1. Design of Experiments (DOE)
Outline
1. Introduction
2. Preparation
3. Components of Experimental Design
4. Purpose of Experimentation
5. Design Guidelines
6. Design Process
7. One Factor Experiments
8. Multi-factor Experiments
9. Taguchi Methods
1. Introduction
The term experiment is defined as the systematic procedure carried out under controlled
conditions in order to discover an unknown effect, to test or establish a hypothesis, or to illustrate a
known effect. When analyzing a process, experiments are often used to evaluate which process
inputs have a significant impact on the process output, and what the target level of those
inputs should be to achieve a desired result (output). Experiments can be designed in many
different ways to collect this information. Design of Experiments (DOE) is also referred to
asDesigned Experiments or Experimental Design - all of the terms have the same meaning.
Experimental design can be used at the point of greatest leverage to reduce design costs by
speeding up the design process, reducing late engineering design changes, and reducing product
material and labor complexity. Designed Experiments are also powerful tools to achieve
manufacturing cost savings by minimizing process variation and reducing rework, scrap, and the
need for inspection.
This Toolbox module includes a general overview of Experimental Design and links and other
resources to assist you in conducting designed experiments. A glossary of terms is also available
at any time through the Help function, and we recommend that you read through it to familiarize
yourself with any unfamiliar terms.
2. Preparation
If you do not have a general knowledge of statistics, review the Histogram, Statistical Process
Control, and Regression and Correlation Analysismodules of the Toolbox prior to working with this
module.
You can use the MoreSteam's data analysis software EngineRoom®
for Excel to create and analyze
many commonly used but powerful experimental designs. Free trials of several other statistical
packages can also be downloaded through the MoreSteam.com Statistical Software module of the
Toolbox. In addition, the book DOE Simplified, by Anderson and Whitcomb, comes with a sample of
excellent DOE software that will work for 180 days after installation.
3. Components of Experimental Design
Consider the following diagram of a cake-baking process (Figure 1). There are three aspects of the
process that are analyzed by a designed experiment:
Factors, or inputs to the process. Factors can be classified as either controllable or uncontrollable
variables. In this case, the controllable factors are the ingredients for the cake and the oven that
the cake is baked in. The controllable variables will be referred to throughout the material as
factors. Note that the ingredients list was shortened for this example - there could be many other
ingredients that have a significant bearing on the end result (oil, water, flavoring, etc). Likewise,
there could be other types of factors, such as the mixing method or tools, the sequence of mixing,

2. or even the people involved. People are generally considered a Noise Factor (see the glossary) -
an uncontrollable factor that causes variability under normal operating conditions, but we can
control it during the experiment using blocking and randomization. Potential factors can be
categorized using the Fishbone Chart (Cause & Effect Diagram) available from the Toolbox.
Levels, or settings of each factor in the study. Examples include the oven temperature setting
and the particular amounts of sugar, flour, and eggs chosen for evaluation.
Response, or output of the experiment. In the case of cake baking, the taste, consistency, and
appearance of the cake are measurable outcomes potentially influenced by the factors and their
respective levels. Experimenters often desire to avoid optimizing the process for one response at
the expense of another. For this reason, important outcomes are measured and analyzed to
determine the factors and their settings that will provide the best overall outcome for the critical-
to-quality characteristics - both measurable variables and assessable attributes.
Figure 1
4. Purpose of Experimentation
Designed experiments have many potential uses in improving processes and products, including:
Comparing Alternatives. In the case of our cake-baking example, we might want to compare
the results from two different types of flour. If it turned out that the flour from different vendors
was not significant, we could select the lowest-cost vendor. If flour were significant, then we
would select the best flour. The experiment(s) should allow us to make an informed decision that
evaluates both quality and cost.

