Statistical Design and Analysis of Experiments - Revision history

Admin: /* Chapter 5: Factorial experiments in completely randomized designs */

2011-06-18T04:47:39Z

Chapter 5: Factorial experiments in completely randomized designs

Admin at 02:29, 31 May 2011

2011-05-31T02:29:24Z

← Older revision		Revision as of 02:29, 31 May 2011
Line 1:		Line 1:
			{{Cite book
			\|last1 = Mason
			\|first1 = Robert
			\|last2 = Gunst
			\|first2 = Richard
			\|last3 = Hess
			\|first3 = James
			\|title = [[Statistical Design and Analysis of Experiments]]
			\|year = 2003
			\|publisher = John Wiley and Sons
			}}

	=Overview=		=Overview=

Admin at 02:13, 31 May 2011

2011-05-31T02:13:48Z

← Older revision		Revision as of 02:13, 31 May 2011
Line 1:		Line 1:

	=Overview=		=Overview=

Line 941:		Line 940:

	[[Category:Reading]]		[[Category:Reading]]
	[[Category:~~Textbook~~]]		[[Category:Textbooks]]

Admin: /* Analysis of Variance (ANOVA) Calculations */

2011-05-16T19:34:35Z

Analysis of Variance (ANOVA) Calculations

@@ Line 510: / Line 510: @@
 <math>
-SS_{ABC} = \sum_i \sum_j \sum_k \frac{ y_{ijk \centerdot}^2 }{ r } - SS_M - SS_A - SS_B - SS_C - SS_{AB} - SS_{AC} - SS_{BC}
+\begin{array}{rcl}
 </math>
@@ Line 517: / Line 528: @@
 <math>
 SS_M = \frac{ y_{\centerdot \centerdot \centerdot \centerdot}^2 }{ abcr }
 </math>

Admin: /* Analysis of Variance (ANOVA) Tables */

2011-05-14T08:20:42Z

Analysis of Variance (ANOVA) Tables

← Older revision		Revision as of 08:20, 14 May 2011
Line 462:		Line 462:

	\|}		\|}

			===Analysis of Variance (ANOVA) Calculations===

			Total sum of squares:

			<math>
			TSS = \sum_i \sum_j \sum_k \sum_l y_{ijkl}^2 - SS_M
			</math>

			Sum of squares for factor A:

			<math>
			SS_A = \sum_i \frac{ y_{i \centerdot \centerdot \centerdot}^2 }{bcr} - SS_M
			</math>

			Sum of squares for factor B:

			<math>
			SS_B = \frac{ \sum_j y_{\centerdot j \centerdot \centerdot}^2 }{ bcr } - SS_M
			</math>

			Sum of squares for factor C:

			<math>
			SS_C = \mbox{etc}
			</math>

			Sum of squares for AB:

			<math>
			SS_{AB} = \sum_i \sum_j \frac{ y_{i j \centerdot \centerdot}^2 }{ cr } - SS_M - SS_A - SS_B
			</math>

			Sum of squares for BC:

			<math>
			SS_{BC} = \sum_j \sum_k \frac{ y_{\centerdot j k \centerdot}^2 }{ ar } - SS_M - SS_B - SS_C
			</math>

			Sum of squares for AC:

			<math>
			SS_{AC} = \mbox{etc}
			</math>

			Sum of squares for ABC:

			<math>
			SS_{ABC} = \sum_i \sum_j \sum_k \frac{ y_{ijk \centerdot}^2 }{ r } - SS_M - SS_A - SS_B - SS_C - SS_{AB} - SS_{AC} - SS_{BC}
			</math>

			Sum of squares for M (mean?):

			<math>
			SS_M = \frac{ y_{\centerdot \centerdot \centerdot \centerdot}^2 }{ abcr }
			</math>

			Averaging of response:

			<math>
			y_{\centerdot \centerdot \centerdot \centerdot} = \sum_i \sum_j \sum_k \sum_l y_{ijkl}
			</math>

			<math>
			y_{i \centerdot \centerdot \centerdot} = \sum_j \sum_k \sum_l y_{ijkl}
			</math>

			<math>
			y_{i j \centerdot \centerdot} = \sum_k \sum_l y_{ijkl}
			</math>

			<math>
			y_{ijk\centerdot} = \sum_l y_{ijkl}
			</math>

	=Chapter 7: fractional factorial experiments=		=Chapter 7: fractional factorial experiments=

