Multilevel Component Analysis applied to
the measurement of a complex product
experience
Boucon, C.A., Petit-Jublot, C.E.F., Groeneschild C., Dijksterhuis, G.B.
Outline
 Background
 Introduction
to Simultaneous Component Analysis (SCA)
and Multi Level Component Analysis (MLCA)
 Technical
 Analysis
details
and results
 Conclusions
& outlook
2
Measuring product experience of food
 Food
is a complex, multisensory experience
Total Product Experience
Multisensory
e.g. preparation,
cooking
Before
 Questionnaire
e.g. aftertaste,
satiety
Usage
After
to capture multisensory experience in consumers
• Based on literature review: Berlyne’s work on exploratory behaviour and aesthetics,
choice/preference theory by Dember and Earl
• Covering different aspects of product experience: manipulation, preparation, consumption
• Evaluative variables related to complexity, aesthetics, usage, novelty
• 33 items, line scale, left and right anchored (example below)
Difficult to open
Old
Easy to open
New
3
Study design
 Evaluation
of product experience in milk tea
• 220 subjects
• Central Location Test (China)
 Stimuli
• Four products following a 2x2 (Packaging, Flavour) factorial design
• Warming up with dummy
• Preparation included in evaluation
 Randomised
 Results
per subject to correct for order and carry over
in multilevel (multiway) multivariate data
Attributes
Attributes
Products
X1
Products
X2
X3
X4
Products
Products
Individuals
Products
4
Analysis of multi-level data in sensory science
 Common
approach to analyse multivariate data from multiple subjects (panel)
in sensory
• Average score per product (or LS estimates after ANOVA)
• PCA (biplot visualisation)
 Multilevel,
multivariate data with consumers
• Not very common
• Averaging does not make sense as consumers are not trained and may vary widely
in their perception or interpretation of the attributes
• Advanced alternatives (e.g. MFA, GPA, STATIS) focus on finding a consensus in
terms of products
 Method that estimate a common component model but would allow to reflect
the individual differences and take into account the hierarchical nature of the
data
5
Introduction to MCA
 Simultaneous
Components Analysis (SCA)
• Generalization of PCA developed (ten Berge, Kiers, van der Stel,1992) for
situations where same variables are measured in two or more populations
• Applied e.g. in social sciences (same questionnaire applied to different
populations)
• Common loadings maximizing explained variance in each groups
 Extension
to model multivariate time series (Timmerman & Kiers, 2003)
• Shows evolution of latent structure in time
• Common loadings
• Different degree of constraints imposed on scores matrices
 Generalisation
of SCA to multi-level data (Timmerman, 2006)
• Decomposition of data into within and between part
• Separate (S)CA to model between and within part
6
Principle
Application to our product experience data
SSoffset
Position of the individual
relative to each other
