Two-factor Factorial MANOVA

Obs	560nm	720nm Format Species Spec. Time Tim.	Amount of Species	Rate of Time
1	12.07	33.03	JL	1
2	11.03	32.37	JL	1
3	12.48	31.31	JL	1
4	12.12	33.33	JL	1
5	15.38	40.00	JL	2
6	14.21	40.48	JL	2
7	9.69	33.90	JL	2
8	14.35	40.15	JL	2
9	38.71	77.14	JL	3
10	44.74	78.57	JL	3
11	36.67	71.43	JL	3
12	37.21	45.00	JL	3
13	8.73	23.27	LP	1
14	7.94	20.87	LP	1
15	8.37	22.16	LP	1
16	7.86	21.78	LP	1
17	8.45	26.32	LP	2
18	6.79	22.73	LP	2
19	8.34	26.67	LP	2
20	7.54	24.87	LP	2
21	14.04	44.44	LP	3
22	13.51	37.93	LP	3
23	13.33	37.93	LP	3
24	12.77	60.87	LP	3
25	9.33	19.14	SS	1
26	8.74	19.55	SS	1
27	9.31	19.24	SS	1
28	8.27	16.37	SS	1
29	10.22	25.00	SS	2
30	10.13	25.32	SS	2
31	10.42	27.12	SS	2
32	10.62	26.28	SS	2
33	15.25	38.89	SS	3
34	16.22	36.67	SS	3
35	17.24	40.74	SS	3
36	12.77	67.50	SS	3

2 Variables:	Y1 Y2

Simple Statistics
Y1	36	14.30139	9.43079	514.85000	6.79000	44.74000	560nm
Y2	36	35.78806	16.53036	1288	16.37000	78.57000	720nm Format Species Spec. Time Tim.

Pearson Correlation Coefficients, N = 36 Prob > \|r\| under H0: Rho=0
Y1 560nm	1.00000	0.81308 <.0001
Y2 720nm Format Species Spec. Time Tim.	0.81308 <.0001	1.00000

Spectral Reflection Data (Page 355, Johnson and Wichern)

Multivariate Analysis of Variance

The GLM Procedure

Class Level Information
Class	Levels	Values
Species	3	JL LP SS
Time	3	1 2 3

Number of Observations Read	36
Number of Observations Used	36

Spectral Reflection Data (Page 355, Johnson and Wichern)

Multivariate Analysis of Variance

The GLM Procedure

Multivariate Analysis of Variance

E = Error SSCP Matrix
	Y1	Y2
Y1	76.658775	37.9299
Y2	37.9299	1769.642225

Partial Correlation Coefficients from the Error SSCP Matrix / Prob > \|r\|
DF = 27	Y1	Y2
Y1	1.000000	0.102981 0.6020
Y2	0.102981 0.6020	1.000000

Spectral Reflection Data (Page 355, Johnson and Wichern)

Multivariate Analysis of Variance

The GLM Procedure

Multivariate Analysis of Variance

H = Type III SSCP Matrix for Species*Time
	Y1	Y2
Y1	795.80794444	375.96311944
Y2	375.96311944	193.54926111

Characteristic Roots and Vectors of: E Inverse * H, where H = Type III SSCP Matrix for Species*Time E = Error SSCP Matrix
Characteristic Root	Percent	Characteristic Vector V'EV=1
Characteristic Root	Percent	Y1	Y2
10.3813996	99.91	0.11426805	-0.00011191
0.0090999	0.09	-0.01128981	0.02389834

MANOVA Test Criteria and F Approximations for the Hypothesis of No Overall SpeciesTime Effect H = Type III SSCP Matrix for SpeciesTime E = Error SSCP Matrix S=2 M=0.5 N=12
Statistic	Value	F Value	Num DF	Den DF	Pr > F
Wilks' Lambda	0.08707032	15.53	8	52	<.0001
Pillai's Trace	0.92115522	5.76	8	54	<.0001
Hotelling-Lawley Trace	10.39049958	33.07	8	34.899	<.0001
Roy's Greatest Root	10.38139964	70.07	4	27	<.0001
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
NOTE: F Statistic for Wilks' Lambda is exact.

H = Type III SSCP Matrix for Time
	Y1	Y2
Y1	1275.2477389	2644.9273639
Y2	2644.9273639	5573.8057056

Characteristic Roots and Vectors of: E Inverse * H, where H = Type III SSCP Matrix for Time E = Error SSCP Matrix
Characteristic Root	Percent	Characteristic Vector V'EV=1
Characteristic Root	Percent	Y1	Y2
18.4568938	99.75	0.10469772	0.00751699
0.0453490	0.25	-0.04714907	0.02268563

MANOVA Test Criteria and F Approximations for the Hypothesis of No Overall Time Effect H = Type III SSCP Matrix for Time E = Error SSCP Matrix S=2 M=-0.5 N=12
Statistic	Value	F Value	Num DF	Den DF	Pr > F
Wilks' Lambda	0.04916603	45.63	4	52	<.0001
Pillai's Trace	0.99198604	13.29	4	54	<.0001
Hotelling-Lawley Trace	18.50224287	118.89	4	30.19	<.0001
Roy's Greatest Root	18.45689384	249.17	2	27	<.0001
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
NOTE: F Statistic for Wilks' Lambda is exact.

