methodological and applicative problems of using pearson ... · methodological and applicative...

Romanian Statistical Review - Supplement nr. 2 / 2017148

Methodological and Applicative Problems of using Pearson Correlation Coeffi cient in the

Analysis of Socio-Economic Variables

Daniela-Emanuela Dănăcică

Faculty of Economics, Constantin Brâncuși University of Târgu-Jiu

Abstract

The aim of this paper is to focus on the methodological and applicative

problems that Pearson correlation coeffi cient may arise when used to

analyzing the relationship between socio-economic variables. Using real data

we empasized the effect of the most important factors infl uencing the size and

interpretation of Pearson coeffi cient and we presented the special cases of this

statistics and their usefulness.

Keywords: correlation, Pearson coeffi cient, size, linearity

JEL Classifi cation: C18, C01

1. Introduction

If we focus on the name of simple linear correlation coeffi cient,

known in the literature as “Pearson product-moment correlation coeffi cient”,

we could erroneously belive that the English mathematician and biostatistician

Karl Pearson developed this coeffi cient himself. In fact, it was Francis Galton

the man who fi rst conceived and formulated the concepts of correlation

and regression (Stanton, 2001). Later on, Bravais and Pearson elaborated a

rigourous mathematical approach for correlation and regression.

Pearson product-moment correlation coeffi cient is widely used in

economics, social sciences, medicine, etc., as a measure of linear relationship

between two variable X and Y. The mathematical formula for this coeffi cient

developed by Pearson in 1895 is:

� �

�

� �

�

��

��

��

�

�

�

��

�

�

�

�

��

��

��

' '

>'

'>

?;9;9@

;;99

��

�

�

(1)

Since :��

��

��

�

�

� ;;99

;� ��9 '

��

�� /��+��, we have

��/��+��

��

��

��

��

;� ��9> � ��9 (2)

Revista Română de Statistică - Supliment nr. 2 / 2017 149

therefore, as Rodgers and Nicewander (1988) underline, Pearson coeffi cient

can be described as a standardized covariance.

In statistics is widely used the following formula:

−

−

⋅−=

∑∑∑ ∑

∑∑ ∑

=== =

== =

n

i

i

n

i

i

n

i

n

i

ii

n

i

i

n

i

n

i

iii

xy

yynxxn

yxyxn

r

1

2

1

2

1 1

22

11 1/

)(

(3)

[ ]1,1/ −∈xyr . The closer the value of xyr / is to +1 or -1, the higher the

association between these two variables is; the closer the value of xyr / is to 0,

the weaker the strengh of the relationship is.

In many statistics textbooks the interval [-1, 1] is divided in subintervals;

the intensity of relationship between X and Y is interpreted depending on the

subintervals of which xyr / belongs. However, a specifi c value of Pearson

coeffi cient is difi cult to interpret. For example, it is diffi cult to interpret a

0.6 value of xyr / compared with a 0.7 value. Rinaman (1993) underlines that

there is no precise interpretation that could be attached to particular values of

Pearson correlation coeffi cient. Also, since ]1,0[/ ∈xyr there is a tendency

to mistakenly interpret its value as the proportion of identical ( ii yx , ) pairs, or

as the probability of a correct prediction (Falk and Well, 1997; Eisenbach and

Falk, 1984).

Although Pearson correlation is widely used by researchers from

many research areas and all the basic statistics textbooks cover the topic of

correlation, there are not so many studies focused on the subtle nuances of this

measure and on the atypical cases of Pearson coeffi cient and their usefulness.

Various analyzes about what is communicated using Pearson coeffi cient must

be carefully presented, because the similarities between some interpretation

are subject to specifi c constraints (Falk and Well, 1997). Each result obtained

using this statistics should be carefully checked if the intepretation is

appropriate to the used model; some interpretation of Pearson correlation

coeffi cient are valid only under certain particular circumstances.