Individual matrices
stacked upon each others
Dim 2
X1
X2
X3
Split variability into
- offset
- between participants
- within participants
+ residual term
CA
Dim 1
SSbetween
X4
Position of the products
relative to each others
Dim 2
SSwithin
(S)CA
Constraint on within models
Dim 1
7
Principle
 Split
the different sources of variability (ANOVA) for each variable j
SS total , j = SSoffset, j + SSbetween participant , j+ SSwithin participant, j + SSerror, j
 Component
model for each of the part
offset per variable
error term
between subject CA
within subject CA
- in model 0 (MLCA): Fiw and Biw differ for each
individual subject
- in model 1 to 4 (MLSCA): Biw= Bw , with different
constraints on the variance-covariance structure of Fiw
8
Within subject model
 Five
alternative with increasing degree of constraint, based on the
alternatives proposed in SCA
Model
Name
Loadings
Covariances
Variances
0
MLCA
free
free
free
4
MLSCA-P
equal
free
free
3
MLSCA-PF2
equal
equal across subjects
free
2
MLSCA-IND
equal
equal to 0
free
1
MLSCA-ECP
equal
equal across subjects
equal across subjects
Compare how the different models fit the data and how they can be
interpreted in terms of consumers’ perception
9
Selecting & comparing models
 Selecting
the right model
• Fit: Variance accounted for within part & between part of the data
• Stability: assessed by means of a split-half procedure
• Degree of complexity and interpretability
 Split-half
procedure
• Random split between participants
• Comparison between models (loadings) for both halves
• Repeat n=100 times
• Average over n repetitions
 Interpretation
• Compare loadings matrices
• Visualisation (biplots)
• Assess agreement between subject by comparing loadings and/or scores
10
Comparing models
 Two
indices quantifying similarities between matrices
• Tucker congruence coefficient (φ)
–
–
–
–
introduced to measure similarity of two factorial configurations
apply to matrices (e.g. loadings or scores matrices) of same dimensions
takes values between -1 and 1 (φ=0 no correlation, |φ|=1 perfect correlation)
applied after (procrustes) rotation and scaling of the factor solution: φrot
• RV-coefficient (Robert & Escoufier, 1976)
where
–
–
–
–
(original)
(modified)
orientation independent
allows for different number of variables
usually used to compare sample configurations (scores)
modified version independent of sample size (Smilde, 2009) and takes values
between -1 and 1 (φ=0 no correlation, |φ|=1 perfect correlation)
11
Results
Fit and model selection
 Fit
and stability of the model
• Between part
• Within part
Number of components
1
2
3
4
5
VAF (%)
21
27
32
34
35
Mean congruency coefficient
0.98
0.95
0.96
0.85
0.88
Number of components
1
2
3
VAF (%)
0 (Unconstrained)
40
50
54
4 (Loadings)
25
32
35
3 (Loadings, cov)
25
31
33
2 (Loadings, cov=0)
25
30
32
1 (Loadings & var-cov)
19
21
22
Number of components
1
2
3
Mean congruency
coefficient
0 (Unconstrained)
-
-
-
4 (Loadings)
0.9920
0.9830
0.9766
