The textbook we are using brings an engineering perspective to the design of experiments. We will bring in other contexts and examples from other fields of study including agriculture (where much of the early research was done) education and nutrition. Surprisingly the service industry has begun using design of experiments as well.

 All experiments are designed experiments, it is just that some are poorly designed and some are well-designed. 

If we had infinite time and resource budgets there probably wouldn't be a big fuss made over designing experiments. In production and quality control we want to control the error and learn as much as we can about the process or the underlying theory with the resources at hand. From an engineering perspective we're trying to use experimentation for the following purposes:

• reduce time to design/develop new products & processes
• improve performance of existing processes
• improve reliability and performance of products
• achieve product & process robustness
• perform evaluation of materials, design alternatives, setting component & system tolerances, etc.

We always want to fine-tune or improve the process. In today's global world this drive for competitiveness affects all of us both as consumers and producers.

Robustness is a concept that enters into statistics at several points. At the analysis, stage robustness refers to a technique that isn't overly influenced by bad data. Even if there is an outlier or bad data you still want to get the right answer. Regardless of who or what is involved in the process - it is still going to work. We will come back to this notion of robustness later in the course (Lesson 12).

Every experiment design has inputs. Back to the cake baking example: we have our ingredients such as flour, sugar, milk, eggs, etc. Regardless of the quality of these ingredients we still want our cake to come out successfully. In every experiment there are inputs and in addition, there are factors (such as time of baking, temperature, geometry of the cake pan, etc.), some of which you can control and others that you can't control. The experimenter must think about factors that affect the outcome. We also talk about the output and the yield or the response to your experiment. For the cake, the output might be measured as texture, flavor, height, size, or flavor.

Here's a quick timeline:

• The agricultural origins, 1918 – 1940s
  • R. A. Fisher & his co-workers
  • Profound impact on agricultural science
  • Factorial designs, ANOVA
• The first industrial era, 1951 – late 1970s
  • Box & Wilson, response surfaces
  • Applications in the chemical & process industries
• The second industrial era, late 1970s – 1990
  • Quality improvement initiatives in many companies
  • CQI and TQM were important ideas and became management goals
  • Taguchi and robust parameter design, process robustness
• The modern era, beginning circa 1990, when economic competitiveness and globalization are driving all sectors of the economy to be more competitive.

Immediately following World War II the first industrial era marked another resurgence in the use of DOE. It was at this time that Box and Wilson (1951) wrote the key paper in response surface designs thinking of the output as a response function and trying to find the optimum conditions for this function. George Box died early in 2013. And, an interesting fact here - he married Fisher's daughter! He worked in the chemical industry in England in his early career and then came to America and worked at the University of Wisconsin for most of his career.

The Second Industrial Era - or the Quality Revolution

W. Edwards Deming

The importance of statistical quality control was taken to Japan in the 1950s by W Edward Deming. This started what Montgomery calls a second Industrial Era, and sometimes the quality revolution. After the second world war, Japanese products were of terrible quality. They were cheaply made and not very good. In the 1960s their quality started improving. The Japanese car industry adopted statistical quality control procedures and conducted experiments which started this new era. Total Quality Management (TQM), Continuous Quality Improvement (CQI) are management techniques that have come out of this statistical quality revolution - statistical quality control and design of experiments.

Taguchi, a Japanese engineer, discovered and published a lot of the techniques that were later brought to the West, using an independent development of what he referred to as orthogonal arrays. In the West, these were referred to as fractional factorial designs. These are both very similar and we will discuss both of these in this course. He came up with the concept of robust parameter design and process robustness.

The Modern Era

Around 1990 Six Sigma, a new way of representing CQI, became popular. Now it is a company and they employ a technique which has been adopted by many of the large manufacturing companies. This is a technique that uses statistics to make decisions based on quality and feedback loops. It incorporates a lot of previous statistical and management techniques.

Clinical Trials

Montgomery omits in this brief history a major part of design of experimentation that evolved - clinical trials. This evolved in the 1960s when medical advances were previously based on anecdotal data; a doctor would examine six patients and from this wrote a paper and published it. The incredible biases resulting from these kinds of anecdotal studies became known. The outcome was a move toward making the randomized double-blind clinical trial the gold standard for approval of any new product, medical device, or procedure. The scientific application of the statistical procedures became very important.

This is an essential component of any experiment that is going to have validity. If you are doing a comparative experiment where you have two treatments, a treatment and a control, for instance, you need to include in your experimental process the assignment of those treatments by some random process. An experiment includes experimental units. You need to have a deliberate process to eliminate potential biases from the conclusions, and random assignment is a critical step.