The aim of this article is to emphasize the methodological and

applicative problems of using Pearson coeffi cient in the analysis of relationship

between socio-economic variables. Pearson coeffi cient is a versatile statistics

and it has many subtile shades. We will focus on the most important factors

affecting the value and interpretation of Pearson coeffi cient and on the

limitation of this statistics. We will discuss about particular cases when one or

both of the analyzed variables are dichotomous and we will present the special


cases of Pearson coeffi cient and their particularities. To our knowledge, there

is no statistics textbooks in our country that cover aspects regarding factors

infl uencing the size and interpretation of Pearson coeffi cient, or aspects

regarding special cases of Pearson coeffi cient (other than Spearman and

Kendall coeffi cient). Thus this paper seeks to fi ll a gap in the existing literature

too.

2. Factors infl uencing the size and interpretation of Pearson correlation

coeffi cient

In this section of paper we will analyze the effect of main factors

infl uencing the size and interpretation of Pearson correlation coeffi cient.

The most important factors that have a signifi cant effect on the value and

interpretation of Pearson correlation coeffi cient are: the shape of distribution,

the size of sample, the aberrant values (outliers), the restriction of range and

nonlinearity, a third variable or more (Chen and Popovich, 2002; Goodwin

and Leech, 2006).

In every statistics textbooks it is written that Pearson correlation

coeffi cient has values within the [-1,1] interval. However, this is true only

if the distribution of exogenous X and endogenous Y are symmetrical and

has the same shape (Goodwin and Leech, 2006; Chen and Popovich, 2002;

Glass and Hopkins, 1996; Hays, 1994; Nunnally and Bernstein, 1994). If

the distribution of the analyzed variables does not share the same shape, the

increase of the exogenous X will not always be accompanied by the increase

of endogenous Y, in case of a positive relationship, or by the decrease of Y in

case of a negative relationship. According to Chen and Popovich (2002) and

Carrol (1961), the less similar the shape of distributions for both analyzed

variables, the smaller the maximum value of || / xyr will be. Goodwin and

Leech (2006) show that if the Pearson coeffi cient has the value xyr / =0,90

for two same-shaped distributions, changes in the form of one distribution

could decrease the value of Pearson coeffi cient from 0.80 to 0.70. However, if

we have a value of =xyr / 0,3 for two same-shaped distribution, even major

changes of the shape of one distribution will have a very low effect on the

estimated value of xyr / .

The size of sample for which Pearson correlation is estimated is

another important factor infl uencing its value and interpretation. The standard

error of xyr / is affected by the size of analyzed sample. According to Chen and

Popovich (2002), when the size of sample is equal with 20, approximately 95% of the correlation coeffi cients have their values in between [-0.47, 0.47]; when

the size of sample is of 102 empirical data, approximately 95% of coeffi cients

have values in between [-0.20, 0.20]. To emphasize the size effect on the


correlation coeffi cient we present the following example: we estimate the

relationship between unemployment duration, expressed in days (endogenous

variable) and the age variable, (exogenous) for the individuals registered

as unemployed at the National Agency of Employment Romania during 1

January 2008-31 December 20101 using Pearson correlation coeffi cient. First

we estimated Pearson correlation coeffi cient using SPSS 17.0 for the whole

dataset, namely 23762531 =N spells; the value is 325,0/1 =

xyr . If we use

a random sample of 10% from the initial dataset, we have 2377752 =N spells and 326,0/2 =

xyr . For a 5% random sample from the second dataset

we have 118443 =N spells and 313,0/3 =xy

r ; for a 5% random sample from the third dataset we have 6364 =N spells and 319,0/4 =

xyr ; for

a 18% random sample of the forth dataset we have 1135 =N spells and 208,0/5 =

xyr and for 3% random sample again from the forth dataset we have 216 =N 21 spells and 452,0/6 =

xyr (table 1). Therefore, we can notice

how the size of sample affects the value of Pearson correlation coeffi cient,

and implicit its interpretation. In conclusion, we must be very careful in

interpreting the Pearson coeffi cient estimated for a small sample data.

