Relationship Between Variables - SKEMA Ph.D programme

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Dec 6, 2012 (4 years and 17 days ago)

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Class 3

Relationship Between Variables

SKEMA Ph.D programme

2010
-
2011

Lionel Nesta

Observatoire Français des Conjonctures Economiques



Lionel.nesta@ofce.sciences
-
po.fr

Qualitative

×


Qualitative

Qualitative

×


Quantitative

Quantitative

×


Quantitative


Which variables are we looking at ?


Relationship Between Variables

ANOVA

ANOVA

ANOVA: ANalysis Of VAriance



ANOVA is a generalization of Student
t

test


Student test applies to two categories only:


H
0
: μ
1

= μ
2


H
1
: μ
1



μ
2


ANOVA is a method to test whether group means are
equal or not.



H
0
:
μ
1

=
μ
2

= μ
3

= ... = μ
n



H
1
: At least one mean differs significantly

ANOVA

This method is called after the fact that it is based on measures of
variance. The F
-
statistics is a ratio comparing the variance due to
group differences (explained variance) with the variance due to other
phenomena (unexplained variance).

explained variance
unexplained variance
F

Higher F means more explanatory power,
thus more significance of groups.

Revenues (in million of US $ )

Sector 1

Sector 2

Sector 3

Firm 1

18.0

21.5

34.8

Firm 2

18.0

21.5

34.8

Firm 3

18.0

21.5

34.8

Firm 4

18.0

21.5

34.8

Firm 5

18.0

21.5

34.8

Revenues (in million of US $ )

Sector 1

Sector 2

Sector 3

Firm 1

18.0

18.0

18.0

Firm 2

21.5

21.5

21.5

Firm 3

25.0

25.0

25.0

Firm 4

28.7

28.7

28.7

Firm 5

34.8

34.8

34.8

Revenues (in million of US $ )

Sector 1

Sector 2

Sector 3

Firm 1

19.6

23.7

30.8

Firm 2

19.4

28.4

32.9

Firm 3

21.9

28.5

35.3

Firm 4

21.2

31.7

31.8

Firm 5

24.6

37.0

35.7

Do sectors differ significantly in their revenues?

H
0

:
μ
1

=
μ
2

=
μ
3

= ... =
μ
n

H
1
: At least one mean differs significantly.

ANOVA







2 2
2
Total Variance Within-group variance Betwe
en-group variance
(Total Sum of Square) (Within sum of Squa
re) (between sum of Square)
SS SS SS
total within between
k k
n n
k k k
ij ij j k j
j i j i j
x x x x n x x
    
  
df = (k


1)

df = n


k

df = n


1

residual

This decomposition produces Fisher’s Statistics as follows:







_
_
1
explained variance
1,
unexplained variance

    

between
df num
df denom
within
SS k
F k N k F
SS N k
Origin of variation

SS

d.f.

MSS

F
-
Stat

Prob>F

SS
-
between

379.1

2

189.6

SS
-
within (residual)

132.5

12

11.0

SS
-
total

511.6

14

36.54

17.7

0.0003

The result tells me that I can reject the null Hypothesis H
0

with 0.03% chances
of rejecting the null Hypothesis H
0

while H
0

holds true (being wrong).

I WILL TAKE THE CHANCE!!!

The ANOVA decomposition on Revenues

Comparison of Means Using Student t
with
STATA

We still use the same command
ttest


ttest var1, by(varcat)


For example:


ttest lnassets, by(type)


ttest lnrd, by(year)


ttest lnrdi, by(type)




Beware!

Unlike
ANOVA
, Student t test can only be perfomed to compare
two categories
.