3 (Loadings, cov)
0.9919
0.5935
0.7576
2 (Loadings, cov=0)
0.9922
0.9719
0.9707
1 (Loadings & var-cov)
0.9935
0.9699
0.9791
 Number of dimensions: between part: Qb=3, within part: Qw=2
12
Comp 2 vs. Comp 3 - Between part
0.8
Between part (rotated)
223
170
146
0.6
27
25
119 40
24
26
166
332910
96
31 16
1759
99
75
28
75
1317
120
158183112
126 172
158 85
209
169107
73
19
224
20749
12
91 201
128
211
32
94
187
35
52
43
186
27
153
171
194
25
93
23
56
21
24
204857
141
203
163 76
219
41 162
77 1 4765188
148 14522
191
16
50 216212
14133
176
124
45
83
9831
132
67
30
66
160
42
26
149
197
105
2169 10 87
193
11 102
114 14
222
178

Component 3
0.4
Interpretation of the questionnaire
scale usage, overall perception of milk tea
0.2
0
74
-0.2
-0.4
• Dim 1: Novelty, Aesthetics & Complexity
1398 79 28
205
108
80 9
103
206
97
110
123
140
38
71
125
82
55
198
122
63
144
109
51174
22
100
118
18
159 164
95
200
192
60
33 121
84
214
221 113
213
185
218 184
3
925864
8854131 90
2 152
168
61
4
116
129
217
30
151
36
204
127
72
182
68 161 180
13029136 53 89
208
147
46
179
189
32
181
135 155
177
17 142
115 81 190
78
34 86
104
210
156
117 199
62
150
101
173 5937 19
143
12
39
44
195
138
215106
20
134 202
220 154 137
196 111
6
13
167
15
-0.8
• Dim 3: Familiarity
-0.6
-0.8
-0.4
Comp 1 vs. Comp 2 - Between part
3
45
2 6
0.5
0.4
0.2
73
0
146
-0.2
3
-0.4
0.4
0.2
0
Component 2
0.8
0.6
190
6 143 101
87
15
152
127
81
202 204 85
208
17
159
144141 24
9125813
10
1938 118 56140 96191
21109
2
84
175
40 35
213 134219
10
198
42
90
120
15
115
160
12828 5522416
111
57
133
161
172177
199
44
43 162
12
206
192
29
12
211
155
48
106
18
119
108
99
69
31 41
60
137
68
32
122
18616
132
24
107
14
32
13
131
26
20
63 25
26 33117
72
76
82 93
22
166
75
10252
27179
20215
205 21
176
182
100
153
5495
19
123
49
169
25
71126
45
145
103
154
196
88
27 30
61
110
194
135
11
4
97
37
78
207
170 214 163
185
5 30 147
7 59 86
46
36
94
9 212
89168
174 66
148
187
64188
22 58112
20151
34220
80
200
181
121
65
197
216
28
138 104
105
92
47
53 91
173129
150
2350156
33
210
183
39124
193
1
14
113
217 142
151
7931158 98
67
8
195
13677
17
221
29
114
116
149139
222 184
130
171
218
178 62
209
83
40
119
166
0.2
73
0.1
0
-0.1
-0.2
-0.3
28
96 17
33
10
93116
13
175
5
8
75 158 5
120
112 172
3 42 85
183 126
209
169
19
107
49201
91
224
35194
211
9452 128
187
43207 171
186
48
93 27153
23 24
56 76 25
21
141
163
203
57
1
65 219
14577 212 191 41
162 47148
188
30
16
124
45
176
98
216
133 2650 67
132
66
83
42
149
160
105
11
31
19319711487
69
222
14
10
102
139
79
80
103
108
205
28
206
38
140
8
123
9
174
97
125
71
110
164100 144
82
55
198
63
122
22 95 51
109
18
60
159
192
33
200
84118
214
185
213
121
221
218
3
58 6113
92 131
184
2 74
64
152
488
61
129
217
30168
90
54
36
151
136204
127116
72
68
182
130
208
147
46
179
161
32 53
89
189 181
29
81
190
142
17
177
135
155
34
86
115
78
104
199
210
156
117
37
62
150 173 101
12
5939 143 44
106
138
195
215
202
20134
137
220
154
196
111
6
13 15
167
15
12
32
20
22
14
21
-0.3
0.5
99
180
26
29
