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John — Fri, 20 Nov 2009 16:03:36 GMT

←Older revision		Revision as of 16:03, 20 November 2009
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	[[Image:TTestMatlab_T.jpg]]		[[Image:TTestMatlab_T.jpg]]
		+
		+
		+
		+
		+	= References =
		+
		+	[http://www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2/pdfs/Ch7.pdf The General Linear Model by S.J. Kiebel and A.P. Holmes]
		+
		+	Neter, J., Wasserman, W., and Kutner, M.H. 1989. Applied Linear Regression Models. IRWIN, Homewood, IL, Second Edition.
		+
		+	Neter, J., Wasserman, W., and Kutner, M.H. Applied Linear Statistical Models. IRWIN, Homewood, IL, Third Edition.
		+
		+	Howell, David C. 2002. Statistical Methods for Psychology. Duxbury, Pacific Grove, CA, Fifth Edition.
		+
		+	[http://www.statsdirect.com/help/statsdirect.htm Stats Direct]

Image:TTestMatlab T.jpg

John — Fri, 20 Nov 2009 15:08:30 GMT

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John — Fri, 20 Nov 2009 15:08:08 GMT

Two-Sample T-Test via GLM:

←Older revision		Revision as of 15:08, 20 November 2009
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	= Two-Sample T-Test via GLM =		= Two-Sample T-Test via GLM =

-	A two-sample T-Test is simply a one-way ANOVA with only two groups.	+	A two-sample T-Test is simply a one-way ANOVA with only two groups. Note that '''T''' is the square root of '''F'''.
		+
		+	== Example Data ==

	Data from [http://www.statsdirect.com/help/parametric_methods/utt.htm http://www.statsdirect.com/help/parametric_methods/utt.htm].		Data from [http://www.statsdirect.com/help/parametric_methods/utt.htm http://www.statsdirect.com/help/parametric_methods/utt.htm].
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	\|		\|
	\|}		\|}
		+
		+	== Solution with Matlab Using the GLM with Matrices ==

	[[Image:TTestMatlabGLM_DataY.jpg]] [[Image:TTestMatlabGLM_DataX.jpg]]		[[Image:TTestMatlabGLM_DataY.jpg]] [[Image:TTestMatlabGLM_DataX.jpg]]
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	[[Image:TTestMatlabGLM_Data_SS_DF_F.jpg]]		[[Image:TTestMatlabGLM_Data_SS_DF_F.jpg]]
		+
		+
		+	== Solution Using Matlab's Two-Sample T-Test ==
		+
		+	[[Image:TTestMatlab_T.jpg]]

Image:TTestMatlabGLM Data SS DF F.jpg

John — Fri, 20 Nov 2009 15:00:19 GMT

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Caret:Documentation:StatisticsGLM

John — Fri, 20 Nov 2009 14:47:41 GMT

←Older revision		Revision as of 14:47, 20 November 2009
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	[[Image:AnovaMatlabGLM_F.jpg]]		[[Image:AnovaMatlabGLM_F.jpg]]
		+
		+	= Two-Sample T-Test via GLM =
		+
		+	A two-sample T-Test is simply a one-way ANOVA with only two groups.
		+
		+	Data from [http://www.statsdirect.com/help/parametric_methods/utt.htm http://www.statsdirect.com/help/parametric_methods/utt.htm].
		+
		+	{\| border="1"
		+	!Hight Protein
		+	!Low Protein
		+	\|-
		+	\|134
		+	\|80
		+	\|-
		+	\|146
		+	\|118
		+	\|-
		+	\|104
		+	\|101
		+	\|-
		+	\|119
		+	\|85
		+	\|-
		+	\|124
		+	\|107
		+	\|-
		+	\|161
		+	\|132
		+	\|-
		+	\|107
		+	\|94
		+	\|-
		+	\|83
		+	\|
		+	\|-
		+	\|113
		+	\|
		+	\|-
		+	\|129
		+	\|
		+	\|-
		+	\|97
		+	\|
		+	\|-
		+	\|123
		+	\|
		+	\|}
		+
		+	[[Image:TTestMatlabGLM_DataY.jpg]] [[Image:TTestMatlabGLM_DataX.jpg]]
		+
		+
		+	[[Image:TTestMatlabGLM_Datab.jpg]]
		+
		+
		+	[[Image:TTestMatlabGLM_DataI_DataJ.jpg]]
		+
		+
		+	[[Image:TTestMatlabGLM_DataH.jpg]]
		+
		+
		+	[[Image:TTestMatlabGLM_Data_SS_DF_F.jpg]]