ANOVA under STATA

We still use the same command
anova


anova var1 varcat


For example
:


anova lnassets isic


anova lnrd isic


anova lnrdi isic



anova cours titype






















T
o
t
a
l




1
.
5
1
7
6
e
+
1
3


1
6
3
3


9
.
2
9
3
1
e
+
0
9

































































































R
e
s
i
d
u
a
l




1
.
0
5
6
4
e
+
1
2


1
4
7
9



7
1
4
2
6
6
3
1
8


















































f
i
d




1
.
4
1
1
9
e
+
1
3



1
5
4


9
.
1
6
8
4
e
+
1
0





1
2
8
.
3
6





0
.
0
0
0
0













































M
o
d
e
l




1
.
4
1
1
9
e
+
1
3



1
5
4


9
.
1
6
8
4
e
+
1
0





1
2
8
.
3
6





0
.
0
0
0
0
































































































S
o
u
r
c
e




P
a
r
t
i
a
l

S
S




d
f







M
S











F





P
r
o
b

>

F



























R
o
o
t

M
S
E






=

2
6
7
2
5
.
8





A
d
j

R
-
s
q
u
a
r
e
d

=


0
.
9
2
3
1



























N
u
m
b
e
r

o
f

o
b
s

=




1
6
3
4





R
-
s
q
u
a
r
e
d





=


0
.
9
3
0
4
.

a
n
o
v
a

l
a
b
o
u
r

f
i
d
Stata Instruction

Sum of Squares

F
-
Stat

P value

STATA Application:
ANOVA

Anova Example in Published Paper


Verify that US companies are larger than those from the rest of the world
with an ANOVA



Are there systematic


Sectoral differences in terms of labour; R&D, sales


Write out H
0

and H
1
for each variables


Analyse


Comparer les moyennes


ANOVA
à

un fateur


What do you conclude at 5% level?


What do you conclude at 1% level?


SPSS Application: ANOVA

SPSS Application: t test comparing means

Descriptives
35
447.4501
182.4318
30.83661
384.78256
510.117613
182.0091
817.9253
32
462.3145
310.5638
54.90044
350.34433
574.284688
19.5265
946.5801
281
32416.80
157435.7
9391.827
13929.247
50904.3542
16.0008
1193810
96
409.9650
453.3413
46.26895
318.10950
501.820453
11.1539
1665.716
100
193.4619
97.58658
9.7586578
174.09856
212.825145
49.3978
558.6539
153
8004.322
30796.25
2489.729
3085.3790
12923.2649
14.1116
184461.8
173
1387.709
1264.239
96.11829
1197.9855
1577.432087
141.0070
5852.729
208
17733.77
124017.6
8599.072
780.78382
34686.7595
123.0168
1664540
74
77161.50
222879.1
25909.17
25524.608
128798.396
281.2427
851216.2
45
1089.904
1240.178
184.8749
717.31279
1462.494371
1.0716
3790.107
155
251.1483
167.9513
13.49017
224.49859
277.797952
27.8838
1432.072
1352
14903.52
103262.3
2808.364
9394.2945
20412.7510
1.0716
1664540
55
50230.05
26169.055
3528.635
43155.57
57304.54
13588
104000
64
133708.02
96812.548
12101.569
109524.96
157891.07
20000
308000
306
55764.62
43392.780
2480.600
50883.36
60645.87
3619
181176
99
63445.73
45073.200
4530.027
54456.04
72435.42
2662
145787
120
36001.37
36324.601
3315.967
29435.42
42567.31
2998
149644
161
101231.85
95716.749
7543.537
86334.11
116129.59
1508
403508
177
128311.31
102126.3
7676.286
113161.90
143460.72
18200
417800
280
140859.11
153239.3
9157.799
122831.96
158886.27
647
876000
76
75601.54
42905.729
4921.625
65797.16
85405.92
11305
165000
65
185022.20
81524.803
10111.907
164821.34
205223.06
30964
317100
231
60497.76
42138.389
2772.502
55035.01
65960.51
1153
173000
1634
91298.87
96400.957
2384.818
86621.25
95976.50
647
876000
55
41423.22
35721.57
4816.696
31766.325
51080.11179
5627.646
121962.6
65
21827.52
15167.33
1881.276
18069.238
25585.80114
2590.539
52380.74
309
565218.4
2146365
122102.5
324957.84
805478.883
2158.768
12400000
99
29890.76
15579.40
1565.789
26783.498
32998.01180
9015.374
69895.68
120
12803.84
6396.795
583.9448
11647.575
13960.11274
2814.375
31224.46
161
821966.6
3180044
250622.6
327011.59
1316921.53
467.169
16600000
178
22379.21
18921.53
1418.229
19580.397
25178.02485
1679.668
79085.95
288
291520.4
1310460
77219.60
139531.82
443508.950
52.365
8071404
77
1522011
3744994
426781.6
672001.30
2372019.91
4679.127
12400000
67
23450.50
18731.51
2288.419
18881.521
28019.47136
38.080
81152.94
231
14908.32
11406.94
750.5212
13429.539
16387.09100
262.905
56015.21
1650
318383.6
1713117
42174.03
235663.33
401103.930
38.080
16600000
13
20
28
29
33
35
36
37
38
48
99
Total
13
20
28
29
33
35
36
37
38
48
99
Total
13
20
28
29
33
35
36
37
38
48
99
Total
rd
labour
sales
N
Moyenne
Ecart-type
Erreur
standard
Borne
inf érieure
Borne
supérieure
Intervalle de conf iance à
95% pour la moyenne
Minimum
Maximum
SPSS Application: t test comparing means