146
0.3
189 203
25
27 24
170
178
Component 3
Component 2
0.4
-0.2
Comp 1 vs. Comp 3 - Between part
223
0.6
167
164
223
6
-0.6
• Dim 2: Usage
0.8
5
2 43
7
180
19
-0.4
-0.6
74
-0.5
-0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
Component 1
0.4
0.6
0.8
-0.5
-0.4
-0.2
-0.1
0
0.1
Component 1
0.2
0.3
0.4
13
Within part
Model 0: unconstrained
 Individual
PCA (rotated)
• Large variability between individual
– Loadings: φrot = 0.34 (median)
– Scores: φrot = 0.76, RV= 0.64, RVM = 0.39 (median)
Comp 1 vs. Comp 2 - RespNr:1
2
0.64
-
3
0.21
0.32
-
7
0.58
0.64
0.18
-
0.5
33
0
-0.5
-1
Scores
1
-
2
0.65
-
3
0.73
0.54
-
7
0.71
0.99
0.55
-
RVM
1
2
3
7
1
-
2
0.42
-
3
0.46
0.20
-
7
0.52
0.98
0.19
-
WSC
1
26
27
25
CCO
12 2 31
32
28
16 13
29 20
15 22
14
103
4
21 9 8175
6
CCC
WSO
-1
19
28
27
0.5
30
12
32 922
2631
8
33
525
13
4
6
10
16
CCC
29 17
24
2021
19 15
14
CCO
23
0
-0.5
WSO
-1
1
0.5
0
-0.5
Component 1
-0.5
-1
Comp 1 vs. Comp 2 - RespNr:3
1
0.5
0
Component 1
Comp 1 vs. Comp 2 - RespNr:7
30
2
1
WSC
Component 2
RV
1
2
3
7
24
1
16
13
CCO
10
28
33
27
8212
14
29
19
17
26
WSO
6322
5
20
21
4
0
32
1531
24
30
25 9
-1
Component 2
1
-
Component 2
φrot
1
2
3
7
WSC 30
1
Component 2
Loadings
Comp 1 vs. Comp 2 - RespNr:2
0.5
WSO
0
-0.5
20
3
CCCCCO
5432
29
19
8
26
31
6
2427 33
21
22 916 14
15
25
13172
10
28
12
-1
-2
WSC
CCC
-2
-1
1
0
Component 1
2
-1
0.5
0
-0.5
Component 1
1
14
Improving interpretation
Unconstrained model
 Segmentation
Cluster analysis (Ward’s method) based on similarity of
individual loading matrices (as measured by φrot)
R-square
•
C5 n=27
phi=0.32
C6 n=21
phi= 0.26
C3 n=23
phi= 0.58
C1 n=35
phi= 0.56
C4 n=73
phi=0.46
C2 n=42
phi=0.62
15
Segment 2, n=42, Phi=0.62
Segmentation
Comp 1 vs. Comp 2 - RespNr:37
 Example
WSC
27
0
-0.5
323
24 1416
6 2 5 25 20
2231
3026
10
4 8
9121513
17
2933
21
19
28
CCC
-1
Visualisation
0.5
-0.5
-1
Comp 1 vs. Comp 2 - RespNr:133
-2
-1
0
Component 1
1
0
Component 1
1
-0.5
28
27
29
26
25
24
WSO
65
432 8 221415
13
20
12
CCC
Component 2
Component 2
30
17
31
33
32
CCO21
10
9 16
0
WSO
-0.5
0
0.5
Component 1
1
Component 2
WSC
WSO
-1.5
-0.5
0
0.5
Component 1
1
-1
-0.5
0
0.5
Component 1
1
Comp 1 vs. Comp 2 - RespNr:49
1
28
25
24
0.5
1
CCO
0
21
17
-0.5
292633 WSC
31
1916
32
2010
22
14
CCC
24
28
27 2 15
13
12
25
59 8
30 6
4
3
0.5
WSO
30
31
WSC
33
22
321
212CCC
10
915
19
20
2722526
28
13
29
24
17 16
148
564
0
-0.5
3
-1
-1
-1.5
WSO
-1
-0.5
0
0.5
Component 1
CCO
-1
1
Comp 1 vs. Comp 2 - RespNr:62
CCC
-0.5
0
Component 1
Segment 3, n=23, Phi=0.58
26 8
624
35
9 29
3033
10
31 17 20
22
13
32 12 CCO
16
19
14 15
21
0
-1
-1
-0.5
1.5
WSC
0.5
-1
-1
16
30
9
10
17 1512
254
6235 28 29
33
14
22
8 21
13
32
26 31
1920
0
WSO
-1
1
27
19
-1
Comp 1 vs. Comp 2 - RespNr:30
1
0.5
5
-0.5
0.5
CCC
2724
-1
WSO
-1
WSC
27
CCO
Comp 1 vs. Comp 2 - RespNr:4
2
Comp 1 vs. Comp 2 - RespNr:16
28
WSC9
29
13CCO
30
20
2510
22
33
26
15
21
131
78 32
19
16
14
43
6
212
19
-1.5
CCC
24
0
20917
21 1322
1610
15
56 3
-0.5
-1
-2
12
CCC
Component 2
0
CCC
WSC 4 2
