Các mẫu nhân tố (Factorial Designs)

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A group of experimental runs conducted under relatively homogeneous conditions. Although every measurement should be taken under consistent experimental conditions (other than those that are being varied as part of the experiment), this is not always possible. Use blocks in experimental design and analysis to minimize bias and error variance due to nuisance factors. For example, you want to test the quality of a new printing press. However, press setup takes several hours and can only be done four times a day. Because the design of the experiment requires at least eight runs, you need at least two days to test the press. You should account for any differences in conditions between days by using "day" as a blocking variable. To distinguish between any block effect (incidental differences between days) and effects due to the experimental factors (temperature, humidity, and press operator), you must account for the block (day) in the experimental design. You should randomize run order within blocks.



Multiple experimental runs with the same factor settings (levels). Replicates are subject to the same sources of variability, independently of one another. You can replicate combinations of factor levels, groups of factor level combinations, or entire designs.

In experimental design, replicate measurements are taken from identical but different experimental runs. This is in contrast to repeats, which are simply repeated observations at the same settings. You can use replicates to estimate the variance (experimental error) caused by slightly different experimental conditions. The experimental error serves as a benchmark to determine whether observed differences in the data are statistically different. To make sure all the experimental variability is observed and quantified, replicates should be randomized to cover the entire range of experimental conditions. If the number of runs is too large to be completed under steady state conditions, you may block on replicates. Blocking allows you to estimate the block effects independently of the experimental error.

For example, if you have three factors with two levels each and you test all combinations of factor levels (full factorial design), one replicate of the entire design would consist of 8 runs (23). You can choose to run the design once or have multiple replicates.

Your experimental design includes the number of replicates you should run. Considerations for replicates:

    Screening designs to reduce a large set of factors usually don't use multiple replicates.

    If you are trying to create a prediction model, multiple replicates may increase the precision of your model.

    If you have more data, you may be able to detect smaller effects or have greater power to detect an effect of fixed size.

    Your resources may dictate the number of replicates you can run. For example, if your experiment is extremely costly, you may be able to run it only once.


Các điểm của mẫu (Design points)

Bao gồm điểm góc ( corner points (or cube points) và điểm trung tâm ( center points). Các điểm trong biểu đồ dưới đây tượng trưng các tổ hợp của các mức độ yếu tố trong mẫu 2 yếu tố .

Corner point


 Center point

    Điểm góc – tượng trưng run thực nghiệm khi tất cả các yếu tố được đặt ở mức độ cao nhất hay thấp nhất. Chẳng hạn, trong mẫu 2 yếu tố , điểm ở góc trên bên phải tượng trưng cho run thực nghiệm khi yếu tố A và B được đặt ở mức cao (1, 1).

   Điểm trung tâm- tượng trưng các run thực nghiệm với các mức độ yếu tố được đặt ở mức trung bình giữa mức cao nhất và thấp.  

Các điểm trung tâm thực sự chỉ có thể được dùng với các yếu tố số (numeric factors) đặt ở điểm giữa hai mức độ được chọn. Nếu có tổ hợp giữa yếu tố ký tự (text factor) và yếu tố số , Minitab sẽ tạo ra các điểm trung tâm giả ( pseudo-center points) . Những điểm này là những điểm trung tâm đối với các yếu tố số ở mỗi tổ hợp với các mức độ của các yếu tố ký tự.

Việc đưa các điểm trung tâm vào mẫu sẽ kiểm tra độ cong trong bề mặt đáp ứng . Nếu độ cong hiện diện, đáp ứng ở điểm trung tâm sẽ hoặc là cao hơn hay thấp hơn đáp ứng trung bình ở các điểm góc. Độ cong thường hiện diện khi các giá trị của yếu tố gần với giá trị cực đại hay cực tiểu của đáp ứng .

Thí dụ về độ cong.

Main effects and main effects plot

Use in conjunction with an analysis of variance and design of experiments to examine differences among level means for one or more factors. A main effect is present when different levels of a factor affect the response differently. A main effects plot graphs the response mean for each factor level connected by a line.

General patterns to look for:

    When the line is horizontal (parallel to the x-axis), then there is no main effect present. Each level of the factor affects the response in the same way, and the response mean is the same across all factor levels.

    When the line is not horizontal, then there is a main effect present. Different levels of the factor affect the response differently. The steeper the slope of the line, the greater the magnitude of the main effect.

For example, fertilizer company B is comparing the plant growth rate measured in plants treated with their product compared to plants treated by company A's fertilizer. They tested the two fertilizers in two locations. Below is the main effects plots of these two factors.

Plant Growth Rate


                     A              B                  1            2


                      Fertilizer                    Location

Fertilizer appears to affect the plant growth rate because the line is not horizontal. Fertilizer B has a higher plant growth rate mean than fertilizer A.  Location also affects the plant growth rate.  Location 1 had a higher plant growth rate mean than location 2. The reference line represents the overall mean.

Main effects plots will not show any interactions that are present. Use the Interactions Plot to view interactions between factors.


To determine if a pattern is statistically significant, you must do an appropriate test.




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