GPA using Senstools v3.3

GPA
using Senstools v3.3
August 2005

Introduction
‣ Generalized Procrustes Analysis
‣ The GPA output
‣ Example

GPA
‣ what is GPA
‣ how does it work
‣ Conventional or Free Choice Profiling
‣ the permutation test
‣ advantage over classical PCA: individual results can
be shown in consensus space
‣ NOTE: GPA will average in case of replicates,
unless you define to treat replicates as different
products

The Procrustean principles
‣ make individual configurations for each subject
(PCA’s)
‣ make these individual configurations fit each other
‣ do this by moving them to a common origin
‣ stretch or shrink each configuration in order to make
it fit as good as possible
‣ if needed, flip them around
‣ Procrustes only allows ‘rigid-body’ transformations
to the datasets
‣ these transformations respect the relative distances
between objects

the problem
‣ consider K configurations of n objects in p-dimensional spaces
‣ how can we represent the K configurations in a common space
while minimizing the goodness of fit criterion?
‣ we do this with the aid of 3 transformations:
‣ translation: move the centroids of each configuration to a common
origin
‣ isotropic scaling: shrink or stretch each configuration isotropically
to make them as similar as possible
‣ rotation/reflection: turn or flip the configurations
‣ with these 3 transformations the individual spaces are made as
similar as possible
‣ next compute a Group-Average-Space of these individual
spaces and compute the difference between the Group and
Individual spaces (the residuals)
‣ minimize the total residual by applying the 3 transformations

Conventional or Free Choice Profiling
‣ in the case of Free Choice Profiling, each assessor is using
different attributes so standard PCA is not possible and GPA is
one of the few solutions
‣ in the case of conventional profiling, we assume that assessors
use the attributes in the same way so we can average over
assessors. However, the following effects occur:
‣ level effect (assessors use different parts of the scale)
‣ range effect (assessors use the whole scale from 0 to 100 or rate
everything between 30-70)
‣ interpretation effects (assessors differ in their interpretation of the
meaning of an attribute)
‣ using GPA, we take these effects into account through the
three Procrustean transformations!

The permutation test in GPA
‣ in contrast with PCA, the amount of variance
explained in itself does not give an indication for the
significance or fit of the final solution
‣ a permutation test is used to estimate the odds that
a ‘random’ dataset would give a similar percentage
consensus variance:
‣ we take the original dataset, permute the rows within each
set and run an analysis
‣ we repeat this 200 times
‣ the 90th and 98th percentile of the percentage of
consensus variance from these permuted sets are
compared to the percentage in the actual dataset

Structuur QDA data
GPA data matrix-conventional profiling
(N ´ M) datamatrix X k
N products
K assessors
M attributes
3-mode data structure conventional
profiling:
N products are rated by K assessors
on M attributes

Structuur QDA data
GPA data matrix-Free Choice Profiling
K-sets data
K assessors
N products
X
1
X
2
X
3
X
K
M 1 attributes
M 2 attributes
M 3 attributes
M K attributes
Data structure representing Free Choice Profiling data: N products
are judged by K assessors each using M k
attributes.

The dataset
‣ 10 assessors (‘sets’), 10 products (‘objects), 2
presentations
‣ 11 attributes: flower, rose, evergreen, wood, burnt,
alcohol, pungent, medicinal, sulphur, grape, bitter
‣ in total 2200 datapoints
dataset with courtesy from
Compusense, data from expert
wine panel, original attribute
names have been recoded,
only part of the data is used

Data example
Replica Assessor Product flower rose evergreen wood burnt alcohol pungent medicinal sulpher
1 1 4 22 12 24 1 2 71 61 8 1
2 1 4 25 7 7 1 1 50 16 8 1
1 1 5 2 1 1 1 1 91 62 10 2
2 1 5 1 1 2 10 1 47 20 10 1
1 1 7 21 0 1 1 1 74 50 10 1
2 1 7 1 6 8 0 2 62 38 7 0
1 1 8 24 1 36 1 1 50 25 6 1
2 1 8 1 1 2 9 1 61 43 7 14
1 1 10 49 17 8 1 1 74 62 6 9
2 1 10 21 2 11 2 1 74 62 8 1
1 1 11 1 1 8 1 1 61 42 1 0
2 1 11 18 1 7 1 1 51 20 6 1
1 1 13 50 25 10 1 2 74 60 13 4
2 1 13 43 15 7 2 0 99 86 2 1
1 1 17 39 20 8 1 8 89 74 7 1
2 1 17 47 19 6 1 1 72 31 7 1
1 1 18 0 1 1 9 15 52 25 9 26
2 1 18 21 1 7 9 13 71 33 8 31
1 1 20 51 18 8 1 4 87 86 10 2
2 1 20 15 1 6 1 0 57 11 7 1
1 4 4 18 0 11 0 0 47 80 0 22
2 4 4 12 25 4 0 0 47 82 0 8
1 4 5 16 0 0 0 0 43 78 0 17
2 4 5 18 0 0 0 5 77 81 0 0
1 4 7 15 0 7 0 0 57 64 0 0
2 4 7 9 0 0 0 4 65 78 34 0
1 4 8 8 0 8 0 2 78 83 0 0
2 4 8 10 0 14 0 0 75 80 14 0
1 4 10 12 28 0 0 0 69 77 0 5
2 4 10 28 25 6 0 0 78 89 0 0
1 4 11 13 15 17 0 0 73 82 0 0
2 4 11 17 24 7 0 0 57 56 0 0
1 4 13 21 28 0 0 2 43 81 6 12
2 4 13 27 31 4 0 0 57 86 0 20

