Two Level Factorial Experiments - Revision history

Lisa Hacker at 23:58, 16 March 2023

2023-03-16T23:58:35Z

Kate Racaza: /* Variability Analysis */

2017-08-10T22:48:39Z

Variability Analysis

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	==Variability Analysis==		==Variability Analysis==

	For replicated two level factorial experiments, DOE++ provides the option of conducting variability analysis (using the Variability Analysis icon under the Data menu). The analysis is used to identify the treatment that results in the least amount of variation in the product or process being investigated. Variability analysis is conducted by treating the standard deviation of the response for each treatment of the experiment as an additional response. The standard deviation for a treatment is obtained by using the replicated response values at that treatment run. As an example, consider the <math>{2}^{3}\,\!</math> design shown in the following figure where each run is replicated four times. A variability analysis can be conducted for this design. DOE++ calculates eight standard deviation values corresponding to each treatment of the design (see second following figure). Then, the design is analyzed as an unreplicated <math>{2}^{3}\,\!</math> design with the standard deviations (displayed as Y Standard Deviation. in second following figure) as the response. The normal probability plot of effects identifies <math>AC\,\!</math> as the effect that influences variability (see third figure following). Based on the effect coefficients obtained in the fourth figure following, the model for Y Std. is:		For replicated two level factorial experiments, the DOE folio provides the option of conducting variability analysis (using the Variability Analysis icon under the Data menu). The analysis is used to identify the treatment that results in the least amount of variation in the product or process being investigated. Variability analysis is conducted by treating the standard deviation of the response for each treatment of the experiment as an additional response. The standard deviation for a treatment is obtained by using the replicated response values at that treatment run. As an example, consider the <math>{2}^{3}\,\!</math> design shown in the following figure where each run is replicated four times. A variability analysis can be conducted for this design. The DOE folio calculates eight standard deviation values corresponding to each treatment of the design (see second following figure). Then, the design is analyzed as an unreplicated <math>{2}^{3}\,\!</math> design with the standard deviations (displayed as Y Standard Deviation. in second following figure) as the response. The normal probability plot of effects identifies <math>AC\,\!</math> as the effect that influences variability (see third figure following). Based on the effect coefficients obtained in the fourth figure following, the model for Y Std. is:

Kate Racaza: /* Example: Two Level Factorial Design with Two Blocks */

2017-08-10T22:47:42Z

Example: Two Level Factorial Design with Two Blocks

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	For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the <math>ABCD\,\!</math> interaction is considered as the sum of squares due to blocks and <math>ABCD\,\!</math>. In DOE++ this is done by displaying this sum of squares as the sum of squares due to the blocks. This is shown in the following figure where the sum of squares in question is obtained as 72.25 and is displayed against Block. The interaction ABCD, which is confounded with the blocks, is not displayed. Since the design is unreplicated, any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks\| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.		For the analysis of this design, the sum of squares for all effects are calculated assuming no blocking. Then, to account for blocking, the sum of squares corresponding to the <math>ABCD\,\!</math> interaction is considered as the sum of squares due to blocks and <math>ABCD\,\!</math>. In the DOE folio, this is done by displaying this sum of squares as the sum of squares due to the blocks. This is shown in the following figure where the sum of squares in question is obtained as 72.25 and is displayed against Block. The interaction ABCD, which is confounded with the blocks, is not displayed. Since the design is unreplicated, any of the methods to analyze unreplicated designs mentioned in [[Two_Level_Factorial_Experiments#Unreplicated_2__Designs_in_2__Blocks\| Unreplicated <math>2^k</math> designs]] have to be used to identify significant effects.