0.5
1
1.5
1
1
12
3231 CCO
33
19
20
30 14
22
21
29
28
26
25 17
10
2 9
24
1513
16
83 4
27
6
5
0.5
0
-0.5
-1
WSC
CCC
-1.5
-1
0
Component 1
1
1
Comp 1 vs. Comp 2 - RespNr:146
WSO
1.5
0
Component 1
CCO
2
Component 2
-1
0.5
8 29
14
Component 2
CCO
22
91528
16
24 25
17
34
12
56 14
20 31
30
1021
13
26
323329
227 8
1
24
27 25
2832
31
26
33
30
Component 2
WSC
19
0
CCO
1.5
Component 2
Comp 1 vs. Comp 2 - RespNr:13
Component 2
Component 2
1
-0.5
0
0.5
Component 1
1
Segment 1, n=35, phi = 0.54
WSO
CCO
1
1
2
WSC
CCO
-0.5
0
0.5
Component 1
17CCC
31
2833
16
62
453269 10
2932
30 1315
20
8 21
14
121922
0
Comp 1 vs. Comp 2 - RespNr:81
Comp 1 vs. Comp 2 - RespNr:6
2425
27
-1
-1
−
1
Component 2
Component 2
of cluster membership
0.5
WSO
WSO
1
Unconstrained model
Comp 1 vs. Comp 2 - RespNr:77
1
27
6423
13
14
1716
19
10
20
15
29
8
12
21
25
22
WSC
933
32
26
28
30
31
24
0
-1
5
WSO
CCC
-2
-2
-1
0
Component 1
1
2
16
Within part
Model 2: constrained loadings and covariance = 0
loadings for all subjects
Dim 2: Novelty/Familiarity
 Agreement between subjects:
– φrot = 0.69, RV=0.50, RVM= 0.28 (median)
Comp 1 vs. Comp 2 - RespNr:1
CCC
0.6
Scores
19
φ
1
3
12
30
1
-
3
0.34
-
12
-0.61
-0.38
-
30
0.81
0.36
-0.69
-
Component 2
0.4
0.2
0
1
-
3
0.42
3
0.12
-
12
0.61
0.23
12
0.35
-0.15
-
27
25
2122
19
20
32
12
13
1531
33
16
83014 10
9
17
28 29
26WSC
0.4
CCO
0.2
62
354
0
-0.2
CCC
WSO
25
-0.4
24
27
-0.6
30
0.81
0.27
0.75
30
0.69
-0.27
0.61
-
20
32
12
13
1531
33
16
83014 10
9
17
28 29
26
WSC
CCO
-0.4
24
-0.6
-0.4
-0.2
0
0.2
Component 1
0.4
-0.6
0.6
Comp 1 vs. Comp 2 - RespNr:12
0.6
19
0.4
Component 2
RVM
1
3
12
30
1
-
2122
-0.2
-0.6
RV
1
3
12
30
62
354
WSO
Comp 1 vs. Comp 2 - RespNr:3
0.6
Component 2
• Dim1: Aesthetics & Complexity
0.2
CCC
CCO
0
WSC6
3542
20
32
12
13
1531
33
16
83014 10
9
17
28 29
26
27
CCC
0.4
WSO
25
-0.4
-0.2
0
0.2
Component 1
0.4
0.6
2122
-0.2
-0.4
0.6
Comp 1 vs. Comp 2 - RespNr:30
Component 2
 Same
19
0.2
62
354
0
25
27
-0.6
20
32
12
13
1531
33
16
83014 10
9
17CCO
28 29
26
-0.2 WSO
-0.4
24
2122
WSC
24
-0.6
-0.6
-0.4
-0.2
0
0.2
Component 1
0.4
0.6
-0.6
-0.4
-0.2
0
0.2
Component 1
0.4
0.6
17
Improving interpretation
Model 2: constrained loadings and covariance = 0
 Segmentation
R-square
• Cluster analysis (Ward’s method) based on similarity of product
configuration in the common space (as measured by unrotated φ)
C2 n=61
phi=0.48
C3 n=39
phi=0.56
C5 n= 41
phi=0.36
C1= n=40
phi=0.48
C4 n=40
phi=0.29
18
C1= n=40 phi=0.48
Cluster 1 - RespNr: 8
0.6
19
12
13 10
33
31
16
83014159
17
28
29
26
0.2
62
354
0
WSC
-0.2
25
-0.4
 Example of cluster
– visualisation
membership
-0.6
0.2
0
-0.2
Component 1
-0.4
25
-0.4
27
0.2
CCO
62
354
0
-0.6
-0.2
0
0.2
Component 1
0.4
-0.4
19
0.4
0.2
WSO
0
CCO
62
354
20
32
12
13 10
33
16
830141531
9
17
28
29
26 WSC
CCC
25
27
19
0.4
WSO
0.2
62
354
0
0.6
24
27
-0.6
-0.4
-0.2
0
0.2
Component 1
0.4
0.6
-0.6
-0.4
CCO
-0.2
0
0.2
Component 1
62
354
0
WSO
25
-0.6
-0.4
27
20
32
12
13 10
33
1415
31
16
830 9
17
28 29
26
-0.4
WSO
25
2122
