Repeated Measures ANOVA
Comparison of Task Completion
Times of 4 Navigation Techniques and
2 Input Methods by 36 Subjects
Source: F.-G. Wu, H. Lin, M. You (2011). "The Enhanced Navigator for the Touch Screen: A
Comparative Study on Navigational Techniques of Web Maps," Displays, Vol. 32, pp. 284-295.
Data Description and Model
• Experiment Conducted to Compare Effects of Navigation
Technique and Input Method on Task Completion Times
 Factor A (Fixed): Navigation Technique: CPB, DPB, ENCC, G&D
• CPB = Combined Panning Buttons, DPB = Distributed Panning Buttons
• ENCC = Enhanced Navigator w/ Continuous Control, G&D = Grab&Drag
 Factor B (Fixed): Input Method: Direct-Touch, Mouse
 Factor C (Random): 36 Subjects measured on all 8 Treatments
• Response is time to complete navigation task.
 Data simulated to match authors’ results (Means, F-tests)
Yijk     i   j   ij   k   ik     jk   ijk
4
2
4
2
            
i 1
i
j 1
j
i 1
 k ~ NID  0,  C2 
ij
j 1
ij
i  1,..., 4; j  1,..., 2; k  1,...,36
0
2
2
 ik ~ NID  0,  AC
    jk ~ NID  0, BC2   ijk ~ NID  0, ABC

 k    ik      jk    ijk 
Data – Multivariate Form
Trt
Trt1
Trt2
Trt3
Trt4
Trt5
Trt6
Trt7
Trt8
Factor_A A1
A1
A2
A2
A3
A3
A4
A4
Factor_B B1
B2
B1
B2
B1
B2
B1
B2
SubjMean
Subject1
163.30
141.23
184.63
127.97
197.05
251.96
132.08
90.58
161.10
Subject2
214.95
112.38
222.46
126.79
54.58
119.24
127.10
50.82
128.54
Subject3
179.73
88.03
183.69
221.24
115.31
145.65
120.18
89.91
142.97
Subject4
164.35
181.68
212.66
125.85
122.03
128.51
91.40
128.94
144.43
Subject5
184.68
144.92
132.57
106.03
156.71
165.00
145.40
133.47
146.10
Subject6
165.21
87.48
119.35
158.34
134.65
107.89
164.91
91.25
128.64
Subject7
171.03
218.02
164.38
175.06
83.26
188.49
124.08
113.43
154.72
Subject8
151.97
148.30
200.06
114.83
43.71
34.78
143.56
112.15
118.67
Subject9
141.27
133.63
190.83
137.32
135.09
133.58
189.17
182.08
155.37
Subject10
146.66
169.38
133.05
224.46
80.78
52.35
191.69
165.22
145.45
Subject11
208.07
110.25
221.84
225.31
90.71
182.57
219.80
159.84
177.30
Subject12
202.50
70.47
230.19
57.34
141.67
82.79
104.17
100.42
123.69
Subject13
221.43
112.14
155.76
110.05
118.50
135.74
175.36
124.65
144.20
Subject14
174.87
111.15
161.20
125.65
163.19
140.89
91.30
55.41
127.96
Subject15
166.08
204.44
159.54
149.55
108.17
107.38
85.44
143.54
140.52
Subject16
177.53
181.48
178.31
91.38
131.60
116.81
162.37
136.37
146.98
Subject17
179.58
202.76
215.61
188.05
110.02
187.60
112.71
200.56
174.61
Subject18
154.37
133.85
188.63
178.62
126.11
111.82
157.39
84.35
141.89
Subject19
243.73
189.98
189.65
137.01
156.90
132.77
202.14
229.90
185.26
Subject20
160.82
136.26
143.35
150.48
119.93
118.32
147.50
181.98
144.83
Subject21
211.94
151.08
190.92
117.70
169.93
101.10
204.37
92.51
154.94
Subject22
178.88
144.35
200.36
122.20
186.91
132.58
152.50
185.77
162.94
Subject23
126.75
182.35
147.85
186.76
127.58
184.29
86.13
114.05
144.47
Subject24
204.81
126.50
142.64
136.08
156.64
128.67
147.29
94.05
142.09
Subject25
152.80
119.68
182.80
127.56
119.07
143.22
99.61
109.20
131.74
Subject26
199.08
159.91
154.04
181.33
110.26
67.74
105.39
144.79
140.32
Subject27
153.48
137.76
185.90
169.92
145.01
90.63
173.21
169.16
153.13
Subject28
164.34
134.61
165.18
164.04
114.98
155.37
141.09
162.04
150.21
Subject29
279.33
207.65
233.47
188.27
136.81
188.10
231.04
179.11
205.47
Subject30
218.29
145.88
169.10
85.59
189.20
103.37
103.26
99.69
139.30
Subject31
213.82
109.13
188.86
183.73
146.53
82.40
181.64
155.26
157.67
Subject32
132.31
176.15
169.81
137.50
64.11
158.56
174.14
172.36
148.12
Subject33
189.12
139.23
182.66
113.12
146.67
110.88
66.90
147.22
136.98
Subject34
225.02
162.85
162.50
126.15
142.59
130.61
104.94
105.90
145.07
Subject35
178.13
95.58
147.25
140.40
153.15
154.66
121.45
73.22
132.98
Subject36
193.50
105.90
210.23
172.75
124.81
127.09
171.96
78.10
148.04
TrtMean
183.16
143.79
178.37
146.79
128.45
130.65
143.13
129.37
147.96
TrtVar
1057.12 1363.08
842.15 1516.83 1242.77 1807.79 1714.76 1846.72
Variance-Covariance Matrix for 8 Navigation
Treatments
1057.12
-8.03
321.31
-144.91
306.43
34.27
390.51
48.87
-8.03
1363.08
10.02
194.72
-204.84
319.82
4.64
783.14
321.31
10.02
842.15
-43.13
-104.03
109.43
212.64
137.75
-144.91
194.72
-43.13
1516.83
-471.53
253.42
485.62
383.97
306.43
-204.84
-104.03
-471.53
1242.77
331.14
-78.18
-78.03
34.27
319.82
109.43
253.42
331.14
1807.79
-7.08
41.39
390.51
4.64
212.64
485.62
-78.18
-7.08
1714.76
676.00
48.87
783.14
137.75
383.97
-78.03
41.39
676.00
1846.72
The Variance of the 36 measurements for NavTrt 1 (NavTech=1, InpMeth=1) is 1057.12
The Covariance of the 36 Measurements for NavTrt’s 1 and 2 (NT=1, IM=2) is -8.03

