Metodi Quantitativi per Economia, Finanza
e Management
Lezione n°9
Analisi fattoriale
I problemi di una analisi di questo tipo sono:
a)-quante componenti considerare
1. rapporto tra numero di componenti e variabili;
2. percentuale di varianza spiegata;
3. le comunalità
4. lo scree plot;
5. interpretabilità delle componenti e loro rilevanza nella
esecuzione dell’analisi successive
b)-come interpretarle
1. correlazioni tra componenti principali e variabili originarie
2. rotazione delle componenti
Analisi Fattoriale

Sono stati individuati 20
attributi caratterizzanti il
prodotto-biscotto

È stato chiesto
all’intervistato di
esprimere un giudizio in
merito all’importanza che
ogni attributo esercita
nell’atto di acquisto
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
Qualità degli ingredienti
Genuinità
Leggerezza
Sapore/Gusto
Caratteristiche Nutrizionali
Attenzione a Bisogni Specifici
Lievitazione Naturale
Produzione Artigianale
Forma/Stampo
Richiamo alla Tradizione
Grandezza della Confezione
(Peso Netto)
Funzionalità della Confezione
Estetica della Confezione
Scadenza
Nome del Biscotto
Pubblicità e Comunicazione
Promozione e Offerte Speciali
Consigli per l’Utilizzo
Prezzo
Notorietà della Marca
Analisi fattoriale
Correlations
Qualità degli ingredienti
Genuinità
Leggerezza
Sapore/gusto
Caratteris tiche nutrizionali
Pears on Correlation
Sig. (2-tailed)
N
Pears on Correlation
Sig. (2-tailed)
N
Pears on Correlation
Sig. (2-tailed)
N
Pears on Correlation
Sig. (2-tailed)
N
Pears on Correlation
Sig. (2-tailed)
N
Qualità degli
ingredienti
1
**. Correlation is s ignificant at the 0.01 level (2-tailed).
220
.629**
.000
220
.299**
.000
218
.232**
.001
220
.234**
.001
214
Caratteris tich
Genuinità Leggerezza Sapore/gusto
e nutrizionali
.629**
.299**
.232**
.234**
.000
.000
.001
.001
220
218
220
214
1
.468**
.090
.354**
.000
.181
.000
220
218
220
214
.468**
1
.030
.460**
.000
.657
.000
218
219
219
213
.090
.030
1
-.015
.181
.657
.823
220
219
221
215
.354**
.460**
-.015
1
.000
.000
.823
214
213
215
215
Total Variance Explained
Component
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Total
4.171
2.678
1.843
1.376
1.129
1.016
.937
.881
.781
.751
.682
.592
.568
.550
.453
.386
.376
.324
.270
.236
Initial Eigenvalues
% of Variance Cumulative %
20.853
20.853
13.389
34.241
9.216
43.457
6.879
50.336
5.643
55.979
5.079
61.057
4.684
65.741
4.405
70.146
3.907
74.054
3.756
77.810
3.412
81.222
2.960
84.183
2.838
87.021
2.750
89.771
2.267
92.038
1.930
93.968
1.880
95.848
1.621
97.470
1.352
98.822
1.178
100.000
Extraction Method: Principal Component Analysis .
1. The ratio between
the number of
components and the
variables:
One out of Three
20 original variables
6-7 Factors
Total Variance Explained
Component
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Total
4.171
2.678
1.843
1.376
1.129
1.016
.937
.881
.781
.751
.682
.592
.568
.550
.453
.386
.376
.324
.270
.236
Initial Eigenvalues
% of Variance Cumulative %
20.853
20.853
13.389
34.241
9.216
43.457
6.879
50.336
5.643
55.979
5.079
61.057
4.684
65.741
4.405
70.146
3.907
74.054
3.756
77.810
3.412
81.222
2.960
84.183
2.838
87.021
2.750
89.771
2.267
92.038
1.930
93.968
1.880
95.848
1.621
97.470
1.352
98.822
1.178
100.000
Extraction Method: Principal Component Analysis .
