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Experimental design and analysis for tree improvement
Ladda ned. Spara som favorit. Laddas ned direkt. Download the free Reader from Adobe. Quality Engineering Your Christmas Tree Abstract: Christmas trees are used as a case study to highlight quality applications in a non-manufacturing industry. You must log in to comment. Please log in or register. However, different criteria can be used in choosing the higher order interactions to be used as generators, with different criteria sometimes leading to different "best" designs.
To make sure that these blocks do not bias your estimates of the effects for the k factors, blocking factors can be added as additional factors in the design. Consequently, you may "sacrifice" additional interaction effects to generate the blocking factors, but these designs often have the advantage of being statistically more powerful, because they allow you to estimate and control the variability in the production process that is due to differences between blocks.
However, a technical description of how fractional factorial designs are constructed is beyond the scope of either introductory overview. Interpreting the design. The design displayed in the Scrollsheet above should be interpreted as follows. So for example, in the first run of the experiment, all factors A through K are set to the higher level, and in the second run, factors A , B , and C are set to the higher level, but factor D is set to the lower level, and so on. Notice that the settings for each experimental run for factor E can be produced by multiplying the respective settings for factors A , B , and C.
The A x B x C interaction effect therefore cannot be estimated independently of the factor E effect in this design because these two effects are confounded. Likewise, the settings for factor F can be produced by multiplying the respective settings for factors B , C , and D. The maximum resolution design criterion.
In general, a design of resolution R is one where no l -way interactions are confounded with any other interaction of order less than R - l. Thus, main effects in this design are unconfounded with each other, but are confounded with two-factor interactions; and consequently, with other higher-order interactions.
One obvious, but nevertheless very important overall design criterion is that the higher-order interactions to be used as generators should be chosen such that the resolution of the design is as high as possible. The maximum unconfounding design criterion. Maximizing the resolution of a design, however, does not by itself ensure that the selected generators produce the "best" design.
Consider, for example, two different resolution IV designs. In both designs, main effects would be unconfounded with each other and 2-factor interactions would be unconfounded with main effects, i. The two designs might be different, however, with regard to the degree of confounding for the 2-factor interactions. For resolution IV designs, the "crucial order," in which confounding of effects first appears, is for 2-factor interactions. In one design, none of the "crucial order," 2-factor interactions might be unconfounded with all other 2-factor interactions, while in the other design, virtually all of the 2-factor interactions might be unconfounded with all of the other 2-factor interactions.
The second "almost resolution V" design would be preferable to the first "just barely resolution IV" design.
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This suggests that even though the maximum resolution design criterion should be the primary criterion, a subsidiary criterion might be that generators should be chosen such that the maximum number of interactions of less than or equal to the crucial order, given the resolution, are unconfounded with all other interactions of the crucial order. The minimum aberration design criterion. In some respects, this criterion is similar to the maximum unconfounding design criterion.
Less technically, the criterion apparently operates by choosing generators that produce the smallest number of pairs of confounded interactions of the crucial order.
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For example, the minimum aberration resolution IV design would have the minimum number of pairs of confounded 2-factor interactions. If you compare these two designs, you will find that in the maximally unconfounded design, 15 of the 36 2-factor interactions are unconfounded with any other 2-factor interactions, while in the minimum aberration design, only 8 of the 36 2-factor interactions are unconfounded with any other 2-factor interactions. The minimum aberration design, however, produces 18 pairs of confounded interactions, while the maximally unconfounded design produces 21 pairs of confounded interactions.
So, the two criteria lead to the selection of generators producing different "best" designs.
For designs with more than 11 factors, the two criteria can lead to the selection of very different designs, and for lack of better advice, we suggest using both criteria, comparing the designs that are produced, and choosing the design that best suits your needs. We will add, editorially, that maximizing the number of totally unconfounded effects often makes more sense than minimizing the number of pairs of confounded effects.
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Also, sometimes some factors may be categorical in nature, with more than 2 categories. For example, you may have three different machines that produce a particular part. Factor C is constructed from the interaction AB of the first two factors. Specifically, the values for factor C are computed as. Here, mod 3 x stands for the so-called modulo-3 operator, which will first find a number y that is less than or equal to x , and that is evenly divisible by 3, and then compute the difference remainder between number y and x.
Fundamental identity. If you apply this function to the sum of columns A and B shown above, you will obtain the third column C. Some of the designs will have fundamental identities that contain the number 2 as a multiplier; e. The next example shows such an identity.
Here is the summary for a 4-factor 3-level fractional factorial design in 9 blocks that requires only 27 runs. This design will allow you to test for linear and quadratic main effects for 4 factors in 27 observations, which can be gathered in 9 blocks of 3 observations each.
The fundamental identity or design generator for the design is ABCD , thus the modulo 3 of the sum of the factor levels across the four factors is equal to 0. The fundamental identity also allows you to determine the confounding of factors and interactions in the design see McLean and Anderson, , for details. Unconfounded Effects experi3. Effects excl. These designs do not have simple design generators they are constructed by combining two-level factorial designs with incomplete block designs , and have complex confounding of interaction.
However, the designs are economical and therefore particularly useful when it is expensive to perform the necessary experimental runs.
For example, when studying the yield of chemical process, then temperature may be related in a non-linear fashion, that is, the maximum yield may be attained when the temperature is set at the medium level. Thus, non-linearity often occurs when a process performs near its optimum. Thus, regardless of the original metric of factor settings e. Factor Effect Std. Main-effect estimates.
By default, the Effect estimate for the linear effects marked by the L next to the factor name can be interpreted as the difference between the average response at the low and high settings for the respective factors.
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The estimate for the quadratic non-linear effect marked by the Q next to the factor name can be interpreted as the difference between the average response at the center medium settings and the combined high and low settings for the respective factors. Interaction effect estimates. Analogously, the interactions by the quadratic components can be interpreted as half the difference between the quadratic main effect of one factor at the respective settings of another; that is, either the high or low setting quadratic by linear interaction , or the medium or high and low settings combined quadratic by quadratic interaction.
For example, a quadratic-by-quadratic A -by- B interaction indicates that the non- linear effect of factor A is modified in a nonlinear fashion by the setting of B. This means that there is a fairly complex interaction between factors present in the data that will make it difficult to understand and optimize the respective process. Sometimes, performing nonlinear transformations e. Centered and non-centered polynomials. As mentioned above, the interpretation of the effect estimates applies only when you use the default parameterization of the model. In that case, you would code the quadratic factor interactions so that they become maximally "untangled" from the linear main effects.
The same diagnostic plots e. Plot of means. When an interaction involves categorical factors e. Surface plot.
When the factors in an interaction are continuous in nature, you may want to look at the surface plot that shows the response surface applied by the fitted model. Note that this graph also contains the prediction equation in terms of the original metric of factors , that produces the respective response surface. You can also generate standard designs with 2 and 3 level factors.
The technical details of the method used to generate these designs are beyond the scope of this introduction. It should be noted however, that, while all of these designs are very efficient, they are not necessarily orthogonal with respect to all main effects.