19
20
32
12
13 10
33
1415
31
16
830 9
17
28 29
26
CCC
0.2
62
354
WSO
0
WSC
-0.2
25
-0.4
24
0.6
0.4
0.2
0
-0.2
Component 1
Cluster 2 - RespNr: 20
0.4
-0.6
CCO 27 24
-0.6
-0.6
-0.4
-0.2
0
0.2
Component 1
0.4
0.6
-0.6
-0.4
Cluster 2 - RespNr: 25
0.6
24
24
0.6
2122
-0.2
WSC
19
0.4
0.4
0.6
WSC
CCO
20
32
12
13 10
33
16
830141531
CCC
9
17
28
29
26
-0.6
-0.6
0.2
25
-0.4
-0.2
CCC
2122
-0.2
WSC
0
27
0.4
19
20
CCO32
12
13 31
33
16
830141510
9
17
28
29
62
26
CCC
34
5
C2 n=61 phi=0.48
24
0.4
0.6
2122
-0.2
-0.4
-0.2
0
0.2
Component 1
0.2
Cluster 2 - RespNr: 1
25
Cluster 3 - RespNr: 9
Component 2
0.6
WSO
0.2
0
-0.2
Component 1
2122
19
-0.6
0.4
-0.6
0.6
-0.4
0.6
0.4
-0.4
24
27
Component 2
-0.4
25
-0.6
WSO
0.2
0
-0.2
Component 1
-0.4
0.4
Component 2
-0.2
-0.4
0.6
Cluster 3 - RespNr: 6
Component 2
20
32
12
13 10
33
16
830141531
9
17
28
29WSC
26
-0.6
-0.6
CCC
CCC
27
-0.6
-0.6
0.6
20
32
12
13 10
31
1415
33
16
830
9
17
28
29
26
2122
WSO
-0.2
-0.4
24
0.6
Component 2
CCC
WSO
0.4
62
354
0
24
27
2122
WSC
0.2
CCC
0.6
Component 2
-0.2
25
-0.6
0.2
WSO
20
32
12
13 31
1415
33
16
830 10
9
17
28 29
26
-0.2
25
27
0.6
20
CCC32
WSO
62
354
0
CCO
-0.2
25
WSC
27
-0.6
12
13 31
1415
33
16
830 10
9
17
28 29
26
0.2
-0.4
24
2122
19
0.4
CCO
-0.4
0.4
Cluster 2 - RespNr: 34
62
354 WSC
0
-0.2
0
0.2
Component 1
0.6
2122
Component 2
62
354
19
Component 2
Component 2
CCO
0
0.6
20
32
12
13 10
33
16
830141531
9
17
28
29
26 WSC
0.4
0.2
Component 2
Cluster 3 - RespNr: 4
Cluster 3 - RespNr: 3
2122
-0.2
Cluster 1 - RespNr: 99
19
CCO
19
62
5
34
0
Cluster 1 - RespNr: 65
0.4
0.6
WSC CCC
CCO
0.2
-0.4
WSO
0.4
0.6
C3 n=39 phi=0.56
20
32
12
13 10
33
31
1415
16
830
9
17
28
29
26
0.4
24
27
-0.6
2122
19
20
CCC 32
CCO
Component 2
Component 2
Segmentation (Model 2)
0.4
Cluster 1 - RespNr: 10
0.6
2122
24
19
-0.6
-0.6
-0.4
-0.2
0
0.2
Component 1
0.4
0.6
-0.6
-0.4
-0.2
0
0.2
Component 1
0.4
0.6
Summary

ML(S)CA proved useful approach to model our data
• Takes account of the hierarchical structure of the data
• Offers the possibility to impose a common factor structure across subjects
• Allows to compare different levels of constraints for the individual models

Congruence & RV-coefficients are useful
• in selecting and comparing models
• in interpreting the solution (measure of similarity between individual configuration & input
for segmentation)
• the best index depends on the purpose of the comparison/segmentation

Unconstrained model provides the best fit but the separate interpretation of the
individual within loadings matrices can be very inefficient and difficult to reveal intraindividual similarities

Imposing SCA constraints on the within part of the data
• Models 4 & 2 perform best: lead to comparable VAF, stability and interpretation; model 2
best VAF/complexity ratio
• Model 3 with 2 components is unstable compared to the rest but model with only one
component has reduced fit and interpretability
• Model 1: drop in fit indicates that same variance not suitable for our data
20