3. Identifying the Significant Inputs (Factors) Affecting an Output (Response) - separating the
vital few from the trivial many. We might ask a question: "What are the significant factors
beyond flour, eggs, sugar and baking?"
Achieving an Optimal Process Output (Response). "What are the necessary factors, and what
are the levels of those factors, to achieve the exact taste and consistency of Mom's chocolate
cake?
Reducing Variability. "Can the recipe be changed so it is more likely to always come out the
same?"
Minimizing, Maximizing, or Targeting an Output (Response). "How can the cake be made
as moist as possible without disintegrating?"
Improving process or product "Robustness" - fitness for use under varying conditions. "Can the
factors and their levels (recipe) be modified so the cake will come out nearly the same no matter
what type of oven is used?"
Balancing Tradeoffs when there are multiple Critical to Quality Characteristics (CTQC's) that
require optimization. "How do you produce the best tasting cake with the simplest recipe (least
number of ingredients) and shortest baking time?"
5. Experiment Design Guidelines
The Design of an experiment addresses the questions outlined above by stipulating the following:
The factors to be tested.
The levels of those factors.
The structure and layout of experimental runs, or conditions.
A well-designed experiment is as simple as possible - obtaining the required information in a cost
effective and reproducible manner.
MoreSteam.com Reminder: Like Statistical Process Control, reliable experiment results are predicated upon
two conditions: a capable measurement system, and a stable process. If the measurement system contributes
excessive error, the experiment results will be muddied. You can use the Measurement Systems Analysis
module from the Toolbox to evaluate the measurement system before you conduct your experiment.
Likewise, you can use the Statistical Process Control module to help you evaluate the statistical stability of the
process being evaluated. Variation impacting the response must be limited to common cause random error -
not special cause variation from specific events.
When designing an experiment, pay particular heed to four potential traps that can create
experimental difficulties:
1. In addition to measurement error (explained above), other sources of error, or unexplained
variation, can obscure the results. Note that the term "error" is not a synonym with "mistakes".
Error refers to all unexplained variation that is either within an experiment run or between
experiment runs and associated with level settings changing. Properly designed experiments can
identify and quantify the sources of error.
2. Uncontrollable factors that induce variation under normal operating conditions are referred to as
"Noise Factors". These factors, such as multiple machines, multiple shifts, raw materials,
humidity, etc., can be built into the experiment so that their variation doesn't get lumped into the
unexplained, or experiment error. A key strength of Designed Experiments is the ability to
determine factors and settings that minimize the effects of the uncontrollable factors.
3. Correlation can often be confused with causation. Two factors that vary together may be highly
correlated without one causing the other - they may both be caused by a third factor. Consider
the example of a porcelain enameling operation that makes bathtubs. The manager notices that
there are intermittent problems with "orange peel" - an unacceptable roughness in the enamel

4. surface. The manager also notices that the orange peel is worse on days with a low production
rate. A plot of orange peel vs. production volume below (Figure 2) illustrates the correlation:
Figure 2
If the data are analyzed without knowledge of the operation, a false conclusion could be reached
that low production rates cause orange peel. In fact, both low production rates and orange peel
are caused by excessive absenteeism - when regular spray booth operators are replaced by
employees with less skill. This example highlights the importance of factoring in operational
knowledge when designing an experiment.Brainstorming exercises and Fishbone Cause & Effect
Diagrams are both excellent techniques available through the Toolbox to capture this operational
knowledge during the design phase of the experiment. The key is to involve the people who live
with the process on a daily basis.
4. The combined effects or interactions between factors demand careful thought prior to
conducting the experiment. For example, consider an experiment to grow plants with two inputs:
water and fertilizer. Increased amounts of water are found to increase growth, but there is a point
where additional water leads to root-rot and has a detrimental impact. Likewise, additional
fertilizer has a beneficial impact up to the point that too much fertilizer burns the roots.
Compounding this complexity of the main effects, there are also interactive effects - too much
water can negate the benefits of fertilizer by washing it away. Factors may generate non-linear
effects that are not additive, but these can only be studied with more complex experiments that
involve more than 2 level settings. Two levels is defined as linear (two points define a line), three
levels are defined as quadratic (three points define a curve), four levels are defined as cubic, and
so on.
6. Experiment Design Process
The flow chart below (Figure 3) illustrates the experiment design process:
Figure 3

5. 7. Test of Means - One Factor Experiment
One of the most common types of experiments is the comparison of two process methods, or two
methods of treatment. There are several ways to analyze such an experiment depending upon the
information available from the population as well as the sample. One of the most straight-forward
methods to evaluate a new process method is to plot the results on an SPC chart that also includes
historical data from the baseline process, with established control limits.
Then apply the standard rules to evaluate out-of-control conditions to see if the process has been
shifted. You may need to collect several sub-groups worth of data in order to make a determination,
although a single sub-group could fall outside of the existing control limits. You can link to
the Statistical Process Control charts module of the Toolbox for help.
An alternative to the control chart approach is to use the F-test (F-ratio) to compare the means of
alternate treatments. This is done automatically by the ANOVA (Analysis of Variance) function of
statistical software, but we will illustrate the calculation using the following example: A commuter
wanted to find a quicker route home from work. There were two alternatives to bypass traffic
bottlenecks. The commuter timed the trip home over a month and a half, recording ten data points
for each alternative.
MoreSteam Reminder: Take care to make sure your experimental runs are randomized - i.e., run in random
order. Randomization is necessary to avoid the impact of lurking variables. Consider the example of measuring
the time to drive home: if a major highway project is started at the end of the sample period increases commute
time, then the highway project could bias the results if a given treatment (route) is sampled during that time
period.
Scheduling the experimental runs is necessary to ensure independence of observations. You can randomize
your runs using pennies - write the reference number for each run on a penny with a pencil, then draw the
pennies from a container and record the order.
The data are shown below along with the mean for each route (treatment), and the variance for
each route:

