Results: Grapefruit1.sas

Oneway ANOVA with repeated measures

Grapefruit data (Applied linear statistical models, 5th ed., Prob 27.6)

The MEANS Procedure

The Means Procedure

Summary statistics


Variable	Label	N	Mean	Std Dev
sales1 sales2 sales3	Sales at Price 1 Sales at Price 2 Sales at Price 3	8 8 8	55.438 53.600 51.338	5.441 6.131 6.374

Oneway ANOVA with repeated measures

Grapefruit data (Applied linear statistical models, 5th ed., Prob 27.6)

Test H0: mu1=mu2=mu3 with proc reg

The REG Procedure

Model: MODEL1

Dependent Variable: d12

The Reg Procedure

MODEL1

Fit

d12

Number of Observations


Number of Observations Read	8
Number of Observations Used	8

Analysis of Variance


Analysis of Variance
Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	0	0	.	.	.
Error	7	8.21875	1.17411
Corrected Total	7	8.21875

Fit Statistics


Root MSE	1.08356	R-Square	0.0000
Dependent Mean	1.83750	Adj R-Sq	0.0000
Coeff Var	58.96937

Parameter Estimates


Parameter Estimates
Variable	DF	Parameter Estimate	Standard Error	t Value	Pr > \|t\|
Intercept	1	1.83750	0.38310	4.80	0.0020

Oneway ANOVA with repeated measures

Grapefruit data (Applied linear statistical models, 5th ed., Prob 27.6)

Test H0: mu1=mu2=mu3 with proc reg

The REG Procedure

Model: MODEL1

Dependent Variable: d13

d13

Number of Observations


Number of Observations Read	8
Number of Observations Used	8

Analysis of Variance


Analysis of Variance
Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	0	0	.	.	.
Error	7	14.80000	2.11429
Corrected Total	7	14.80000

Fit Statistics


Root MSE	1.45406	R-Square	0.0000
Dependent Mean	4.10000	Adj R-Sq	0.0000
Coeff Var	35.46484

Parameter Estimates


Parameter Estimates
Variable	DF	Parameter Estimate	Standard Error	t Value	Pr > \|t\|
Intercept	1	4.10000	0.51409	7.98	<.0001

Oneway ANOVA with repeated measures

Grapefruit data (Applied linear statistical models, 5th ed., Prob 27.6)

Test H0: mu1=mu2=mu3 with proc reg

The REG Procedure

Model: MODEL1

Multivariate Test: Price

Multivariate Test Price

Multivariate Statistics


Multivariate Statistics and Exact F Statistics
S=1 M=0 N=2
Statistic	Value	F Value	Num DF	Den DF	Pr > F
Wilks' Lambda	0.09185282	29.66	2	6	0.0008
Pillai's Trace	0.90814718	29.66	2	6	0.0008
Hotelling-Lawley Trace	9.88698145	29.66	2	6	0.0008
Roy's Greatest Root	9.88698145	29.66	2	6	0.0008

Oneway ANOVA with repeated measures

Grapefruit data (Applied linear statistical models, 5th ed., Prob 27.6)

Specifying linear combinations within proc reg

The REG Procedure

Model: MODEL1

Multivariate Test: Price

The Reg Procedure

MODEL1

Multivariate Test Price

Multivariate Statistics


Multivariate Statistics and Exact F Statistics
S=1 M=0 N=2
Statistic	Value	F Value	Num DF	Den DF	Pr > F
Wilks' Lambda	0.09185282	29.66	2	6	0.0008
Pillai's Trace	0.90814718	29.66	2	6	0.0008
Hotelling-Lawley Trace	9.88698145	29.66	2	6	0.0008
Roy's Greatest Root	9.88698145	29.66	2	6	0.0008

Oneway ANOVA with repeated measures

Grapefruit data (Applied linear statistical models, 5th ed., Prob 27.6)

Test H0: mu1=mu2=mu3 with proc glm

The GLM Procedure

Data

Number of Observations


Number of Observations Read	8
Number of Observations Used	8

Oneway ANOVA with repeated measures

Grapefruit data (Applied linear statistical models, 5th ed., Prob 27.6)

Test H0: mu1=mu2=mu3 with proc glm

The GLM Procedure

Dependent Variable: sales1 Sales at Price 1

Analysis of Variance

sales1

Overall ANOVA


Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	1	24586.53125	24586.53125	830.39	<.0001
Error	7	207.25875	29.60839
Uncorrected Total	8	24793.79000

Fit Statistics


R-Square	Coeff Var	Root MSE	sales1 Mean
0.000000	9.815305	5.441359	55.43750

Type I Model ANOVA


Source	DF	Type I SS	Mean Square	F Value	Pr > F
Intercept	1	24586.53125	24586.53125	830.39	<.0001

Type III Model ANOVA


Source	DF	Type III SS	Mean Square	F Value	Pr > F
Intercept	1	24586.53125	24586.53125	830.39	<.0001

Solution


Parameter	Estimate	Standard Error	t Value	Pr > \|t\|
Intercept	55.43750000	1.92381109	28.82	<.0001

Oneway ANOVA with repeated measures

Grapefruit data (Applied linear statistical models, 5th ed., Prob 27.6)

