Section 3.1 A One-way ANOVA Example
Section 3.4 Model Adequacy Checking
Section 3.8.1 Single Factor Experiment
Section 3.8.2 Application of a Designed Experiment
Section 3.8.3 Discovering Dispersion Effects
In this chapter, the t-test is generalized to accommodate factors with more than two levels. The method of analysis of variance (ANOVA) introduced here allows us to study the equality of the means of three or more factor levels. ANOVA partitions the total sample variance into two parts: the variance explained by the factor under study, and the remaining, unexplained variance.
The method makes several assumptions about the distribution of the random error term in the model. If the model structure represents the true structure of the process, the model residuals may be thought of as random numbers generated from the distribution of the random error term, which is typically assumed to be a normal distribution. Several diagnostics are available for the residuals. They may be plotted on a normal quantile plot to check the assumption of normality of the random error term. They may also be plotted against the predicted values: the residuals and predicted values ought to be independent, and no patterns should be present in the plot. ANOVA also assumes that the error terms are independent and identically distributed. This chapter considers two formal tests, Bartlett’s and Levene’s, for the homogeneity of residual variance across factor levels. If any of the residual diagnostics show abnormalities, a transformation of the response variable is often useful for improving the model fit.
When the ANOVA test rejects the null hypothesis that all treatment means are equal, it is often necessary to know which factor levels are significantly different from each other. Special techniques are necessary for multiple comparisons of different linear combinations of factor level means in order to control the so-called experimentwise error rate. Examples are presented for Tukey’s HSD (honestly significant difference) test and the Fisher (Student’s t) least significant difference method. If one of the factors represents a control group, Dunnett’s test may be used to compare the control group with each of the other factor levels.
Other topics covered include power analysis for ANOVA to determine a required sample size, an introduction to the random effects models that are useful when the factor levels are only a sample of a larger population, and an example of a nonparametric method. The Kruskal-Wallis test relaxes the assumption that the response distribution is normal in each factor level, though it does require that the distributions across factor levels have the same shape.
The first example will illustrate how to build an ANOVA model from data imported into JMP. This entails specifying the response column, the factor column, and ensuring that the factor column is set to the nominal modeling type. Afterward, we will show how models may be designed in JMP, and how the appropriate modeling options are saved as scripts attached to the data table. For the remainder of the text, we will assume that the data tables are created in JMP.
Section 3.1 A One-way ANOVA Example
1. Open Etch-Rate-Import.jmp.
2. Click the blue icon (triangle) next to Power in the Columns panel and select Nominal.
Even though the levels of Power are ordinal, we are not incorporating that information into the current analysis. This distinction is not critical since Power is only a factor, and not the response. Treating an ordinal factor as nominal yields the same model fit. For a response variable, a nomial modeling type prompts a multinomial logistic regression, while an ordinal modeling type prompts an ordered logistic regression.
3. Select Analyze > Fit Y by X.
4. Select Etch Rate and click Y, Response.
5. Select Power and click X, Factor.
6. Click OK.
7. To produce a box plot, click the red triangle next to One-way Analysis of Etch Rate By Power and select Quantiles.
The box plots show that the etch rate increases as power increases, and that the variability of the etch rate is roughly the same for each power setting.
8. Select Window > Close All. We will now demonstrate how this model can be created in JMP.
9. Select DOE > Full Factorial Design.
10. Under Response Name, double click Y and change the response name to Etch Rate.
11. In the Factors section, select Categorical > 4 Level.
12. Double-click the name of the new factor, X1, and change it to Power.
13. Likewise, change the Values of the new factor from L1, L2, L3, and L4 to 160, 180, 200, and 220, respectively.
14. Click Continue.
15. Leave Run Order set to Randomize. Then, the experiment should be run in the order in which the rows appear in the resulting JMP table.
Number of Runs: 4 indicates that the design requires four different runs. Once Make Table has been clicked, Number of Runs will change to 20, reflecting the runs needed for the four replicates.
16. Enter 4 for Number of Replicates. This indicates the original 4 runs will be replicated 4 times for a total of 20 runs.
17. Click Make Table.
A new data table has been created with three columns. The Pattern column indicates which combination of factor levels are being used for the current row. Since there is only one factor, Power, the Pattern column simply indicates which level of Power is being run. The Power column has automatically been set to the Nominal modeling type. Additional metadata about the columns has been included from the Full Factorial platform, as indicated by the
All of the JMP platforms demonstrated in this book are capable of fitting models in the presence of missing data. That is, if it is not possible to perform the 20th run, which is at the Power setting of 220, it would still be possible to analyze the first 19 runs. However, missing observations can affect the aliasing structure of a design, which will be discussed in later chapters. In addition, if the cause of the missing values is related to the response (missing not at random), then the resulting estimates could be biased.
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