36
1
S12 
Y11k  Y 11

36  1 k 1

2


1 36
S12 
Y11k  Y 11 Y12 k  Y 12

36  1 k 1

Sphericity Assumption
Let y1 be an arbitrary measurement from Treatment 1 and y2 from Trt 2 on same individual:
The Variance of difference is:  y21  y2   y21   y22  2 y1 , y2
Sphericity Assumption:  y2i  y j   y2i '  y j '
where  y1 , y2 is their Covariance
 i, j , i ', j '
Case with 3 Treatments:  y21  y2   y21  y3   y22  y3
  y21

   y1 , y2

 y1 , y3
 y ,y
1
2
 y2
2
1
3
3
1 0 
 CC '  

0 1 

 y ,y 

 y2 
2
 y ,y
2
 y ,y 
3
3
1
2
1
6

1
2
1
6

 y21   y1 , y2

2
and C    2
  
y1 , y2  2 y1 , y3
 y1
6



 y21   y1 , y2


 C C '  
2
  y1   y1 , y2  2 y1 , y3




Let C  



  
 1


 2
0 
 1
  C '  
2 
 2


6

 0

 y ,y
  y22
y1 , y2
   2 y2 , y3
2
  
12
1
2
2
   2 y2 , y3
 y ,y
2
y2
1
6

 
2
y1


2

2 
  y2 , y3  2 y3

6

1
2

y1 , y2
2
y2
 y ,y  y ,y
 y , y   y22
1
1 

6 
1 
6 
2 

6 
2
y1
3
3
2
3
 
 
  y1 , y2   y1 , y2   y22  2  y1 , y3   y2 , y3
 
12
 

  y1 , y2  2 y1 , y3   y1 , y2    2 y2 , y3  2  y1 , y3   y2 , y3  2 y23
2
y2
6







Sphericity Assumption Continued


C



1
2
1
6

1
2
1
6

 1


 2
0 
 1
  C '  
2 
 2


6

 0


 y21   y1 , y2


 2
  y1   y1 , y2  2 y1 , y3


  
y1 , y2
  y22

2
  
y1 , y2
  y22  2 y2 , y3
 
 C C ' 