Image:AnovaMatlabGLM F.jpg

John — Thu, 19 Nov 2009 21:07:15 GMT

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John — Thu, 19 Nov 2009 21:06:57 GMT

Solving One-Way ANOVA Using GLM With Matrices in Matlab:

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	\|}		\|}

-	~~In the case of ANOVA, '''X''' is a~~ Design Matrix that indicates group membership of each subject. In the matrix '''X''', there is one row for each subject and ('''K + 1''') columns. The first column always contains ones. The remaining columns are used to indicate membership in a group with a one in column M indicating membership in group (M - 1) and zero indicating not a member of the group.	+	== Design Matrix ==

		+	In the case of ANOVA, '''X''' is a Design Matrix that indicates group membership of each subject. In the matrix '''X''', there is one row for each subject and ('''K + 1''') columns. The first column always contains ones. The remaining columns are used to indicate membership in a group with a value of one in column M indicating membership in group (M - 1) and a value of zero indicating not a member of the group.
		+
		+	The matrix '''Y''' contains the value from each subject, one per row.

	Data from Statistical Methods for Psychology, page 608.		Data from Statistical Methods for Psychology, page 608.
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	</math>		</math>

		+	== Computing Sums of Squares using General Linear Model ==

	<math>SS_{Total} = \mathbf{Y'} * (\mathbf{I} - \frac{1}{n} * \mathbf{J}) * \mathbf{Y}</math>		<math>SS_{Total} = \mathbf{Y'} * (\mathbf{I} - \frac{1}{n} * \mathbf{J}) * \mathbf{Y}</math>
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	<math>\mathbf{H}</math> is the Hat matrix equal to <math>\mathbf{X} * {(\mathbf{X'} * \mathbf{X})}^{-1} * \mathbf{X'} </math>		<math>\mathbf{H}</math> is the Hat matrix equal to <math>\mathbf{X} * {(\mathbf{X'} * \mathbf{X})}^{-1} * \mathbf{X'} </math>
		+
		+	== Solving One-Way ANOVA Using GLM With Matrices in Matlab ==

	[[Image:AnovaMatlabGLM_Data1.jpg]]		[[Image:AnovaMatlabGLM_Data1.jpg]]
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	[[Image:AnovaMatlabGLM_SS.jpg]]		[[Image:AnovaMatlabGLM_SS.jpg]]
		+
		+	[[Image:AnovaMatlabGLM_F.jpg]]

Image:AnovaMatlabGLM Data1.5.jpg

John — Thu, 19 Nov 2009 18:46:29 GMT

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Image:AnovaMatlabGLM SS.jpg

John — Thu, 19 Nov 2009 18:29:43 GMT

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John — Thu, 19 Nov 2009 18:26:46 GMT

One-Way ANOVA Via GLM:

←Older revision		Revision as of 18:26, 19 November 2009
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	<math>SS_{Treatment} = \mathbf{Y'} * (\mathbf{H} - \frac{1}{n} * \mathbf{J}) * \mathbf{Y} </math>		<math>SS_{Treatment} = \mathbf{Y'} * (\mathbf{H} - \frac{1}{n} * \mathbf{J}) * \mathbf{Y} </math>
		+
		+	<math>\mathbf{b} = {(\mathbf{X'} * \mathbf{X})}^{-1} * \mathbf{X'} * \mathbf{Y} </math>

