SPECIAL CORRELATION

Introduction… The Pearson correlation specifically

measures the degree of linear relationship between two variables.

It is most commonly used measure of relationship and is used with data from an interval or a ratio scale of measurement

However, other correlation measures have been developed for nonlinear relationship and for other types of data (scale of measurement)

Other correlation measures Spearman Rank Order

Correlation Biserial Correlation Point Biserial Correlation Tetrachoric Phi Coefficient

Spearman Rank Order Correlation

Spearman (Rank Order)

Correlation Spearman Correlation is designed to

measure the relationship between variables measured on an ordinal scale of measurement

Also can be used as a valuable alternative to the Pearson correlation, even the original raw scores are on an interval or ratio scale

The Spearman correlation measures consistency rather than form: ‘When two variables are consistently related, their rank will be linearly related.

Spearman’s Rank-Difference Correlation Method

Especially, when samples are small It can be applied as a quick substitute

when the number of pairs, or N, is less than 30

It should be applied when the data are already in terms of rank orders rather than interval measurement

X Y X – MX Y – MY (X-MX)2 (Y-MY)2 (X-MX)(Y-MY)7

4

6

3

5

11

3

5

4

7

2

-1

1

-2

0

5

-3

-1

-2

1

10

3

-1

4

0

4

1

1

4

0

r =√(SSX)(SSY)

SP

25

9

1

4

1

Scores DeviationsSquared

Deviations Products

=√ (10)(40)

+16 =

20

+16 = +0,8

The Computation of Spearman (Rank Order) Correlation

rρ = 1

-

6 Σ D2 n(n2 –

1)

X

74635

Y

113547

X

14253

Y

15342

RX – RY

0-1-111

(RX – RY)2

01111

Scores Rank Different

= 1 -6 (4) 5(25 –

1)

rρ = 1 -

6 Σ D2

n(n2 – 1)

= 0,8

INTERPRETATION OF A RANK DIFFERENCE COEFFICIENT

The rho coefficient is closely to the Pearson r that would be computed from the original measurement.

The rρ values are systematically a bit lower than the corresponding Pearson-r values, but the maximum difference, which occurs when both coefficient are near .50

To measure the relationship between anxiety level and test performance, a psychologist obtains a sample of n = 6 college students from introductory statistics course. The students are asked to come to the laboratory 15 minutes before the final exam. In the lab, the psychologist records psychological measure of anxiety (heart rate, skin resistance, blood pressure, etc) for each student. In addition, the psychologist obtains the exam score for each student.

LEARNING CHECK

Student

Anxiety

Rating

Exam Scores

A 5 80

B 2 88

C 7 80

D 7 79

E 4 86

F 5 85

Compute the Pearson and Spearman

correlation for the following

data.

Test the correlation with α = .05

The BISERIAL Coefficient of Correlation

The BISERIAL Coefficient of

Correlation The biserial r is especially designed for

the situation in which both of the variables correlated are continuously measurable, BUT one of the two is for some reason reduced to two categories

This reduction to two categories may be a consequence of the only way in which the data can be obtained, as, for example, when one variable is whether or not a student passes or fails a certain standard

The COMPUTATION The principle upon which the formula

for biserial r is based is that with zero correlation

There would no difference means for the continuous variable, and the larger the difference between means, the larger the correlation

THE

BISERIAL CORRELATIO

NWhere:Mp = mean of X values for the higher group in

the dichotomized variable, the one having ability on which sample is divided into two subgroups

Mq = mean of X values for the lower groupp = proportion of cases in the higher groupq = proportion of cases in the lower groupy = ordinate of the unit normal-distribution

curve at the point of division between segments containing p and q proportion of the cases

St = standard deviation of the total sample in the continuously measured variable X

rb =Mp – Mq

StX

pqy

THE

BISERIAL CORRELATIO

N

OR

rb =Mp – Mq

St

Xpqy

rb =Mp – MT

St

Xp

y

y

q area p area

The Standard Error of rb

If the obtained rb is greater than 1.96 times its standard error, we conclude that at .05 level the obtained correlation would not very probably have arisen by chance from a population in which the correlation is zero

Srb =√

N

pq

y√

AN EVALUATION OF THE BISERIAL r Before computing r, of course we need to

dichotomize each Y distribution. In adopting a division point, it is well to

come as near the median as possible, why? In all these special instances, however, we

are not relieve of the responsibility of defending the assumption of the normal population distribution of Y

It may seem contradictory to suggest that when the obtained Y distribution is skewed, we resort the biserial r, but note that is the sample distribution that is skewed and the population distribution that must be assumed to be normal

THE BISERIAL r IS LESS RELIABLE THAN THE PEARSON r

Whenever there is a real choices between computing a Pearson r or a biserial r, however, one should favor the former, unless the sample is very large and computation time is an important consideration

The standard error for a biserial r is considerably larger than that for a Pearson r derived from the same sample