ANOVA
5.11E+011
10
5.11E+010
4.934
.000
1.39E+013
1341
1.04E+010
1.44E+013
1351
2.79E+012
10
2.79E+011
36.607
.000
1.24E+013
1623
7.63E+009
1.52E+013
1633
2.43E+014
10
2.43E+013
8.683
.000
4.60E+015
1639
2.80E+012
4.84E+015
1649
Inter-groupes
Intra-groupes
Total
Inter-groupes
Intra-groupes
Total
Inter-groupes
Intra-groupes
Total
rd
labour
sales
Somme
des carrés
ddl
Moyenne
des carrés
F
Signif ication
Qualitative

×


Qualitative

Qualitative

×


Quantitative

Quantitative

×


Quantitative


Which variables are we looking at ?


Relationship Between Variables

Chi
-
Square
Independence Test


Chi
-
Square Independence Test

Introduction to
Chi
-
Square


This part devoted to the study of whether two
qualitative (categorical) variables are
independent
:



H
0
: Independent:

the two qualitative variables do not
exhibit any systematic association.


H
1
: Dependent:

the category of one qualitative
variable is associated with the category of another
qualitative variable in some systematic way which
departs significantly from randomness.


The Four Steps Towards The Test

1.
Build the cross tabulation to compute observed joint
frequencies

2.
Compute expected joint frequencies under the
assumption of independence

3.
Compute the Chi
-
square (
χ
²)

distance between
observed and expected joint frequencies

4.
Compute the significance of the
χ
²

distance and
conclude on H
0

and H
1

1. Cross Tabulation


A cross tabulation displays the
joint distribution

of two
or more variables. They are usually referred to as a
contingency tables.


A
contingency table

describes the distribution of two (or
more) variables simultaneously. Each cell shows the
number of respondents that gave a specific
combination of responses, that is, each cell contains a
single cross tabulation.


1. Cross Tabulation


We have data on two qualitative and
categorical dimensions and we wish to know
whether they are related


Region (AM, ASIA, EUR)


Type of company (DBF, LDF)


1. Cross Tabulation


We have data on two qualitative and
categorical dimensions and we wish to know
whether they are related


Region (AM, ASIA, EUR)




Type of company (DBF, LDF)







T
o
t
a
l










4
3
1






1
0
0
.
0
0

























































J
P










1
1
7







2
7
.
1
5






1
0
0
.
0
0








E
U
R











5
1







1
1
.
8
3







7
2
.
8
5







A
M
E
R










2
6
3







6
1
.
0
2







6
1
.
0
2


















































c
o
n
t
i
n
e
n
t








F
r
e
q
.





P
e
r
c
e
n
t








C
u
m
.
.

t
a
b
u
l
a
t
e

c
o
n
t
i
n
e
n
t
1. Cross Tabulation


We have data on two qualitative and
categorical dimensions and we wish to know
whether they are related


Region (AM, ASIA, EUR)


Type of company (DBF, LDF)










T
o
t
a
l










4
3
1






1
0
0
.
0
0



















































p
h
a
r
m
a
c
e
u
t
i
q
u
e










2
6
4







6
1
.
2
5






1
0
0
.
0
0
b
i
o
t
e
c
h
n
o
l
o
g
i
e










1
6
7







3
8
.
7
5







3
8
.
7
5





























































t
y
p
e








F
r
e
q
.