2. The percentage of the
explained variance:
Between 60%-75%
Factor Analysis
3. The scree plot :
The point at which
the scree begins
Total Variance Explained
Component
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Total
4.171
2.678
1.843
1.376
1.129
1.016
.937
.881
.781
.751
.682
.592
.568
.550
.453
.386
.376
.324
.270
.236
Initial Eigenvalues
% of Variance Cumulative %
20.853
20.853
13.389
34.241
9.216
43.457
6.879
50.336
5.643
55.979
5.079
61.057
4.684
65.741
4.405
70.146
3.907
74.054
3.756
77.810
3.412
81.222
2.960
84.183
2.838
87.021
2.750
89.771
2.267
92.038
1.930
93.968
1.880
95.848
1.621
97.470
1.352
98.822
1.178
100.000
Extraction Method: Principal Component Analysis .
4. Eigenvalue:
Eigenvalues>1
Factor Analysis
Total Variance Explained
Component
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Total
4.171
2.678
1.843
1.376
1.129
1.016
.937
.881
.781
.751
.682
.592
.568
.550
.453
.386
.376
.324
.270
.236
Initial Eigenvalues
% of Variance Cumulative %
20.853
20.853
13.389
34.241
9.216
43.457
6.879
50.336
5.643
55.979
5.079
61.057
4.684
65.741
4.405
70.146
3.907
74.054
3.756
77.810
3.412
81.222
2.960
84.183
2.838
87.021
2.750
89.771
2.267
92.038
1.930
93.968
1.880
95.848
1.621
97.470
1.352
98.822
1.178
100.000
Extraction Method: Principal Component Analysis .
Total Variance Explained
Analisi Fattoriale
Component
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Total
4.171
2.678
1.843
1.376
1.129
1.016
.937
.881
.781
.751
.682
.592
.568
.550
.453
.386
.376
.324
.270
.236
Initial Eigenvalues
% of Variance Cumulative %
20.853
20.853
13.389
34.241
9.216
43.457
6.879
50.336
5.643
55.979
5.079
61.057
4.684
65.741
4.405
70.146
3.907
74.054
3.756
77.810
3.412
81.222
2.960
84.183
2.838
87.021
2.750
89.771
2.267
92.038
1.930
93.968
1.880
95.848
1.621
97.470
1.352
98.822
1.178
100.000
Extraction Method: Principal Component Analysis.
Extraction Sums of Squared Loadings
Total
% of Variance Cumulative %
4.171
20.853
20.853
2.678
13.389
34.241
1.843
9.216
43.457
1.376
6.879
50.336
1.129
5.643
55.979
1.016
5.079
61.057
Communalities
Qualità degli ingredienti
Genuinità
Leggerezza
Sapore/gusto
Caratteristiche nutrizionali
Attenzione a bisogni
s pecifici
Lievitazione naturale
Produzione artigianale
Forma e s tampo
Richiamo alla tradizione
Grandezza della
confezione (peso netto)
Funzionalità della
confezione
Estetica della confezione
Scadenza
Nome del biscotto
Pubblicità e
comunicazione
Promozioni e offerte
s peciali
Consigli per l'utilizzo
Prezzo
Notorietà della marca
Initial
1.000
1.000
1.000
1.000
1.000
Extraction
.717
.746
.588
.670
.631
1.000
.332
1.000
1.000
1.000
1.000
.674
.762
.689
.600
1.000
.579
1.000
.414
1.000
1.000
1.000
.599
.432
.494
1.000
.717
1.000
.736
1.000
1.000
1.000
.463
.653
.716
Extraction Method: Principal Component Analysis.