Replication is some in sense the heart of all of statistics. To make this point... Remember what the standard error of the mean is? It is the square root of the estimate of the variance of the sample mean, i.e., \(\sqrt{\dfrac{s^2}{n}}\). The width of the confidence interval is determined by this statistic. Our estimates of the mean become less variable as the sample size increases.

Replication is the basic issue behind every method we will use in order to get a handle on how precise our estimates are at the end. We always want to estimate or control the uncertainty in our results. We achieve this estimate through replication. Another way we can achieve short confidence intervals is by reducing the error variance itself. However, when that isn't possible, we can reduce the error in our estimate of the mean by increasing n.

Another way is to reduce the size or the length of the confidence interval is to reduce the error variance - which brings us to blocking.

Blocking is a technique to include other factors in our experiment which contribute to undesirable variation. Much of the focus in this class will be to creatively use various blocking techniques to control sources of variation that will reduce error variance. For example, in human studies, the gender of the subjects is often an important factor.  Age is another factor affecting the response.  Age and gender are often considered nuisance factors which contribute to variability and make it difficult to assess systematic effects of a treatment.  By using these as blocking factors, you can avoid biases that might occur due to differences between the allocation of subjects to the treatments, and as a way of accounting for some noise in the experiment. We want the unknown error variance at the end of the experiment to be as small as possible. Our goal is usually to find out something about a treatment factor (or a factor of primary interest), but in addition to this, we want to include any blocking factors that will explain variation.

We will spend at least half of this course talking about multi-factor experimental designs: \(2^k\) designs, \(3^k\) designs, response surface designs, etc. The point to all of these multi-factor designs is contrary to the scientific method where everything is held constant except one factor which is varied. The one factor at a time method is a very inefficient way of making scientific advances. It is much better to design an experiment that simultaneously includes combinations of multiple factors that may affect the outcome. Then you learn not only about the primary factors of interest but also about these other factors. These may be blocking factors which deal with nuisance parameters or they may just help you understand the interactions or the relationships between the factors that influence the response.

Confounding is something that is usually considered bad! Here is an example. Let's say we are doing a medical study with drugs A and B. We put 10 subjects on drug A and 10 on drug B. If we categorize our subjects by gender, how should we allocate our drugs to our subjects? Let's make it easy and say that there are 10 male and 10 female subjects. A balanced way of doing this study would be to put five males on drug A and five males on drug B, five females on drug A and five females on drug B. This is a perfectly balanced experiment such that if there is a difference between male and female at least it will equally influence the results from drug A and the results from drug B.

An alternative scenario might occur if patients were randomly assigned treatments as they came in the door. At the end of the study, they might realize that drug A had only been given to the male subjects and drug B was only given to the female subjects. We would call this design totally confounded. This refers to the fact that if you analyze the difference between the average response of the subjects on A and the average response of the subjects on B, this is exactly the same as the average response on males and the average response on females. You would not have any reliable conclusion from this study at all. The difference between the two drugs A and B, might just as well be due to the gender of the subjects since the two factors are totally confounded.

Confounding is something we typically want to avoid but when we are building complex experiments we sometimes can use confounding to our advantage. We will confound things we are not interested in order to have more efficient experiments for the things we are interested in. This will come up in multiple factor experiments later on. We may be interested in main effects but not interactions so we will confound the interactions in this way in order to reduce the sample size, and thus the cost of the experiment, but still have good information on the main effects.

The practical steps needed for planning and conducting an experiment include: recognizing the goal of the experiment, choice of factors, choice of response, choice of the design, analysis and then drawing conclusions. This pretty much covers the steps involved in the scientific method.

1. Recognition and statement of the problem
2. Choice of factors, levels, and ranges
3. Selection of the response variable(s)
4. Choice of design
5. Conducting the experiment
6. Statistical analysis
7. Drawing conclusions, and making recommendations

What this course will deal with primarily is the choice of the design. This focus includes all the related issues about how we handle these factors in conducting our experiments.

We usually talk about "treatment" factors, which are the factors of primary interest to you. In addition to treatment factors, there are nuisance factors which are not your primary focus, but you have to deal with them. Sometimes these are called blocking factors, mainly because we will try to block on these factors to prevent them from influencing the results.

There are other ways that we can categorize factors:

Experimental vs. Classification Factors

Quantitative vs. Qualitative Factors