Admin: /* Chapter 6: analysis of completely randomized designs */

2011-05-14T07:26:08Z

Chapter 6: analysis of completely randomized designs

@@ Line 228: / Line 228: @@
 \sum_i ( \alpha \beta )_{ij} &=& 0 \\
 \sum_j ( \alpha \beta )_{ij} &=& 0 \\
 \mbox{etc.} &&
 \end{array}
 </math>
@@ Line 257: / Line 257: @@
 + \overline{\mu}_{\centerdot \centerdot k}
 - \overline{\mu}_{\centerdot \centerdot \centerdot}
 \end{array}
@@ Line 361: / Line 358: @@
 and the last term measures contribution of the joint effects of all three factos
 =Chapter 7: fractional factorial experiments=

Admin: /* Chapter 6: analysis of completely randomized designs */

2011-05-14T02:19:08Z

Chapter 6: analysis of completely randomized designs

← Older revision		Revision as of 02:19, 14 May 2011
Line 305:		Line 305:

	===Analysis of Variance (ANOVA) Tables===		===Analysis of Variance (ANOVA) Tables===

			Example given for three-factor experiment

			* A with a levels
			* B with b levels
			* C with c levels
			* r repeats

			Previous sections of book: sample variance (or its square root, standard deviation) used to measure variability

			in ANOVA procedure, first step is to define measure of variation for which partitioning into different assignable causes and random variations can be done

			Measure of variability: total sum of squares (TSS)

			<math>
			\mbox{TSS} = \sum_{i=1}^{n} ( y_i - \overline{y} )^2
			</math>

			For three-factor experiment, adjust for the overall average by subtracting it from each individual response

			Thus it is also called the total adjusted sum of squares TSS(adj)

			TSS(adj) consists of squared differences of observations <math>y_{ijkl}</math> from overall oaverage response <math>\overline{y}_{\centerdot \centerdot \centerdot \centerdot}</math>:

			<math>
			\mbox{TSS} = \sum_i \sum_j \sum_k \sum_l ( y_{ijkl} - \overline{y}_{\centerdot \centerdot \centerdot \centerdot} )^2
			</math>

			and to partition into ''assignable causes'' and ''random variation'', add/subtract factor-level ocmbination averages <math>\overline{y}_{ijk \centerdot}</math> from each term in the summation for the total sum of squares:

			<math>
			\begin{array}{rcl}
			\mbox{TSS} &=& \sum_i \sum_j \sum_k \sum_l ( y_{ijkl} - \overline{y}_{ijk\centerdot} + \overline{y}_{ijk\centerdot} - \overline{y}_{\centerdot \centerdot \centerdot \centerdot} )^2 \\
			&=& r \sum_i \sum_j \sum_k ( \overline{y}_{ijk\centerdot} - \overline{y}_{\centerdot \centerdot \centerdot \centerdot} )^2 \\
			&& + \sum_i \sum_j \sum_k \sum_l ( y_{ijkl} - \overline{y}_{ijk\centerdot} )^2 \\
			&=& \mbox{MMS} + \mbox{SS}_{E}
			\end{array}
			</math>

			where the first component of the above equation is the contribution to the total sum of squares of all the variability attributable to assignable cuases (model sum of squares):

			<math>
			\mbox{MMS} = r \sum_i \sum_j \sum_k ( y_{ijk\centerdot} - \overline{y}_{\centerdot \centerdot \centerdot \centerdot} )^2
			</math>

			and it can be partitioned into the following components:

			<math>
			\mbox{MSS} = \mbox{SS}_A + \mbox{SS}_B + \mbox{SS}_C + \mbox{SS}_{AB} + \mbox{SS}_{AC} + \mbox{SS}_{BC} + \mbox{SS}_{ABC}
			</math>

			where the first three terms measure individual factor contributions (main effects of 3 experimental factors)

			the next three terms measure contributiuons of two-factor joint effects

			and the last term measures contribution of the joint effects of all three factos

	=Chapter 7: fractional factorial experiments=		=Chapter 7: fractional factorial experiments=

