Randomization and Blocking in DOE - Revision history

Richard House: /* Calculation of the Test Statistics */

2017-08-10T16:24:36Z

Calculation of the Test Statistics

← Older revision		Revision as of 16:24, 10 August 2017
Line 305:		Line 305:


	Assuming that the desired significance level is 0.1, since <math>p\,\!</math> value > 0.1, we fail to reject <math>{{H}_{0}}:{{\zeta }_{i}}=0\,\!</math> and conclude that there is no significant variation in the mileage from one vehicle to the other. Statistics to test the significance of other factors can be calculated in a similar manner. The complete analysis results obtained from DOE++ for this experiment are presented in the following figure.		Assuming that the desired significance level is 0.1, since <math>p\,\!</math> value > 0.1, we fail to reject <math>{{H}_{0}}:{{\zeta }_{i}}=0\,\!</math> and conclude that there is no significant variation in the mileage from one vehicle to the other. Statistics to test the significance of other factors can be calculated in a similar manner. The complete analysis results obtained from the DOE folio for this experiment are presented in the following figure.


	[[Image:doe6_14.png\|center\|644px\|Analysis results for the experiment in the example.]]		[[Image:doe6_14.png\|center\|644px\|Analysis results for the experiment in the example.]]

Richard House: /* Expression of the ANOVA Model as y = ΧΒ + ε */

2017-08-10T16:24:17Z

Expression of the ANOVA Model as y = ΧΒ + ε

← Older revision		Revision as of 16:24, 10 August 2017
Line 67:		Line 67:


	Therefore, only two of the <math>{{\zeta }_{i}}\,\!</math> effects are independent. Assuming that <math>{{\zeta }_{1}}\,\!</math> and <math>{{\zeta }_{2}}\,\!</math> are independent, <math>{{\zeta }_{3}}=-({{\zeta }_{1}}+{{\zeta }_{2}})\,\!</math>. (The null hypothesis to test the significance of the blocks can be rewritten using only the independent effects as <math>{{H}_{0}}:{{\zeta }_{1}}={{\zeta }_{2}}=0\,\!</math>.) In DOE++, the independent block effects, <math>{{\zeta }_{1}}\,\!</math> and <math>{{\zeta }_{2}}\,\!</math>, are displayed as Block[1] and Block[2], respectively.		Therefore, only two of the <math>{{\zeta }_{i}}\,\!</math> effects are independent. Assuming that <math>{{\zeta }_{1}}\,\!</math> and <math>{{\zeta }_{2}}\,\!</math> are independent, <math>{{\zeta }_{3}}=-({{\zeta }_{1}}+{{\zeta }_{2}})\,\!</math>. (The null hypothesis to test the significance of the blocks can be rewritten using only the independent effects as <math>{{H}_{0}}:{{\zeta }_{1}}={{\zeta }_{2}}=0\,\!</math>.) In DOE folios, the independent block effects, <math>{{\zeta }_{1}}\,\!</math> and <math>{{\zeta }_{2}}\,\!</math>, are displayed as Block[1] and Block[2], respectively.

	Constraints on <math>{{\tau }_{j}}\,\!</math> are:		Constraints on <math>{{\tau }_{j}}\,\!</math> are:

Richard House: /* Example */

2016-02-12T17:54:33Z

Example

← Older revision		Revision as of 17:54, 12 February 2016
Line 17:		Line 17:


	[[Image:doe6_13.png\|center\|643px\|Randomized complete block design for the mileage test using three blocks.]]		[[Image:doe6_13.png\|center\|643px\|Randomized complete block design for the mileage test using three blocks.\|link=]]

Richard House: /* Calculation of the Test Statistics */

2016-01-06T22:14:12Z

Calculation of the Test Statistics

← Older revision		Revision as of 22:14, 6 January 2016
Line 308:		Line 308:


	[[Image:doe6_14.png~~\|thumb~~\|center\|644px\|Analysis results for the experiment in the example.]]		[[Image:doe6_14.png\|center\|644px\|Analysis results for the experiment in the example.]]

Richard House: /* Example */

2016-01-06T22:13:54Z

Example

← Older revision		Revision as of 22:13, 6 January 2016
Line 17:		Line 17:


	[[Image:doe6_13.png~~\|thumb~~\|center\|643px\|Randomized complete block design for the mileage test using three blocks.]]		[[Image:doe6_13.png\|center\|643px\|Randomized complete block design for the mileage test using three blocks.]]

Harry Guo at 21:21, 27 August 2013

2013-08-27T21:21:50Z

← Older revision		Revision as of 21:21, 27 August 2013
Line 17:		Line 17:


	[[Image:~~doe6.13~~.png\|thumb\|center\|643px\|Randomized complete block design for the ~~experiment in the fifth table~~ using three blocks.]]		[[Image:doe6_13.png\|thumb\|center\|643px\|Randomized complete block design for the mileage test using three blocks.]]


Line 308:		Line 308:


	[[Image:~~doe6.14~~.png\|thumb\|center\|644px\|Analysis results for the experiment in the example.]]		[[Image:doe6_14.png\|thumb\|center\|644px\|Analysis results for the experiment in the example.]]

