Multivariate analysis

SAS Results for Problem 1:

Factor (IV) in ANCOVA: 3 levels

The GLM Procedure

Class Level Information

Class Levels Values

STRATEGY 3 ADDSPAC INSTORE SPECIAL

Overal model F-test

Number of Observations Read 45

Number of Observations Used 45

Dependent Variable (DV) analyzed

The GLM Procedure

Dependent Variable: POSTSALE

Sum of

Source DF Squares Mean Square F Value Pr > F

Model 3 2286.700121 762.233374 119.40 <.0001

Error 41 261.744324 6.384008

F-test for testing equality of IV means, not adjusting for covariate

Corrected Total 44 2548.444444

Covariate in ANCOVA

R-Square Coeff Var Root MSE POSTSALE Mean

0.897293 7.630851 2.526659 33.11111

Source DF Type I SS Mean Square F Value Pr > F

2nd DV analyzed

STRATEGY 2 824.711111 412.355556 64.59 <.0001

PRESALE 1 1461.989010 1461.989010 229.01 <.0001

Source DF Type III SS Mean Square F Value Pr > F

STRATEGY 2 962.622709 481.311355 75.39 <.0001

PRESALE 1 1461.989010 1461.989010 229.01 <.0001

Estimates of betas in ANCOVA model

Standard

Parameter Estimate Error t Value Pr > |t|

Intercept 4.54895560 B 1.63278951 2.79 0.0080

STRATEGY ADDSPAC 7.18970827 B 0.92330468 7.79 <.0001

STRATEGY INSTORE 11.20805570 B 0.92605278 12.10 <.0001

STRATEGY SPECIAL 0.00000000 B . . .

PRESALE 0.90604178 0.05987188 15.13 <.0001

NOTE: The X'X matrix has been found to be singular, and a generalized inverse was used to solve the normal equations. Terms whose estimates are followed by the letter 'B' are not uniquely estimable.

Type III SS F-test for testing equality of IV means, after accounting for covariate

This note simply means that SAS selected base level for IV in ANCOVA model.

Type III SS F-test for testing covariate

The GLM Procedure

Least Squares Means

Adjustment for Multiple Comparisons: Bonferroni

IV levels numbered for identification

Least Squares Means of IV levels (adjusted for covariate)

POSTSALE LSMEAN

STRATEGY LSMEAN Number

ADDSPAC 34.1682314 1

INSTORE 38.1865788 2

SPECIAL 26.9785231 3

Least Squares Means for effect STRATEGY

Pr > |t| for H0: LSMean(i)=LSMean(j)

p-values for pairwise comparisons of IV least squares means

Dependent Variable: POSTSALE

i/j 1 2 3

1 0.0003 <.0001

2 0.0003 <.0001

3 <.0001 <.0001

95% confidence interval for each IV mean adjusted for covariate

POSTSALE

STRATEGY LSMEAN 95% Confidence Limits

ADDSPAC 34.168231 32.846771 35.489691

INSTORE 38.186579 36.862507 39.510651

SPECIAL 26.978523 25.660682 28.296364

95% confidence interval for difference between each pair of adjusted IV means

Least Squares Means for Effect STRATEGY

Difference Simultaneous 95%

Between Confidence Limits for

i j Means LSMean(i)-LSMean(j)

1 2 -4.018347 -6.339406 -1.697288

1 3 7.189708 4.884959 9.494458

2 3 11.208056 8.896447 13.519665