CORRELATION
• We have studied the major statistical proprties
we investigate in univariate data, i.e. data
observed on a single variable. However,in
practical situations,we consider several
characteristics/variables( interval or scale type
) simultaneously.
• Ex: age,height,weight,BP,blood
sugar,cholesterol,mothers weight, weight of
the new born baby, etc.
• Consider the pairs of variables: (height,weight) , (
age, weight ) , (weight , BP )
• How does weight vary with height? or what kind
of association is shown between the two
variables?
• By association we mean the nature of change in
the values of the two variables. Do they change in
the same direction – both increase or both
decrease together,or as one variable increases
the other variable decreases?
• or no such association is found.
• This kind of association existing between two
variables is called correlation.
• When the two variables are associated in the
same direction,we say that they are positively
correlated. If the association is in opposite
directions , it is called negative correlation.
• If the two variables do not show clear and
consistent association , then they are
uncorrelated.
Some examples
• Degree of overcrowding and prevalence of
pulmonary tuberculosis
• Severity of malnutrition and prevalence of
tuberculosis
• Economic status and health status
• Ageing and blood pressure
Understanding correlation graphically
• The type of association between two variables
can be graphically examined by plotting the
values of one variable,say height ( X ) against
weight ( Y ) in a graph. Such a plot is called
scatter plot.
• As height increases, if weight also increases ( it
is so ,generally ) then height and weight are
positively correlated.
• As X increases Y decreases , then X and Y are
negatively correlated.
• If no such clear pattern of variation is found
then there is no correlation or zero
correlation.
Scatter plot
zero correlation
+ ve corr
-ve corr
+ve corre
Fig.1 SCATTER PLOT SHOWING
DIFFERENT CORRELATION TYPES
Fig.2 Positive correlation
-ve correlation
Fig.3 Negative correlation
Zero correlation
Fig. 4 Zero correlation
How to use scatter plot?
• If the points are forming an elliptic shape
starting from the left bottom and ascending to
the right top then the variables are positively
correlated. ( Fig. 1 , Fig. 2 )
• If the points are forming an elliptic shape
starting from the left top and descending to
the right bottom then the variables are
negatively correlated. ( Fig. 3 )
• If the points plotted do not show increasing or
decreasing pattern but spread out in the
scatter plot, then there is no correlation/zero
correlation between the variables.
• Presence of correlation between the two
variables is indicative of possible linear
relationship between the two variables.
• Hence some authors/books call it ‘ linear
correlation ‘
Measurement of correlation
• Scatter plot is useful to identify the
type/nature of correlation. A quantitative
measure of correlation between two variables
is correlation coefficient.
• Karl Pearson’s Product Moment Correlation
Coefficient ( or simply Pearson’s correlation
coefficient ) is one such measure.
• Pearson’s correlation coefft. tells us how
much one variable changes as the values
of another changes – their covariation.
• Covariation is measured by calculating the
amount by which each value of X varies
from the mean of X, and the amount by
which each value of Y varies from the
mean of Y and multiplying the differences
together and finding the average ( dividing
by (n-1) in case of sample.). This is called
covariance between X and Y.
• The covariance between X and Y is :
•
( x  x)( y  y )
Cov( X , Y ) 
n 1
• Pearson’s correlation coefficient is given by
the following formula
cov(X, Y)
r
sd(X).sd(Y )
• Where ‘r’ is Pearson’s correlation coefficient
between X and Y , based on a sample of size n
• The formula for computing Pearson’s
correlation coefficient can be simplified to the
following form:
r
(x  x )(y  y)
(x  x ) (y  y)
2
2
• Which can be further simplified to the
following formula:
(xy)  (n x y)
r
2
2
{(x )  n x }{(y )  n y }
2
2
• Here x and y are sample means of X and Y
•  xy is the sum of products of X and Y values
2
 x is the sum squares of X values
•
2
•  y is the sum of squares of Y values
• n is the sample size
PROPERTIES OF CORRELATION COEFFICIENT
• Pearson’s correlation coefficient lies between
-1 and +1. r=+1 implies perfect positive
correlation between the variables
• r = -1 implies perfect negative correlation
between the variables
• r = 0 implies no correlation or zero
correlation between the two variables. We say
that the two variables are uncorrelated.
• ‘ r ‘ is free from unit of measurement
• Note 1: correlation does not mean
causation. We can only investigate
causation by reference to our problem of
study. However (thinking about it the other
way round) there is unlikely to be
causation if there is no correlation.
• What is causation? Explain with examples.
• Note 2: What is good amount of correlation or significant
correlation between two variables?
• THUMB RULES:
• If r> = + .70
->
Very strong positive relationship
+.40 to +.69 ->
Strong positive relationship
+.30 to +.39 ->
Moderate positive relationship
+.20 to +.29 ->
Weak positive relationship
+.01 to +.19 ->
No or negligible relationship
-.01 to -.19 ->
No or negligible relationship
-.20 to -.29
->
Weak negative relationship
-.30 to -.39
->
Moderate negative relationship
-.40 to -.69
->
Strong negative relationship
-.70 or less
->
Very strong negative relationship
• The statistical significance of computed value
of ‘r’ from a sample of size n has to be
decided using appropriate statistical test
procedure.
• The thumb rule is only for understanding the
extent of correlation present and help to
proceed with further analysis.
Some precautions
• The two variables of interest are measured on
the same entity. If we consider height and
weight, these two variables are measured on
the same individual. It does not make sense to
speak of the correlation between the heights
of one group of individuals and the weights of
another group.
• No matter how strong is the correlation between
two variables , it should not be interpreted as one
of cause and effect. A significant correlation
between X and Y may mean:
• X causes Y
• Y causes X
• Some third factor inducing association between X
and Y
• The correlation could be ‘spurious’ or
‘nonsensical’
For self learning………
• When data are given in the form of grouped
bivariate with frequency values how to
compute correlation coefficient?
• Learn from text books or other sources.
• Since in most of the cases data are organised
in computer files and calculation is done using
statistical packages, the above situation does
not arise.
SPEARMAN’S RANK CORRELATION
• Spearman's rank correlation coefficient or Spearman's
rho, named after Charles Spearman and often denoted
by the Greek letter ρ , is a nonparametric measure of
statistical dependence between two variables.
• Spearman rank correlation measures the association
between two ranked variables, or one ranked variable
and one measurement variable. You can also use
Spearman rank correlation instead of Pearson’s
correlation for two measurement variables if you're
worried about non-normality, but this is not usually
necessary.
• To compute rank correlation , we first convert
the measured values into ranks and then
compute the difference between ranks in each
pair of data d
i
• . If there are n sample observations and d i is
the difference between ranks of i th pair
(assuming no tied ranks ) , then Spearman’s
rank correlation is given by:
•
2
i
s
2
6d
r  1
n(n  1)
Example
• X : 34,54,46,43,57,78,25,60,48,56
• Y: 46,54,57,55,65,50, 67,59,52, 56
• Let X and Y denote the scores given to a
candidate by two judges/examiners. We want
to know the degree of association in the
assessment by the two examiners.
• Rank the scores and find the difference in
ranks. Use the formula to compute rank
correlation between X and Y.