# Design of Experiments

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Course Number: |
ENGRG 7510 |

Course Name: |
Design of Experiments (Online) |

Course Description: |
This course on Design of Experiments (DOE) provides experiences in planning, conducting, and analyzing statistically designed experiments. The methods of DOE may be applied to design or improve products and processes. Analysis of variance (ANOVA), test of hypothesis, confidence interval estimation, response surface methods, and other statistical methods are applied in this course to set values for design, process, or control factors so that one or more responses will be optimized, even when noise factors are present in the system. This course is designed to teach the nuts and bolts of DOE as simply as possible. |

Prerequisites: |
None |

Level: |
Graduate |

Credits: |
3 |

Format: |
Online |

NOTE: The information below is representative of the course and is subject to change. The specific details of the course will be available in the Desire2Learn course instance for the course in which a student registers.

## Additional Information

**Learning Outcomes**

Upon successful completion of this course, you should be able to function competently in applied research involving the design and analysis of experiments.

More specifically, after completing this course, you should be able to implement, formulate, and analyze the resulting data for:

- Complete randomized design
- Randomized blocks and related designs
- 2k factorial design
- Factorial design (fixed, random, and mixed effects models)
- Nested design
- Split-Plot design
- Response surface methods
- Regression models
- Use multiple comparison techniques to draw simultaneous inference about parameters
- Use residual analysis to check for violation of the model assumptions

**Unit Descriptions (courses are divided in lessons and not in units)**

**Lesson 1: Introduction to Design of Experiments**

The purpose of this lesson is to introduce the thinking involved in the design phase of the experiment. Experimentation through the use of statistical design concepts is an efficient and relatively inexpensive means of increasing our knowledge about a subject of interest. The benefits are enormous if sufficient effort is exerted in the planning stage of the experiment. Available subject-matter knowledge must be used in this stage so that all potential factors that could cause change in the response variable are considered. Regrettably, too often investigators seek sophisticated statistical techniques post facto to analyze deficient and poorly acquired data. There is virtually no way that a sophisticated statistical technique can compensate for poorly acquired data. In this lesson, we present specific procedures for planning a successful experiment involving multiple factors. These procedures can then be used to design experiments involving even a single factor.

**Lesson 2: Simple Comparative Experiments**

This lesson considers experiments which compare two conditions. We often refer to these two conditions as two treatments, or as two levels of a factor of interest. These types of experiments are often called simple comparative experiments. When our interest is focused on potential difference between the means of these two differences, the appropriate statistical technique is the t-test. A t-test is any statistical hypothesis test in which the test statistic follows a Student’s t distribution if the null hypothesis is supported. Other topics in this lesson include the Z-test, which is used in situations where it is reasonable to assume that the population variances are known; statistical tests on variances of normal distributions; and methods for determining sample sizes based on power analysis. This lesson also includes an introduction to Minitab via webcast.

**Lesson 3:.Experiments with a Single Factor: The Analysis of Variance**

This lesson focuses on methods for the design and analysis of single-factor experiments with an arbitrary number a levels of the factor (or a treatments).

**Lesson 4: Randomized Blocks, Latin Squares, and Related Designs**

This lesson focuses on methods for dealing with nuisance factors in an experiment. A nuisance factor is a factor that probably has an effect on the outcome of the experiment, but which is of no real interest to the experimenter. Ideally, the experimenter would like to minimize or eliminate the variability transmitted into the response variable from the nuisance factor.

Nuisance factors are common in experimental work. Common nuisance factors are different operators, different batches of raw material, and different time periods. Essentially, anything that prohibits conducting the experiment under homogeneous conditions is a potential nuisance and presents a situation where blocking may need to be used. Different blocking designs will be discussed including Latin Square and Graeco-Latin square design.

**Lesson 5: Introduction to Factorial Designs**

This lesson introduces the general factorial experiment. These are multifactor experiments in which every combination of the factor levels are run. For example, chapter five of the Montgomery text includes a battery life example with two factors: material type and temperature. Both factors have three levels, so the example could be called a 3 x 3 factorial design. Factorials are useful for studying the main effects of the design factors. The main effects provide information about what happens when the factor is changed and other factors are held constant and information about interactions between the factors. An interaction occurs when the effect of one main factor depends on the levels of another factor (or factors). Two-factor interactions are very common in multifactor experimental problems.

**Lesson 6: The 2k Factorial Design**

Factorial designs are widely used in experiments that involve several factors so that it is necessary to study the joint effect of the factors on a response. A very important, special case of the general factorial design is that of k factors, each at only two levels. These levels may be quantitative, such as two values of temperature, pressure, or time; or they may be qualitative, such as two machines, two operators, the "high" and "low" levels of a factor, or perhaps the presence or absence of a factor. A complete replicate of this design requires 2k observations and is called a 2k factorial design. This lesson focuses on this extremely important class of designs. We assume that the factors are fixed, the designs are completely randomized, and the usual normality assumptions are satisfied.