2
y1
 
 
  y1 , y2   y1 , y2   y22  2  y1 , y3   y2 , y3
12
 
 
6
 


 

  y23  2 y1 , y3   y22   y23  2 y2 , y3 

12


 y21   y22  2 y1 , y2  4 y23  4 y1 , y3  4 y2 , y3 

6
2
y1
Note that under the null hypothesis:  y21   y22  2 y1 , y2   y21   y23  2 y1 , y3   y22   y23  2 y2 , y3

 
 
 

Note that: 2   y21   y23  2 y1 , y3   y22   y23  2 y2 , y3    y21   y22  2 y1 , y2  3  y21   y22  2 y1 , y2 


2
2
2
  y1   y2  2 y1 , y2  4 y3  4 y1 , y3  4 y2 , y3
 C C '   I

 y2   y2  2 y , y
1
2
2
1
2

 y2   y2  2 y , y
1

  y1 , y2  2 y1 , y3   y1 , y2   y22  2 y2 , y3  2  y1 , y3   y2 , y3  2 y23
2
y1
12

 y21   y22  2 y1 , y2

2


2
2
2
2
  y1   y3  2 y1 , y3   y2   y3  2 y2 , y3

12

1 

6 
1 
6 
2 

6 
3
2
1
3

 y2   y2  2 y , y
2
3
2
2
3

 y2  y
i
2
j
i j







Mauchley Test
H 0 : CΣC'   I (Sphericity)
H A : CΣC'   I

1
The k row of C : 
 k  k  1
1
th
Note: The first k elements are
k  k  1

k
k  k  1
1
k  k  1
When there are t repeated measures, the matrix C is  t  1  t
t  1 CSC'

W
t 1
tr  CSC' 
t 1
Compute:
df Mauchley 
t (t  1)
1
2
dfSubjects(Trt)  # Subjects - # Between Subjects Factor Levels

2t 2  3t  3 
Test Statistic: X    dfSubjects(Trt) 
 ln W 
6(t  1) 

which is approximately  2 with df Mauchley degrees of freedom
2
0

0

C Matrix and CSC’ Matrix
C
1
2
3
4
5
6
7
1
0.707107
0.408248
0.288675
0.223607
0.182574
0.154303
0.133631
2
-0.707107
0.408248
0.288675
0.223607
0.182574
0.154303
0.133631
3
0.000000
-0.816497
0.288675
0.223607
0.182574
0.154303
0.133631
4
0.000000
0.000000
-0.866025
0.223607
0.182574
0.154303
0.133631
5
0.000000
0.000000
0.000000
-0.894427
0.182574
0.154303
0.133631
6
0.000000
0.000000
0.000000
0.000000
-0.912871
0.154303
0.133631
7
0.000000
0.000000
0.000000
0.000000
0.000000
-0.925820
0.133631
8
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
-0.935414
CSC'
1218.13
-268.045
209.0658
-376.216
207.1655
-264.458
511.8739
-268.045
741.2398
-2.31998
-65.1789
11.78635
72.45961
-161.281
209.0658
-2.31998
1460.035
-406.888
118.1084
247.4369
34.19751
-376.216
-65.1789
-406.888
1455.757
201.1838
-246.855
-268.101
207.1655
11.78635
118.1084
201.1838
1348.258
-388.797
-339.81
-264.458
72.45961
247.4369
-246.855
-388.797
1411.223
277.9774
511.8739
-161.281
34.19751
-268.101
-339.81
277.9774
1356.336
Mauchley Test
H 0 : CΣC'   I (Sphericity)
CSC'  1.94935E  21
tr  CSC'  8990.978494
 t  1 CSC' = 8  1 1.94935E  21 =0.338009
W
t 1
8 1
tr  CSC' 
8990.978494
t 1
Compute:
df Mauchley 
dfSubjects(Trt)
H A : CΣC'   I
8 1
t (t  1)
8(8  1)
1 
 1  27
2
2
 36  1  35 (No Between Subjects Factors)

