11.1.
(a) Ho: The means of Y1 and Y2 for the two groups are equal..
Ha: The means of Y1 and Y2 for the two groups are not equal.
(b) The following two tables give the computations for the two data sets.
Data Set 1
X1 = (3.25 6.25) X2 = (7.5 15.75) (X1 - X2) = (-4.25 -9.50)
SSCP1 = 8.750 16.750 SSCP2 = 3.000 23.500
16.750 34.750 23.500 42.750
SSCPw = 21.750 40.250 SSCPt = 57.875 121.000
40.250 77.500 121.000 258.000
SSCPb = 36.174 80.750
80.750 180.500
Multivariate Analysis
· MD2 = (X1 - X2) * (SSCPw /6) -1 *(X1 - X2)’= 10.300
· T2 = (4*4)/(4+4) * MD2 = 20.600
· Eigenvalue of SSCPb SSCPw-1 = 3.435
· Wilks’ L = .225
· Pillai’s trace = .775
· Hotelling’s trace = 3.435
· Roy’s largest root = 0.805(textbook formula) 3.435(SAS formula)
· F-ratio = 8.59
· Effect Size - Partial eta square = .775
Univariate Analysis
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Statistic Y1 Y2
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(a) Statistical Analysis
MD2 4.983 6.987
T2 = F 9.966 13.974
t 3.157 3.738
(b) Effect Size
Partial eta square 0.624 0.700
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Data Set 2
X1=(3.333 3.167) X2=(8.333 7.333) (X1 - X2) = (-5.000 -4.164)
SSCP1 = 13.333 -13.833 SSCP2 = 13.333 -12.167
-13.833 15.973 -12.167 27.473
SSCPw = 26.667 -26 SSCPt = 101.667 36.500
-26 43.446 36.500 95.530
SSCPb = 75 62.5
62.5 52.083
Multivariate Analysis
· MD2 = (X1 - X2) * (SSCPw /10) -1 *(X1 - X2)’=54.524
· T2 = (4*4)/(4+4) * MD2 = 65.428
· Eigenvalue of SSCPb SSCPw-1 = 16.365
· Wilks’ L = 0.058
· Pillai’s trace = 0.942
· Hotelling’s trace = 16.365
· Roy’s largest root = 0.942 (textbook formula) 16.365 (SAS formula)
· F-ratio = 73.64
· Effect Size -- Partial eta square = 0.942
Univariate Analysis
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Statistic Y1 Y2
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(a) Statistical Analysis
MD2 9.375 3.991
T2 = F 28.125 11.973
t 5.303 3.460
(b) Effect Size
Partial eta square 0.738 0.545
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(c) The multivariate significance tests indicate that for both data sets the centroids of the two groups are significantly different. Also, the univariate tests indicate that for both the data sets the two groups are significantly different from each other with respect to each of the variables. The effect size is quite large as indicated by the partial eta squares.
(d) The within-group correlation between Y1 and Y2 for data set 1 is 0.98 and for data set 2 it is equal to 0.018. Consequently, there could be a difference in multivariate and univariate significance for data set 1, but not for data set 2.
(e) The (maximum) eigenvalue would be, respectively, 3.435 and 5.646 for data sets 1 and 2.
(f) The results and the conclusions are the same.
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