Different values of Pearson correlation coeffi cient for the relationship

between unemployment duration (days) and age, estimated for different

size of sample

Table 1Size of sample (number of spells) Pearson correlation Sig. (2-tailed)

2376253 of which 2079882 completed spells 0.325** 0.000237775 of which 207847 completed spells 0.326** 0.00011844 of which 10333 completed spells 0.313** 0.000636 of which 559 completed spells 0.319** 0.000113 of which 96 completed spells 0.208** 0.00021 of which 21 completed spells 0.452* 0.000**Correlation is signifi cant at the 0.01 level (2-tailed)* Correlation is signifi cant at the 0.05 level (2-tailed)

Outliers are extreme values of empirical data which can signifi cantly affect the value of Pearson correlation coeffi cient, especially if the size of sample is small (Chen and Popovich, 2002). An outlier is a value of a variable very small or very large than the rest of sample. Outliers can be found in the values of exogenous variable, of endogenous variable, or both analyzed

1. This dataset was gathered from National Agency of Employment Romania during the author postdoctoral stage and used for research in the thesis Dănăcică D.E., (2013), Cercetări privind

impactul factorilor ce infl uenţează durata șomajului şi probabilitatea (re)angajării în România,

Romanian Academy Publishing House, Bucharest.


variables. The presence of outliers in the samples is due to the data collection errors, errors in registration or simply the occurrence of unusual value compared with the average of one variable values (Goodwin and Leech, 2006; Brase and Brase, 1999). To illustrate the effect of outliers on the size of Pearson coeffi cient we use a random sample of 39 records of registered

unemployed from the initial dataset above presented; 37 individuals are aged

in between 16 and 23 years, one individual is 56 years old and another one is

65 years old (table 2).

Unemployment duration (days) and age of 39 registered unemployed

Table 2Unemployment

duration (days)Age

Unemployment

duration (days)Age

Unemployment

duration (days)Age

44 16 79 18 87 2350 16 90 18 90 2367 18 90 23 181 2370 18 106 23 181 2374 18 110 23 181 2329 18 90 23 181 2350 18 97 23 182 2357 18 110 23 182 2370 18 68 23 100 2367 18 90 23 100 2391 18 98 23 104 2354 18 158 23 443 5657 18 89 23 365 65

We estimate Pearson coeffi cient as a measure of relationship between

unemployment duration (days) and age of each individual. In fi gure 1 is

presented the scatterplot of the empirical data. We can notice the two outliers,

age=56 years and age=65 years.

Scatterplot of unemployment duration (Y) and age (X)

Figure 1


Pearson correlation coeffi cient estimated with SPSS 17.0 for all the

39 cases has the value 890,0/ =xyr , which suggests a strong correlation

between two analyzed variables. If we drop the record of 56 years old person,

we have 827,0/ =xyr ; if we drop the record of 65 years old person, we have

890,0/ =xyr and if we drop to the both cases we have 669,0/ =xyr . As we can

notice from this example, the existence of „outliers” can easly change the size

of Pearson coeffi cient, especially for small sample. It can be possible to affect

even the direction of this statistics. In general, large samples enable a more

accurat estimation of Pearson coeffi cient, being less affected by outliers. As a

solution for „outliers” problem some authors suggest the removal of unusual

data from the sample and restoration of the regression model. However, this

option may lead to loss of important information, perhaps even decisive for

the examined process. Because of this, we must fi rst check if the outliers

occured as a reasult of measurement errors, if the outliers are irrelevant for the

case study, or if the outliers have a major effect on the regression model and

on the regression coeffi cients. Until now we don’t have a clear answer to the

question “what should we do with the outliers”. Several statistical techniques

for outliers identifi cation can be found in the literature (Neter, Wasserman şi

Kutner, 1986 or Orr, Sachett şi Dubois, 1991). However, we can not appreciate

which of these techniques are the best choice. It is diffi cult to know whether

the outliers are real, and must be taken into account, or are simply errors and

should be removed from the analysis. Kruskal (1990) suggest an analysis with

and without outliers in the sample.