Relation to other methods

Framework for comparison in van Deun et al (2009)
• MFA: common object model i.e. look for a common configuration of the product , preprocessing (scaling by individual), weight individual matrices by amount of redundant
information (1st eigenvalue)
• STATIS: common object mode, no specific pre-processing, larger weight on matrices
with cross-products(RV) most similar to others (compromise)
• GPA: common object mode, pre-processing taken into account by translation or
scaling transformation, all individual matrices equally weighted in consensus
• ML(S)CA: common variable mode i.e. seeks for a common set of underlying
components, pre-processing: normalising per respondent taken care by offset &
between part of model, all individual matrices equally weighted in solution

Timmermans (2006) also makes the parallel with multiway methods and multilevel SEM
• Tucker-1 model equivalent to SCA-P model
• Tucker-2, -3 & PARAFAC more than one mode is reduced into a component matrix;
possible alternatives for within part of the model
• Existing multilevel SEM constrain within covariance matrices to be equal for all
participants
21
References
Multilevel components analysis, M.E. Timmerman, British Journal of
Mathematical and Statistical Psychology (2006), 59, 301-320
Four simultaneous component models for the analysis of multivariate time
series from more than one subject to model intraindividual and
interindividual differences, M.E. Timmerman, H.K. Kiers, Psychometrika, 2003,
68 (1), 105-121
Simultaneous Components Analysis, J.M.F. Ten Berge, H.A.L. Kiers and V van
der Stel, Statistica Applicata, 1992, 4(4), 277-392
A unifying tool for linear multivariate statistical methods: the RVCoefficient, P. Robert and Y. Escouffier, Applied Statistics, 1976, 25(3), 257-265
Matrix correlations for high-dimensional data: the modified RV-coefficient,
A.K. Smilde et al., Bioinformatics, 2009, 25(3),401-405
A structured overview of simultaneous component based data integration,
K. Van Deun et al., BMC Bioinformatics, 2009, 10:246
22
Acknowledgements
Henk Kiers and Marieke Timmermans
Ivana Stanimirova
My colleagues & co-authors
Backup slide: MLCA algorithm
 Minimizing
the SSres using a OLS approach
where
 Offset,
(1)
between and within part solved separately by minimizing
where Ysup denotes a supermatrix
with the Yi stacked upon each others
Solved by taking m = vector containing the observed mean scores
computed for all participants and products
(2)
where Fsup denotes a supermatrix
with the 1Kif’ib stacked upon each others
(3)
Both (2) and (3) solved based on singular value decomposition
24
Backup slide: MLSCA algorithm
 Minimizing
 Offset,
the SSres using an OLS approach
between and within part solved separately by minimizing
• Offset and between part, see previous slide
• Within part solved based on ALS algorithm described in Kiers, ten Berge &
Bro (1999)
MLSCA-P:
subject to constraint on covariance of