Kate Racaza: /* Example: Two Level Factorial Design with Two Blocks */

2017-08-10T22:46:52Z

Example: Two Level Factorial Design with Two Blocks

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	Therefore, <math>ab\,\!</math> should be assigned to Block 2 or the second operator. Other treatments can be allocated to the two operators in a similar manner to arrive at the allocation scheme shown in the figure below.		Therefore, <math>ab\,\!</math> should be assigned to Block 2 or the second operator. Other treatments can be allocated to the two operators in a similar manner to arrive at the allocation scheme shown in the figure below.
	In DOE++, to confound the <math>ABCD\,\!</math> interaction for the <math>{2}^{4}\,\!</math> design into two blocks, the number of blocks are specified as shown in the figure below. Then the interaction <math>ABCD\,\!</math> is entered in the Block Generator window (second following figure) which is available using the Block Generator button in the following figure. The design generated by the Weibull++ DOE folio is shown in the third of the following figures. This design matches the allocation scheme of the preceding figure.		In a DOE folio, to confound the <math>ABCD\,\!</math> interaction for the <math>{2}^{4}\,\!</math> design into two blocks, the number of blocks are specified as shown in the figure below. Then the interaction <math>ABCD\,\!</math> is entered in the Block Generator window (second following figure) which is available using the Block Generator button in the following figure. The design generated by the Weibull++ DOE folio is shown in the third of the following figures. This design matches the allocation scheme of the preceding figure.

	[[Image:doe7_23.png\|center\|700px\| Adding block properties for the experiment in the example.\|link=]]		[[Image:doe7_23.png\|center\|700px\| Adding block properties for the experiment in the example.\|link=]]

Kate Racaza: /* Using Center Point Replicates to Test Curvature */

2017-08-10T22:45:37Z

Using Center Point Replicates to Test Curvature

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	===Using Center Point Replicates to Test Curvature===		===Using Center Point Replicates to Test Curvature===

	Center point replicates can also be used to check for curvature in replicated or unreplicated <math>{2}^{k}\,\!</math> designs. The test for curvature investigates whether the model between the response and the factors is linear. The way DOE++ handles center point replicates is similar to its handling of blocks. The center point replicates are treated as an additional factor in the model. The factor is labeled as Curvature in the results of the DOE folio. If Curvature turns out to be a significant factor in the results, then this indicates the presence of curvature in the model.		Center point replicates can also be used to check for curvature in replicated or unreplicated <math>{2}^{k}\,\!</math> designs. The test for curvature investigates whether the model between the response and the factors is linear. The way the DOE folio handles center point replicates is similar to its handling of blocks. The center point replicates are treated as an additional factor in the model. The factor is labeled as Curvature in the results of the DOE folio. If Curvature turns out to be a significant factor in the results, then this indicates the presence of curvature in the model.

Richard House: /* Example */

2017-08-10T17:42:11Z

Example

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	====Example====		====Example====

	The design of an automobile fuel cone is thought to be affected by six factors in the manufacturing process: cavity temperature (factor <math>A\,\!</math>), core temperature (factor <math>B\,\!</math>), melt temperature (factor <math>C\,\!</math>), hold pressure (factor <math>D\,\!</math>), injection speed (factor <math>E\,\!</math>) and cool time (factor <math>F\,\!</math>). The manufacturer of the fuel cone is unable to run the <math>{2}^{6}=64\,\!</math> runs required to complete one replicate for a two level full factorial experiment with six factors. Instead, they decide to run a fractional factorial design. Considering that three factor and higher order interactions are likely to be inactive, the manufacturer selects a <math>{2}^{6-2}\,\!</math> design that will require only 16 runs. The manufacturer chooses the resolution IV design which will ensure that all main effects are free from aliasing (assuming three factor and higher order interactions are absent). However, in this design the two factor interactions may be aliased with each other. It is decided that, if important two factor interactions are found to be present, additional experiment trials may be conducted to separate the aliased effects. The performance of the fuel cone is measured on a scale of 1 to 15. In ~~DOE~~++, the design for this experiment is set up using the properties shown in the following figure. The Fraction Generators for the design, <math>E=ABC\,\!</math> and <math>F=BCD\,\!</math>, are the same as the defaults used in DOE++. The resulting <math>{2}^{6-2}\,\!</math> design and the corresponding response values are shown in the following two figures.		The design of an automobile fuel cone is thought to be affected by six factors in the manufacturing process: cavity temperature (factor <math>A\,\!</math>), core temperature (factor <math>B\,\!</math>), melt temperature (factor <math>C\,\!</math>), hold pressure (factor <math>D\,\!</math>), injection speed (factor <math>E\,\!</math>) and cool time (factor <math>F\,\!</math>). The manufacturer of the fuel cone is unable to run the <math>{2}^{6}=64\,\!</math> runs required to complete one replicate for a two level full factorial experiment with six factors. Instead, they decide to run a fractional factorial design. Considering that three factor and higher order interactions are likely to be inactive, the manufacturer selects a <math>{2}^{6-2}\,\!</math> design that will require only 16 runs. The manufacturer chooses the resolution IV design which will ensure that all main effects are free from aliasing (assuming three factor and higher order interactions are absent). However, in this design the two factor interactions may be aliased with each other. It is decided that, if important two factor interactions are found to be present, additional experiment trials may be conducted to separate the aliased effects. The performance of the fuel cone is measured on a scale of 1 to 15. In a Weibull++ DOE folio, the design for this experiment is set up using the properties shown in the following figure. The Fraction Generators for the design, <math>E=ABC\,\!</math> and <math>F=BCD\,\!</math>, are the same as the defaults used in a DOE folio. The resulting <math>{2}^{6-2}\,\!</math> design and the corresponding response values are shown in the following two figures.