	<math>\mathbf{I}</math> is an NxN identity matrix.		<math>\mathbf{I}</math> is an NxN identity matrix.
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	[[Image:AnovaMatlabGLM_Data1.jpg]]		[[Image:AnovaMatlabGLM_Data1.jpg]]
		+
		+	[[Image:AnovaMatlabGLM_Data1.5.jpg]]

	[[Image:AnovaMatlabGLM_Data2.jpg]]		[[Image:AnovaMatlabGLM_Data2.jpg]]

	[[Image:AnovaMatlabGLM_SS.jpg]]		[[Image:AnovaMatlabGLM_SS.jpg]]

Image:AnovaMatlabGLM Data2.jpg

John — Thu, 19 Nov 2009 18:23:55 GMT

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John — Thu, 19 Nov 2009 18:22:04 GMT

One-Way ANOVA Via GLM:

←Older revision		Revision as of 18:22, 19 November 2009
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	In the case of ANOVA, '''X''' is a Design Matrix that indicates group membership of each subject. In the matrix '''X''', there is one row for each subject and ('''K + 1''') columns. The first column always contains ones. The remaining columns are used to indicate membership in a group with a one in column M indicating membership in group (M - 1) and zero indicating not a member of the group.		In the case of ANOVA, '''X''' is a Design Matrix that indicates group membership of each subject. In the matrix '''X''', there is one row for each subject and ('''K + 1''') columns. The first column always contains ones. The remaining columns are used to indicate membership in a group with a one in column M indicating membership in group (M - 1) and zero indicating not a member of the group.
		+
		+
		+	Data from Statistical Methods for Psychology, page 608.
		+
		+	{\| border="1"
		+	!Treatment 1
		+	!Treatment 2
		+	!Treatment 3
		+	!Treatment 4
		+	\|-
		+	\|8
		+	\|5
		+	\|3
		+	\|6
		+	\|-
		+	\|9
		+	\|7
		+	\|4
		+	\|4
		+	\|-
		+	\|7
		+	\|3
		+	\|1
		+	\|9
		+	\|}
		+
		+
		+	<math>
		+	\mathbf{Y} =
		+	\begin{vmatrix}
		+	8 \\
		+	9 \\
		+	7 \\
		+	5 \\
		+	7 \\
		+	3 \\
		+	3 \\
		+	4 \\
		+	1 \\
		+	6 \\
		+	4 \\
		+	9
		+	\end{vmatrix}
		+	</math>
		+	<math>
		+	\mathbf{X} =
		+	\begin{vmatrix}
		+	1 & 1 & 0 & 0 & 0 \\
		+	1 & 1 & 0 & 0 & 0 \\
		+	1 & 1 & 0 & 0 & 0 \\
		+	1 & 0 & 1 & 0 & 0 \\
		+	1 & 0 & 1 & 0 & 0 \\
		+	1 & 0 & 1 & 0 & 0 \\
		+	1 & 0 & 0 & 1 & 0 \\
		+	1 & 0 & 0 & 1 & 0 \\
		+	1 & 0 & 0 & 1 & 0 \\
		+	1 & 0 & 0 & 0 & 1 \\
		+	1 & 0 & 0 & 0 & 1 \\
		+	1 & 0 & 0 & 0 & 1
		+	\end{vmatrix}
		+	</math>
		+
		+
		+	Recall from the General Linear Model that
		+	<math>
		+	\mathbf{b} = {(\mathbf{X'} * \mathbf{X})}^{-1} * \mathbf{X'} * \mathbf{Y}
		+	</math>
		+
		+	However <math>(\mathbf{X'} * \mathbf{X})</math> is singular so an inverse cannot be computed. To resolve this problem, the design matrix '''X''' is modified in two ways. First, the last column, indicating membership in the last group, is removed. Recall that a value of one in a column indicates membership in a group and a value of zero in a column indicates NOT membership in a group. The second modification is to use a value of negative one in a column to indicate membership in the last group (the column that was removed).
		+
		+	<math>
		+	\mathbf{X} =
		+	\begin{vmatrix}
		+	1 & 1 & 0 & 0 \\
		+	1 & 1 & 0 & 0 \\
		+	1 & 1 & 0 & 0 \\
		+	1 & 0 & 1 & 0 \\
		+	1 & 0 & 1 & 0 \\
		+	1 & 0 & 1 & 0 \\
		+	1 & 0 & 0 & 1 \\
		+	1 & 0 & 0 & 1 \\
		+	1 & 0 & 0 & 1 \\
		+	1 & -1 & -1 & -1 \\
		+	1 & -1 & -1 & -1 \\
		+	1 & -1 & -1 & -1
		+	\end{vmatrix}
		+	</math>
		+
		+
		+	<math>SS_{Total} = \mathbf{Y'} * (\mathbf{I} - \frac{1}{n} * \mathbf{J}) * \mathbf{Y}</math>
		+
		+	<math>SS_{Error} = \mathbf{Y'} * \mathbf{Y} - \mathbf{b'} * \mathbf{X'} * \mathbf{Y} </math>
		+
		+	<math>SS_{Treatment} = \mathbf{Y'} * (\mathbf{H} - \frac{1}{n} * \mathbf{J}) * \mathbf{Y} </math>
		+
		+	<math>\mathbf{I}</math> is an NxN identity matrix.
		+
		+	<math>\mathbf{J}</math> is an NxN matrix with all elements having the value 1.
		+
		+	<math>\mathbf{H}</math> is the Hat matrix equal to <math>\mathbf{X} * {(\mathbf{X'} * \mathbf{X})}^{-1} * \mathbf{X'} </math>
		+
		+	[[Image:AnovaMatlabGLM_Data1.jpg]]
		+
		+	[[Image:AnovaMatlabGLM_Data2.jpg]]
		+
		+	[[Image:AnovaMatlabGLM_SS.jpg]]

