Các mẫu nhân tố (Factorial Designs)


Mẫu trực giao ( Orthogonal design)



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Mẫu trực giao ( Orthogonal design)


Mẫu trực giao chứa mỗi tổ hợp mức độ (các cặp yếu tố) một số lần bằng nhau.

Orthogonal design

A

B

C

-1

-1

1

1

-1

-1

-1

1

-1

1

1

1




Chẳng hạn, đối với cặp AB , ta có (-1, -1); (1, -1); (-1, 1); và (1,1). Tương tự khi xét các cặp khác , AC và BC,

 

Mẫu không trực giao không chứa mỗi tổ hợp mức độ (các cặp yếu tố) số lần bằng nhau .



Non-orthogonal design

A

B

C

-1

-1

1

1

-1

-1

-1

1

-1

1

1

-1




Đối với cặp AB ta có (-1, -1); (1, -1); (-1, 1); và (1,1). Nhưng với hai cặp khác AC và BC, ta thấy chúng có tổ hợp lập lại và mất tổ hợp . Đối với AC, ta có (-1, 1); (1, -1); (-1, -1); (1,-1); và mất tổ hợp (1,1) . Đối với BC, ta có (-1, 1); (-1, -1); (1, -1); (1,-1); và mất tổ hợp (1,1) .

 

 

Sum of squares


Represents a measure of variation or deviation from the mean. It is calculated as a summation of the squares of the differences from the mean. The calculation of the total sum of squares considers both the sum of squares from the factors and from random chance or error.

Regression


In regression, the total sum of squares helps express the total variation of the y's. For example, you collect data to determine a model explaining overall sales as a function of your advertising budget.

The total sum of squares =  regression sum of squares (SSR) + residual sum of squares (SSE)

 (y  )2 =  (  )2 +   (y  )2

The regression sum of squares is the variation attributed to the relationship between the x's and y's, or in this case between the advertising budget and your sales. The residual sum of squares is the variation attributed to the error.

By comparing the regression sum of squares to the total sum of squares, you find the proportion of the total variation that is explained by the regression model (R2, the coefficient of determination). The larger this value is, the better the relationship explaining sales as a function of advertising budget.

ANOVA


In analysis of variance (ANOVA), the total sum of squares helps express the total variation that can be attributed to various factors. For example, you run an experiment to test the effectiveness of three laundry detergents.

The total sum of squares =  treatment sum of squares (SST) + error sum of squares (SSE)

The treatment sum of squares is the variation attributed to, or in this case between the laundry detergents. The residual sum of squares is the variation attributed to the error.

Converting the sum of squares into mean squares by dividing by the degrees of freedom will allow you to compare these ratios and determine whether there is a significant difference due to detergent. The larger this ratio is, the more the treatments affect the outcome.  



Sequential sum of squares vs. adjusted sum of squares

Minitab breaks down the SS Regression or Treatments component of variance into sums of squares for each factor.

    Sequential sums of squares depend on the order the factors are entered into the model. It is the unique portion of SS Regression explained by a factor, given any previously entered factors.

For example, if you have a model with three factors, X1, X2, and X3, the sequential sum of squares for X2 shows how much of the remaining variation X2 explains, given that X1 is already in the model. To obtain a different sequence of factors, repeat the regression procedure entering the factors in a different order.

    Adjusted sums of squares does not depend on the order the factors are entered into the model. It is the unique portion of SS Regression explained by a factor, given all other factors in the model, regardless of the order they were entered into the model.

For example, if you have a model with three factors, X1, X2, and X3, the adjusted sum of squares for X2 shows how much of the remaining variation X2 explains, given that X1 and X3 are also in the model.

 

 

Alpha ()


Used in hypothesis testing, alpha () is the maximum acceptable level of risk for rejecting a true null hypothesis (type I error) and is expressed as a probability ranging between 0 and 1. Alpha is frequently referred to as the level of significance. You should set before beginning the analysis then compare p-values to  to determine significance:   

    If your p-value is less than or equal to the -level, reject the null hypothesis in favor of the alternative hypothesis.

    If your p-value is greater than the -level, fail to reject the null hypothesis.

The most commonly used -level is 0.05.  At this level, your chance of finding an effect that does not really exist is only 5%. The smaller the value, the less likely you are to incorrectly reject the null hypothesis. However, a smaller value for  also means a decreased chance of detecting an effect if one truly exists (lower power).

Sometimes it may be better to choose a smaller value for .  For example, you are testing samples from a new milling machine to decide whether to purchase a dozen for your plant.  You stand to save a large amount of money due to fewer defective products if the new machine is more accurate.  However, the cost of purchasing and installing a dozen machines is very high. You want to be sure that the new machine is more accurate before making the purchase. In this case, you might want to select a lower value for , such as 0.001. That way, you have only a 0.1% chance of concluding the new machine is more accurate, if in fact no difference exists.

On the other hand, sometimes choosing a larger value for  is better. For example, suppose you are a jet engine manufacturer and you are testing the strength of cheaper ball bearings. Saving a small amount of money does not outweigh the potentially disastrous effects if the bearings are weaker. Therefore, you might want to select a higher value for , such as 0.1. Although this means you will be more likely to reject a true null hypothesis, more importantly, you will also be more likely to detect a real weakness in the cheaper bearings.

 

 



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