Another two factors infl uencing the values of Pearson correlation coeffi cient and the intensity estimation of correlation between variables are

range restriction and nonlinearity. According to Chen and Popovich (2002),

range restriction of empirical data may occur when the measurement methods

are not sensitive enough to capture all the particularities of analyzed variables,

or when researchers are using relatively homogeneous samples for their

studies. This type of range restriction is known in the literature as “incidental

selection” (Glass and Hopkins, 1996). The size of Pearson coeffi cient may

increase or deacrese function of the type of empirical data and of their range;

generally they have the tendency to decrease if the range is restricted (a lower

value) and the sample is homogenous.

Using Pearson coeffi cient when the relationship between two variables

is not linear is a wrong decision; the resulted values led to wrong interpretation.

First is recommended to visualize the distribution of ),( ii yx pairs using the

scatterplot. If the relationship between variables is not linear, then we must

use the R correlation coeffi cient, to transform the data or to use the polinomial

regression (see Abrami et. al., 2001; Bobko, 1995; Pedhazur, 1973).


Sometimes two variables are statistically correlated, but in reality there is no causal relationship between them. In the literature this relationship is called „spourious correlation” or „illusory correlation”. „Spuorious correlation” was in attention of well-known statisticians like Yule, Hans Zeisel, Kendall or Lazarfeld. The occurence of spourious correlation may be the result of the effect of one or more so-called „a third variable”. For example we can fi nd mentioned in the literature the positive relationship between

age and job satisfaction of a person. Studies show that people satisfi ed with

their job within a company stay a longer time in their company. However,

the positive link between the two variables above mentioned may occur due

to the infl uence of a third variable, namely job tenure (Chen and Popovich,

2002). Another example widely presented in the literature is the one refered

to the link between religion and the height of European population. If we

move from northern Europe to the south, the proportion of Roman Catholic

population increases, while there is a decrease in the average height of the

inhabitants. The empirical data may suggest a negative correlation between

the Roman Catholic population and the height of individuals that belong to

this population. In reality this relationship is an illusory one, the height of

population depending on entirely other factors. The conclusion is that the

association does not alway imply causation between variables. A carefull

statistician should always keep that in mind.

3. Special cases of Pearson correlation coeffi cient xyr /

In this section of paper we will focus on special cases of Pearson

correlation coeffi cients, namely the point-bisearial correlation coeffi cient, phi

coeffi cient, biserial coeffi cient, tetrachoric coeffi cient and eta coeffi cient.

Point-biserial coeffi cient (

�� )�� #��

�� 9 �� ;��,� �� .��

��+��,��+��)�� )��+��

) is a special case of Pearson correlation

coeffi cient, used to assess the type, the direction and the intensity of a

relationship between a countinous variable and a true dichotomous one. An

interesting fact is that the inference of point-biserial coeffi cient

�� )�� #��

�� 9 �� ;��,� �� .��

��+��,��+��)�� )��+��

gives the

same information as an independent two sample t test, even they have different

focus (Chen and Popovich, 2002).

In the following, using the point-biserial coeffi cient we will analyze

the potential differences between male and female students regarding

the given importance to study of a second foreign language during their

undergraduate studies; we use a sample of 30 students from the second and

third year undergraduate program from Faculty of Economics, Constantin

Brâncuși University of Târgu-Jiu, Romania. The score given by students is the endogenous variable (Y) and the gender is the exogenous one (X). We will highlight the link between the two sample t test and the point biserial


coeffi cient. We have in the sample 15 male students and 15 female students.