P
e
r
c
e
n
t








C
u
m
.
.

t
a
b
u
l
a
t
e

t
y
p
e

1. Cross Tabulation


Crossing Region (AM, ASIA, EUR)
×

Type of
company (DBF, LDF)


tabulate continent type






T
o
t
a
l









1
6
7








2
6
4









4
3
1






















































J
P











0








1
1
7









1
1
7








E
U
R










1
1









4
0










5
1







A
M
E
R









1
5
6








1
0
7









2
6
3















































c
o
n
t
i
n
e
n
t



b
i
o
t
e
c
h
n
o


p
h
a
r
m
a
c
e
u







T
o
t
a
l





















t
y
p
e
.

t
a
b


c
o
n
t
i
n
e
n
t

t
y
p
e
2. Expected Joint Frequencies


In order to say something on the relationship between
two categorical variables, it would be nice to produce
expected
, also called

theoretical
,

frequencies

under the
assumption of independence between the two
variables.

Total line Total Column
Overall Sample Size
ij
E



tabulate continent type , expected

2. Expected Joint Frequencies


















1
6
7
.
0






2
6
4
.
0







4
3
1
.
0






T
o
t
a
l









1
6
7








2
6
4









4
3
1
































































4
5
.
3







7
1
.
7







1
1
7
.
0









J
P











0








1
1
7









1
1
7
































































1
9
.
8







3
1
.
2








5
1
.
0








E
U
R










1
1









4
0










5
1































































1
0
1
.
9






1
6
1
.
1







2
6
3
.
0







A
M
E
R









1
5
6








1
0
7









2
6
3















































c
o
n
t
i
n
e
n
t



b
i
o
t
e
c
h
n
o


p
h
a
r
m
a
c
e
u







T
o
t
a
l





















t
y
p
e
























e
x
p
e
c
t
e
d

f
r
e
q
u
e
n
c
y








f
r
e
q
u
e
n
c
y































K
e
y







































.

t
a
b
u
l
a
t
e


c
o
n
t
i
n
e
n
t

t
y
p
e
,

e
x
p
e
c
t
e
d
3. Computing the
χ
²
statistics



We can now compare what we observe with what we
should observe, would the two variables be
independent. The larger the difference, the less
independent the two variables. This difference is
termed a Chi
-
Square distance.



2
2
ij ij
i j
ij
O E
E




With a contingency table of

n

lines and
m

columns, the
statistics follows a
χ²

distribution with (
n
-
1)
×
(
m
-
1) degree of
freedom, with the lowest expected frequency being at least 5.

3. Computing the
χ
²
statistics











P
e
a
r
s
o
n

c
h
i
2
(
2
)

=

1
2
7
.
2
3
3
4



P
r

=

0
.
0
0
0

















1
6
7
.
0






2
6
4
.
0







4
3
1
.
0






T
o
t
a
l









1
6
7








2
6
4









4
3
1
































































4
5
.
3







7
1
.
7







1
1
7
.
0









J
P











0








1
1
7









1
1
7
































































1
9
.
8







3
1
.
2








5
1
.
0








E
U
R










1
1









4
0










5
1































































1
0
1
.
9






1
6
1
.
1







2
6
3
.
0







A
M
E
R









1
5
6








1
0
7









2
6
3















































c
o
n
t
i
n
e
n
t



b
i
o
t
e
c
h
n
o


p
h
a
r
m
a
c
e
u







T
o
t
a
l





















t
y
p
e
























e
x
p
e
c
t
e
d

f
r
e
q
u
e
n
c
y








f
r
e
q
u
e
n
c
y































K
e
y







































.

t
a
b
u
l
a
t
e


c
o
n
t
i
n
e
n
t

t
y
p
e
,

e
x
p
e
c
t
e
d

c
h
i
2

tabulate

continent type ,
expected

chi2

4. Conclusion on H
0

versus H
1



We reject H
0

with 0.00% chances of being wrong


I will take the chance, and I tentatively conclude
that the type of companies and the regional origins
are not independent.


Using our appreciative knowledge on
biotechnology, it makes sense: biotechnology was
first born in the USA, with European companies
following and Asian (i.e. Japanese) companies
being mainly large pharmaceutical companies.


Most DBFs are found in the US, then in Europe.
This is less true now.


