Response Surface Methodology - Revision history

Admin: /* T-statistic */

2011-07-01T19:30:51Z

T-statistic

← Older revision		Revision as of 19:30, 1 July 2011
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	<math>		<math>
	t = \frac{ b_0 - \beta_0 }{ s_e \left( n^{-1} + \overline{x}^2 / s_{xx} \right)^2 }		t = \frac{ b_0 - \beta_0 }{ s_e \left( n^{-1} + \overline{x}^2 / s_{xx} \right)^2 }
			</math>

	'''Multiple Linear Model'''		'''Multiple Linear Model'''

Admin: /* F-Statistic */

2011-07-01T18:31:04Z

F-Statistic

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	===F-Statistic===		===F-Statistic===

			{{Main\|F-Statistic}}

	I found this short YouTube video very helpful for illustrating what the F-statistic means physically: http://www.youtube.com/watch?v=TMwSS8DAVYk		I found this short YouTube video very helpful for illustrating what the F-statistic means physically: http://www.youtube.com/watch?v=TMwSS8DAVYk
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	If <math>k = p-1</math> then the F-statistic is the square of the t-statistic from the full model corresponding to the term left out of the reduced model		If <math>k = p-1</math> then the F-statistic is the square of the t-statistic from the full model corresponding to the term left out of the reduced model

	To determine if F-statistic is highly significant, use p value (significant if <math>p < 0.~~001~~</math>)		To determine if F-statistic is highly significant, use p value (95% likelihood of being significant if <math>p < 0.05</math>, 99% likelihood if <math>p < 0.01</math>)

	Example: use to determine if interaction effect is important to surrogate model		Example: use to determine if interaction effect is important to surrogate model

	Using this procedure and using the t-statistic to test the significance of a given term are ''equivalent''!		Using this procedure and using the t-statistic to test the significance of a given term are ''equivalent''!





	===T-statistic===		===T-statistic===

Admin: /* See Also */

2011-07-01T18:28:49Z

Admin: /* T-statistic */

2011-07-01T18:28:08Z

T-statistic

← Older revision		Revision as of 18:28, 1 July 2011
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	Using this procedure and using the t-statistic to test the significance of a given term are ''equivalent''!		Using this procedure and using the t-statistic to test the significance of a given term are ''equivalent''!





	===T-statistic===		===T-statistic===
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	e.g. "Each individual predictor variable contributes significantly to the given fits, given that the other two predictor variables are also included in the model"		e.g. "Each individual predictor variable contributes significantly to the given fits, given that the other two predictor variables are also included in the model"










	===Response Confidence Intervals===		===Response Confidence Intervals===

Admin: /* F-Statistic */

2011-07-01T18:26:09Z

F-Statistic

← Older revision		Revision as of 18:26, 1 July 2011
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	I found this short YouTube video very helpful for illustrating what the F-statistic means physically: http://www.youtube.com/watch?v=TMwSS8DAVYk		I found this short YouTube video very helpful for illustrating what the F-statistic means physically: http://www.youtube.com/watch?v=TMwSS8DAVYk

	~~This~~ is a ~~quantity that~~ is ~~part of ANOVA tables~~		The F-statistic can be thought of as a frequentist metric for hypothesis-testing. Once an F-statistic and corresponding p-value is calculated from the ANOVA table quantities, you can determine how confident you can be in a given hypothesis test (where the hypothesis is the model onto which you've chosen to regress your data).

	~~Allows testing~~ of ~~significance~~ of ~~slope parameter~~		Mason: Different from tests for significance of factor effects in analysis of designed experiments

	~~Different from tests for significance~~ of ~~factor effects in analysis of designed experiments~~		Mason: example of acid-content data...