The POINT BISERIAL Coefficient of Correlation

The POINT BISERIAL Coefficient of Correlation

When one of the two variables in a correlation problem is genuine dichotomy, the appropriate type of coefficient to use is point biserial r

Examples of genuine dichotomies are male vs female, being a farmer vs not being a farmer

Bimodal or other peculiar distributions, although not representating entirely discrete categories, are sufficiently discontinuous to call for the point biserial rather than biserial r

The COMPUTATION A product-moment r could be

computed with Pearson’s basic formula

If rpbi were computed from data that actually justified the use of rb, the coefficient computed would be markly smaller than rb obtained from the same data

rb is √pq/y times as large as rpbi

THE POINT BISERIAL

CORRELATION

Where:Mp = mean of X values for the higher group in the

dichotomized variable, the one having ability on which sample is divided into two subgroups

Mq = mean of X values for the lower groupp = proportion of cases in the higher groupq = proportion of cases in the higher groupSt = standard deviation of the total sample in the

continuously measured variable X

rpbi =Mp – Mq

St

pq

ALTERNATIVE

METHODS OF

COMPUTATIO

NFOR THE

POINT

BISERIAL

CORRELATIO

N

rpbi =Mp – MT

St

pq

rpbi =Mp – MT

St

NpNq

rpbi =Mp – Mq

St

NpNq

POINT-BISERIAL vs BISERIAL When the dichotomous variable is

normally distributed without reasonable doubt, it is recommended that rb be computed and interpreted

If there is little doubt that the distribution is a genuine dichotomy, rpbi should be computed and interpreted

When in doubt, the rpbi is probably the safer choice

Mathematical relation of rpbi to rb

rb = rpbi√ pq

y

rpbi = rb√ pq

y

The TETRACHORIC Correlation

TETRACHORIC CORRELATION A tetrachoric r is computed from data

in which both X and Y have been reduced artificially to two categories

Under the appropriate condition it gives a coefficient that is numerically equivalent to a Pearson r and may be regard as an approximation to it

Where:a = frequency of cases in the higher group in both

variablesb = frequency of cases in (the higher group in

variables X and the lower group in variable Y)c = frequency of cases in (the lower group in

variables X and the higher group in variable Y)d = frequency of cases in the lower group in both

variablesy = ordinate of the unit normal-distribution curve at

the point of division between segments containing p and q proportion of the cases

y’ = ordinate of the unit normal-distribution curve at the point of division between segments containing p’ and q’ proportion of the cases

rcos-pi = cos π √ad

bc

√ bc√+

Variable X

Var

iabl

e Y b a

d c

p’q’

p

q

y

y’

ALTERNATIVE METHODS OF

COMPUTATION FOR THE TETRACHORIC CORRELATION

rcos-pi = cos π √ad

bc

√ bc√+

rcos-pi = cos 1800√

ad

bc

√ bc√+

1800

ad/bc1+ √= cos

EXAMPLE for THE TETRACHORIC CORRELATIONStude

ntHeigh

tWeig

ht

A 165 61

B 162 54

C 170 60

D 174 58

E 168 50

F 172 63

G 163 52

H 159 49

Category

Light(<60kg)

Heavy(>60kg)

Tall(>170cm

)

b a

Short(<170cm

)

d c

rcos-pi =ad - bcyy’N2

The Standard Error of rt

If the obtained rt is greater than 1.96 times its standard error, we conclude that at .05 level the obtained correlation would not very probably have arisen by chance from a population in which the correlation is zero

Srt =√

N

pp’qq’

yy’√

TETRACHORIC CORRELATION The tetrachoric r requires that both X

and Y represent continuous, normally distributed, and linearly related variables

The tetrachoric r is less reliable than the Pearson r.

It is more reliable when:○ N is large, as is true of all statistic

○ rt is large, as is true of other r’s

○ the division in the two categories are near the medians

The Phi Coefficient rΦ

THE Phi COEFFICIENT rФ related to the chi square from 2 x 2 table

When two distributions correlated are genuinely dichotomous– when the two classes are separated by real gap between them, and previously discussed correlational method do not apply– we may resort to the phi coefficient

This coefficient was designed for so-called point distributions, which implies that the two classes have two point values and merely represent some qualitative attribute

THE PHI COEFFICIENT

rФ rФ =

αδ - βγ

√ pqp’q’Categor

y

Normal Color Vision

Color Blind BOTH

Male42β

18α

60p

Female26δ

14γ

40q

BOTH 68q’

32p’

1001,00

Compute the

Phi Coefficient

rΦ for the

following data.

PRTentukan teknik korelasi yang tepat untuk mengetahui gambaharan hubungan antara variabel di bawah ini. Kemudian hitung koefisien korelasinya 1.IPK dengan Tinggi Badan2.Jenis Kelamin dengan IPK3.Jenis Kelamin dengan Jumlah HP>1