5. Communalities:
The quote of
explained
variability for each
input variable
must be
satisfactory
In the example
the overall
explained
variability (which
represents the
mean value) is
0.61057
Factor Analysis
 6. Interpretation: Component Matrix (factor loadings)
 The most relevant output of a factorial analysis is the
so called “component matrix”, which shows the
correlations between the original input variables and
the obtained components (factor loadings)
 Each variable is associated specifically to the factors
(components) with which there is the highest
correlation
 The interpretation of the each factor has to be guided
considering the variables with the highest correlations
related to single factor
Component Matrixa
Qualità degli ingredienti
Genuinità
Leggerezza
Sapore/gusto
Caratteristiche nutrizionali
Attenzione a bisogni
s pecifici
Lievitazione naturale
Produzione artigianale
Forma e s tampo
Richiamo alla tradizione
Grandezza della
confezione (peso netto)
Funzionalità della
confezione
Estetica della confezione
Scadenza
Nome del biscotto
Pubblicità e
comunicazione
Promozioni e offerte
s peciali
Consigli per l'utilizzo
Prezzo
Notorietà della marca
1
.418
.383
.426
.163
.410
2
-.513
-.717
-.478
-.079
-.364
Component
3
4
.072
.099
.082
-.080
.136
-.349
.195
.671
.298
-.417
.410
-.220
-.214
-.197
-.032
-.172
.624
.573
.482
.615
-.360
-.339
.320
.046
-.309
-.160
-.272
-.269
.019
.377
.202
.372
-.228
-.374
.430
-.082
-.083
-.109
-.234
-.045
.403
.287
.461
.196
.209
-.197
.483
.131
.162
-.123
.081
-.340
.463
.390
.416
.439
-.158
.306
-.383
.100
-.383
-.026
.088
-.126
.174
-.473
.252
-.118
-.118
.032
.421
.525
-.145
-.331
-.062
.361
.340
.419
.660
-.062
-.025
.073
.629
.429
.413
.123
.265
.434
.093
.594
-.115
-.173
.129
-.121
-.058
-.166
-.305
.104
-.047
.486
Extraction Method: Principal Component Analysis.
a. 6 components extracted.
5
.375
.137
.162
.229
.100
6
.353
.231
.105
.310
-.240
6. Interpretation:
Correlation
between
Input Vars
&
Factors
The new Factors
must have a
meaning based
on the
correlation
structure
Rotated Component Matrixa
Genuinità
Leggerezza
Qualità degli ingredienti
Caratteristiche nutrizionali
Attenzione a bisogni
s pecifici
Promozioni e offerte
s peciali
Prezzo
Grandezza della
confezione (peso netto)
Funzionalità della
confezione
Forma e s tampo
Estetica della confezione
Nome del biscotto
Produzione artigianale
Lievitazione naturale
Scadenza
Richiamo alla tradizione
Notorietà della marca
Pubblicità e
comunicazione
Consigli per l'utilizzo
Sapore/gusto
Component
3
4
-.123
.237
-.007
.096
.078
.080
.009
.111
1
.795
.748
.716
.619
2
-.089
.072
-.026
.312
5
-.051
.050
.007
-.127
6
.178
-.104
.437
-.349
.327
-.054
.243
.324
.020
-.239
.002
.799
-.052
-.111
.286
.035
-.015
.764
-.063
.180
.154
.092
.017
.697
.250
.006
-.067
.159
.158
.448
.334
.165
-.028
-.219
-.011
-.096
.071
.158
.369
.066
.023
-.083
.163
.065
-.040
.028
-.103
.211
.082
.108
.799
.704
.624
.083
.224
-.137
.439
.103
.070
.107
.005
.836
.681
.593
.566
.161
-.024
.268
.309
-.023
.094
.078
.132
.811
.137
-.076
-.047
.172
-.065
-.086
.251
.051
-.002
.139
.310
-.055
.764
-.119
.282
.048
.342
.163
.228
.025
.234
.083
.394
-.074
-.064
.793
Extraction Method: Principal Component Analysis.
Rotation Method: Varimax with Kais er Normalization.
a. Rotation converged in 6 iterations.
6. Interpretation:
The correlation
structure
between
Input Vars
&
Factors
In this case the
correlation
structure is well
defined and the
interpretation
phase is easier
Factor Analysis
Issues of the Factor Analysis are the following:
a) How many Factors (or components) need to be considered
6. The degree of the interpretation of the components and how they
affect the next analyses
b) How to interpret
1. The correlation between the principal components and the original
variables
2. The rotation of the principal components
Factor Analysis
 6. Interpretation: The rotation of factors
 There are numerous outputs of factorial analysis
which can be produced through the same input data
 These numerous outputs don’t provide interpretation
that are remarkably different from one another, as
matter of fact they differ only slightly and there are
areas of ambiguity
Factor Analysis
CF*j
CFj
x1
x4
CF*i
x2
CFi
x3
The coordinates of the graph
are the factor loadings
Interpretation of the
factors
Factor Analysis
 6. Interpretation: The rotation of factors
 The Varimax method of rotation, suggested by Kaiser, has the
purpose of minimizing the number of variables with high
saturations (correlations) for each factor
 The Quartimax method attempts to minimize the number of
factors tightly correlated to each variable
 The Equimax method is a cross between the Varimax and the
Quartimax
 The percentage of the overall variance of the rotated factors
doesn’t change, whereas the percentage of the variance
explained by each factors shifts
Total Variance Explained
Analisi Fattoriale
Component
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Total
4.171
2.678
1.843
1.376
1.129
1.016
.937
.881
.781
.751
.682
.592
.568
.550
.453
.386
.376
.324
.270
.236
Initial Eigenvalues
% of Variance Cumulative %
20.853
20.853
13.389
34.241
9.216
43.457
6.879
50.336
5.643
55.979
5.079
61.057
4.684
65.741
4.405
70.146
3.907
74.054
3.756
77.810
3.412
81.222
2.960
84.183
2.838
87.021
2.750
89.771
2.267
92.038
1.930
93.968
1.880
95.848
1.621
97.470
1.352
98.822
1.178
100.000
Extraction Method: Principal Component Analysis.