The gender variable was codifi ed as follows: 0- female students, 1- male

students. Students were asked to give a scor for the importance that they

perceive for studing a secong foreign language during their undergraduated

studies, using a scale from 1 to 10. The data are presented in table 3.

On a scale of 1-10 how inportant is for you to study a second foreign

language during the undergraduate program?

Table 3Female Male Female Male Female Male

7 8 10 8 10 108 5 5 10 9 10

10 10 10 9 7 77 10 8 10 10 97 8 9 10 10 9

First we will determine the value of t-two sample test, using the

formula:

2

2

1

2

21 /)(nn

YYtpp σσ

+−= ,with degree of freedom /��+�)��)�� ; 9 ' �� 9!;��

�

�

�

%'(#'

�

(4)

where, 22

12 σσ and represent the estimated variance of the female

scores sample, respectively male scores sample, 21 YandY are the means of

the two samples, and 1n respectively 2n represent the size of female and male

sample.

We have the variance of the total sample scores:

)1()1(

)1()1(

21

22

212

12

−+−

−+−=

nn

nnp

σσσ (5)

where 22

12 σσ and are the variance of Y for female and male

sample.

Using the data from table 3 we have 81 =Y 8.47,

15598.1,552.2 1112 === nσσ 15 and '%!%F#'�' !# �2F#2

�� #�

�

%'(#'

�

.

We have 338.22 =pσ and t(28)=-0.7164, the calculated value of t is lower

that the critical value of t (28)=2.05 (� 9 2;K #$%�9 $%#$�� ;�� )��,�� ,��-��+��

�

�

%'(#'

�

), thus we conclude that there

is not statistical signifi cance for gender differences in student perception

about having or not a second foreign language in the curricula during their

undergraduate program.

To estimate the point-biserial correlation coeffi cient we will use the

following formula:

�-

��

��

�' �� (6)


where 1Y and 2Y have the above presented meaning, σ is the standard deviation of the whole sample, p and q represents the proportion of female and male students in the total sample.

For our study we have ��)��/��+�� %#$%#$%'(#'

!F#22F#2��

�� K$#'& ��,�� +��+��

�� +��/��+� �+��5�� 6��)��9��$��';�� +�� +�� )�

=0.132, a positive

association that suggest the fact that with the “increase” of gender (from 0 to 1), increase also the score for having a second foreign language in the undergraduate curricula. However, the difference is not statistical signifi cant.

The relationship between the point-biserial coeffi cient and the

estimated t statistics derived from the two-sample t test is given by the

relations (Chen and Popovich, 2002, p.28):

#; 9

'

JJ

'

' ��

�'��

� ��

��

��

��

�� (7)

Indeed we have 3�)��)�/�� +�� F'#$ '%'%;'& #$9'

'& #$

' '

��

��

��

��

�� /+� +� �+��

;F'(!#$9�

�

which the value is of || t . And �#�G�)�

'

;F'(!#$9; '%'%9

;F'(!#$9

; 9 ��

��

�K$#$'F==(�/+� +��+��

�

which is the value of K$#$'F==(�/+� +��+�� #�

�

.

Phi coeffi cient (φ ) is another special case of Pearson correlation used

when both analyzed variables are naturally dichotomous. We will analyze in

this part of paper the association between gender of an individual (X) and his/

her employment status (employed/unemployed, Y), at the end of the observed

period. From the dataset gathered from National Agency of Romania, above

presented, we extracted a random sample of 30 individuals, from which 11

women (3 employed at the end of period) and 19 men (9 employed at the end

of period). Data are presented in table 4 and were coded as follows: 1- male

subjects, 0- female subjects, 1- employed, 0- unemployed.