Test H0: mu1=mu2=mu3 with proc glm

The GLM Procedure

Dependent Variable: sales2 Sales at Price 2

sales2

Overall ANOVA


Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	1	22983.68000	22983.68000	611.45	<.0001
Error	7	263.12000	37.58857
Uncorrected Total	8	23246.80000

Fit Statistics


R-Square	Coeff Var	Root MSE	sales2 Mean
0.000000	11.43834	6.130952	53.60000

Type I Model ANOVA


Source	DF	Type I SS	Mean Square	F Value	Pr > F
Intercept	1	22983.68000	22983.68000	611.45	<.0001

Type III Model ANOVA


Source	DF	Type III SS	Mean Square	F Value	Pr > F
Intercept	1	22983.68000	22983.68000	611.45	<.0001

Solution


Parameter	Estimate	Standard Error	t Value	Pr > \|t\|
Intercept	53.60000000	2.16761884	24.73	<.0001

Oneway ANOVA with repeated measures

Grapefruit data (Applied linear statistical models, 5th ed., Prob 27.6)

Test H0: mu1=mu2=mu3 with proc glm

The GLM Procedure

Dependent Variable: sales3 Sales at Price 3

sales3

Overall ANOVA


Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	1	21084.31125	21084.31125	518.99	<.0001
Error	7	284.37875	40.62554
Uncorrected Total	8	21368.69000

Fit Statistics


R-Square	Coeff Var	Root MSE	sales3 Mean
0.000000	12.41552	6.373816	51.33750

Type I Model ANOVA


Source	DF	Type I SS	Mean Square	F Value	Pr > F
Intercept	1	21084.31125	21084.31125	518.99	<.0001

Type III Model ANOVA


Source	DF	Type III SS	Mean Square	F Value	Pr > F
Intercept	1	21084.31125	21084.31125	518.99	<.0001

Solution


Parameter	Estimate	Standard Error	t Value	Pr > \|t\|
Intercept	51.33750000	2.25348441	22.78	<.0001

Oneway ANOVA with repeated measures

Grapefruit data (Applied linear statistical models, 5th ed., Prob 27.6)

Test H0: mu1=mu2=mu3 with proc glm

The GLM Procedure

Repeated Measures Analysis of Variance

Repeated Measures Analysis

Level Information


Repeated Measures Level Information
Dependent Variable	sales1	sales2	sales3
Level of price	1	2	3

MANOVA

Model

price

Multivariate Tests


MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no price Effect H = Type III SSCP Matrix for price E = Error SSCP Matrix S=1 M=0 N=2
Statistic	Value	F Value	Num DF	Den DF	Pr > F
Wilks' Lambda	0.09185282	29.66	2	6	0.0008
Pillai's Trace	0.90814718	29.66	2	6	0.0008
Hotelling-Lawley Trace	9.88698145	29.66	2	6	0.0008
Roy's Greatest Root	9.88698145	29.66	2	6	0.0008

Oneway ANOVA with repeated measures

Grapefruit data (Applied linear statistical models, 5th ed., Prob 27.6)

Test H0: mu1=mu2=mu3 with proc glm

The GLM Procedure

Repeated Measures Analysis of Variance

Univariate Tests of Hypotheses for Within Subject Effects

Within

Type III Model ANOVA


Source	DF	Type III SS	Mean Square	F Value	Pr > F	Adj Pr > F
Source	DF	Type III SS	Mean Square	F Value	Pr > F	G - G	H - F
price	2	67.48083333	33.74041667	49.35	<.0001	<.0001	<.0001
Error(price)	14	9.57250000	0.68375000

Epsilons


Greenhouse-Geisser Epsilon	0.7568
Huynh-Feldt Epsilon	0.9213

Summary

Oneway ANOVA with repeated measures

Grapefruit data (Applied linear statistical models, 5th ed., Prob 27.6)

Test H0: mu1=mu2=mu3 with proc glm

The GLM Procedure

Repeated Measures Analysis of Variance

Analysis of Variance of Contrast Variables

price_N represents the contrast between the nth level of price and the last

Contrast Variable: price_1

price_1

Type III Model ANOVA


Source	DF	Type III SS	Mean Square	F Value	Pr > F
Mean	1	134.4800000	134.4800000	63.61	<.0001
Error	7	14.8000000	2.1142857

Contrast Variable: price_2

price_2

Type III Model ANOVA


Source	DF	Type III SS	Mean Square	F Value	Pr > F
Mean	1	40.95125000	40.95125000	50.30	0.0002
Error	7	5.69875000	0.81410714

Oneway ANOVA with repeated measures

Grapefruit data (Applied linear statistical models, 5th ed., Prob 27.6)

Test H0: mu1=mu2=mu3 with proc glm

The GLM Procedure

Repeated Measures Analysis of Variance

Means of Within Subjects Effects


Level of price	N	Mean	Std Dev
1	8	55.43750000	5.44135947
2	8	53.60000000	6.13095192
3	8	51.33750000	6.37381642

Oneway ANOVA with repeated measures

Grapefruit data (Applied linear statistical models, 5th ed., Prob 27.6)

Pairwise matched t-tests

The MEANS Procedure

The Means Procedure

Summary statistics


Variable	Mean	t Value	Pr > \|t\|
d12 d13 d23	1.838 4.100 2.263	4.80 7.98 7.09	0.0020 <.0001 0.0002