Image:Anova SS Result Matlab.jpg

John — Thu, 19 Nov 2009 16:36:07 GMT

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Image:Anova SS Data Matlab.jpg

John — Thu, 19 Nov 2009 16:34:15 GMT

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John — Thu, 19 Nov 2009 16:32:04 GMT

←Older revision		Revision as of 16:32, 19 November 2009
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	Y = 10.0 + 2.0X		Y = 10.0 + 2.0X
		+
		+	== Matlab Solution ==

	[[Image:Regression_LS_Matlab.jpg]]		[[Image:Regression_LS_Matlab.jpg]]
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-	=== GLM ===	+	== GLM ==

	<math>		<math>
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-	=== Least Squares Estimation ===	+	== Least Squares Estimation ==

	<math>		<math>
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-	=== Least Squares Estimation Using Example Data ===	+	== Least Squares Estimation Using Example Data ==
	<math>		<math>
	\mathbf{X'} =		\mathbf{X'} =
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-	=== Result ===	+	== Result ==
	<math>		<math>
	\mathbf{B} =		\mathbf{B} =
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	</math>		</math>

-	=== Matlab Solution ===	+	== Matlab Solution ==

	[[Image:Regression_GLM_Matlab.jpg]]		[[Image:Regression_GLM_Matlab.jpg]]
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	== Sum Of Squares Example ==		== Sum Of Squares Example ==

-	Data from Statistical Methods for Psychology, page ~~327~~.	+	Data from Statistical Methods for Psychology, page 608.