Extraction Sums of Squared Loadings
Total
% of Variance Cumulative %
4.171
20.853
20.853
2.678
13.389
34.241
1.843
9.216
43.457
1.376
6.879
50.336
1.129
5.643
55.979
1.016
5.079
61.057
Before the rotation step
Total Variance Explained
Analisi Fattoriale
Component
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Total
4.171
2.678
1.843
1.376
1.129
1.016
.937
.881
.781
.751
.682
.592
.568
.550
.453
.386
.376
.324
.270
.236
Initial Eigenvalues
% of Variance Cumulative %
20.853
20.853
13.389
34.241
9.216
43.457
6.879
50.336
5.643
55.979
5.079
61.057
4.684
65.741
4.405
70.146
3.907
74.054
3.756
77.810
3.412
81.222
2.960
84.183
2.838
87.021
2.750
89.771
2.267
92.038
1.930
93.968
1.880
95.848
1.621
97.470
1.352
98.822
1.178
100.000
Extraction Method: Principal Component Analysis.
Rotation Sums of Squared Loadings
Total
% of Variance Cumulative %
2.490
12.448
12.448
2.294
11.468
23.917
2.214
11.068
34.984
2.203
11.016
46.000
1.736
8.680
54.680
1.276
6.378
61.057
After the rotation step
Communalities
Qualità degli ingredienti
Genuinità
Leggerezza
Sapore/gusto
Caratteristiche nutrizionali
Attenzione a bisogni
s pecifici
Lievitazione naturale
Produzione artigianale
Forma e s tampo
Richiamo alla tradizione
Grandezza della
confezione (peso netto)
Funzionalità della
confezione
Estetica della confezione
Scadenza
Nome del biscotto
Pubblicità e
comunicazione
Promozioni e offerte
s peciali
Consigli per l'utilizzo
Prezzo
Notorietà della marca
Initial
1.000
1.000
1.000
1.000
1.000
Extraction
.717
.746
.588
.670
.631
1.000
.332
1.000
1.000
1.000
1.000
.674
.762
.689
.600
1.000
.579
1.000
.414
1.000
1.000
1.000
.599
.432
.494
1.000
.717
1.000
.736
1.000
1.000
1.000
.463
.653
.716
Extraction Method: Principal Component Analysis.
5. Communalities:
The
communalities
don’t change after
the Rotation Step
Rotated Component Matrixa
Genuinità
Leggerezza
Qualità degli ingredienti
Caratteristiche nutrizionali
Attenzione a bisogni
s pecifici
Promozioni e offerte
s peciali
Prezzo
Grandezza della
confezione (peso netto)
Funzionalità della
confezione
Forma e s tampo
Estetica della confezione
Nome del biscotto
Produzione artigianale
Lievitazione naturale
Scadenza
Richiamo alla tradizione
Notorietà della marca
Pubblicità e
comunicazione
Consigli per l'utilizzo
Sapore/gusto
Component
3
4
-.123
.237
-.007
.096
.078
.080
.009
.111
1
.795
.748
.716
.619
2
-.089
.072
-.026
.312
5
-.051
.050
.007
-.127
6
.178
-.104
.437
-.349
.327
-.054
.243
.324
.020
-.239
.002
.799
-.052
-.111
.286
.035
-.015
.764
-.063
.180
.154
.092
.017
.697
.250
.006
-.067
.159
.158
.448
.334
.165
-.028
-.219
-.011
-.096
.071
.158
.369
.066
.023
-.083
.163
.065
-.040
.028
-.103
.211
.082
.108
.799
.704
.624
.083
.224
-.137
.439
.103
.070
.107
.005
.836
.681
.593
.566
.161
-.024
.268
.309
-.023
.094
.078
.132
.811
.137
-.076
-.047
.172
-.065
-.086
.251
.051
-.002
.139
.310
-.055
.764
-.119
.282
.048
.342
.163
.228
.025
.234
.083
.394
-.074
-.064
.793
Extraction Method: Principal Component Analysis.