Analyse

Statistiques descriptives

Tableaux
Croisés

Cellule

Observé & Théorique

2. SPSS : Expected Joint Frequencies

Tableau croisé continent * type
156
107
263
101.9
161.1
263.0
59.3%
40.7%
100.0%
93.4%
40.5%
61.0%
36.2%
24.8%
61.0%
11
40
51
19.8
31.2
51.0
21.6%
78.4%
100.0%
6.6%
15.2%
11.8%
2.6%
9.3%
11.8%
0
117
117
45.3
71.7
117.0
.0%
100.0%
100.0%
.0%
44.3%
27.1%
.0%
27.1%
27.1%
167
264
431
167.0
264.0
431.0
38.7%
61.3%
100.0%
100.0%
100.0%
100.0%
38.7%
61.3%
100.0%
Ef f ectif
Ef f ectif théorique
% dans continent
% dans type
% du total
Ef f ectif
Ef f ectif théorique
% dans continent
% dans type
% du total
Ef f ectif
Ef f ectif théorique
% dans continent
% dans type
% du total
Ef f ectif
Ef f ectif théorique
% dans continent
% dans type
% du total
AMER
EUR
JP
continent
Total
DBF
LDF
type
Total

Analyse

Statistiques descriptives

Tableaux
Croisés

Statistique

Chi
-
deux

Tests du Khi-deux
127.233
a
2
.000
166.879
2
.000
431
Khi-deux de Pearson
Rapport de
vraisemblance
Nombre d'observations
valides
Valeur
ddl
Signif ication
asymptotique
(bilatérale)
0 cellules (.0%) ont un ef f ectif théorique inf érieur à 5.
L'ef f ectif théorique minimum est de 19.76.
a.
3. SPSS : Computing the
χ
²
statistics

Qualitative

×


Qualitative

Qualitative

×


Quantitative

Quantitative

×


Quantitative


Which variables are we looking at ?


Relationship Between Variables

Correlations


Correlations

Introduction to Correlations


This part is devoted to the study of
whether



and the
extent to which



two or more quantitative variables are
related:





Positively correlated:
the values of one variable “varying somewhat
in step” with the values of another variable




Negatively correlated:

the values of one continuous variable
“varying somewhat in opposite step” with the values of another
variable




Not correlated:

the values of one continuous variable “varying
randomly” when the values of another variable vary.

Scatter Plot of R&D and Patents (log)

Scatter Plot of R&D and Patents (log)

-20
-15
-10
-5
lpat_assets
-6
-4
-2
0
lrdi

The Pearson product
-
moment correlation coefficient is
a measure of the co
-
relation between two variables x
and y.



Pearson's
r

reflects the
intensity

of linear relationship
between two variables. It ranges from +1 to
-
1.







r


near 1 :

Positive Correlation





r


near
-
1 :

Negative Correlation





r


near 0 :

No or poor correlation

,
1 1
  
x y
r
Pearson’s Linear Correlation Coefficient
r












1
,
2 2
1 1
,

 
 
 
 
 

 
n
i i
i
x y
n n
x y
i i
i i
x x y y
Cov x y
r
x x y y
Cov(x,y)

: Covariance between x and y


x

et

y

: Standard deviation of x and Standard deviation of y

n

: Number of observations

Pearson’s Linear Correlation Coefficient
r





Pearson’s Linear Correlation Coefficient
r



corr

lpat

lassets

lrd

lrdi

lpat_assets


l
p
a
t
_
a
s
s
e
t
s





0
.
3
8
2
1


-
0
.
8
2
4
9


-
0
.
6
9
1
9



0
.
6
4
1
6



1
.
0
0
0
0









l
r
d
i





0
.
1
4
5
0


-
0
.
5
9
0
5


-
0
.
2
4
2
8



1
.
0
0
0
0










l
r
d





0
.
3
1
6
7



0
.
9
2
6
3



1
.
0
0
0
0






l
a
s
s
e
t
s





0
.
2
0
7
1



1
.
0
0
0
0









l
p
a
t





1
.
0
0
0
0















































































l
p
a
t


l
a
s
s
e
t
s






l
r
d





l
r
d
i

l
p
a
t
_
a
~
s
.

p
w
c
o
r
r

l
p
a
t

l
a
s
s
e
t
s

l
r
d

l
r
d
i

l
p
a
t
_
a
s
s
e
t
s

Is


significantly different from 0 ?