	{\| border="1"		{\| border="1"
-	!~~Counting~~	+	!Treatment 1
-	!~~Rhyming~~	+	!Treatment 2
-	!~~Adjective~~	+	!Treatment 3
-	!~~Imagery~~	+	!Treatment 4
-	~~!Intentional~~	+
-	\|-	+
-	\|9	+
-	\|7	+
-	~~\|11~~	+
-	~~\|12~~	+
-	~~\|10~~	+
	\|-		\|-
	\|8		\|8
-	\|9	+	\|5
-	\|13	+	\|3
-	\|11	+	\|6
-	~~\|19~~	+
	\|-		\|-
-	\|6
-	\|6
-	\|8
-	~~\|16~~
-	~~\|14~~
-	\|-
-	\|8
-	\|6
-	\|6
-	~~\|11~~
-	\|5
-	\|-
-	~~\|10~~
-	\|6
-	~~\|14~~
	\|9		\|9
-	\|10	+	\|7
-	\|-	+	\|4
	\|4		\|4
-	~~\|11~~
-	~~\|11~~
-	~~\|23~~
-	~~\|11~~
	\|-		\|-
-	\|6	+	\|7
-	\|6	+
-	~~\|13~~	+
-	~~\|12~~	+
-	~~\|14~~	+
-	\|-	+
-	\|5	+
	\|3		\|3
-	\|13	+	\|1
-	\|10	+	\|9
-	~~\|15~~	+
-	\|-	+
-	\|7	+
-	\|8	+
-	~~\|10~~	+
-	~~\|19~~	+
-	~~\|11~~	+
-	\|-	+
-	\|7	+
-	\|7	+
-	~~\|11~~	+
-	~~\|11~~	+
-	~~\|11~~	+
	\|}		\|}

-	<math>SS_{Total} = ~~786~~.82</math>	+	<math>SS_{Total} = 73.00</math>

-	<math>SS_{Error} = ~~435~~.3</math>	+	<math>SS_{Error} = 27.333</math>

-	<math>SS_{Treatment} = ~~351~~.52</math>	+	<math>SS_{Treatment} = 45.667</math>

-	<math>df_{Total} = 49</math>	+	<math>df_{Total} = 11</math>

-	<math>df_{Error} = 45</math>	+	<math>df_{Error} = 8</math>

-	<math>df_{Treatment} = 4</math>	+	<math>df_{Treatment} = 3</math>

-	<math> MS_{Treatment} = \frac{SS_{Treatment}} {df_{Treatment}} = \frac{~~351~~.52}{4} = 87.88</math>	+	<math> MS_{Treatment} = \frac{SS_{Treatment}} {df_{Treatment}} = \frac{45.667}{3} = 15.222</math>

-	<math> MS_{Error} = \frac{SS_{Error}} {df_{Error}} = \frac{~~435~~.3}{45} = 9.~~6733~~</math>	+	<math> MS_{Error} = \frac{SS_{Error}} {df_{Error}} = \frac{27.333}{8} = 3.417</math>

-	<math> F = \frac{MS_{Treatment}} {MS_{Error}} = \frac{87.88}{9.~~6733~~} = 9.08</math>	+	<math> F = \frac{MS_{Treatment}} {MS_{Error}} = \frac{15.222}{3.417} = 4.46</math>

	== Sum of Squares Example in Matlab ==		== Sum of Squares Example in Matlab ==

Image:Regression GLM Matlab.jpg

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John — Wed, 18 Nov 2009 22:36:43 GMT

←Older revision		Revision as of 22:36, 18 November 2009
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	\end{vmatrix}		\end{vmatrix}
	</math>		</math>
		+
		+	=== Matlab Solution ===
		+
		+	[[Image:Regression_GLM_Matlab.jpg]]

	= One-Way ANOVA Via Sums Of Squares =		= One-Way ANOVA Via Sums Of Squares =

Image:Regression LS Matlab.jpg

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John — Wed, 18 Nov 2009 22:33:09 GMT

←Older revision		Revision as of 22:33, 18 November 2009
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	Y = 10.0 + 2.0X		Y = 10.0 + 2.0X
		+
		+	[[Image:Regression_LS_Matlab.jpg]]

	= Matrices =		= Matrices =

Image:Anova SS Result Matlab.jpg

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John — Wed, 18 Nov 2009 22:26:14 GMT

←Older revision		Revision as of 22:26, 18 November 2009
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	= General Linear Model (GLM) =		= General Linear Model (GLM) =
		+
		+	The equation of the General Linear Model is '''Y = Xb + e''' where '''Y''', '''X''', '''b''', and '''e''' are matrices. Statistical test such as Analysis of Variance and Regression are special cases of the General Linear Model. These statistical tests can be formulated and solved using the General Linear Model.