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	From the alias structure, it can be seen that for the present design interaction effect, <math>BF,\,\!</math> is confounded with <math>CD\,\!</math>. Therefore, the actual source of this effect cannot be known on the basis of the present experiment. However because neither factor <math>B\,\!</math> nor <math>F\,\!</math> is found to be significant there is an indication the observed effect is likely due to interaction, <math>CD\,\!</math>. To confirm this, a follow-up <math>{2}^{2}\,\!</math> experiment is run involving only factors <math>B\,\!</math> and <math>F\,\!</math>. The interaction, <math>BF\,\!</math>, is found to be inactive, leading to the conclusion that the interaction effect in the original experiment is effect, <math>CD\,\!</math>. Given these results, the fitted regression model for the fuel cone design as per the coefficients obtained from DOE++ is shown next.		From the alias structure, it can be seen that for the present design interaction effect, <math>BF,\,\!</math> is confounded with <math>CD\,\!</math>. Therefore, the actual source of this effect cannot be known on the basis of the present experiment. However because neither factor <math>B\,\!</math> nor <math>F\,\!</math> is found to be significant there is an indication the observed effect is likely due to interaction, <math>CD\,\!</math>. To confirm this, a follow-up <math>{2}^{2}\,\!</math> experiment is run involving only factors <math>B\,\!</math> and <math>F\,\!</math>. The interaction, <math>BF\,\!</math>, is found to be inactive, leading to the conclusion that the interaction effect in the original experiment is effect, <math>CD\,\!</math>. Given these results, the fitted regression model for the fuel cone design as per the coefficients obtained from the DOE folio is shown next.

	::<math>\hat{y}=7.6875+C+2\cdot D+2.1875\cdot CD\,\!</math>		::<math>\hat{y}=7.6875+C+2\cdot D+2.1875\cdot CD\,\!</math>

Richard House: /* Example */

2017-08-10T17:39:20Z

Example

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	\end{align}\,\!</math></center>		\end{align}\,\!</math></center>

	In DOE++, the alias structure is displayed in the Design Summary and as part of the Design Evaluation result, as shown next:		In a DOE folio, the alias structure is displayed in the Design Summary and as part of the Design Evaluation result, as shown next:

	[[Image:doe7_41.png\|center\|700px\|Alias structure for the experiment design in the [[Two_Level_Factorial_Experiments#Example_7\| example]].\|link=]]		[[Image:doe7_41.png\|center\|700px\|Alias structure for the experiment design in the [[Two_Level_Factorial_Experiments#Example_7\| example]].\|link=]]

Richard House: /* Design Resolution */

2017-08-10T17:38:25Z

Design Resolution

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	All of the two level fractional factorial designs available in DOE++ are shown next.		All of the two level fractional factorial designs available in a DOE folio are shown next.


	[[Image:doe7.38.png\|center\|471px\|Two level fractional factorial designs available in ~~DOE~~++ and their resolutions.]]		[[Image:doe7.38.png\|center\|471px\|Two level fractional factorial designs available in Weibull++ and their resolutions.]]

