SAS Output

T-square Examples

Sweat Data: Table 5.1 Johnson & Wichern

The CORR Procedure

3 Variables:	y1 y2 y3

Simple Statistics
Variable	N	Mean	Std Dev	Sum	Minimum	Maximum
y1	20	0.64000	1.69687	12.80000	-2.50000	4.50000
y2	20	-4.60000	14.13465	-92.00000	-36.50000	21.60000
y3	20	-0.03500	1.90464	-0.70000	-2.90000	4.00000

y1

1.00000

0.41735

0.0671

-0.55974

0.0103

y2

0.41735

0.0671

1.00000

-0.20950

0.3754

y3

-0.55974

0.0103

-0.20950

0.3754

1.00000

T-square Examples

Sweat Data: Table 5.1 Johnson & Wichern

The GLM Procedure

Multivariate Analysis of Variance

E = Error SSCP Matrix
	y1	y2	y3
y1	54.708	190.19	-34.372
y2	190.19	3795.98	-107.16
y3	-34.372	-107.16	68.9255

y1

1.000000

0.417350

0.0671

-0.559744

0.0103

y2

0.417350

0.0671

1.000000

-0.209498

0.3754

y3

-0.559744

0.0103

-0.209498

0.3754

1.000000

T-square Examples

Sweat Data: Table 5.1 Johnson & Wichern

The GLM Procedure

Multivariate Analysis of Variance

H = Type III SSCP Matrix for Intercept
	y1	y2	y3
y1	8.192	-58.88	-0.448
y2	-58.88	423.2	3.22
y3	-0.448	3.22	0.0245

Canonical Analysis H = Type III SSCP Matrix for Intercept E = Error SSCP Matrix
	Canonical Correlation	Adjusted Canonical Correlation	Approximate Standard Error	Squared Canonical Correlation	Eigenvalues of Inv(E)*H = CanRsq/(1-CanRsq)				Test of H0: The canonical correlations in the current row and all that follow are zero
	Canonical Correlation	Adjusted Canonical Correlation	Approximate Standard Error	Squared Canonical Correlation	Eigenvalue	Difference	Proportion	Cumulative	Likelihood Ratio	Approximate F Value	Num DF	Den DF	Pr > F
1	0.582127	0.534963	0.147833	0.338872	0.5126		1.0000	1.0000	0.66112774	2.90	3	17	0.0649

The F statistic is exact.

Canonical Structure
	Total	Between	Within
	Can1	Can1	Can1
y1	0.5405	1.0000	0.5405
y2	-0.4664	-1.0000	-0.4664
y3	-0.0263	-1.0000	-0.0263

Canonical Coefficients
	Standardized	Raw
	Can1	Can1
y1	1.13732169	0.67024673
y2	-0.85051345	-0.06017222
y3	0.43209366	0.22686356

MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall Intercept Effect H = Type III SSCP Matrix for Intercept E = Error SSCP Matrix S=1 M=0.5 N=7.5
Statistic	Value	F Value	Num DF	Den DF	Pr > F
Wilks' Lambda	0.66112774	2.90	3	17	0.0649
Pillai's Trace	0.33887226	2.90	3	17	0.0649
Hotelling-Lawley Trace	0.51256699	2.90	3	17	0.0649
Roy's Greatest Root	0.51256699	2.90	3	17	0.0649

T-square Examples

Obs	_NAME_	_SOURCE_	_TYPE_	y1	y2	y3	DF	SS	F	PROB
1	y1	ERROR	ERROR	54.708	190.19	-34.372	19	54.71	.	.
2	y2	ERROR	ERROR	190.190	3795.98	-107.160	19	3795.98	.	.
3	y3	ERROR	ERROR	-34.372	-107.16	68.926	19	68.93	.	.
4	y1	Intercept	SS1	8.192	-58.88	-0.448	1	8.19	2.84507	0.10801
5	y2	Intercept	SS1	-58.880	423.20	3.220	1	423.20	2.11824	0.16188
6	y3	Intercept	SS1	-0.448	3.22	0.025	1	0.02	0.00675	0.93536
7	y1	Intercept	SS3	8.192	-58.88	-0.448	1	8.19	2.84507	0.10801
8	y2	Intercept	SS3	-58.880	423.20	3.220	1	423.20	2.11824	0.16188
9	y3	Intercept	SS3	-0.448	3.22	0.024	1	0.02	0.00675	0.93536
10		Intercept	CANCORR	0.582	0.00	0.000	.	.	.	.
11	CAN1	Intercept	STRUCTUR	0.540	-0.47	-0.026	.	.	.	.
12	CAN1	Intercept	SCORE	0.670	-0.06	0.227	.	.	.	.