Data about the link between gender (X) and employment status (Y) for

30 registered unemployed

Table 4Gender (X) Employment status (Y) Gender (X) Employment status (Y)

1 1 1 11 1 1 01 1 1 11 1 1 01 1 0 11 1 0 01 1 0 01 0 0 01 0 0 01 0 0 01 0 0 01 0 0 11 0 0 01 0 0 01 0 0 1

We have:

−

−

⋅−=

∑∑∑ ∑

∑∑ ∑

=== =

== =

n

i

i

n

i

i

n

i

n

i

ii

n

i

i

n

i

n

i

iii

yynxxn

yxyxn

1

2

1

2

1 1

22

11 1

)(

φ

='

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� �� '='

��

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��

�

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�� '='

��

�

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�� ' '

��

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��

� Thus we have:

�

� �� '=2#$

!F# '

!

' ' &$'='=&$

' '==&$

��

��

��

�

��

� �

�

��

�

�

The obtained results show a positive association between analyzed variables; as gender „increases” from 0-women to 1- men, the probability of employment at the end of study increase too. We eill check the null hypothesis using the 2χ test of independence. We have 22 φχ n= with df=1, thus we obtain )�K'��+��/��-�� 'F(#''=2#$&$ �� /+� +��/��+��+�� 2!

��

�

��

�

, which is lower than the critical value 32 =χ 3.84 for .0=α 0.05, or even 705.22 =χ for 0=α 0.10. Therefore we will conclude that there are no signifi cant differences between men and women

from this sample regarding the employment probability. We can also use the

contingency table 5 to estimate the phi-coeffi cient.


Contingency table for estimating phi coeffi cient

Table 5Unemployed (Y=0) Employed (Y=1) Total

Male (X=1) 10 a 9 b 19Female (X=0) 8 c 3 d 11Total 18 12 30

We estimate the phi correlation coeffi cient using the formula:

;;9;9;99 ��

��

��

��

&$F �

, if ;

�� $;;9;9;99 �� 92;�

�

�

(8)

We have: H�� +��1� '=2#$;'2;9' ;9'';9'=9

&$F ��

�� )� �+�� ,�� +�� and the interpretation

is similar to the one presented above.

Biserial correlation coeffi cient ( bisr ) is used when the exogenous

variable X has a normal distribution, but is artifi cially measured as a

dichotomous variable (e.g. accepted versus rejected, effi cient versus ineffi cient)

in its relationship with a continous endogenous Y variable (e.g. the results of an

admission exam, labor productivity of a sample of workers, etc). In oder words,

biserial coeffi cient is a statistic measure that allow us to estimate the strenght of

relationship between the exogenous X and the endogenous Y would be if X is not

artifi cially converted in a dichotomous variable (Chen and Popovich, 2002).

Biserial coeffi cient ( bisr ) has the following formula:

))(

(2

21

1212

nn

nnYYr

Y

bis +

−=

λσ (9)

where 2Y and 1Y are the mean of the two samples of endogenous

Y, 1n and 2n is the size of the samples and Yσ is the standard deviation of Y;

λ represents the ordinate of the standard normal distribution at )/( 212 nnn +proportion of subjects from the second sample.

The relationship between bisr and pbr is described by Chen and

Popovich (2002) as follows:

.)(

)1(3

21

2

2112

nn

nnnnrr pbbis +

−+=

λ (10)

Since .1)(

)1(3

21

2

2112 ≥+

−+

nn

nnnn

λ1.25, the value of biserial coeffi cient is

always higher than the value of point-biserial coeffi cient (Chen and Popovich,

2002, Glass and Hopkins, 1996).


Tetrachoric coeffi cient, noted with tetr is used for estimating the relationship between two countinous variables X and Y, both of them artifi cially

transformed in dihotomous variables. In order to highlights how to use this

coeefi cient of correlation, we will estimate the intensity of relationship between

education (exogenous variable X) and unemployment duration (endogenous

variable Y) for a random sample of 30 unemployed subjects. The variable

education is expressed in number of study for each unemployed person. The

sample is extracted from the initial dataset of 2376253 unemployment spells.