	~~Test for significance of slope parameter conduced using large confidence level~~

	~~Mason: example of acid-content data...~~

	For multiple linear regression/model, can use F-statistic to simultaneously test whether all <math>\beta_j = 0</math> versus alternative, that at least one <math>\beta_j</math> is not zero		For multiple linear regression/model, can use F-statistic to simultaneously test hypothesis: whether all <math>\beta_j = 0</math> versus the alternative, that at least one <math>\beta_j</math> is not zero

	i.e. <math>\beta_1 = \beta_2 = \dots = \beta_p = 0</math> versus <math>\beta_j \neq 0</math>		i.e. <math>\beta_1 = \beta_2 = \dots = \beta_p = 0</math> versus <math>\beta_j \neq 0</math>

	(~~Personally~~, it's ~~not clear to me that~~ this ~~is even important to do~~..~~. I mean, if F-statistic is significant, that basically tells you that at least one~~ of ~~your input parameters can~~ be ~~used to predict response~~		(while this seems silly, it's much more useful if you're doing this for a subset of coefficients - i.e. testing the hypothesis of whether any of a subset of coefficients should be non-zero)

	'''Lack of Fit F-Statistic'''		'''Lack of Fit F-Statistic'''
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	Consider full regresion model with <math>p</math> predictor variables		Consider full regresion model with <math>p</math> predictor variables

	Now consider reduced regression model with <math>k < p</math> predictor variables		Now consider a ''reduced'' regression model with <math>k < p</math> predictor variables

	Full model = <math>M_1</math>		Full model = <math>M_1</math>

Admin: /* F-Statistic */

2011-07-01T18:17:10Z

F-Statistic

← Older revision		Revision as of 18:17, 1 July 2011
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	large <math>s_e</math>: large random error, or poor selection of model form		large <math>s_e</math>: large random error, or poor selection of model form




	===F-Statistic===		===F-Statistic===

			I found this short YouTube video very helpful for illustrating what the F-statistic means physically: http://www.youtube.com/watch?v=TMwSS8DAVYk

	This is a quantity that is part of ANOVA tables		This is a quantity that is part of ANOVA tables
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	Using this procedure and using the t-statistic to test the significance of a given term are ''equivalent''!		Using this procedure and using the t-statistic to test the significance of a given term are ''equivalent''!



	===T-statistic===		===T-statistic===

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

2011-07-01T18:16:11Z

Analysis of Variance (ANOVA) Table

← Older revision		Revision as of 18:16, 1 July 2011
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	\|<math>		\|<math>
	MSE = \frac{SSE}{df} = \frac{SSE}{n - p - 1}		MSE = \frac{SSE}{df} = \frac{SSE}{n - p - 1}
			</math>
	\|		\|

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

2011-07-01T18:15:01Z

Analysis of Variance (ANOVA) Table

← Older revision		Revision as of 18:15, 1 July 2011
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	\|p		\|p
	\|SSR		\|SSR
	\|MSR = SSR/p		\|<math>
	\|F = MSR/~~MSE~~		MSR = \frac{SSR}{df} = \frac{SSR}{p}
			</math>
			\|<math>
			F = \frac{MSR}{MSE}
			</math>

	\|-		\|-
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	\|n-p-1		\|n-p-1
	\|SSE		\|SSE
	\|MSE = SSE/(n-p-1)		\|<math>
			MSE = \frac{SSE}{df} = \frac{SSE}{n - p - 1}
	\|		\|

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

2011-07-01T18:13:53Z

Analysis of Variance (ANOVA) Table

← Older revision		Revision as of 18:13, 1 July 2011
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	\|1		\|1
	\|SSR		\|SSR
	\|MSR = SSR		\|<math>
	\|F = MSR/~~MSE~~		MSR = \frac{SSR}{df} = SSR
	\|p		</math>
			\|<math>
			F = \frac{MSR}{MSE}
			</math>
			\|p-value is obtained from F-statistic

	\|-		\|-
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	\|n-2		\|n-2
	\|SSE		\|SSE
	\|MSE = SSE/(n-2)		\|<math>
			MSE = \frac{SSE}{df} = \frac{SSE}{n-2}
			</math>
	\|		\|
	\|		\|
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	large <math>s_e</math>: large random error, or poor selection of model form		large <math>s_e</math>: large random error, or poor selection of model form



	===F-Statistic===		===F-Statistic===

Admin: /* Comparison Metrics */

2011-07-01T02:06:17Z

Comparison Metrics

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 Caution should be used in relying on single measure of fit (e.g. <math>R^2</math>
@@ Line 543: / Line 541: @@
 For multiple responses, a contour plot for each response can be made, infeasible regions shaded gray, and the plots overlaid to yield the feasible region.
 ==Other Things to Look At==

← Older revision		Revision as of 18:28, 1 July 2011
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	* [[Experimental Design Lecture]]		* [[Experimental Design Lecture]]



			{{ExperimentalDesign}}