Richard House: /* Calculation of Aliases */

2017-08-10T17:37:27Z

Calculation of Aliases

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	Other aliases can be obtained in a similar way. It can be seen that each effect in this design has three aliases. In general, each effect in a <math>{2}^{k-p}\,\!</math> design has <math>{2}^{p-1}\,\!</math> aliases.		Other aliases can be obtained in a similar way. It can be seen that each effect in this design has three aliases. In general, each effect in a <math>{2}^{k-p}\,\!</math> design has <math>{2}^{p-1}\,\!</math> aliases.
	The aliases for the <math>{2}^{4-2}\,\!</math> design show that in this design the main effects are aliased with each other (<math>A\,\!</math> is aliased with <math>D\,\!</math> and <math>B\,\!</math> is aliased with <math>C\,\!</math>). Therefore, this design is not a useful design and is not available in ~~DOE~~++. It is important to ensure that main effects and lower order interactions of interest are not aliased in a fractional factorial design. This is known by looking at the resolution of the fractional factorial design.		The aliases for the <math>{2}^{4-2}\,\!</math> design show that in this design the main effects are aliased with each other (<math>A\,\!</math> is aliased with <math>D\,\!</math> and <math>B\,\!</math> is aliased with <math>C\,\!</math>). Therefore, this design is not a useful design and is not available in a Weibull++ DOE folio. It is important to ensure that main effects and lower order interactions of interest are not aliased in a fractional factorial design. This is known by looking at the resolution of the fractional factorial design.

	==Design Resolution==		==Design Resolution==

Richard House: /* Half-fraction Designs */

2017-08-10T17:36:34Z

Half-fraction Designs

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	[[Image:doe7.34.png\|center\|330px\|Half-fractions of the <math>2^3\,\!</math> design. (a) shows the full factorial <math>2^3\,\!</math> design, (b) shows the <math>2^{3-1}\,\!</math> design with the defining relation <math>I=ABC\,\!</math> and (c) shows the <math>{2}^{3-1}\,\!</math> design with the defining relation <math>I=-ABC\,\!</math>.]]		[[Image:doe7.34.png\|center\|330px\|Half-fractions of the <math>2^3\,\!</math> design. (a) shows the full factorial <math>2^3\,\!</math> design, (b) shows the <math>2^{3-1}\,\!</math> design with the defining relation <math>I=ABC\,\!</math> and (c) shows the <math>{2}^{3-1}\,\!</math> design with the defining relation <math>I=-ABC\,\!</math>.]]

	In the <math>{2}^{3-1}\,\!</math> design of figure (b), since the interaction <math>ABC\,\!</math> is always included at the same level (the high level represented by 1), it is not possible to measure this interaction effect. The effect, <math>ABC\,\!</math>, is called the ''generator'' or ''word'' for this design. It can be noted that, in the design matrix of the following figure (b), the column corresponding to the intercept, <math>I\,\!</math>, and column corresponding to the interaction <math>ABC\,\!</math>, are identical. The identical columns are written as <math>I=ABC\,\!</math> and this equation is called the ''defining relation'' for the design. In DOE++, the present <math>{2}^{3-1}\,\!</math> design can be obtained by specifying the design properties as shown in the following figure.		In the <math>{2}^{3-1}\,\!</math> design of figure (b), since the interaction <math>ABC\,\!</math> is always included at the same level (the high level represented by 1), it is not possible to measure this interaction effect. The effect, <math>ABC\,\!</math>, is called the ''generator'' or ''word'' for this design. It can be noted that, in the design matrix of the following figure (b), the column corresponding to the intercept, <math>I\,\!</math>, and column corresponding to the interaction <math>ABC\,\!</math>, are identical. The identical columns are written as <math>I=ABC\,\!</math> and this equation is called the ''defining relation'' for the design. In a DOE folio, the present <math>{2}^{3-1}\,\!</math> design can be obtained by specifying the design properties as shown in the following figure.

	[[Image:doe7_35.png\|center\|700px\|Design properties for the <math>2^{3-1}\,\!</math> design.\|link=]]		[[Image:doe7_35.png\|center\|700px\|Design properties for the <math>2^{3-1}\,\!</math> design.\|link=]]