H = Type III SSCP Matrix for Species
	Y1	Y2
Y1	965.18117222	1377.6019139
Y2	1377.6019139	2026.8563722

Characteristic Roots and Vectors of: E Inverse * H, where H = Type III SSCP Matrix for Species E = Error SSCP Matrix
Characteristic Root	Percent	Characteristic Vector V'EV=1
Characteristic Root	Percent	Y1	Y2
13.0712473	99.75	0.10981860	0.00458854
0.0333424	0.25	-0.03353388	0.02345396

MANOVA Test Criteria and F Approximations for the Hypothesis of No Overall Species Effect H = Type III SSCP Matrix for Species E = Error SSCP Matrix S=2 M=-0.5 N=12
Statistic	Value	F Value	Num DF	Den DF	Pr > F
Wilks' Lambda	0.06877382	36.57	4	52	<.0001
Pillai's Trace	0.96119962	12.49	4	54	<.0001
Hotelling-Lawley Trace	13.10458966	84.21	4	30.19	<.0001
Roy's Greatest Root	13.07124729	176.46	2	27	<.0001
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
NOTE: F Statistic for Wilks' Lambda is exact.

Spectral Reflection Data (Page 355, Johnson and Wichern)

Multivariate Analysis of Variance

The GLM Procedure

Least Squares Means

Adjustment for Multiple Comparisons: Bonferroni

Species	Y1 LSMEAN	LSMEAN Number
JL	21.5550000	1
LP	9.8058333	2
SS	11.5433333	3

Least Squares Means for effect Species Pr > \|t\| for H0: LSMean(i)=LSMean(j) Dependent Variable: Y1
i/j	1	2	3
1		<.0001	<.0001
2	<.0001		0.0532
3	<.0001	0.0532

Species	Y1 LSMEAN	97.5% Confidence Limits
JL	21.555000	20.400530	22.709470
LP	9.805833	8.651364	10.960303
SS	11.543333	10.388864	12.697803

Least Squares Means for Effect Species
i	j	Difference Between Means	Simultaneous 97.5% Confidence Limits for LSMean(i)-LSMean(j)
1	2	11.749167	9.790794	13.707539
1	3	10.011667	8.053294	11.970039
2	3	-1.737500	-3.695873	0.220873

Species	Y2 LSMEAN	LSMEAN Number
JL	46.3925000	1
LP	30.8200000	2
SS	30.1516667	3

Least Squares Means for effect Species Pr > \|t\| for H0: LSMean(i)=LSMean(j) Dependent Variable: Y2
i/j	1	2	3
1		0.0002	0.0001
2	0.0002		1.0000
3	0.0001	1.0000

Species	Y2 LSMEAN	97.5% Confidence Limits
JL	46.392500	40.845677	51.939323
LP	30.820000	25.273177	36.366823
SS	30.151667	24.604844	35.698490

Least Squares Means for Effect Species
i	j	Difference Between Means	Simultaneous 97.5% Confidence Limits for LSMean(i)-LSMean(j)
1	2	15.572500	6.163204	24.981796
1	3	16.240833	6.831537	25.650130
2	3	0.668333	-8.740963	10.077630

Spectral Reflection Data (Page 355, Johnson and Wichern)

Multivariate Analysis of Variance

The GLM Procedure

Least Squares Means

Adjustment for Multiple Comparisons: Bonferroni

Time	Y1 LSMEAN	LSMEAN Number
1	9.6875000	1
2	10.5116667	2
3	22.7050000	3

Least Squares Means for effect Time Pr > \|t\| for H0: LSMean(i)=LSMean(j) Dependent Variable: Y1
i/j	1	2	3
1		0.7239	<.0001
2	0.7239		<.0001
3	<.0001	<.0001

Time	Y1 LSMEAN	97.5% Confidence Limits
1	9.687500	8.533030	10.841970
2	10.511667	9.357197	11.666136
3	22.705000	21.550530	23.859470

Least Squares Means for Effect Time
i	j	Difference Between Means	Simultaneous 97.5% Confidence Limits for LSMean(i)-LSMean(j)
1	2	-0.824167	-2.782539	1.134206
1	3	-13.017500	-14.975873	-11.059127
2	3	-12.193333	-14.151706	-10.234961

Time	Y2 LSMEAN	LSMEAN Number
1	24.3683333	1
2	29.9033333	2
3	53.0925000	3

Least Squares Means for effect Time Pr > \|t\| for H0: LSMean(i)=LSMean(j) Dependent Variable: Y2
i/j	1	2	3
1		0.3166	<.0001
2	0.3166		<.0001
3	<.0001	<.0001

Time	Y2 LSMEAN	97.5% Confidence Limits
1	24.368333	18.821510	29.915156
2	29.903333	24.356510	35.450156
3	53.092500	47.545677	58.639323

Least Squares Means for Effect Time
i	j	Difference Between Means	Simultaneous 97.5% Confidence Limits for LSMean(i)-LSMean(j)
1	2	-5.535000	-14.944296	3.874296
1	3	-28.724167	-38.133463	-19.314870
2	3	-23.189167	-32.598463	-13.779870

Simple Statistics
Variable	N	Mean	Std Dev	Sum	Minimum	Maximum	Label
Y1	36	14.30139	9.43079	514.85000	6.79000	44.74000	560nm
Y2	36	35.78806	16.53036	1288	16.37000	78.57000	720nm Format Species Spec. Time Tim.