Design and Analysis of Experiments by Douglas Montgomery. Heath Rushing
of a mixture design must be incorporated; non-normal responses should either be transformed or modeled with a generalized linear model; correlation between repeated observations on an experimental unit may be modeled with random effects; non-constant variance in the response variable across the design factors may be modeled, etc. Software for analyzing designed experiments should provide all of these capabilities in an accessible interface.
JMP offers an outstanding software solution for both designing and analyzing experiments. In terms of design, all of the classic designs that are presented in the textbook may be created in JMP. Optimal designs are available from the JMP Custom Design platform. These designs are extremely useful for cases where a constrained design space or a restriction on the number of experimental runs eliminates classical designs from consideration. Multiple designs may be created and compared with methods described in the textbook, including the Fraction of Design Space plot. Once a design is chosen, JMP will randomize the run order and produce a data table, which the researcher may use to store results. Metadata for the experimental factors and response variables is attached to the data table, which simplifies the analysis of these results.
The impressive graphical analysis functionality of JMP accelerates the discovery process particularly well with the dynamic and interactive profilers and plots. If labels for plotted points overlap, can by clicking and dragging the labels. Selecting points in a plot produced from a table selects the appropriate rows in the table and highlights the points corresponding to those rows in all other graphs produced from the table. Plots can be shifted and rescaled by clicking and dragging the axes. In many other software packages, these changes are either unavailable or require regenerating the graphical output.
An additional benefit of JMP is the ease with which it permits users to manipulate data tables. Data table operations such as sub-setting, joining, and concatenating are available via intuitive graphical interfaces. The relatively short learning curve for data table manipulation enables new users to prepare their data without remembering an extensive syntax. Although no command-line knowledge is necessary, the underlying JMP scripting language (JSL) scripts for data manipulation (and any other JMP procedure) may be saved and edited to repeat the analysis in the future or to combine with other scripts to automate a process.
This supplement to Design and Analysis of Experiments follows the chapter topics of the textbook and provides complete instructions and useful screenshots to use JMP to solve every example problem. As might be expected, there are often multiple ways to perform the same operation within JMP. In many of these cases, the different possibilities are illustrated across different examples involving the relevant operation. Some theoretical results are discussed in this supplement, but the emphasis is on the practical application of the methods developed in the textbook. The JMP DOE functionality detailed here represents a fraction of the software’s features for not only DOE, but also for most other areas of applied statistics. The platforms for reliability and survival, quality and process control, time series, multivariate methods, and nonlinear analysis procedures are beyond the scope of this supplement.
2
Simple Comparative Experiments
Section 2.2 Basic Statistical Concepts
Section 2.4.1 Hypothesis Testing
Section 2.4.3 Choice of Sample Size
Section 2.5.1 The Paired Comparison Problem
Section 2.5.2 Advantages of the Paired Comparison Design
The problem of testing the effect of a single experimental factor with only two levels provides a useful introduction to the statistical techniques that will later be generalized for the analysis of more complex experimental designs. In this chapter, we develop techniques that will allow us to determine the level of statistical significance associated with the difference in the mean responses of two treatment levels. Rather than only considering the difference between the mean responses across the treatments, we also consider the variation in the responses and the number of runs performed in the experiment. Using a t-test, we are able to quantify the likelihood (expressed as a p-value) that the observed treatment effect is merely noise. A “small” p-value (typically taken to be one smaller than α = 0.05) suggests that the observed data are not likely to have occurred if the null hypothesis (of no treatment effect) were true.
A related question involves the likelihood that the null hypothesis is rejected given that it is false (the power of the test). Given a fixed significance level, α (our definition of what constitutes a “small” p-value), theorized values for the pooled standard deviation, and a minimum threshold difference in treatment means, it is possible to solve for the minimum sample size that is necessary to achieve a desired power. This procedure is useful for determining the number of runs that must be included in a designed experiment.
In the first example presented in this chapter, a scientist has developed a modified cement mortar formulation that has a shorter cure time than the unmodified formulation. The scientist would like to test if the modification has affected the bond strength of the mortar. To study whether the two formulations, on average, produce bonds of different strengths, a two-sided t-test is used to analyze the observations from a randomized experiment with 10 measurements from each formulation. The null hypothesis of this test is that the mean bond strengths produced by the two formulations are equal; the alternative hypothesis is that mean bond strengths are not equal.
We also consider the advantages of a paired t-test, which provides an introduction to the notion of blocking. This test is demonstrated using data from an experiment to test for similar performance of two different tips that are placed on a rod in a machine and pressed into metal test coupons. A fixed pressure is applied to the tip, and the depth of the resulting depression is measured. A completely randomized design would apply the tips in a random order to the test coupons (making only one measurement on each coupon). While this design would produce valid results, the power of the test could be increased by removing noise from the coupon-to-coupon variation. This may be achieved by applying both tips to each coupon (in a random order) and measuring the difference in the depth of the depressions. A one-sample t-test is then used for the null hypothesis that the mean difference across the coupons is equal to 0. This procedure reduces experimental error by eliminating a noise factor.
This chapter also includes an example of procedures for testing the equality of treatment variances, and a demonstration of the t-test in the presence of potentially unequal group variances. This final test is still valid when the group variances are equal, but it is not as powerful as the pooled t-test in such situations.
Section 2.2 Basic Statistical Concepts
1. Open Tension-Bond.jmp.
2. Select Analyze > Distribution.
3. Select Strength for Y, Columns.
4. Select Mortar for By. As we will see in later chapters, these fields will be automatically populated for data tables that were created in JMP.
5. Click OK.
6. Click the red triangle next to Distributions Mortar=Modified and select Uniform Scaling.
7. Repeat step 6 for Distributions Mortar=Unmodified.
8. Click the red triangle next to Distributions Mortar=Modified and select Stack.
9. Repeat step 8 for Distributions Mortar=Unmodified.
10. Hold down the Ctrl key and click the red triangle next to Strength. Select Histogram Options > Show Counts. Holding down Ctrl applies the command to all of the histograms created by the Distribution platform; it essentially “broadcasts” the command.