Admin: /* Chapter 6: analysis of completely randomized designs */

2011-05-14T00:28:43Z

Chapter 6: analysis of completely randomized designs

@@ Line 263: / Line 263: @@
 \end{array}
 </math>
 =Chapter 7: fractional factorial experiments=

Admin: /* Chapter 6: analysis of completely randomized designs */

2011-05-14T00:19:26Z

Chapter 6: analysis of completely randomized designs

@@ Line 148: / Line 148: @@
 * statistical tests of factor effects
 * multiple comparison techniques for comparing means/groups of means
 =Chapter 7: fractional factorial experiments=

Admin: /* Confounding of effects */

2011-05-03T15:33:01Z

Confounding of effects

← Older revision		Revision as of 15:33, 3 May 2011
Line 230:		Line 230:

	<math>		<math>
	\begin{~~align~~}		\begin{array}{rcl}
	M(B) &=& \frac{1}{4} ~~\left~~[ (y_{121} + s_1) + (y_{122} + s_1) + (y_{221} + s_2) \\		M(B) &=& \frac{1}{4} [ (y_{121} + s_1) + (y_{122} + s_1) + (y_{221} + s_2) \\
	&+& (y_{222} + s_2) - (y_{111} + s_1) - (y_{112} + s_1) - (y_{211} + s_2) - (y_{212} + s_2) ~~\right~~] \\		& & + (y_{222} + s_2) - (y_{111} + s_1) - (y_{112} + s_1) - (y_{211} + s_2) - (y_{212} + s_2) ] \\
	&=& \overline{y}_{\cdot 2 \cdot} - \overline{y}_{\cdot 1 \cdot}		&=& \overline{y}_{\cdot 2 \cdot} - \overline{y}_{\cdot 1 \cdot}
	\end{~~align~~}		\end{array}
	</math>		</math>

← Older revision		Revision as of 04:47, 18 June 2011
Line 104:		Line 104:

	construction of factorial experiments in randomized designs:		construction of factorial experiments in randomized designs:

	1. enumerate all factor-level combinations		1. enumerate all factor-level combinations

	2. number factor-level combinations sequentially, 1 to N		2. number factor-level combinations sequentially, 1 to N

	3. from a random number table or sequence of random numbers, obtain random sequence of 1 to N		3. from a random number table or sequence of random numbers, obtain random sequence of 1 to N

	4. assign factor-level combinations to the experimental units in order specified by random number sequence		4. assign factor-level combinations to the experimental units in order specified by random number sequence

Line 129:		Line 133:

	effects coding of factor levels:		effects coding of factor levels:

	1. designate one level of each factor as -1 and the other level as +1		1. designate one level of each factor as -1 and the other level as +1

	2. lay out a table with column headings for each of the factors A, B, ..., K		2. lay out a table with column headings for each of the factors A, B, ..., K

	3. let n = 2^k, where k is the number of factors		3. let n = 2^k, where k is the number of factors

	4. set the first n/2 of the levels for factor A equal to -1 and the last n/2 equal to +1. Set the first n/4 levels of factor B equal to -1, the next n/4 equal to +1, the next n/4 equal to -1, etc. Continue in this fashion until the last column (for factor K) has alternating -1 and +1 sign		4. set the first n/2 of the levels for factor A equal to -1 and the last n/2 equal to +1. Set the first n/4 levels of factor B equal to -1, the next n/4 equal to +1, the next n/4 equal to -1, etc. Continue in this fashion until the last column (for factor K) has alternating -1 and +1 sign

Line 143:		Line 151:

	calculation of effects for two-level factors:		calculation of effects for two-level factors:

	1. construct effects representation for each main effect and each interaction		1. construct effects representation for each main effect and each interaction

	2. calculate linear combinations of the average responses using the signs in the effects column for each main effect and for each interaction		2. calculate linear combinations of the average responses using the signs in the effects column for each main effect and for each interaction

	3. divide respective linear combinations of average responses by 2^{k-1}, where k is number of factors in experiment		3. divide respective linear combinations of average responses by 2^{k-1}, where k is number of factors in experiment

Line 151:		Line 162:
	example of how to do this		example of how to do this
	yield as function of temperature, concentration, catalyst		yield as function of temperature, concentration, catalyst


	=Chapter 6: analysis of completely randomized designs=		=Chapter 6: analysis of completely randomized designs=