Rotation Method: Varimax with Kais er Normalization.
a. Rotation converged in 6 iterations.
6. Interpretation:
The correlation
structure
between
Input Vars
&
Factors
improves
after the rotation
step
Rotated Component Matrixa
Genuinità
Leggerezza
Qualità degli ingredienti
Caratteristiche nutrizionali
Attenzione a bisogni
s pecifici
Promozioni e offerte
s peciali
Prezzo
Grandezza della
confezione (peso netto)
Funzionalità della
confezione
Forma e s tampo
Estetica della confezione
Nome del biscotto
Produzione artigianale
Lievitazione naturale
Scadenza
Richiamo alla tradizione
Notorietà della marca
Pubblicità e
comunicazione
Consigli per l'utilizzo
Sapore/gusto
Component
3
4
-.123
.237
-.007
.096
.078
.080
.009
.111
1
.795
.748
.716
.619
2
-.089
.072
-.026
.312
5
-.051
.050
.007
-.127
6
.178
-.104
.437
-.349
.327
-.054
.243
.324
.020
-.239
.002
.799
-.052
-.111
.286
.035
-.015
.764
-.063
.180
.154
.092
.017
.697
.250
.006
-.067
.159
.158
.448
.334
.165
-.028
-.219
-.011
-.096
.071
.158
.369
.066
.023
-.083
.163
.065
-.040
.028
-.103
.211
.082
.108
.799
.704
.624
.083
.224
-.137
.439
.103
.070
.107
.005
.836
.681
.593
.566
.161
-.024
.268
.309
-.023
.094
.078
.132
.811
.137
-.076
-.047
.172
-.065
-.086
.251
.051
-.002
.139
.310
-.055
.764
-.119
.282
.048
.342
.163
.228
.025
.234
.083
.394
-.074
-.064
.793
Extraction Method: Principal Component Analysis.
Rotation Method: Varimax with Kais er Normalization.
a. Rotation converged in 6 iterations.
6. Interpretation:
The correlation
structure
between
Input Vars
&
Factors
The variable with
the lowest
communality is
not well
explained by this
solution
Factor Analysis
 Once an adequate solution is found, it is possible to use
the obtained factors as new macro variables to consider
for further analyses on the phenomenon under
investigation, thus replacing the original variables;
 Again taking into consideration the example, we may add
six new variables into the data file, as follows:
 Health,
 Convenience & Practicality,
 Image,
 Handicraft,
 Communication,
 Taste.
 They are standardized variables: zero mean and variance
equal to one.
 They will be the input for further analyses of Dependence
or/and Interdependence.
Factor Analysis
Indentification of
the input variables
Standardization
P.C. methods first
findings
Number of factors
Rotation
Interpretation
8062 Quantitative Methods for Marketing
-Final Project –
Submission date: 5th of June 2009
Sushi Fever Team:
Bancheva, Kamelia
Bettinali, Francesco
Bonchev, Dimitar
Pasca, Ruxandra
Petranovic, Marija
&
Coffee Consumption in Italy
6. Factor Analysis
We ran a Factor Analysis on two numerical questions from
the survey that we felt might have correlated variables: Q15 (“What
are you general coffee preferences?”) and Q16 (“If you drink your
coffee outside (in a bar/coffee place) which are the main factors
that, in general, influence your decision on where you drink your
coffee?”).
• We used the Principal Components Method that was
supposed to solve the multicollinearity problem among our
variables and provide us with summarized number of
variables/factors which are not correlated (standardized by
definition, with mean 0, standard deviation 1) to better explain
and understand the specific situation of coffee consumption.
• This represents a preliminary phase for cluster analysis and
regression analysis.
6.1. Initial Variables used for
analysis
On the right, there are our
initial 21 variables (taken
from Q15 and Q16) that we
selected for running the
factor analysis.