2t 2  3t  3 
Test Statistic: X    dfSubjects(Trt) 
 ln W  
6(t  1) 

2

2 8  3 8  3 
   35 
 ln  0.338009   35.200563

6
8

1
  

2
P   272  35.200563  0.133858
 2  0.05; 27   40.113
Degrees of Freedom Adjustments
When Sphericity Assumption is Rejected, Degrees of Freedom Adjustments Applied:
 a11
a
21
Greenhouse-Geisser  : A  CSC'  


 at 1,1
t 1
a
i 1
ii
 tr  CSC'  8990.978494
a12
a22
at 1,2
t 1 t 1
 a
i 1 j 1
2
ij
2
a1,t 1 
 t 1 
  aii 
a2,t 1  ^
   i 1t 1 t1

 t  1  aij2

at 1,t 1 
i 1 j 1
 14786640 
^
~
Huynh-Feldt Adjustment:  
N  t  1   2
^

 t  1  dfSubjects(Trts)   t  1  



8990.978494   0.780992

8  114786640 
^
2
36  8  1 0.781  2
8  1  35  8  1 0.781
 0.942333
Multivariate Tests for Within-Subjects Factor(s)
Case 1: Treatments  8 Distinct Combinations of NavTech and InpMeth (No Between Treatment Factors)
 Y1 '   Y11 Y21
Y ' Y
Y22
12
Y  2 
36 x 8

 

 
 Y36 ' Y1,36 Y2,36
Y81 
Y82 


Y8,36 
1
1
 
X

36 x1

1
β   1
where:
Yij  Measurement for Treatment i by Subject j
i  Population Mean for Treatment i
H 0 : 1  ...  8
for L  1
 1  8  0, ..., 7  8  0  LβM  0
1 0 0 0 0 0 0
0 1 0 0 0 0 0


0 0 1 0 0 0 0


0
0
0
1
0
0
0

M
0 0 0 0 1 0 0


0
0
0
0
0
1
0


0 0 0 0 0 0 1


 1 1 1 1 1 1 1
2
3  4
5  6
7
8 
Multivariate Tests for Within Subjects Factor(s)
Residual Sum of Squares and Cross-Products Matrix:
S E  dfSubjects(Trts) M'SM where S  Sample Variance-Covariance Matrix
Hypothesis Sum of Squares Matrix for Testing H 0 : L'βM  0 :
1
S H   LBM  ' L  X'X  L'


1
 LBM 
Let  1 ,  2 ,... be ordered eigenvalues of
B   X'X  X'Y
1
S E1S H
(Note these do not need to be computed directly, Except for Roy's Largest Root):
B
183.16
143.79
178.37
146.79
128.45
LBM
53.7897222 14.42056 49.00083 17.42028
L(X'X)^-1L'
0.02777778
130.65
143.13
129.37
-0.91889 1.280833 13.76028
SE
SH
S_E
98213.2
35233.65
69349.18
44413.67
76380.63
62675.52
52932.22
35233.65
57523.18
32754.81
30601.31
32786.69
46970.27
13727.41
S_H
104160.03
27924.40
94886.68
33733.15
-1779.36
2480.24
26645.81
27924.40
7486.29
25438.29
9043.56
-477.03
664.93
7143.51
INV(S_E)*S_H
2.021077 0.541833
-0.10194 -0.02733
1.158596 0.310609
0.095487 0.025599
-1.37209 -0.36785
-0.48884 -0.13105
-0.75368 -0.20205
(SE)-1SH Matrices
69349.18
32754.81
84468.16
44865.43
58903.97
62195.67
43596.3
94886.68
25438.29
86438.94
30729.89
-1620.95
2259.43
24273.54
1.841141
-0.09286
1.055446
0.086986
-1.24994
-0.44532
-0.68658
44413.67
30601.31
44865.43
90846.38
37423.61
58617.35
44532.72
33733.15
9043.56
30729.89
10924.78
-576.26
803.25
8629.48
0.654544
-0.03301
0.375222
0.030924
-0.44436
-0.15831
-0.24409
76380.63
32786.69
58903.97
37423.61
113593.9
77507.32
40969.45
62675.52
46970.27
62195.67
58617.35
77507.32
125010.5
39278.73
52932.22
13727.41
43596.3
44532.72
40969.45
39278.73
77331.28
-1779.36
-477.03
-1620.95
-576.26
30.40
-42.37
-455.19
2480.24
664.93
2259.43
803.25
-42.37
59.06
634.49
26645.81
7143.51
24273.54
8629.48
-455.19
634.49
6816.43
-0.03453
0.001741
-0.01979
-0.00163
0.023439
0.008351
0.012875
0.048126
-0.00243
0.027588
0.002274
-0.03267
-0.01164
-0.01795
0.517024
-0.02608
0.296387
0.024427
-0.351
-0.12505
-0.1928
Wilk’s L
SE
1  L
1
L

