The document discusses Yates’ algorithm for 2n factorial experiments, a statistical method for analyzing data from designed experiments utilizing factorial designs. Named after statistician Frank Yates, the algorithm aims to generate least squares estimates for factor effects and interactions, providing insights such as ranked lists of factors and goodness-of-fit metrics. The document emphasizes the importance of arranging data in Yates order prior to analysis and explains the outputs generated from the analysis.

1. Statistical Methods for Engineering Research
Yates’ algorithm for 2n factorial experiment

2. Prepared By
Dr. Manu Melwin Joy
Assistant Professor
School of Management Studies
Cochin University of Science and Technology
Kerala, India.
Phone – 9744551114
Mail – manumelwinjoy@cusat.ac.in
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3. Yates’ algorithm for 2n factorial
experiment
• In statistics, a Yates analysis is an approach to
analyzing data obtained from a designed experiment,
where a factorial design has been used.
• This algorithm was named after the English
statistician Frank Yates and is called Yates' algorithm.

4. Yates’ algorithm for 2n factorial
experiment
• In statistics, a Yates analysis is an approach to
analyzing data obtained from a designed experiment,
where a factorial design has been used.
• This algorithm was named after the English
statistician Frank Yates and is called Yates' algorithm.

5. Yates’ algorithm for 2n factorial
experiment
• Full- and fractional-factorial designs are common in
designed experiments for engineering and scientific
applications. In these designs, each factor is assigned
two levels. These are typically called the low and
high levels.
• For computational purposes, the factors are scaled
so that the low level is assigned a value of -1 and the
high level is assigned a value of +1. These are also
commonly referred to as "-" and "+".

6. Yates’ algorithm for 2n factorial
experiment
• A full factorial design contains all possible
combinations of low/high levels for all the factors. A
fractional factorial design contains a carefully chosen
subset of these combinations.
• Formalized by Frank Yates, a Yates analysis exploits
the special structure of these designs to generate
least squares estimates for factor effects for all
factors and all relevant interactions.

7. Yates’ algorithm for 2n factorial
experiment
• The Yates analysis can be used to answer the
following questions:
– What is the ranked list of factors?
– What is the goodness-of-fit (as measured by the residual
standard deviation) for the various models?

8. Yates Order
• Before performing a Yates analysis, the data should
be arranged in "Yates order". That is, given k factors,
the kth column consists of 2(k - 1) minus signs (i.e., the
low level of the factor) followed by 2(k - 1) plus signs
(i.e., the high level of the factor).

9. Yates Order
• For example, for a full factorial design with three
factors, the design matrix is

10. Output
• A Yates analysis generates the following output.
• A factor identifier (from Yates order). The specific
identifier will vary depending on the program
used to generate the Yates analysis. Dataplot for
example, uses the following for a 3-factor model.

11. Output
• 1 = factor 1
• 2 = factor 2
• 3 = factor 3
• 12 = interaction of factor 1 and factor 2
• 13 = interaction of factor 1 and factor 3
• 23 = interaction of factor 2 and factor 3
• 123 = interaction of factors 1, 2, and 3

12. Output
• A ranked list of important factors. That is, least
squares estimated factor effects ordered from
largest in magnitude (most significant) to smallest
in magnitude (least significant).
• A t value for the individual factor effect estimates.
The t-value is computed as
• where e is the estimated factor effect and se is the
standard deviation of the estimated factor effect.