H
0

:
r
x,y
= 0


H
1

:
r
x,y


0



,
*
2
,
1
2



x y
x y
r
t
r
n


t*

: if t* > t


with (n


2) degree of freedom and critical
probability
α

(5%), we reject H
0

and conclude that
r

significantly different from 0.

Pearson’s Linear Correlation Coefficient
r





Pearson’s Linear Correlation Coefficient
r



pwcorr

lpat

lassets

lrd

lrdi

lpat_assets
,
sig
































0
.
0
0
0
0



0
.
0
0
0
0



0
.
0
0
0
0



0
.
0
0
0
0

l
p
a
t
_
a
s
s
e
t
s





0
.
3
8
2
1


-
0
.
8
2
4
9


-
0
.
6
9
1
9



0
.
6
4
1
6



1
.
0
0
0
0
































0
.
0
0
2
5



0
.
0
0
0
0



0
.
0
0
0
0








l
r
d
i





0
.
1
4
5
0


-
0
.
5
9
0
5


-
0
.
2
4
2
8



1
.
0
0
0
0
































0
.
0
0
0
0



0
.
0
0
0
0









l
r
d





0
.
3
1
6
7



0
.
9
2
6
3



1
.
0
0
0
0
































0
.
0
0
0
0





l
a
s
s
e
t
s





0
.
2
0
7
1



1
.
0
0
0
0





































l
p
a
t





1
.
0
0
0
0















































































l
p
a
t


l
a
s
s
e
t
s






l
r
d





l
r
d
i

l
p
a
t
_
a
~
s
.

p
w
c
o
r
r

l
p
a
t

l
a
s
s
e
t
s

l
r
d

l
r
d
i

l
p
a
t
_
a
s
s
e
t
s
,

s
i
g

Assumptions of Pearson’s
r


There is a linear relationships between
x

and
y


Both x and y are continuous random variables


Both variables are normally distributed


Equal differences between measurements represent
equivalent intervals.


We may want to relax (one of) these assumptions

Pearson’s Linear Correlation Coefficient
r

Spearman’s Rank Correlation Coefficient
ρ


Spearman's rank correlation is a
non parametric

measure of the
intensity

of a
correlation

between two
variables, without making any assumptions about the
distribution of the variables, i.e. about the linearity,
normality or scale of the relationship.











near 1

: Positive Correlation








near
-
1

: Negative Correlation








near 0

: No or poor correlation

x,y
1 1
   


n
2
i 1
x,y x,y
2
6 d
Rho 1
n n 1


   




: the difference between ranks of paired values of x and y

n

: Number of observations



ρ is simply a special case of the Pearson product
-
moment
coefficient in which the data are converted to ranks before
calculating the coefficient.


Spearman’s Rank Correlation Coefficient
ρ



Spearman’s Rank Correlation Coefficient
ρ


l
p
a
t
_
a
s
s
e
t
s





0
.
3
7
0
9


-
0
.
8
0
0
6


-
0
.
6
9
0
1



0
.
6
0
9
3



1
.
0
0
0
0









l
r
d
i





0
.
1
1
7
2


-
0
.
5
5
6
4


-
0
.
2
9
1
9



1
.
0
0
0
0










l
r
d





0
.
3
2
0
2



0
.
9
3
5
3



1
.
0
0
0
0






l
a
s
s
e
t
s





0
.
2
2
5
7



1
.
0
0
0
0









l
p
a
t





1
.
0
0
0
0















































































l
p
a
t


l
a
s
s
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t
s






l
r
d





l
r
d
i

l
p
a
t
_
a
~
s
(
o
b
s
=
4
3
1
)
.

s
p
e
a
r
m
a
n

l
p
a
t

l
a
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s
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t
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l
r
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l
r
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i

l
p
a
t
_
a
s
s
e
t
s















spearman

lpat

lassets

lrd

lrdi

lpat_assets

Spearman’s Rank Correlation Coefficient
ρ


spearman

lpat

lassets

lrd

lrdi

lpat_assets
,
stats
(rho p)


