which can be converted to F-Statistic: FW 
SE  SH
1  i
L1/ d
1/ d
  rd  2u 


t *q


with degrees of freedom:  1  t *q  2  rd  2u
where:
t *  rank  S E 
(7 in this case)
g  # of Between Subjects Treatment groups (1 in this case)

q  rank L  X'X  L'
r  N  g
1
t

*

 q  1
2
(1 in this case)
(35-3.5=31.5 in this case)
t *q  2
u
(1.25 in this case)
4
 t *2 q 2  4

2
d   t *2  q 2  5
if t *  q 2  5  0

1 otherwise
|S_E|
|S_E+S_H|
L
1.00336E+33 3.912E+33 0.2564685
(1 in this case)
 1  0.2565   31.5(1)  2(1.25) 
FW  
  12.0086

7(1)
 0.2565  

 1  7,  2  29
F  0.05;7, 29   2.346
Pillai’s Trace

V  trace S H  S H  S E 
1
   1  
which can be converted to F-Statistic: FP 
i
i
 2n  s  1  V 
 2m  s  1  s  V 
with degrees of freedom:  1  s  2m  s  1  2  s  2n  s  1
where:
t *  rank  S E 
(7 in this case)
g  # of Between Subjects Treatment groups (1 in this case)

q  rank L  X'X  L'
1

N  g  t* 1
n
2
t*  q 1
m
2
s  min  t * , q 
S_H*INV(S_H+S_E)
0.518343 -0.02614
0.138963 -0.00701
0.472195 -0.02382
0.16787 -0.00847
-0.00885 0.000447
0.012343 -0.00062
0.1326 -0.00669
0.297143
0.079662
0.270689
0.096232
-0.00508
0.007076
0.076014
trace(S_H*INV(S_H+S_E))
0.743532
(1 in this case)
(13.5 in this case)
(2.5 in this case)
(1 in this case)
0.024489
0.006565
0.022309
0.007931
-0.00042
0.000583
0.006265
-0.3519
-0.09434
-0.32057
-0.11397
0.006011
-0.00838
-0.09002
-0.12537
-0.03361
-0.11421
-0.0406
0.002142
-0.00299
-0.03207
-0.19329
-0.05182
-0.17609
-0.0626
0.003302
-0.0046
-0.04945
 2(13.5)  1  1   0.7435 
FP  

  12.0086
 2(2.5)  1  1   1  0.7435 
 1  7,  2  29
F  0.05;7, 29   2.346
Hotelling-Lawley Trace
U  trace  S E1S H     i which can be converted to F-Statistic: FHL 
2  sn  1 U
s 2  2m  s  1
with degrees of freedom:  1  s  2m  s  1  2  2  sn  1
where:
t *  rank  S E 
(7 in this case)
g  # of Between Subjects Treatment groups (1 in this case)

q  rank L  X'X  L'
1

N  g  t* 1
n
2
t*  q 1
m
2
s  min  t * , q 
INV(S_E)*S_H
2.021077 0.541833
-0.10194 -0.02733
1.158596 0.310609
0.095487 0.025599
-1.37209 -0.36785
-0.48884 -0.13105
-0.75368 -0.20205
trace(INV(S_E)*S_H)
2.899115
1.841141
-0.09286
1.055446
0.086986
-1.24994
-0.44532
-0.68658
0.654544
-0.03301
0.375222
0.030924
-0.44436
-0.15831
-0.24409
(1 in this case)
(13.5 in this case)
(2.5 in this case)
(1 in this case)
-0.03453
0.001741
-0.01979
-0.00163
0.023439
0.008351
0.012875
0.048126
-0.00243
0.027588
0.002274
-0.03267
-0.01164
-0.01795
0.517024
-0.02608
0.296387
0.024427
-0.351
-0.12505
-0.1928
FHL
2 1(13.5)  1 2.8991
 2
 12.0106
1  2(2.5)  1  1
 1  7,  2  29
F  0.05;7, 29   2.346
Roy’s Largest Root
  max   i  which can be converted to F-Statistic: FR 
with degrees of freedom:  1  r  2   N  g  r  q 
 N  g  r  q
r
where:
t *  rank  S E 
g  # of Between Subjects Treatment groups (1 in this case)