T-square Examples

Table 6.1 Effluent Data -- Paired Comparisons

Analysis on Differences

The GLM Procedure

Multivariate Analysis of Variance

E = Error SSCP Matrix
	d1	d2
d1	1992.5454545	883.09090909
d2	883.09090909	4186.1818182

d1

1.000000

0.305768

0.3605

d2

0.305768

0.3605

1.000000

T-square Examples

Table 6.1 Effluent Data -- Paired Comparisons

Analysis on Differences

The GLM Procedure

Multivariate Analysis of Variance

H = Type III SSCP Matrix for Intercept
	d1	d2
d1	964.45454545	-1367.090909
d2	-1367.090909	1937.8181818

Canonical Analysis H = Type III SSCP Matrix for Intercept E = Error SSCP Matrix
	Canonical Correlation	Adjusted Canonical Correlation	Approximate Standard Error	Squared Canonical Correlation	Eigenvalues of Inv(E)*H = CanRsq/(1-CanRsq)				Test of H0: The canonical correlations in the current row and all that follow are zero
	Canonical Correlation	Adjusted Canonical Correlation	Approximate Standard Error	Squared Canonical Correlation	Eigenvalue	Difference	Proportion	Cumulative	Likelihood Ratio	Approximate F Value	Num DF	Den DF	Pr > F
1	0.759589	0.748881	0.127547	0.576976	1.3639		1.0000	1.0000	0.42302415	6.14	2	9	0.0208

The F statistic is exact.

Canonical Structure
	Total	Between	Within
	Can1	Can1	Can1
d1	0.5957	1.0000	0.5957
d2	-0.5826	-1.0000	-0.5826

Canonical Coefficients
	Standardized	Raw
	Can1	Can1
d1	0.85366286	0.06047589
d2	-0.84359710	-0.04123121

MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall Intercept Effect H = Type III SSCP Matrix for Intercept E = Error SSCP Matrix S=1 M=0 N=3.5
Statistic	Value	F Value	Num DF	Den DF	Pr > F
Wilks' Lambda	0.42302415	6.14	2	9	0.0208
Pillai's Trace	0.57697585	6.14	2	9	0.0208
Hotelling-Lawley Trace	1.36393121	6.14	2	9	0.0208
Roy's Greatest Root	1.36393121	6.14	2	9	0.0208

T-square Examples

Table 6.1 Effluent Data -- Paired Comparisons

Analysis Using a Linear Transformation

The GLM Procedure

Multivariate Analysis of Variance

M Matrix Describing Transformed Variables
	x11	x12	x21	x22
BOD_DIFF	1	0	-1	0
SS_DIFF	0	1	0	-1

E = Error SSCP Matrix
	BOD_DIFF	SS_DIFF
BOD_DIFF	1992.5454545	883.09090909
SS_DIFF	883.09090909	4186.1818182

BOD_DIFF

1.000000

0.305768

0.3605

SS_DIFF

0.305768

0.3605

1.000000

T-square Examples

Table 6.1 Effluent Data -- Paired Comparisons

Analysis Using a Linear Transformation

The GLM Procedure

Multivariate Analysis of Variance

H = Type III SSCP Matrix for Intercept
	BOD_DIFF	SS_DIFF
BOD_DIFF	964.45454545	-1367.090909
SS_DIFF	-1367.090909	1937.8181818

Canonical Analysis H = Type III SSCP Matrix for Intercept E = Error SSCP Matrix Variables have been transformed by the M Matrix
	Canonical Correlation	Adjusted Canonical Correlation	Approximate Standard Error	Squared Canonical Correlation	Eigenvalues of Inv(E)*H = CanRsq/(1-CanRsq)				Test of H0: The canonical correlations in the current row and all that follow are zero
	Canonical Correlation	Adjusted Canonical Correlation	Approximate Standard Error	Squared Canonical Correlation	Eigenvalue	Difference	Proportion	Cumulative	Likelihood Ratio	Approximate F Value	Num DF	Den DF	Pr > F
1	0.759589	0.748881	0.127547	0.576976	1.3639		1.0000	1.0000	0.42302415	6.14	2	9	0.0208

The F statistic is exact.