6. As shown on the table above, both new routes home (B&C) appear to be quicker than the existing
route A. To determine whether the difference in treatment means is due to random chance or a
statistically significant different process, an ANOVA F-test is performed.
The F-test analysis is the basis for model evaluation of both single factor and multi-factor
experiments. This analysis is commonly output as an ANOVA table by statistical analysis software, as
illustrated by the table below:
The most important output of the table is the F-ratio (3.61). The F-ratio is equivalent to the Mean
Square (variation) between the groups (treatments, or routes home in our example) of 19.9 divided
by the Mean Square error within the groups (variation within the given route samples) of 5.51.
The Model F-ratio of 3.61 implies the model is significant.The p-value ('Probability of exceeding the
observed F-ratio assuming no significant differences among the means') of 0.0408 indicates that
there is only a 4.08% probability that a Model F-ratio this large could occur due to noise (random
chance). In other words, the three routes differ significantly in terms of the time taken to reach
home from work.
The following graph (Figure 4) shows 'Simultaneous Pairwise Difference' Confidence Intervals for
each pair of differences among the treatment means. If an interval includes the value of zero
(meaning 'zero difference'), the corresponding pair of means do NOT differ significantly. You can use
these intervals to identify which of the three routes is different and by how much. The intervals

7. contain the likely values of differences of treatment means (1-2), (1-3) and (2-3) respectively, each
of which is likely to contain the true (population) mean difference in 95 out of 100 samples. Notice
the second interval (1-3) does not include the value of zero; the means of routes 1 (A) and 3 (C)
differ significantly. In fact, all values included in the (1, 3) interval are positive, so we can say that
route 1 (A) has a longer commute time associated with it compared to route 3 (C).
Figure 4
Other statistical approaches to the comparison of two or more treatments are available through the
online statistics handbook - Chapter 7:
Statistics Handbook
8. Multi-Factor Experiments
Multi-factor experiments are designed to evaluate multiple factors set at multiple levels. One
approach is called a Full Factorial experiment, in which each factor is tested at each level in every
possible combination with the other factors and their levels. Full factorial experiments that study all
paired interactions can be economic and practical if there are few factors and only 2 or 3 levels per
factor. The advantage is that all paired interactions can be studied. However, the number of runs
goes up exponentially as additional factors are added. Experiments with many factors can quickly
become unwieldy and costly to execute, as shown by the chart below:

8. To study higher numbers of factors and interactions, Fractional Factorial designs can be used to
reduce the number of runs by evaluating only a subset of all possible combinations of the factors.
These designs are very cost effective, but the study of interactions between factors is limited, so the
experimental layout must be decided before the experiment can be run (during the experiment
design phase).
MoreSteam Reminder: When selecting the factor levels for an experiment, it is critical to capture
the natural variation of the process. Levels that are close to the process mean may hide the
significance of factor over its likely range of values. For factors that are measured on a variable
scale, try to select levels at plus/minus three standard deviations from the mean value.
You can also download a 30-day free trial EngineRoom for Excel, MoreSteams statistical data
analysis software (an Excel add-in), to design and analyze several popular designed experiments.
The software includes tutorials on planning and executing full, fractional and general factorial
designs.
9. Advanced Topic - Taguchi Methods
Dr. Genichi Taguchi is a Japanese statistician and Deming prize winner who pioneered techniques to
improve quality through Robust Design of products and production processes. Dr. Taguchi
developed fractional factorial experimental designs that use a very limited number of experimental
runs. The specifics of Taguchi experimental design are beyond the scope of this tutorial, however, it
is useful to understand Taguchi's Loss Function, which is the foundation of his quality improvement
philosophy.
Traditional thinking is that any part or product within specification is equally fit for use. In that case,
loss (cost) from poor quality occurs only outside the specification (Figure 5). However, Taguchi
makes the point that a part marginally within the specification is really little better than a part
marginally outside the specification.
As such, Taguchi describes a continuous Loss Function that increases as a part deviates from the
target, or nominal value (Figure 6). The Loss Function stipulates that society's loss due to
poorly performing products is proportional to the square of the deviation of the
performance characteristic from its target value.

9. Taguchi adds this cost to society (consumers) of poor quality to the production cost of the product
to arrive at the total loss (cost). Taguchi uses designed experiments to produce product and process
designs that are more robust - less sensitive to part/process variation.