The data are presented in table 6.

Data about the association between unemployment duration (days) and

education (years of study) for 30 registered unemployed persons

Table 6Nr. crt

Unemployment duration (days)

Education (years)

Nr.crt.


Education (years)

Nr.crt


Education (years)

1 365 8 11 260 8 21 140 132 20 16 12 300 10 22 156 123 270 10 13 240 12 23 30 164 190 12 14 190 14 24 32 165 60 18 15 195 16 25 67 186 360 4 16 40 18 26 12 207 90 6 17 10 16 27 300 98 180 12 18 185 12 28 23 129 181 12 19 360 4 29 180 13

10 182 14 20 365 12 30 181 14

As we can notice from table 6, both variables are continonus. We will

transform the education variable in a dichomotous variable with two categories,

„low-educated” (education<=12 years) and „good-educated” (education>12

years) and the duration of unemployment variable in a dichomotous variable

too, with the categories „short-term unemployment” (duration<180 days)

and „long-term unemployment” (duration>=180 zile). The initial data are

rearanged in table 7.

Contingency table for the link between unemployment duration and

education, 30 subjects sample

Table 7

Long-term

unemployment (Y=0)

Short-term

unemployment (Y=1)Total

Good-educated (X=1) 5 a 9 b 14 (a+b)Poor-educated (X=0) 13 c 3 d 16 (c+d)Total 18 (a+c) 12 (b+d) 30 (a+b+c+d)


Tetrachoric coeffi cient is estimated as follows (Chen and Popovich,

2002, p.38):

�

��

��

��

�� , (11)

where n is the size of sample, Xλ is the ordinate of the standardized

normal distribution at (a+b)/(a+b+c+d) proportion of subjects with X=1, Yλis the ordinate of the standardized normal distribution at (b+d)/(a+b+c+d)

proportion of subjects with Y=1.

Therefore we have: �+�� /�� +��1� ($F#$

&$N&$

'

&$

'!

'%''F

�

�

�� #� �+�� +��

�

S�� S��+�

��,��

��)�� S��+�

��,��

$$$� &#$$� '� '(#F$� �

$$'� $#2$� '� '%#=$� �

$$ � F#!$� '� '=#($� �

$$&� %#'$� '� '(#%$� �

$$!� =#$$� '� 'F#2$� �

$$%� %#$$� '� 'F#!$� �

$$(� =#'$� '� 'F#F$� �

$$F� F#!$� '� 'F#&$� �

$$2� 2#&$� '� '%#'$� �

$$=� &'#($� '� 'F#&$� �

$'$� F#'$� '� '#'$� �

$''� &$#F$� '� #%$� �

$' � &'#'$� '� $#2$� �

$'&� &'#%$� '� #'$� �

$'!� &!# $� '� '#2$� �

$'%� =#!$� '� '=#F$� �

. This value of

tetrachoric coeffi cient suggests a positive and intense relationship between

education and unemployment duration. With the „increase” of X values, from

0 to 1 (from poor-educated to good-educated), the value of Y „increase” too,

from 0 to 1 (from long-term unemployment to short-term unemployment).

However, as Glass and Hopkins (1996) underline, the tetrachoric

coeffi cient is not a robust measure of correlation for smaller than 400

registration samples. Both biserial and tetrachoric coeffi cients are rarely used

in practice; their use and interpretation requires a special attention (Chen and

Popovich, 2002; Glass and Hopkins, 1996).

Eta coeffi cient of correlation, noted with η , is another special case

of Pearson coeffi cient used for estimate the intensity of association between

a multichotomous exogenous variable (X) with n distinct categories, and

an endogenous variable (Y) with intervals or ratio scores. Eta correlation

coeffi cient can be used for nonlinear relationship too. Eta coeffi cient is used

in the context of ANOVA analysis.