Judging by the SPSS
Correlation Matrix (that is
not present in the slide
because of its big size –
please see the output for the
check), we have many
variables which are
significantly correlated.
Need for FACTOR
REDUCTION! Start real
Factor Analysis!
6.2. Choosing the right number of
factors
1.
2.
3.
4.
1/3 criteria: 21/3= 7 factors
Variance explained (60%-75%): 7, 8,
9, 10 factors
Scree Plot: 6, 8, 10 factors
Eigenvalues: 6, 7, 8 factors
The optimal
values seem
to be 7 or 8
factors.
6.2. Choosing the right number of
factors – continued -
The present
Scree Plot
represents the
number 3
criteria of
number of
factors selection
from the
previous slide.
6.3. Factor Analysis with 8 Factors
After analyzing the Communalities
table, we identified one variable
that is not properly explained by
our 8 selected factors (0.387 is not
satisfying)! This variable is Price
which we consider an important
variable in our analysis!
Decreasing the number of factors to 7,
will not improve the explanatory power
of the variables for the price!
We decided to exclude the Price
variable from this factor analysis and
consider it as a separate factor
(given its very high importance from
our qualitative point view) in the
future analysis: cluster & regression
analysis.
6.4. Factor Analysis with 20 Factors
After elimination of the Price variable
1.
2.
3.
4.
1/3 criteria: 20/3= 6 factors
Variance explained (60%-75%):
factors
Scree Plot: 6, 7, 9 factors
Eigenvalues:
6, 7, 8 factors
7, 8, 9
The optimal
choice seems to
be 7 factors.
6.4. Factor Analysis with 20 Factors
After elimination of the Price variable
-continued-
The present
Scree Plot
represents the
number 3
criteria of
number of
factors selection
from the
previous slide.
6.5. Factor Analysis with 7 Factors
After
analyzing
the
Communalities table, we that
so far the 7 factors properly
explain the initial variables. All
communalities are over 0.400,
which is a good result.
We are ready to take a look at the
Rotated Component Matrix to see
if the factors make sense/can be
explained!
6.6. Factors - explained
•
•
•
1.
2.
3.
4.
5.
6.
7.
The method used for
rotation was
Varimax.
After closely
analyzing the
Rotated Component
Matrix, we tried to
give meaning to our
7 factors.
The names of the
respective factors
are the following:
Socialization factor
Internet/
Trendiness factor
Close meeting
place factor
Intellectual/ nonsmoking factor
Familiarity factor
Variety/To Go factor
Traditionality &
Addiction factor
6.6. Factors – explained
- continued 1. Socialization Factor
Socialize, sit down, being with
friends, cozy atmosphere
2. Internet/Trendiness Factor
Wi-Fi availability, internet, trendy
place
3. Close meeting place Factor
Close to home/work/school,
ability to meet people, quality of
coffee not important
4. Intellectual/Non-smoking Factor
Non-smokers, usually snack,
love to read
5. Familiarity Factor
Go to the same bar, do not like
trying new places, concerned
about quality of coffee
6. Variety/To-go Factor
Variety and coffee to go, non
traditional Italian coffee,
preference for taking coffee
alone
7. Traditionality/Addiction Factor
Italian coffee preference, addicts
Quantitative Methods
class 22
The consumption of Digital Music
and its impact on the Music
Industry
Simona Bara
Claudio Boccassini
Danilo Broseghini
Himashu Chikara
Isabella Rossi
Federico Salvaggio
Alessandro Tiso
Factor Analysis
 We have taken into
consideration questions n° 4,9,10
and therefore we have 24
variables
We asked interviewees to give a
score from 1 to 9 (1: “I don’t like
it” 9: “I love it”)
or to use percentages
Quest.n.4: score
Quest.n.9: score
Quest.n.10: %
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
Home
Car
Outside in general
Office/University
Shops
Restaurants
Bars/discoteque
Record player
Cassette player
CD player
Digital player
Car stereo
House stereo
Radio
Mobile phone
USE record player
USE cassette player
USE CD Player
USE digital player
USE car stereo
USE house stereo
USE radio
USE PC
USE mobile phone
Factor Analysis
Number of factors: 9
First
hypothesis:
Extraction: Principal Component Analysis
Max number of interaction: 25
Rotation : Varimax
Total Vari ance Ex pla ined
Component
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Total
3,389
2,768
1,970
1,542
1,539
1,388
1,355
1,164
1,058
,956
,919
,823
,714
,689
,600
,565
,504
,464
,455
,355
,321
,253
,211
-7, 9E-017
Initial Eigenvalues
% of Variance Cumulative %
14,120
14,120
11,533
25,653
8,209
33,862
6,425
40,287
6,411
46,698
5,782
52,480
5,646
58,126
4,850
62,976
4,408
67,385
3,985
71,369
3,831
75,201
3,427
78,628
2,975
81,603
2,872
84,475
2,498
86,973
2,353
89,326
2,098
91,424
1,935
93,359
1,894
95,254
1,480
96,733
1,336
98,070
1,053
99,122
,878
100,000
-3, 31E-016
100,000
Ex trac tion Met hod: Principal Component Analys is.