q  rank L  X'X  L'
1

r  max  t * , q 
(1 in this case)
(7 in this case)
S -1ES H
Largest eigenvalue is 2.899115
(Computed in R)
2.899115(36  1  7  1)
FR 
 12.0106
7
 1  7,  2  29
F  0.05;7, 29   2.346
Multivariate Tests for Within-Subjects Factor(s)
Case 2: Treatments  NavTech (df = 3) ImpMeth (df = 1) Interaction (df = 3)
 Y1 '   Y111 Y121
Y ' Y
Y122
112
Y  2 
36 x 8

 

 
 Y36 ' Y11,36 Y12,36
Y421 
Y422 


Y42,36 
1
1
 
X

36 x1

1
β   11
12
21 22
31 32
41 42 
where:
Yijk  Measurement for NavTech i, InpMeth j by Subject k
H 0A : 1  2  3  4
H 0B : 1  2
H
AB
0
 1  4  0, 2  4  0, 3   4  0  LβM A  0
 1  2  0  LβM B  0
:  11  12    41  42   0,
for L  1
ij     i   j   ij  Population Mean for Navtech i, InpMeth j
1 0 0
1 0 0


0 1 0


0 1 0

MA 
0 0 1


0 0 1
 1 1 1


 1 1 1
 21  22    41  42   0,  31  32    41  42   0
1
 1
 
1
 
1
MB   
1
 
 1
1
 
 1
M AB
1 0 0
 1 0 0 


0 1 0


0 1 0 


0 0 1


 0 0 1
 1 1 1


 1 1 1 
 LβM AB  0
Test For Navigation Technique
H 0A : 1  2  3  4
 1  4  0,  2   4  0, 3   4  0  LβM A  0 L  1
B
183.16
143.79
178.37
146.79
128.45
130.65
143.13
1 0 0
1 0 0


0 1 0


0 1 0

MA 
0 0 1


0 0 1
 1 1 1


 1 1 1
129.37
LBM_A
54.45
52.66083333 -13.3983333
LXXIL'
0.027777778
S_E
170215.7 99661.6 149236.6
99661.6 166118.6 126094.7
149236.6 126094.7 310453.9
S_H*INV(S_H+S_E)
0.338807 0.239652 -0.29048
0.327675 0.231777 -0.28094
-0.08337 -0.05897 0.071479
S_H
106732.9 103225.8 -26263.413
103225.8 99833.88 -25400.4264
-26263.4 -25400.4
6462.5521
|S_E|
|S_E+S_H|
L
3.04E+15 8.49E+15 0.357937
Wilks’ L
INV(S_E)*S_H
0.946557427 0.9154546 -0.232916
0.669538178 0.6475379 -0.164751
-0.81155237 -0.784886 0.199696
V=trace(S_H*INV(S_H+S_E))
0.642063464
Pillai’s Trace
U=trace(INV(S_E)S_H)
1.793791359
Hotelling-Lawley Trace
& Roy’s Largest Root
F-Tests for Navigation Technique Effects
H 0A : 1  2  3  4
 1  4  0, 2  4  0, 3  4  0
Wilks' L
L
0.357937
t*
3
g
1
q
1
r
33.5
u
0.25
d1
1
F_W
19.7317
df1
3
df2
33
F(.05)
2.891564
Pillai's Trace
V
0.642063
t*
3
g
1
q
1
n
15.5
m
0.5
s
1
F_P
19.7317
df1
3
df2
33
F(.05)
2.891564
U
Hotelling- Lawley Trace 1.793791
t*
3
g
1
q
1
n
15.5
m
0.5
s
1
F_P
19.7317
df1
3
df2
33
F(.05)
2.891564

