special-correlation.ppt

© aSup -2007
Statistics II – SPECIAL CORRELATION  

1
SPECIAL
CORRELATION

© aSup -2007

2
The SPEARMAN Correlation
 The Pearson correlation specially measures
the degree of linear relationship between
two variables
 Other correlation measures have been
developed for nonlinear relationship and of
other types of data
 One of these useful measures is called the
Spearman correlation

© aSup -2007

3
The SPEARMAN Correlation
 Measure the relationship between variables
measured on an ordinal scale of
measurement
 The reason that the Spearman correlation
measures consistency, rather than form,
comes from a simple observation: when two
variables are consistently related, their
ranks will be linearly related

© aSup -2007

4
INTRODUCTION
 Pearson product-moment coefficient is the
standard index of the amount of correlation
between two variables, and we prefer it
whenever its use is possible and
convenient.
 But there are data to which this kind of
correlation method cannot be applied, and
there are instances in which can be applied
but in which, for practical purpose, other
procedures are more expedient

© aSup -2007

5
Person X Y Rank X Rank Y
A
B
C
D
E
3
4
8
10
13
12
10
11
9
3
1
2
3
4
5
5
3
4
2
1
rs =
SP
√(SSx) (SSy)
SP = ΣXY
(ΣX)(ΣY)
n
SSX = ΣX2
(ΣX)2
n
XY
5
6
12
8
5

© aSup -2007

6
The COMPUTATION
1. Rank the individual in the (two)
variables
2. For every pair of rank (for each
individual), determine the difference
(d) in the two ranks
3. Square each d to find d2

© aSup -2007

7
Person X Y Rank X Rank Y
A
B
C
D
E
3
4
8
10
13
12
10
11
9
3
1
2
3
4
5
5
3
4
2
1
rs = 1 -
6 Σ D2
n(n2 – 1)
D2
16
1
1
4
16

© aSup -2007

8
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

© aSup -2007

9
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

© aSup -2007

10
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

© aSup -2007

11
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

© aSup -2007

12
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

© aSup -2007

13
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

© aSup -2007

14
AN EVALUATION OF THE BISERIAL r
 Before computing rb, 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

© aSup -2007

15
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

© aSup -2007

16
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

© aSup -2007

17
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

© aSup -2007

18
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

© aSup -2007

19
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

© aSup -2007

20
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
a. N is large, as is true of all statistic
b. rt is large, as is true of other r’s
c. the division in the two categories are near the
medians

© aSup -2007

21
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

© aSup -2007

22
DEFINITION
 A partial correlation between two variables
is one that nullifies the effects of a third
variable (or a number of other variables)
upon both the variables being correlated

© aSup -2007

23
EXAMPLE
 The correlation between height and weight
of boys in a group where age is permitted to
vary would be higher than the correlation
between height and weight in a group at
constant age
 The reason is obvious. Because certain boys
are older, they are both heavier and taller.
Age is a factor that enhances the strength
of correspondence between height and
weight

© aSup -2007

24
THE GENERAL FORMULA
r12.3 =
r12 – r13r23
√(1 – r2
13)(1 – r2
23)
When only one variable is held constant,
we speak of a first-order partial correlation

© aSup -2007

26
THE 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 group
p = proportion of cases in the higher group
q = proportion of cases in the higher group
Y = 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 continously
measured variable X
rb =
Mp – Mq
St
X
pq
y

© aSup -2007

27
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 group
p = proportion of cases in the higher group
q = proportion of cases in the higher group
St = standard deviation of the total sample in the continously
measured variable X
rpbi =
Mp – Mq
St
pq

© aSup -2007

30
THE GENERAL FORMULA
r12.3 =
r12 – r13r23
√(1 – r2
13)(1 – r2
23)
When two variables is held constant, we
speak of a second-order partial correlation

special-correlation.ppt

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to special-correlation.ppt

Similar to special-correlation.ppt (20)

Recently uploaded

Recently uploaded (20)

special-correlation.ppt