Metodi Quantitativi per Economia, Finanza e Management Lezione n

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Metodi Quantitativi per Economia, Finanza e
ManagementLezione n9

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Analisi fattoriale

• I problemi di una analisi di questo tipo sono
• a)-quante componenti considerare
• rapporto tra numero di componenti e variabili
• percentuale di varianza spiegata
• le comunalità
• lo scree plot
• interpretabilità delle componenti e loro
  rilevanza nella esecuzione dellanalisi
  successive
• b)-come interpretarle
• correlazioni tra componenti principali e
  variabili originarie
• rotazione delle componenti

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Analisi Fattoriale

1. Qualità degli ingredienti
2. Genuinità
3. Leggerezza
4. Sapore/Gusto
5. Caratteristiche Nutrizionali
6. Attenzione a Bisogni Specifici
7. Lievitazione Naturale
8. Produzione Artigianale
9. Forma/Stampo
10. Richiamo alla Tradizione
11. Grandezza della Confezione (Peso Netto)
12. Funzionalità della Confezione
13. Estetica della Confezione
14. Scadenza
15. Nome del Biscotto
16. Pubblicità e Comunicazione
17. Promozione e Offerte Speciali
18. Consigli per lUtilizzo
19. Prezzo

• Sono stati individuati 20 attributi
  caratterizzanti il prodotto-biscotto
• È stato chiesto allintervistato di esprimere un
  giudizio in merito allimportanza che ogni
  attributo esercita nellatto di acquisto

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Analisi fattoriale
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1. The ratio between the number of components and
the variables One out of Three 20 original
variables 6-7 Factors
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2. The percentage of the explained variance
Between 60-75
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Factor Analysis
3. The scree plot The point at which the scree
begins
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4. Eigenvalue Eigenvaluesgt1
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Factor Analysis
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Analisi Fattoriale
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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
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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

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6. Interpretation Correlation between Input
Vars Factors The new Factors must have a
meaning based on the correlation structure
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6. Interpretation The correlation structure
between Input Vars Factors In this case
the correlation structure is well defined and the
interpretation phase is easier
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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
• The correlation between the principal components
  and the original variables
• The rotation of the principal components

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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 dont provide
  interpretation that are remarkably different from
  one another, as matter of fact they differ only
  slightly and there are areas of ambiguity

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Factor Analysis
x4
x3
Interpretation of the factors
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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 doesnt change, whereas the
  percentage of the variance explained by each
  factors shifts

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Analisi Fattoriale
Before the rotation step
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Analisi Fattoriale
After the rotation step
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5. Communalities The communalities dont change
after the Rotation Step
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6. Interpretation The correlation structure
between Input Vars Factors improves after
the rotation step
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6. Interpretation The correlation structure
between Input Vars Factors The variable
with the lowest communality is not well explained
by this solution
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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.

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Factor Analysis

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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
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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.

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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!
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6.2. Choosing the right number of factors

1. 1/3 criteria 21/3 7 factors
2. Variance explained (60-75) 7, 8, 9, 10
   factors
3. Scree Plot 6, 8, 10 factors
4. Eigenvalues 6, 7, 8 factors

The optimal values seem to be 7 or 8 factors.
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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.
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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.
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6.4. Factor Analysis with 20 FactorsAfter
elimination of the Price variable

1. 1/3 criteria 20/3 6 factors
2. Variance explained (60-75) 7, 8, 9 factors
3. Scree Plot 6, 7, 9 factors
4. Eigenvalues 6, 7, 8 factors

The optimal choice seems to be 7 factors.
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6.4. Factor Analysis with 20 FactorsAfter
elimination of the Price variable-continued-
The present Scree Plot represents the number 3
criteria of number of factors selection from the
previous slide.
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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!
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6.6. Factors - explained

• 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/ non-smoking factor
• Familiarity factor
• Variety/To Go factor
• Traditionality Addiction factor

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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
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Quantitative Methodsclass 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

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Factor Analysis

1. Home
2. Car
3. Outside in general
4. Office/University
5. Shops
6. Restaurants
7. Bars/discoteque
8. Record player
9. Cassette player
10. CD player
11. Digital player
12. Car stereo
13. House stereo
14. Radio
15. Mobile phone
16. USE record player
17. USE cassette player
18. USE CD Player
19. USE digital player

• 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 dont like it 9 I love it)
• or to use percentages
• Quest.n.4 score
• Quest.n.9 score
• Quest.n.10

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Factor Analysis
Number of factors 9 Extraction Principal
Component Analysis Max number of interaction
25 Rotation Varimax
First hypothesis
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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
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Factor Analysis
Number of factors 8 Extraction Principal
Component Analysis Max number of interaction
25 Rotation Varimax
Second hypothesis
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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 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
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Factor Analysis
Scree plot ADEQUATE Quite linear slope 
From the 9th component , there is little increase
in significance explained.
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Factor Analysis Interpretation

• Problems with the 9th component its over.
• We choosed Varimax option to minimize the number
  of variables that have elevated saturations for
  each factor

WE CHOOSE THE SECOND HYPOTHESIS
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(No Transcript)
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Factor Analysis Interpretation
Office/University Shops Restaurants Bars/Discoteque OUTSIDE LISTENING
Record player Use record player Cassette player Use cassette player STEREO
Digital player Use digital player DIGITAL PLAYER
Radio Use radio RADIO
Car Car stereo CD player Use CD player CAR LISTENING
Home House stereo Use house stereo HOUSE LISTENING
Outside in general Use PC PC
Mobile phone Use mobile phone MOBILE PHONE