0
.
0
0
0
0



0
.
0
0
0
0



0
.
0
0
0
0



0
.
0
0
0
0


l
p
a
t
_
a
s
s
e
t
s





0
.
3
7
0
9


-
0
.
8
0
0
6


-
0
.
6
9
0
1



0
.
6
0
9
3



1
.
0
0
0
0
































0
.
0
1
5
0



0
.
0
0
0
0



0
.
0
0
0
0









l
r
d
i





0
.
1
1
7
2


-
0
.
5
5
6
4


-
0
.
2
9
1
9



1
.
0
0
0
0
































0
.
0
0
0
0



0
.
0
0
0
0










l
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0
.
3
2
0
2



0
.
9
3
5
3



1
.
0
0
0
0
































0
.
0
0
0
0






l
a
s
s
e
t
s





0
.
2
2
5
7



1
.
0
0
0
0






































l
p
a
t





1
.
0
0
0
0















































































l
p
a
t


l
a
s
s
e
t
s






l
r
d





l
r
d
i

l
p
a
t
_
a
~
s























S
i
g
.

l
e
v
e
l









r
h
o


































K
e
y
































(
o
b
s
=
4
3
1
)
.

s
p
e
a
r
m
a
n

l
p
a
t

l
a
s
s
e
t
s

l
r
d

l
r
d
i

l
p
a
t
_
a
s
s
e
t
s
,

s
t
a
t
s
(
r
h
o

p
)
Pearson’s
r

or Spearman’s
ρ
?


Relationship between tastes and levels of
consumption on a large sample?
(
ρ
)


Relationship between income and
Consumption on a large sample?
(
r
)


Relationship between income and
Consumption on a small sample? Both
(
ρ
)
and

(
r
)



Analyse


Corrélation


Bivariée



Click on Pearson

Cor rélations
1
.217
**
.146
**
.389
**
.326
**
.000
.002
.000
.000
457
457
457
457
457
.217
**
1
-.588
**
-.815
**
.929
**
.000
.000
.000
.000
457
457
457
457
457
.146
**
-.588
**
1
.642
**
-.248
**
.002
.000
.000
.000
457
457
457
457
457
.389
**
-.815
**
.642
**
1
-.684
**
.000
.000
.000
.000
457
457
457
457
457
.326
**
.929
**
-.248
**
-.684
**
1
.000
.000
.000
.000
457
457
457
457
457
Corrélation de Pearson
Sig. (bilatérale)
N
Corrélation de Pearson
Sig. (bilatérale)
N
Corrélation de Pearson
Sig. (bilatérale)
N
Corrélation de Pearson
Sig. (bilatérale)
N
Corrélation de Pearson
Sig. (bilatérale)
N
lnpatent
lnassets
lnrd_assets
lnpat_assets
lnrd
lnpatent
lnassets
lnrd_assets
lnpat_assets
lnrd
La corrélation est signif icative au niveau 0.01 (bilatéral).
**.
Pearson’s Linear Correlation Coefficient
r




Analyse


Corrélation


Bivariée



Click on “Spearman”

Spearman’s Rank Correlation Coefficient
ρ

Cor rélations
1.000
.243
**
.130
**
.385
**
.335
**
.
.000
.005
.000
.000
457
457
457
457
457
.243
**
1.000
-.536
**
-.774
**
.941
**
.000
.
.000
.000
.000
457
457
457
457
457
.130
**
-.536
**
1.000
.604
**
-.282
**
.005
.000
.
.000
.000
457
457
457
457
457
.385
**
-.774
**
.604
**
1.000
-.669
**
.000
.000
.000
.
.000
457
457
457
457
457
.335
**
.941
**
-.282
**
-.669
**
1.000
.000
.000
.000
.000
.
457
457
457
457
457
Coef f icient de corrélation
Sig. (bilatérale)
N
Coef f icient de corrélation
Sig. (bilatérale)
N
Coef f icient de corrélation
Sig. (bilatérale)
N
Coef f icient de corrélation
Sig. (bilatérale)
N
Coef f icient de corrélation
Sig. (bilatérale)
N
lnpatent
lnassets
lnrd_assets
lnpat_assets
lnrd
lnpatent
lnassets
lnrd_assets
lnpat_assets
lnrd
La corrélation est signif icative au niveau 0,01 (bilatéral).
**.