Repeated measures
Table of means
obj 4
obj 5
obj 7
obj 8
obj 10
obj 11
obj 13
obj 17
obj 18
obj 20
flowers
17
15
19
18
26
16
28
22
13
24
rose
13
13
18
13
18
15
23
16
10
17
evergreen
17
11
10
16
12
13
13
15
10
15
wood
8
3
6
4
4
3
4
3
12
2
burnt
9
3
4
7
3
3
5
3
9
4
alcohol
51
54
53
55
62
51
61
62
54
56
pungent
55
55
54
59
65
53
67
62
59
60
medicinal
4
3
9
5
5
2
6
4
6
4
sulphur
20
3
5
4
3
2
7
2
25
6
grape
37
39
41
39
41
41
39
39
30
41
bitter
49
43
49
43
51
47
52
49
47
50
red: significant at 1%
blue: significant at 5%

Running GPA
Step 2
Step 3
Step 1
(click Permutation
test)

Repeated measures
GPA output: permutation test
Permutation Results
Total Variance Accounted (TVA) for in the Real Data Set : 70.8 at 0 %
Upper 10 % of the TVA in the Permutated Data Sets : 67.21
Upper 5 % of the TVA in the Permutated Data Sets : 67.78
Conclusion: the ‘real’ dataset explains significantly more variance than
the permutated datasets

Repeated measures
GPA output: Procrustes Anova
PANOVA Results per Dimension
Real Resid Total
Dim 1 23.59 7.22 30.81
Dim 2 14.40 5.53 19.93
Dim 3 9.56 3.40 12.96
Total 47.55 16.15 63.70
47.55% of the total variance
is explained by the data
after averaging the individual
configurations into the group
configuration, 36.3% of the
original variance is lost

Repeated measures
GPA output: individual weights
amount of stretching or shrinking of individual assessors
2.0
Weights by Set
1.5
1.0
0.5
0.0
set 1 set 2 set 3 set 4 set 5 set 6 set 7 set 8 set 9 set 10
Configurations of assessors with weights below 1 are shrunk and
above 1 are stretched, these assessors differ in their scaling behavior

Repeated measures
GPA output: Procrustes Anova
0.15
amount of variance explained by object for each dimension
Real Variance by Object
Dim3
0.10
Dim2
0.05
0.00
object 4 object 7 object 10 object 13 object 18
object 5 object 8 object 11 object 17 object 20
Dim1
Object 18 is mainly characterized by the attributes which make up
dimension 1 and object 5 by the attributes which make up dimension 2;
object 11 is located in the center of the product space

Repeated measures
GPA output: Group average space
GPA Group Average : dimension 1 versus 2
1.57
object 18 and 17 are the most
different objects on dimension 1
object 18
object 17
object 20
object 10
object 13
-1.57
object 8
1.57
object 11
object 4
object 7
object 5
-1.57
object 5 and 7 are
discriminated on
dim. 2 but not on dim. 1
object 8 and 11 have no
variance on the first
two dimensions

Repeated measures
GPA output: Interpretation of axis
GPA Group Average : dimension 1 versus 2
this axis represents sulpher and not-grape,
object 18 is rated very high on this aspect,
objects 5, 10 and 14 are rated low
1.35
this axis represents pungent,
alcohol, objects 20, 13, 10 and 17
are rated high on these aspect,
objects 4, 7 and 5 are rated low
object 18 pungent object 17
object 20 alcohol
sulphur
bitter object 10
medicinal object 13
wood
-1.35
flowers
1.35
burnt
object
object evergreen
8
11 rose
object 4
grape
object 7
object 5
-1.35

Repeated measures
GPA output: Individual ratings
in GPA, the ratings of each individual can
be shown in the product space or the
individual ratings for each attribute
i.e. set 1 for all attributes or an attribute
for all sets
selecting individuals will show the product
location of all individuals in the space