1.793791
t*
3
g
1
q
1
n
15
m
0.5
r
3
F_P
19.7317
df1
3
df2
33
F(.05)
2.891564
Roy's Largest Root
Test For Input Method
H 0B : 1  2
 1  2  0  LβM B  0
for L  1
1
 1
 
1
 
1
MB   
1
 
 1
1
 
 1
LBM_B
82.51027778
B
183.16
143.79
178.37
S_E
548837.43
|S_E|
|S_E+S_H|
L
548837.4 793923.49 0.691298
Wilks’ L
146.79
128.45
S_H
245086.05
130.65
143.13
INV(S_E)*S_H
0.446555
129.37
LXXIL'
0.027777778
S_H*INV(S_H+S_E)
0.308702
V=trace(S_H*INV(S_H+S_E))
0.308702
U=trace(INV(S_E)S_H)
0.446555
Pillai’s Trace
Hotelling-Lawley Trace
& Roy’s Largest Root
F-Tests for Input Method Effects
H 0B : 1  2
 1  2  0
Wilks' L
L
0.691298
t*
1
g
1
q
1
r
34.5
u
-0.25
d1
1
F_W
15.62942
df1
1
df2
35
F(.05)
4.121338
Pillai's Trace
V
0.308702
t*
1
g
1
q
1
n
16.5
m
-0.5
s
1
F_P
15.62942
df1
1
df2
35
F(.05)
4.121338
Hotelling-Lawley Trace
U
0.446555
t*
1
g
1
q
1
n
16.5
m
-0.5
s
1
F_P
15.62942
df1
1
df2
35
F(.05)
4.121338
Roy's Largest Root

0.446555
t*
1
g
1
q
1
n
16
m
-0.5
r
1
F_P
15.62942
df1
1
df2
35
F(.05)
4.121338
Test For NavTech/InpMeth Interaction
H 0AB :  11  12    41  42   0,
 21  22    41  42   0,  31  32    41  42   0
 LβM AB  0 for L  1 M AB
1 0 0
 1 0 0 


0 1 0


0 1 0 


0 0 1


0
0

1


 1 1 1


 1 1 1 
LBM_AB
25.60888889 17.82027778
B
183.16
143.79
178.37
S_E
84190.74 61844.9 64324.43
61844.9 164787.8 94479.01
64324.43 94479.01 157539.5
|S_E|
|S_E+S_H|
L
9.01E+14 1.6E+15 0.563035
Wilks’ L
146.79
128.45
S_H
23609.35 16428.87 -14713.8432
16428.87 11432.24 -10238.8188
-14713.8 -10238.8
9169.9776
130.65
143.13
129.37
LXXIL'
0.027777778
INV(S_E)*S_H
0.456973819 0.3179912 -0.284796
0.13521027 0.0940878 -0.084266
-0.36107098 -0.251256 0.2250271
V=trace(S_H*INV(S_H+S_E))
0.436965055
Pillai’s Trace
-15.96
S_H*INV(S_H+S_E)
0.257292 0.076128 -0.2033
0.17904 0.052975 -0.14147
-0.16035 -0.04744 0.126698
U=trace(INV(S_E)S_H)
0.776088693
Hotelling-Lawley Trace
& Roy’s Largest Root
F-Tests for Navigation Technique Effects
H 0AB :  11  12    41  42   0,
 21  22    41  42   0,
 31  32    41  42   0
L
0.563035
t*
3
g
1
q
1
r
33.5
u
0.25
d1
1
F_W
8.536976
df1
3
df2
33
F(.05)
2.891564
V
0.436965
t*
3
g
1
q
1
n
15.5
m
0.5
s
1
F_P
8.536976
df1
3
df2
33
F(.05)
2.891564
U
Hotelling-Lawley Trace 0.776089
t*
3
g
1
q
1
n
15.5
m
0.5
s
1
F_P
8.536976
df1
3
df2
33
F(.05)
2.891564

0.776089
t*
3
g
1
q
1
n
15
m
0.5
r
3
F_P
8.536976
df1
3
df2
33
F(.05)
2.891564
E+S_H|
Wilks' L
Pillai's Trace
Roy's Largest Root