We will use eta coeffi cient to analyze if there are signifi cant statistical

difference between the dynamic of the ILO unemployment rate for of young

individuals aged in between 15 and 19 years (fi rst group), 20-24 years (second

group) and 25-29 years (third group), for the period 2000-2015. We use the

data offered by the National Statistical Institute, Tempo-Online database. The

unemployment rate of youth is coded as follows: 1 for 15-19 group, 2- for 20-

24 group and 3- for 25-29 group. The data are presented in table 8.


The dynamic of youth unemployment rate in between 2000-2015 period

function of age intervals

Table 8

YearYouth

unemployment rate (15-19)

CodeYouth


CodeYouth


Code

2000 23.00 1 16.70 2 8.60 32001 20.80 1 15.90 2 7.90 32002 27.40 1 19.60 2 9.80 32003 25.10 1 16.50 2 8.80 32004 29.00 1 17.80 2 9.20 32005 25.00 1 17.40 2 8.40 32006 29.10 1 17.70 2 9.20 32007 27.40 1 17.30 2 7.50 32008 28.30 1 15.10 2 6.50 32009 31.60 1 17.30 2 8.30 32010 27.10 1 21.10 2 9.60 32011 30.70 1 22.50 2 9.60 32012 31.10 1 20.80 2 10.30 32013 31.50 1 22.10 2 10.60 32014 34.20 1 21.80 2 10.50 32015 29.40 1 19.70 2 10.20 3�

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Using analysis of variance ANOVA we have: Unemployment rate

Sum of Squares df Mean Square F Sig.Between Groups 2815.031 2 1407.516 162.277 0.000Within Groups 390.308 45 8.674 * *Total 3205.339 47 * * *

The results of ANOVA show that there are statistical signifi cant

differences between the three age groups regarding the dynamics of

unemployment rates. The association between age and the dynamics of

unemployment rate can be measured using eta correlation coeffi cient,

estimated with the following formula:

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Therefore we have: �+�� /�� +��1� �!��&&�� K 2'%#$&'�� & $%#&&=��&& � ��)�

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,

and 937.0=η . Since F=162.277 and Sig.=0.000, we conclude that eta

coeffi cient is signifi cantly different than 0, meaning that there are signifi cant

differences between unemployment rate dynamics of these three youth

age groups during the analyzed period, and a strong inverse correlation


between age and unemployment rate of young people. If we use the mean values of unemployment rates instead of the codes 1, 2 and 3 (mean [15-19] unemployment rate= 28.1688 , mean [20-24] unemployment rate= 18.7063 and mean [25-34] unemployment rate=9.0625 ) and use the Pearson coeffi cient we get r= - 0,937, we observe that 937.0|| =r , which is exactly

the value of eta coeffi cient.

4. Conclusions

Pearson correlation coeffi cient is widely used in statistic analysis to

estimate the relationship between different variables; however its subtle nuances

and versatile nature are not generally well-known and are insuffi ciently investigated

in the literature. The aim of this paper was to analyze this complex measure and

to present its problems and limitations. We focused on the most important factors

infl uencing the size, direction and interpretation of Pearson coeffi cient; using real

data we proved that the interpretation of the values of Pearson coeffi cient must be

presented with caution. For each case we must check if the interpretation is in line

with the particularities of data and the used model. In all basic statistics textbooks

we can fi nd a chapter or a section devoted to simple linear correlation and Pearson

coeffi cient. However, only a few cover the topic of factors infl uencing the size

and interpretation of r (mostly the lack of linearity). To our knowledge, there is no

approach of these aspects in the Romanian basic statistics textbooks.

We focused also on particular cases when one or both of the analyzed

variables are naturally dichotomous or artifi cially converted as dichotomous

variable. Using real data we illustrated how to use the point-biserial coeffi cient,

phi coeffi cient, tetrachoric coeffi cient and eta coeffi cient and how to interpret

the obtained results. The issues presented in this paper can be useful to

researchers, tutors who teach statistics and students.

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