Repeated measures
GPA output: Individual product locations
GPA Group Average : dimension 1 versus 2
1.50
set 10
set 9
set 5
set 10
set 10
set 4
set 5 set 6
set 8object 18
set 6
set set 9
set
set 3
set
2
6
set set 1 6
1
set 3 set38
object 17
object set 5set 20 10
set 7
set 8 set set 2 1
set 2 set 4 set
set 83
9
set
set
7
2set 7
set object
1 45 29
set 10 6
set 9 object 13set 3
-1.50 7
set 7set set
10 set 7
set 7
4set 2
set set 8 4
set 5
1.50
set 3
set object
object 6 8
set 11
8 set 9 set 4set 3 9
set 2 set 4
set 10
object set 5
set 7
set
4 set
set 1
6 2 set 7
set 8
set
1
18 set 10 set 5
set 8
set set object set 5 5 set 9
1 7 set 7
set 9
set
set 3 set 4
set 2 10 set 4
set 6
3
set set 6 2 set set 3 8
set 1set 10
object 5 set 5
set 6
set 10
set 9
-1.50
this graph shows the product locations for the individual assessors (blue)
and the location of the product for the group (averaged over assessors)

Repeated measures
GPA output: attribute ratings of an individual
all attributes for assessor 8 (set 8) in the group space
GPA Group Average : dimension 1 versus 2
1.50
object 18
object 17
object 20
set 8 sulphur
set 8 grape object 10
-1.50
object 13
set 8 flowers
set object set 8object medicinal 8 set wood 8
811
burnt
object 4
1.50
object 7 set
set
8 alcohol
8 pungent
set 8 rose
set 8 evergreen
set 8 bitter
object 5
-1.50

Repeated measures
GPA output: attribute ratings of an individual
all assessors for attribute sulpher (products differ significantly for this attribute)
GPA Group Average : dimension 1 versus 2
1.50
set 4 sulphur
object 18 set 5 sulphur
set 1 sulphur
object 17
set 9 sulphur
set 2 sulphur
object 20
set 10 8 sulphur
set 3 sulphur
object 10
set 6 sulphur
object 13
-1.50 1.50
object
object
8
11
set 7 sulphur object 4
object 7
object 5
good agreement
-1.50

Repeated measures
GPA output: attribute ratings of an individual
all assessors for attribute bitter (products do not differ from each other for this
attribute)
GPA Group Average : dimension 1 versus 2
1.50
set 9 bitter
object 18
set 1 bitter set 5 bitter
set 6 bitter
object 17
object 20
set 4 bitter object 10
object set 132 bitter
set 3 bitter
-1.50 1.50
object
object
8
11
object 4
set 7 bitter
object 7
set 8 bitter
object 5
set 10 bitter
-1.50

Repeated measures
GPA with replicates as individual products
define replicates as
individual products in
selection menu (sel
button)
in this way, you can
visualize the variability
between the replicates

Repeated measures
GPA with replicates as individual products
GPA Group Average : dimension 1 versus 2
object 4(2)
object 5(2)
0.60
large differences for object
4, 5 and 7
object 7(1)
object 11(2)
object 7(2) object 8(1) object 11(1) object 5(1)
object 13(1) 8(2) object object 13(2) 10(2)
-0.60
object 10(1)
0.60
object 20(1)
object 17(1)
object 20(2) object 17(2)
small difference
object 18(1)
object 4(1)
object 18(2)
small difference
-0.60
objects n (1) and (2) are
replicates

Repeated measures
GPA with replicates as individual products
Ratings of objects on dim 1 and dim 2
0.8
0.6
0.4
0.2
0
-0.2
-0.4
obj 4
obj 5
obj 7
Dim 1-rep 1 Dim 1-rep 2
obj 13
obj 11
obj 10
obj 8
obj 17
obj 18
obj 20
objects 4 and 5
are rated quite
differently in rep 1
and 2, the other
objects are rated
almost identical
-0.6
-0.8
0.6
Dim 2-rep 1 Dim 2-rep 2
0.4
0.2
0
-0.2
-0.4
obj 4
obj 5
obj 7
obj 8
obj 10
obj 11
obj 13
obj 17
obj 18
obj 20
-0.6
-0.8

Repeated measures
GPA for each replicate seperately
GPA Group Average : dimension 1 versus 2
1.40
replicate 1
GPA Group Average : dimension 1 versus 2
1.40
object 18
replicate 2
object 17
object sulphur 4 object 20
pungent bitter alcohol
object 18
evergreen rose flowers
-1.40
wood burnt medicinal
object 13 10
1.40
object 11 grape
object 8 object 5
object 7
sulphur
object 20
wood
burnt medicinal object 17
-1.40
bitter pungent
1.40
object 4 object object 7 object 13
alcohol 11
evergreen object
rose flowers object 8 10
grape
object 5
-1.40
-1.40
objects 4 and 5 are rated quite differently in rep 1 and 2, the other
objects are rated almost identical

Repeated measures
Running different scenarios
One of the advantages of Senstools
is that you can run different scenarios
without much difficulty.
It is very easy to temporary remove
objects, attributes or subjects from the
analysis using the selection menu
(left)
Furthermore, it is also useful to look
at the dataset from different perspectives
(compare GPA results with PCA
results)

Repeated measures
To summarize
‣ GPA is an useful tool for the inspection of profiling data, especially
because it shows the individuals in the group space
‣ the use of the permutation test is strongly advised
Questions about GPA or Senstools v3.3?
Contact Pieter Punter at pieter@opp.nl
see also: Example-panelmonitoring using Senstools v3.3