**Lesson 7: Blocking and Confounding in the 2k Factorial Design**

This lesson introduces blocking and confounding schemes for the 2k factorial design. Two general cases are considered: replicated designs and unreplicated designs. In the replicated design case, the usual approach is to make each replicate a block, leading to the 2k factorial design run in a RCBD. The analysis is identical to the case of a general factorial experiment in a RCBD which was discussed in lesson 5. The second case, where the design is unreplicated, is slighted more complicated.

**Lesson 8: Two-Level Fractional Factorial Design**

This lesson introduces two-level fractional factorial designs. As the number of factors in a 2kfactorial design increases, the number of runs required for a complete replication of the design quickly outgrows the resources of most experimenters. For example, a complete replication of the 26 design requires 64 runs. In this design, only six of the 63 degrees of freedom correspond to main effects, and only 15 degrees of freedom correspond to two-factor interactions. Since main effects and two-factor interactions are of primary interest, only 21 degrees of freedom are associated with these effects. The remaining 42 degrees of freedom are associated with three-factor and higher interactions which are likely to be of much less interest. Fractional factorial designs are among the most widely used types of designs for product and process design, and for process development and improvement.

**Lesson 9: Three-Level and Mixed-Level Factorial and Fractional Factorial Designs**

This lesson considers several topics of factorial designs. Considerable information is provided on factorial designs where all factors are at three levels. These designs are called 3k factorial designs. Like the designs in the 2k series, the 3k factorial design can be confounded in blocks. The natural confounding scheme produces a design in three blocks, nine blocks, etc. Because main effects have two degrees of freedom, two-factor interactions have four degrees of freedom; three-factor interactions have eight degrees of freedom and so on. The blocking schemes confound two-degrees-of-freedom components of interaction with blocks. Other topics include mixed-level fractions,and mixed-level fractional factorials. Design with two-level and three-level factors and designs with two-level and four-level factors are special cases of some practical interest.

**Lesson 10: Fitting Regression Models**

This lesson presents the details of using the method of least square for fitting regression models. Least squares were used for model fitting in a previous lesson, but the details were omitted until now. This lesson focuses on regression methods and models of interest in experimental design. Regression models are useful in analyzing designed experiments when some of the observations are missing or when the experimenter has been unable to achieve the exact levels of the factors that are required by the design.

**Lesson 11: Response Surface Methods and Designs**

This lesson introduces response surface methods and designs. Response surface methods (RSM) are used for process and product optimization experiments. The designs and methods of RSM are the focus of this lesson. RSM is a sequential procedure. We assume that at the outset of experimentation the experimenter has screened the factors and knows which ones are important, but the process may be operating at a set of conditions that are far from optimum. The method of steepest ascent is used to move quickly and efficiently from the original region of experimentation to a region that is more likely to contain the optimum. Usually, two-level factorial and factorial designs are used in the method of steepest ascent.

**Lesson 12: Experiments with Random Factors**

This lesson introduces experiments with random factors. Previous lessons have concentrated on fixed factors, which are factors for which the levels are chosen specifically by the experimenter because they are of direct interest and/or the experimenter wants to predict the response at those points or to interpolate between them. A factor is random if its levels are chosen at random from a large population (theoretically, an infinitely large population) of levels. There are many experiments that involve random factors. Experiments in which all of the factors are random are called random models. Experiments with both fixed and random factors are called mixed models.

**Lesson 13: Nested and Split-Plot Designs**

Nested designs are multifactor designs used in situations where the levels of one factor are different from, but similar to each other at different levels of a second factor. Because a nested factor cannot be crossed with the factor it is nested within, this is not a factorial structure, so there cannot be an interaction between these two factors. Nested designs often involve random factors. For example, it is not unusual to find that the higher-level factor is fixed and that the levels of the factor nested within that factor are random. Consequently, the topics from the previous lesson on estimating the variance components are important. Split-plot designs are multifactor designs in which the order of the runs cannot be completely randomized because some factors are hard to change or require large experimental units. Generally, the hard-to-change factors are assigned to whole plots, which are divided into subplots (or split-plots), and the easy to change factors are assigned to the subplots.

**Grading Information**

The breakdown of points is as follows:

Assignments | % Weight |

Two Tests (10% Each) | 20% |

Homework | 25% |

Discussion | 15% |

Term Project | 25% |

Final Exam | 15% |

TOTAL | 100% |

**Grading Scale:**

A | B | C | D | F |

90 to 100% | 80 to less than90% | 70 to less than 80% | 60 to less than 70% | Less than 60% |

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## Contact

Distance Learning Center

University of Wisconsin-Platteville

1 University Plaza

Platteville, WI 53818-3099

**Phone:** 608.342.1468**Toll Free:** 800.362.5460**Fax:** 608.342.1071**E-Mail:** DistEd@uwplatt.edu