A.
Shewhart.
Shewhart. While working at the SQC Unit of ISI, he was introduced to the orthogonal
arrays invented by C. R. Rao - a topic which was to be instrumental in enabling him to
develop the foundation blocks of what is now known as Rao-Taguchi methods
Rao-

16. Taguchi’s Works at Indian Statistical
Institute
Visiting professor at ISI from 1954
Taguchi Worked under C. R. Rao , Ronald Fisher and Walter A.
Shewhart at Indian Statistical Institute.
He was introduced to the orthogonal arrays invented by C. R. Rao
from SQC & OR unit of ISI which is the foundation block of well
known Taguchi methods
C.R. Rao &G. Taguchi made extensive use of OA’s in industrial
experimentation.
(described by Forbes Magazine as “new mantra” for industries),

17. L.H.C. Tippet, C.R. Rao, W.R. Deming (With Mrs. Deming) and
G. Taguchi

18. Letter from Dr. Krishna kumar-
kumar- 2 Trail House Court
Consultant in Business Analytics and Economist Rockville,MD. 20850, U.S.A.
Rockville,MD.
Dear ISI Friends,
One of my relatives recently visited us here in Rockville, MD. She is Nalini Prakash Magal. Duringa our
conversation she mentioned that her father HS Narayana Rao, former CEO of Hindustan Photo Films, Ooty,
and before that Chief Engineer, HAL, knew Prof PC Mahalanobis very well. She said she must have a
picture of them together some place. Here is one picture she and her husband Prakash (my wife, Nirmala's
cousin)sent me. This photo may be of some interest to ISI, Karnataka Branch
and its Life Members like me.I also recall here the speech delivered in Bangalore around 1990-95 by Dr
me. 1990-
Taguchi at the International Conference on Statistical Quality Control. He said that he conceived of optimal
Control.
manufacturing designs for quality improvement using Orthogonal Latin Squares and Dr CR Rao's papers
when he was visiting ISI, Calcutta in 50s and when Prof PC Mahalanobis sent him to Mysore Lamps for
50s
SQC projects. Some of you perhaps have the history of ISI, Karnataka Branch and this photograph may
projects.
have been taken during the visit of Prof PC Mahalanobis to Bangalore to establish that branch with the
assistance of Prof Ranganath of Information Science fame. Taguchi's work must be one of the earliest
works at Bangalore centre. Are there any other similar pictures that may be of historical interest to ISI
centre.
Karnataka Branch? Can others in my contact list above throw some light on this history?

19. ORTHOGONAL ARRAYS
Orthogonal Arrays (referred to Rao-Taguchi Methods) are often employed in industrial
experiments to study the effect of several control factors.An orthogonal array is a type of
experiment where the columns for the independent variables are “orthogonal” to one another.
Benefits:
1. Conclusions valid over the entire region spanned by the control factors
and their settings
2. Large saving in the experimental effort
3. Analysis is easy
To define an orthogonal array, one must identify:
1. Number of factors to be studied
2. Levels for each factor
3. The specific 2-factor interactions to be estimated
4. The special difficulties that would be encountered in running the
Experiment
After the visit to ISI, working under prof. C.R.Rao, the sensitivity and
Use of OA’s was followed in the following example in Arc welding process in Japan

20. An arc welding experiment performed by national railway cooperation of
Japan in 1959. Nine variables were studied as shown in the table:

21. OA design

22. CONCLUSION
• Idea of publishing a standard set of designs (orthogonal arrays is a
great idea) – C.R. Rao and Taguchi brought the field of DOE to the
masses
• Select variables that don’t interact -- interesting!! Assuming that
interactions
do not exist doesn’t mean they aren’t there!
• As a precaution select design generators that give best design
Design resolution and that provide desirable confounding structure.

23. C. R. Rao ’s orthogonal polynomial (RMM table)

24. Taguchi Methods
Taguchi's work includes three principal contributions to statistics:
A specific loss function — Taguchi loss function
The philosophy of off-line quality control and
Innovations in the design of experiments.
Taguchi’s Product development three stages:
System design stage
Non statistical stage for engineering, marketing and customer knowledge.
Parameter stage: How the product should perform against defined parameters. The
robust solution of cost effective manufacturing irrespective of the operating
parameters.
Tolerance design stage: Tolerance round the desired settings. Finding the balance
between manufacturing cost and loss.

25. Honors
Indigo Ribbon from the Emperor of Japan
Willard F. Rockwell Medal of the International Technology Institute
Honorary member of the Japanese Society of Quality Control and of the
American Society for Quality
Honorary member of American society of Mechanical engineers
3 times winner of Deming Application prize
Shewhart Medal of the American Society for Quality (1995)
Honored as a Quality Guru by the British Department of Trade and Industry
(1990)

26. Prof. P.C. Mahalanobis (1893 – 1972)
Prasanta Chandra Mahalanobis FRS
founder of Indian Statistical Institute
scientist and applied statistician
Mahalanobis distance, a statistical measure

27. Honours
Weldon Medal from Oxford University (1944)
Fellow of the Royal Society, London (1945)
President of Indian Science Congress (1950)
Fellow of the Econometric Society, U.S.A. (1951)
Fellow of the Pakistan Statistical Association (1952)
Honorary Fellow of the Royal Statistical Society, U.K. (1954)
Foreign member of the Academy of Sciences of the USSR (1958)
Honorary Fellow of King's College, Cambridge (1959)
Fellow of the American Statistical Association (1961)
Padma Vibhushan (1968)
Srinivasa Ramanujam Gold Medal (1968)

28. The government of India decided in 2006 to celebrate his birthday,
29 June, as National Statistical Day