Chris Kahn at 22:17, 30 January 2013

2013-01-30T22:17:29Z

← Older revision		Revision as of 22:17, 30 January 2013
Line 35:		Line 35:


	where <math>M{{S}_{Block}}\,\!</math> represents the mean square for the block main effect and <math>M{{S}_{E}}\,\!</math> is the error mean square. The hypothesis statements and test statistics to test the significance of factors <math>A\,\!</math> (speed), <math>B\,\!</math> (fuel additive) and the interaction <math>AB\,\!</math> (speed-fuel additive interaction) can be obtained as explained in the [[~~ANOVA_for_Designed_Experiments~~#~~Example_2~~\| example]]. The ANOVA model for this example can be written as:		where <math>M{{S}_{Block}}\,\!</math> represents the mean square for the block main effect and <math>M{{S}_{E}}\,\!</math> is the error mean square. The hypothesis statements and test statistics to test the significance of factors <math>A\,\!</math> (speed), <math>B\,\!</math> (fuel additive) and the interaction <math>AB\,\!</math> (speed-fuel additive interaction) can be obtained as explained in the [[General_Full_Factorial_Designs#Example\| example]]. The ANOVA model for this example can be written as:


Line 308:		Line 308:


	[[Image:doe6.14.png\|thumb\|center\|644px\|Analysis results for the experiment in the ~~[[ANOVA_for_Designed_Experiments#Example_3\|~~ example]].]]		[[Image:doe6.14.png\|thumb\|center\|644px\|Analysis results for the experiment in the example.]]

Chris Kahn: /* Example */

2013-01-30T21:17:23Z

Example

← Older revision		Revision as of 21:17, 30 January 2013
Line 14:		Line 14:
	====Example====		====Example====

	Consider the ~~experiment~~ discussed in [[General_Full_Factorial_Designs#Example\|General Full Factorial Design]] where the mileage of a sports utility vehicle was investigated for the effects of speed and fuel additive type. Now assume that the three replicates for this experiment were carried out on three different vehicles. To ensure that the variation from one vehicle to another does not have an effect on the analysis, each vehicle is considered as one block. See the experiment design in the following figure.		Consider the example discussed in [[General_Full_Factorial_Designs#Example\|General Full Factorial Design]] where the mileage of a sports utility vehicle was investigated for the effects of speed and fuel additive type. Now assume that the three replicates for this experiment were carried out on three different vehicles. To ensure that the variation from one vehicle to another does not have an effect on the analysis, each vehicle is considered as one block. See the experiment design in the following figure.

Chris Kahn: /* Example */

2013-01-30T21:17:03Z

Example

← Older revision		Revision as of 21:17, 30 January 2013
Line 14:		Line 14:
	====Example====		====Example====

	Consider the experiment ~~of the fifth table~~ where the mileage of a sports utility vehicle was investigated for the effects of speed and fuel additive type. Now assume that the three replicates for this experiment were carried out on three different vehicles. To ensure that the variation from one vehicle to another does not have an effect on the analysis, each vehicle is considered as one block. See the experiment design in the following figure.		Consider the experiment discussed in [[General_Full_Factorial_Designs#Example\|General Full Factorial Design]] where the mileage of a sports utility vehicle was investigated for the effects of speed and fuel additive type. Now assume that the three replicates for this experiment were carried out on three different vehicles. To ensure that the variation from one vehicle to another does not have an effect on the analysis, each vehicle is considered as one block. See the experiment design in the following figure.

Chris Kahn: /* Expression of the ANOVA Model as y=X\beta +\epsilon */

2013-01-30T20:51:34Z

Expression of the ANOVA Model as y=X\beta +\epsilon

← Older revision		Revision as of 20:51, 30 January 2013
Line 54:		Line 54:
	In order to calculate the test statistics, it is convenient to express the ANOVA model of the equation given above in the form <math>y=X\beta +\epsilon \,\!</math>. This can be done as explained next.		In order to calculate the test statistics, it is convenient to express the ANOVA model of the equation given above in the form <math>y=X\beta +\epsilon \,\!</math>. This can be done as explained next.

	====Expression of the ANOVA Model as <~~math~~>y=~~X\beta~~ +\epsilon </~~math~~>====		====Expression of the ANOVA Model as <i>y</i> = <i>ΧΒ</i> + <i>ε</i>====

	Since the effects <math>{{\zeta }_{i}}\,\!</math>, <math>{{\tau }_{j}}\,\!</math>, <math>{{\delta }_{k}}\,\!</math>, and <math>{{(\tau \delta )}_{jk}}\,\!</math> are defined as deviations from the overall mean, the following constraints exist.		Since the effects <math>{{\zeta }_{i}}\,\!</math>, <math>{{\tau }_{j}}\,\!</math>, <math>{{\delta }_{k}}\,\!</math>, and <math>{{(\tau \delta )}_{jk}}\,\!</math> are defined as deviations from the overall mean, the following constraints exist.