Ex trac tion Sums of Squared Loadings
Total
% of Variance Cumulative %
3,389
14,120
14,120
2,768
11,533
25,653
1,970
8,209
33,862
1,542
6,425
40,287
1,539
6,411
46,698
1,388
5,782
52,480
1,355
5,646
58,126
1,164
4,850
62,976
1,058
4,408
67,385
Rotation Sums of Squared Loadings
Total
% of Variance Cumulative %
2,427
10,114
10,114
2,126
8,857
18,972
1,991
8,297
27,268
1,877
7,820
35,088
1,659
6,912
42,000
1,647
6,861
48,861
1,568
6,535
55,396
1,457
6,072
61,469
1,420
5,916
67,385
Factor Analysis
Ratio between
component number
and variable
number
ADEQUATE
For a set of 17 variables,
the ideal number of
components is 4-5.
In this case for a set of 24
variables, we have
considered 9 components
% global explained
variance
OK
About 68% - the optimal
range is 60% - 70%
Communalities
ADEQUATE
The values vary among
0,456 and 0,917
We found a problem looking at the rotated component matrix:
CORRELATION AMONG COMPONENTS AND ORIGINAL
VARIABLES
NON OPTIMAL
problematic 9th
component
Factor Analysis
Number of factors: 8
Second hypothesis:
Extraction: Principal Component Analysis
Max number of interaction: 25
Total Vari ance Ex pla ined
Component
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Total
3,389
2,768
1,970
1,542
1,539
1,388
1,355
1,164
1,058
,956
,919
,823
,714
,689
,600
,565
,504
,464
,455
,355
,321
,253
,211
-3, 1E-016
Initial Eigenvalues
% of Variance Cumulative %
14,120
14,120
11,533
25,653
8,209
33,862
6,425
40,287
6,411
46,698
5,782
52,480
5,646
58,126
4,850
62,976
4,408
67,385
3,985
71,369
3,831
75,201
3,427
78,628
2,975
81,603
2,872
84,475
2,498
86,973
2,353
89,326
2,098
91,424
1,935
93,359
1,894
95,254
1,480
96,733
1,336
98,070
1,053
99,122
,878
100,000
-1, 30E-015
100,000
Ex trac tion Met hod: Principal Component Analys is.
Ex trac tion Sums of Squared Loadings
Total
% of Variance Cumulative %
3,389
14,120
14,120
2,768
11,533
25,653
1,970
8,209
33,862
1,542
6,425
40,287
1,539
6,411
46,698
1,388
5,782
52,480
1,355
5,646
58,126
1,164
4,850
62,976
Rotation Sums of Squared Loadings
Total
% of Variance Cumulative %
2,634
10,974
10,974
2,339
9,744
20,718
1,891
7,880
28,598
1,810
7,541
36,139
1,776
7,399
43,538
1,721
7,171
50,709
1,486
6,191
56,900
1,458
6,077
62,976
Rotation : Varimax
Factor Analysis
Comm una litie s
"Casa"
"A utomobile"
"Fuori in generale"
"Ufficio/Università"
"Negoz i"
"Ristoranti"
"B ar/Discoteche"
"Regis tratore audio"
"Cassette player"
"CD player"
"Digital player"
"A utoradio"
"S tereo di c asa"
"Radio"
"Cellulare"
"USO Registrat ore audio"
"USO Cass ette play er"
"USO CD player"
"USO Digit al player"
"USO A utoradio"
"USO S tereo di cas a"
"USO Radio"
"USO Computer"
"USO Cellulare"
Initial
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
1,000
Ex trac tion
,626
,546
,522
,450
,623
,630
,450
,657
,797
,545
,670
,664
,646
,516
,736
,355
,431
,632
,870
,828
,679
,685
,814
,747
Ex trac tion Method: Principal Component Analysis .