matrices F iw of the specific SCA model
MLSCA-PF2:
MLSCA-IND:
MLSCA-ECP:
25
MLSCA
Within subject model

Model 0 (MLCA): individual PCA for each subject; the space describing the products
and their position therein is different for each subject; similar to like GPA (Paey’s
approach) except that we are not trying to rotate the results to get a consensus (might
be an idea)

Model 4 (SCA-P): the space describing the position of the products is the same but the
position of the product may vary per individual; this is similar to Tucker-1 and performing
PCA on the stacked matrices (under certain conditions)

Model 3 (SCA-PF2): the space describing the position of the products is the same and
the relative position is constrained to be the same for each subject; the variance may
differ per subject; related to PARAFAC

Model 2 (SCA-IND): the components are constrained to be uncorrelated for each
individual

Model 1 (SCA-ECP): most constrained model where variance is constrained to be the
same for all subjects; might be less relevant to our data
 Interesting to compare how the different models fit the data and how they can be
interpreted in terms of consumers’ perception
26
Additional issues
 Pre-processing
• Centring across or per subject not necessary since offset and between
subject terms are modelled explicitly
• Normalisation per variable (over other modes) recommended
– Eliminate artificial scale differences between variables
– No further lost of source of variability, factor model preserved
– Arguable in our situation: might choose not to standardised at all, since
difference in variability between variable might reflect perceived differences
 Rotational
freedom
• Between part: insensitive to orthogonal and oblique rotation
• Within part:
– Model 0, 1 & 4: insensitive to orthogonal and oblique rotation
– Model 2 & 3: unique solutions
• Normalisation of component scores to facilitate comparisons
RV =
tr ( XX ' YY ' )
tr[( XX ' ) 2 ]tr[(YY ' ) 2 ]
ϕ=
tr ( XY ' )
27' )
tr ( XX ' )tr (YY
Results
Agreement between subjects

Overview (median)
Model 0
Model 4
Model 3
Model 2
Model 1
Description
Unconstrained
Constrained loadings
Constrained loadings
and cov
Constrained
loadings and cov=0
Constrained loadings
and var-cov matrices
Loadings
φ, φrot
0.14, 0.34
-
-
-
-
Scores
φ, φrot, RV, RVM
0.12,0.76,0.64,0.39
φ, φrot, RV, RVM
0.07,0.63,0.41,0.19
φ, φrot, RV, RVM
0.10, 0.58, 0.31, 0.10
φ, φrot, RV, RVM
0.05,0.69,0.50,0.28
φ, φrot, RV, RVM
0.03, 0.75, 0.63, 0.36

Most suitable index depends on objective
– Model 0: compare rotated configuration since individual models unconstrained
• moderate agreement on loadings
• seemingly high agreement on scores but not higher than chance given the small number of samples
– Model 1 to 0: compare scores directly (unrotated) makes sense since loadings are constrained
to be equal
• very low agreement
• higher level of constraint improves agreement on relative position of products
• model 3 falls out of this trend
28