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# General Full Factorial Designs

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❶In the position shown, the spring A is stretched 0.

## Independence of Factors   Therefore, only two out of the six effects are independent. Assuming that and are independent, the other four effects can be expressed in terms of these effects.

The null hypothesis to test the significance of interaction can be rewritten using only the independent effects as. Since factor has three levels, two indicator variables, and , are required which need to be coded as shown next:.

Factor has two levels and can be represented using one indicator variable, , as follows:. The interaction will be represented by all possible terms resulting from the product of the indicator variables representing factors and. There are two such terms here - and. The vector can be substituted with the response values from the above table to get:. Knowing , and , the sum of squares for the ANOVA model and the extra sum of squares for each of the factors can be calculated.

These are used to calculate the mean squares that are used to obtain the test statistics. Since five effect terms , , , and are used in the model, the number of degrees of freedom associated with is five.

The total sum of squares, , can be calculated as:. Since there are 18 observed response values, the number of degrees of freedom associated with the total sum of squares is The error sum of squares can now be obtained:.

Since there are three replicates of the full factorial experiment, all of the error sum of squares is pure error. This can also be seen from the preceding figure, where each treatment combination of the full factorial design is repeated three times. The number of degrees of freedom associated with the error sum of squares is:. The sequential sum of squares for factor can be calculated as:. Since there are two independent effects , for factor , the degrees of freedom associated with are two.

Similarly, the sum of squares for factor can be calculated as:. Since there is one independent effect, , for factor , the number of degrees of freedom associated with is one.

The sum of squares for the interaction is:. Since there are two independent interaction effects, and , the number of degrees of freedom associated with is two. Knowing the sum of squares, the test statistic for each of the factors can be calculated. Analyzing the interaction first, the test statistic for interaction is:. The value corresponding to this statistic, based on the distribution with 2 degrees of freedom in the numerator and 12 degrees of freedom in the denominator, is:.

Assuming that the desired significance level is 0. In the absence of the interaction, the analysis of main effects becomes important. The test statistic for factor is:. The value corresponding to this statistic based on the distribution with 2 degrees of freedom in the numerator and 12 degrees of freedom in the denominator is:.

Therefore, it can be concluded that speed and fuel additive type affect the mileage of the vehicle significantly. Results for the effect coefficients of the model of the regression version of the ANOVA model are displayed in the Regression Information table in the following figure. Calculations of the results in this table are discussed next. The effect coefficients can be calculated as follows:.

Therefore, , , etc. As mentioned previously, these coefficients are displayed as Intercept, A and A respectively depending on the name of the factor used in the experimental design. The standard error for each of these estimates is obtained using the diagonal elements of the variance-covariance matrix. For example, the standard error for is:. Then the statistic for can be obtained as:.

The value corresponding to this statistic is:. We know that there is no interaction between the factors when we can talk about the effect of one factor without mentioning the other factor. In the above example, there are three hypotheses to be tested. Interaction effect is not present. For main effect gender, the null hypothesis means that there is no significant difference in reduction of hypertension in males and females. The null hypothesis for the main effect quantity means that there is no significant difference in reduction of hypertension whether the patients are given mg or mg of the drug.

For the interaction effect, the null hypothesis means that the two main effects gender and quantity are independent. The computational aspect involves computing F-statistic for each hypothesis. Factorial design has several important features. The assumptions remain the same as with other designs - normality, independence and equality of variance. Check out our quiz-page with tests about:. Retrieved Sep 10, from Explorable. The text in this article is licensed under the Creative Commons-License Attribution 4.

No problem, save it as a course and come back to it later. Share this page on your website: This article is a part of the guide: When you insert this tool into the Roadmap, you can use this form to record the data analysis from your experiment. Provides a cost-effective methodology for conducting controlled experiments DOEs where all of the factors process inputs are held at one of two levels settings during each run of the experiment plus optional center points.

Which process inputs factors have the largest effects on the process output which inputs are the key inputs? Do any important interactions between factors exist? Is the current testing space near an optimal condition for the process output?

If no, what direction do you need to move to get closer to the area where the optimal condition can be found? If yes, what settings of the key inputs will result in the optimal process output? If I change a factor from its low setting to its high setting, how much will the process output change? How much of the variation in the process output can be explained by varying the process inputs?

When to Use Purpose Mid-project Low resolution III or IV 2K fractional-factorial DOEs can be used as an early screening tool to perform a first-pass elimination of noncritical inputs, especially when you have many inputs for example, more than five and cost or time is a significant issue.

Mid-project You can use 2K full-factorial DOEs especially for 3 or 4 factors and resolution V or higher 2K factorial DOEs for 5 or more factors to model 2-way interactions and determine the settings for the key variables that result in the optimal process output.

Mid-project If all factors are numeric and no significant curvature is present, these designs can be used to determine the direction in which to continue experimenting to locate an area closer to the optimal solution. Mid-project If all factors are continuous and significant curvature is present, you can expand the 2K full-factorial DOE and resolution V or higher 2K fractional-factorial DOEs to allow the fitting of a quadratic model 3-dimensional modeling using central-composite designs to find optimal settings.

How-To State your factors typically less than eight factors and their levels of interest only two levels plus an optional center point allowed. If you are using a 2K fractional-factorial DOE, determine your fraction one-half, one-fourth, and so on based on your budget and desired resolution.

Verify that the measurement systems for the Y data and the inputs factors are adequate. ## Main Topics

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Fractional Factorial Experiments Stats Homework, assignment and Project Help, Fractional Factorial Experiments The 2k factorial experiment can become quite demanding, in terms of the number. of experimental units required, when.

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Analysis of Fractional Factorial Experiments Stats Homework, assignment and Project Help, Analysis of Fractional Factorial Experiments The difficulty of making formal significance tests using data from fractional factorial experiments lies. View Notes - Introduction to Factorial Experiments Lecture from STAT at Virginia Tech. 2k Factorial Experiments Introduction Chapter 15 1 Introduction Designed experiments intentionally.

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2^k Factorial Design 2^ k factorial designs consist of k factors, each of which has two levels. A key use of such designs to identify which of many variables is most important and should be considered for further analysis in more details. k Factorial Experiments 1 Purpose After this section you will understand how to: Describe the overall concepts of.