Ratio between
component number
and variable
number
ADEQUATE
For a set of 17 variables,
the ideal number of
components is 4-5.
In this case for a set of 24
variables, we have
considered 8 components
% global explained
variance
OK
About 63% - the optimal
range is 60% - 70%
Communalities
ACCEPTABLE
The values vary among
0,431 and 0,870
Factor Analysis
Scree plot
ADEQUATE
From the 9th component , there is little increase in significance
explained.
“Quite linear
slope”
Factor Analysis
Interpretation
Rotated Component Matrixa
1
"Casa"
"Automobile"
"Fuori in generale"
"Ufficio/Università"
"Negozi"
"Ristoranti"
"Bar/Discoteche"
"Regis tratore audio"
"Cassette player"
"CD player"
"Digital player"
"Autoradio"
"Stereo di casa"
"Radio"
"Cellulare"
"USO Regis tratore audio"
"USO Cass ette player"
"USO CD player"
"USO Digital player"
"USO Autoradio"
"USO Stereo di casa"
"USO Radio"
"USO Computer"
"USO Cellulare"
,157
,547
,638
,760
,726
,536
,179
,242
,289
,242
2
3
,252
6
,409
-,195
,306
7
8
,296
-,168
,218
,310
,757
,827
,433
-,203
-,375
,448
-,279
,246
,211
-,264
Component
4
5
-,364
,499
,164
,668
,150
-,570
,570
,348
,654
,206
,332
,164
,180
,151
-,198
,198
,156
,742
1. Problems with the 9th
component it’s over.
2. We choosed Varimax
option to minimize the
number of variables that
have elevated saturations
for each factor
,825
,527
,584
,264
-,294
-,206
-,171
,665
-,857
,277
-,365
,360
-,248
-,190
-,207
,773
-,245
-,490
-,371
,816
,892
-,160
,824
Extraction Method: Principal Component Analys is.
Rotation Method: Varimax with Kaiser Normalization.
a. Rotation converged in 10 iterations.
WE CHOOSE THE SECOND
HYPOTHESIS
Rotated Component Matrixa
1
"Casa"
"Automobile"
"Fuori in generale"
"Ufficio/Università"
"Negozi"
"Ristoranti"
"Bar/Discoteche"
"Regis tratore audio"
"Cassette player"
"CD player"
"Digital player"
"Autoradio"
"Stereo di casa"
"Radio"
"Cellulare"
"USO Regis tratore audio"
"USO Cass ette player"
"USO CD player"
"USO Digital player"
"USO Autoradio"
"USO Stereo di casa"
"USO Radio"
"USO Computer"
"USO Cellulare"
,157
,547
,638
,760
,726
,536
,179
,242
,289
,242
2
3
,252
6
,409
-,195
,306
7
8
,296
-,168
,218
,310
,757
,827
,433
-,203
-,375
,448
-,279
,246
,211
-,264
Component
4
5
-,364
,499
,164
,668
,150
-,570
,570
,348
,654
,206
,332
,164
,180
,151
-,198
,198
,156
,742
,825
,527
,584
,264
-,294
-,206
-,171
,665
-,857
,277
-,365
,360
-,248
-,190
-,207
,773
-,245
-,490
-,371
,816
,892
-,160
Extraction Method: Principal Component Analys is.
Rotation Method: Varimax with Kaiser Normalization.
a. Rotation converged in 10 iterations.
,824
Factor Analysis
Interpretation
Office/University
Shops
Restaurants
Bars/Discoteque
Record player
Use record player
Cassette player
Use cassette player
Digital player
Use digital player
Radio
Use radio
Car
Car stereo
CD player
Use CD player
Home
House stereo
Use house stereo
OUTSIDE LISTENING
STEREO
DIGITAL PLAYER
RADIO
CAR LISTENING
HOUSE LISTENING
Outside in general
Use PC
PC
Mobile phone
Use mobile phone
MOBILE PHONE