# Cấu trúc trùng hợp (Alias structure)

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## Cấu trúc trùng hợp (Alias structure)

Các ảnh hưởng không thể được ước lượng riêng biệt với nhau được gọi là trùng hợp với nhau . Sự trùng hợp xảy ra khi sử dụng mẫu nhân tố từng phần vì không thể thực hiện tất cả các tổ hợp giữa các mức độ của các yếu tố . Chẳng hạn, nếu yếu tố A bị trùng hợp tương tác 3 chiều BCD khi đó ảnh hưởng dự đoán của yếu tố A là tổng ảnh hưởng của A và của BCD . Những ảnh hưởng này được gọi là trùng hợp nhau (aliased) . Cấu trúc trùng hợp mô tả sự trùng hợp xảy ra trong mẫu. Các ảnh hưởng bị trùng hợp không thể được ước lượng một cách riêng biệt nhau. Chẳng hạn, nếu tương tác 2 chiếu BC bị trùng hợp với tương tác 3 chiều ADE chúng ta không thể nói ảnh hưởng có ý nghĩa là do tương tác BC hay ADE.

Chìa khóa của cấu trúc trùng hợp là phát biểu đơn vị ( identity statement) , chẳng hạn, I + ABCDE. Để xác định ảnh hưởng nào là bị trùng hợp , ta nhân thành phần quan tâm với phát biểu đơn vị và sau đó loại bỏ các thành phần bình phương. Chẳng hạn, để tìm thành phần trùng hợp với BC :

(BC)(I ABCDE) =  BC  AB2C2DE = BC  ADE

Như vậy , BC và ADE trùng hợp với nhau.

Interaction

When the effect of a one factor depends on the level of the other factor. You can use an interaction plot to visualize possible interactions.

For example, a food scientist is looking for the combination of whey and supplements for a pancake mix that yields the best quality. She runs an experiment with four levels of whey content (0%, 10%, 20%, 30%) and two different supplements (1 and 2), and draws an interactions plot of the results. Parallel lines in an interaction plot indicate no interaction. The greater the difference in slope between the lines, the higher the degree of interaction. However, the interaction plot doesn't tell you if the interaction is statistically significant. This plot indicates an interaction between the supplement and the whey content. The supplement with the highest quality level depends on the whey content. Specifically, supplement 1 is better when the whey content is 0 and 10%, while supplement 2 is better when the whey content is 20 and 30%.

Interaction plots are most often used to visualize interactions during ANOVA or DOE.

Minitab draws a single interaction plot if you enter two factors, or a matrix of interaction plots if you enter more than two factors.

## Statistically significant

A sample statistic is statistically significant if a hypothesis test proves it to be too unlikely to have occurred by chance. You can assess statistical significance by looking at a test's p-value, which is the probability of obtaining a test statistic at least as extreme as the value you actually calculated from your sample, if the null hypothesis is true.  If the p-value is below a specified significance – or alpha () – level (typically 0.10, 0.05, or 0.01), you can declare the statistic to be statistically significant and reject the test's null hypothesis.

For example, suppose you want to determine whether the thickness of car windshields exceeds 4mm, as required by safety rules. You take a sample of windshields and conduct a 1-sample t-test with an -level of 0.05 and the following hypotheses: H0:  = 4 and H1:  > 4. If the test produces a p-value of 0.001, you declare statistical significance and reject the null hypothesis because the p-value is less than your chosen -level. You conclude in favor of the alternative hypothesis: that the windshield thickness does exceed 4mm. But if the p-value equals 0.50, you cannot claim statistical significance because the p-value is greater than your chosen -level. Therefore, you fail to reject the null hypothesis, and conclude that windshield thickness does not exceed 4mm.

Occasionally, because of sample limitations, a hypothesis will incorrectly assess statistical significance in one of two ways:

    Type 1 error: the hypothesis test declares statistical significance when none exists; a false positive

    Type 2 error: the hypothesis fails to detect statistical significance that does exist; a false negative

Increasing the power of a hypothesis test improves its ability to correctly detect statistical significance. One way to gain power is to increase sample size.

Statistical significance does not necessarily imply practical significance.  A test with extremely high power can declare the slightest difference from the test's hypothesized value to be statistically significant, although such a small difference may be meaningless in practice. For example, if a mixed nuts company advertises that their product contains no more than 50% peanuts, and you find 50.01% percent peanuts in your jar, a sample size of 100,000,000 will declare that result to be statistically significant. Therefore, use your specialized knowledge in conjunction with hypothesis tests to draw meaningful conclusions.

Example of specifying generators

main topic

Suppose you want to add two factors to a base design with three factors and eight runs.

1    Choose Stat > DOE > Factorial > Create Factorial Design.

2    Choose 2-level factorial (specify generators).

3    From Number of factors, choose 3.

4    Click Designs.

5    In the Designs box at the top, highlight the row for a full factorial. This design will serve as the base design.

6    Click Generators. In Add factors to the base design by listing their generators, enter D = AB     E = AC. Click OK in each dialog box.

Session window output
 Fractional Factorial Design     Factors:  5   Base Design:         3, 8   Resolution:  III Runs:     8   Replicates:             1   Fraction:    1/4 Blocks:   1   Center pts (total):     0   * NOTE * Some main effects are confounded with two-way interactions.     Design Generators: D = AB, E = AC     Alias Structure (up to order 3)   I + ABD + ACE   A + BD + CE B + AD + CDE C + AE + BDE D + AB + BCE E + AC + BCD BC + DE + ABE + ACD BE + CD + ABC + ADE

Interpreting the results

The base design has three factors labeled A, B, and C. Then Minitab adds factors D and E. Because of the generators selected, D is confounded with the AB interaction and E is confounded with the AC interaction. This gives a 2(5-2) or resolution III design. Look at the alias structure to see how the other effects are confounded.

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