Canonical Structure
	Total	Between	Within
	Can1	Can1	Can1
BOD_DIFF	0.5957	1.0000	0.5957
SS_DIFF	-0.5826	-1.0000	-0.5826

Canonical Coefficients
	Standardized	Raw
	Can1	Can1
BOD_DIFF	0.85366286	0.06047589
SS_DIFF	-0.84359710	-0.04123121

MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall Intercept Effect on the Variables Defined by the M Matrix Transformation H = Type III SSCP Matrix for Intercept E = Error SSCP Matrix S=1 M=0 N=3.5
Statistic	Value	F Value	Num DF	Den DF	Pr > F
Wilks' Lambda	0.42302415	6.14	2	9	0.0208
Pillai's Trace	0.57697585	6.14	2	9	0.0208
Hotelling-Lawley Trace	1.36393121	6.14	2	9	0.0208
Roy's Greatest Root	1.36393121	6.14	2	9	0.0208

T-square Examples

Table 6.1 Effluent Data -- Paired Comparisons

Analysis as a Randomized Block Design

The GLM Procedure

Class Level Information
Class	Levels	Values
Lab	2	Commercial State
Sample	11	1 2 3 4 5 6 7 8 9 10 11

T-square Examples

Table 6.1 Effluent Data -- Paired Comparisons

Analysis as a Randomized Block Design

The GLM Procedure

Multivariate Analysis of Variance

E = Error SSCP Matrix
	BOD	SS
BOD	996.27272727	441.54545455
SS	441.54545455	2093.0909091

BOD

1.000000

0.305768

0.3605

SS

0.305768

0.3605

1.000000

T-square Examples

Table 6.1 Effluent Data -- Paired Comparisons

Analysis as a Randomized Block Design

The GLM Procedure

Multivariate Analysis of Variance

H = Type III SSCP Matrix for Lab
	BOD	SS
BOD	482.22727273	-683.5454545
SS	-683.5454545	968.90909091

Canonical Analysis H = Type III SSCP Matrix for Lab E = Error SSCP Matrix
	Canonical Correlation	Adjusted Canonical Correlation	Approximate Standard Error	Squared Canonical Correlation	Eigenvalues of Inv(E)*H = CanRsq/(1-CanRsq)				Test of H0: The canonical correlations in the current row and all that follow are zero
	Canonical Correlation	Adjusted Canonical Correlation	Approximate Standard Error	Squared Canonical Correlation	Eigenvalue	Difference	Proportion	Cumulative	Likelihood Ratio	Approximate F Value	Num DF	Den DF	Pr > F
1	0.759589	0.748881	0.127547	0.576976	1.3639		1.0000	1.0000	0.42302415	6.14	2	9	0.0208

The F statistic is exact.

Canonical Structure
	Total	Between	Within
	Can1	Can1	Can1
BOD	0.3395	1.0000	0.5957
SS	-0.5447	-1.0000	-0.5826

Canonical Coefficients
	Standardized	Raw
	Can1	Can1
BOD	1.37766686	0.08552582
SS	-1.54520905	-0.05830973

MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall Lab Effect H = Type III SSCP Matrix for Lab E = Error SSCP Matrix S=1 M=0 N=3.5
Statistic	Value	F Value	Num DF	Den DF	Pr > F
Wilks' Lambda	0.42302415	6.14	2	9	0.0208
Pillai's Trace	0.57697585	6.14	2	9	0.0208
Hotelling-Lawley Trace	1.36393121	6.14	2	9	0.0208
Roy's Greatest Root	1.36393121	6.14	2	9	0.0208

T-square Examples

Tables 5.11 and 6.9 -- Hook-billed Kites

The GLM Procedure

Class Level Information
Class	Levels	Values
Sex	2	Female Male

T-square Examples

Tables 5.11 and 6.9 -- Hook-billed Kites

The GLM Procedure

Dependent Variable: x1 Tail Length

Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	1	960.40000	960.40000	9.27	0.0031
Error	88	9120.22222	103.63889
Corrected Total	89	10080.62222