	= Regression Via Least Squares =		= Regression Via Least Squares =
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	\|10		\|10
	\|-		\|-
-	\|8\|9\|13\|11\|19	+	\|8
		+	\|9
		+	\|13
		+	\|11
		+	\|19
	\|-		\|-
-	\|6\|6\|8\|16\|14	+	\|6
		+	\|6
		+	\|8
		+	\|16
		+	\|14
	\|-		\|-
-	\|8\|6\|6\|11\|5	+	\|8
		+	\|6
		+	\|6
		+	\|11
		+	\|5
	\|-		\|-
-	\|10\|6\|14\|9\|6	+	\|10
		+	\|6
		+	\|14
		+	\|9
		+	\|10
	\|-		\|-
-	\|4\|11\|11\|23\|11	+	\|4
		+	\|11
		+	\|11
		+	\|23
		+	\|11
	\|-		\|-
-	\|6\|6\|13\|12\|14	+	\|6
		+	\|6
		+	\|13
		+	\|12
		+	\|14
	\|-		\|-
-	\|5\|3\|13\|10\|15	+	\|5
		+	\|3
		+	\|13
		+	\|10
		+	\|15
	\|-		\|-
-	\|7\|8\|10\|19\|11	+	\|7
		+	\|8
		+	\|10
		+	\|19
		+	\|11
	\|-		\|-
-	\|7\|7\|11\|11\|11	+	\|7
		+	\|7
		+	\|11
		+	\|11
		+	\|11
	\|}		\|}
		+
		+	<math>SS_{Total} = 786.82</math>
		+
		+	<math>SS_{Error} = 435.3</math>
		+
		+	<math>SS_{Treatment} = 351.52</math>
		+
		+	<math>df_{Total} = 49</math>
		+
		+	<math>df_{Error} = 45</math>
		+
		+	<math>df_{Treatment} = 4</math>
		+
		+	<math> MS_{Treatment} = \frac{SS_{Treatment}} {df_{Treatment}} = \frac{351.52}{4} = 87.88</math>
		+
		+	<math> MS_{Error} = \frac{SS_{Error}} {df_{Error}} = \frac{435.3}{45} = 9.6733</math>
		+
		+	<math> F = \frac{MS_{Treatment}} {MS_{Error}} = \frac{87.88}{9.6733} = 9.08</math>
		+
		+	== Sum of Squares Example in Matlab ==
		+
		+	[[Image:Anova_SS_Data_Matlab.jpg]]
		+
		+	[[Image:Anova_SS_Result_Matlab.jpg]]

	= One-Way ANOVA Via GLM =		= One-Way ANOVA Via GLM =
		+
		+	Recall that the equation of the General Linear Model is '''Y = Xb + e''' where '''Y''', '''X''', '''b''', and '''e''' are matrices.
		+
		+	In the case of a One-Way ANOVA with '''N''' total subjects from '''K''' groups:
		+
		+	{\| border="1"
		+	!Matrix
		+	!Dimensions
		+	\|-
		+	\|Y
		+	\| N x 1
		+	\|-
		+	\|X
		+	\| N x (K + 1)
		+	\|-
		+	\|b
		+	\| (K + 1) x 1
		+	\|-
		+	\|e
		+	\| N x 1
		+	\|-
		+	\|}
		+
		+	In the case of ANOVA, '''X''' is a Design Matrix that indicates group membership of each subject. In the matrix '''X''', there is one row for each subject and ('''K + 1''') columns. The first column always contains ones. The remaining columns are used to indicate membership in a group with a one in column M indicating membership in group (M - 1) and zero indicating not a member of the group.