R-Square	Coeff Var	Root MSE	x1 Mean
0.095272	5.348054	10.18032	190.3556

Source	DF	Type I SS	Mean Square	F Value	Pr > F
Sex	1	960.4000000	960.4000000	9.27	0.0031

Source	DF	Type III SS	Mean Square	F Value	Pr > F
Sex	1	960.4000000	960.4000000	9.27	0.0031

T-square Examples

Tables 5.11 and 6.9 -- Hook-billed Kites

The GLM Procedure

Dependent Variable: x2 Wing Length

Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	1	26.67778	26.67778	0.14	0.7080
Error	88	16622.97778	188.89747
Corrected Total	89	16649.65556

R-Square	Coeff Var	Root MSE	x2 Mean
0.001602	4.902928	13.74400	280.3222

Source	DF	Type I SS	Mean Square	F Value	Pr > F
Sex	1	26.67777778	26.67777778	0.14	0.7080

Source	DF	Type III SS	Mean Square	F Value	Pr > F
Sex	1	26.67777778	26.67777778	0.14	0.7080

T-square Examples

Tables 5.11 and 6.9 -- Hook-billed Kites

The GLM Procedure

Multivariate Analysis of Variance

E = Error SSCP Matrix
	x1	x2
x1	9120.2222222	9265.7555556
x2	9265.7555556	16622.977778

x1

1.000000

0.752530

<.0001

x2

0.752530

<.0001

1.000000

T-square Examples

Tables 5.11 and 6.9 -- Hook-billed Kites

The GLM Procedure

Multivariate Analysis of Variance

H = Type III SSCP Matrix for Sex
	x1	x2
x1	960.4	-160.0666667
x2	-160.0666667	26.677777778

Canonical Analysis H = Type III SSCP Matrix for Sex E = Error SSCP Matrix
	Canonical Correlation	Adjusted Canonical Correlation	Approximate Standard Error	Squared Canonical Correlation	Eigenvalues of Inv(E)*H = CanRsq/(1-CanRsq)				Test of H0: The canonical correlations in the current row and all that follow are zero
	Canonical Correlation	Adjusted Canonical Correlation	Approximate Standard Error	Squared Canonical Correlation	Eigenvalue	Difference	Proportion	Cumulative	Likelihood Ratio	Approximate F Value	Num DF	Den DF	Pr > F
1	0.475161	0.468074	0.082067	0.225778	0.2916		1.0000	1.0000	0.77422172	12.69	2	87	<.0001

The F statistic is exact.

Canonical Structure
	Total	Between	Within
	Can1	Can1	Can1
x1	-0.6496	-1.0000	-0.6009
x2	0.0842	1.0000	0.0742

Canonical Coefficients
	Standardized	Raw
	Can1	Can1
x1	-1.58305069	-0.14874629
x2	1.20785879	0.08830973

MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall Sex Effect H = Type III SSCP Matrix for Sex E = Error SSCP Matrix S=1 M=0 N=42.5
Statistic	Value	F Value	Num DF	Den DF	Pr > F
Wilks' Lambda	0.77422172	12.69	2	87	<.0001
Pillai's Trace	0.22577828	12.69	2	87	<.0001
Hotelling-Lawley Trace	0.29161967	12.69	2	87	<.0001
Roy's Greatest Root	0.29161967	12.69	2	87	<.0001

T-square Examples

Tables 5.11 and 6.9 -- Hook-billed Kites

The GLM Procedure

Multivariate Analysis of Variance

Dependent Variable: x1 Tail Length

Source	DF	Type III SS	Mean Square	F Value	Pr > F
Sex	1	960.400000	960.400000	9.27	0.0031
Error	88	9120.222222	103.638889

T-square Examples

Tables 5.11 and 6.9 -- Hook-billed Kites

The GLM Procedure

Multivariate Analysis of Variance

Dependent Variable: x2 Wing Length

Source	DF	Type III SS	Mean Square	F Value	Pr > F
Sex	1	26.67778	26.67778	0.14	0.7080
Error	88	16622.97778	188.89747