This document discusses correlation analysis and different correlation coefficients. It defines correlation as a linear association between two random variables. Correlation can be positive, negative, or zero. There are three main types of correlation: between two variables, linear vs nonlinear, and simple vs multiple vs partial. Methods for studying correlation include scatter plots, Karl Pearson's coefficient of correlation r, and Spearman's rank correlation coefficient. The coefficient of correlation r ranges from -1 to 1, while the coefficient of determination r^2 ranges from 0 to 1. Higher absolute r values indicate stronger correlation, while r^2 represents the percent of variation explained by the linear relationship.
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2. Introduction
Correlation a LINEAR association between two
random variables
Correlation analysis show us how to determine
both the nature and strength of relationship
between two variables
When variables are dependent on time correlation
is applied
Correlation lies between +1 to -1
3. A zero correlation indicates that there is no
relationship between the variables
A correlation of –1 indicates a perfect negative
correlation
A correlation of +1 indicates a perfect positive
correlation
5. Type1
Positive Negative No Perfect
If two related variables are such that when
one increases (decreases), the other also
increases (decreases).
If two variables are such that when one
increases (decreases), the other decreases
(increases)
If both the variables are independent
6. When plotted on a graph it tends to be a perfect
line
When plotted on a graph it is not a straight line
Type 2
Linear Non – linear
8. Two independent and one dependent variable
One dependent and more than one independent
variables
One dependent variable and more than one
independent variable but only one independent
variable is considered and other independent
variables are considered constant
Type 3
Simple Multiple Partial
10. Methods of Studying Correlation
Scatter Diagram Method
Karl Pearson Coefficient Correlation of
Method
Spearman’s Rank Correlation Method
11. 0
20
40
60
80
100
120
140
160
180
0 50 100 150 200 250
Drug A (dose in mg)
Symptom
Index
0
20
40
60
80
100
120
140
160
0 50 100 150 200 250
Drug B (dose in mg)
S
ymptom
Index
Very good fit Moderate fit
Correlation: Linear
Relationships
Strong relationship = good linear fit
Points clustered closely around a line show a strong correlation.
The line is a good predictor (good fit) with the data. The more
spread out the points, the weaker the correlation, and the less
good the fit. The line is a REGRESSSION line (Y = bX + a)
12. Coefficient of Correlation
A measure of the strength of the linear relationship
between two variables that is defined in terms of the
(sample) covariance of the variables divided by their
(sample) standard deviations
Represented by “r”
r lies between +1 to -1
Magnitude and Direction
13. -1 < r < +1
The + and – signs are used for positive linear
correlations and negative linear
correlations, respectively
15. Interpreting Correlation
Coefficient r
strong correlation: r > .70 or r < –.70
moderate correlation: r is between .30 &
.70
or r is between –.30 and –.70
weak correlation: r is between 0 and .30
or r is between 0 and –.30 .
16. Coefficient of Determination
Coefficient of determination lies between 0 to 1
Represented by r2
The coefficient of determination is a measure of
how well the regression line represents the data
If the regression line passes exactly through
every point on the scatter plot, it would be able
to explain all of the variation
The further the line is away from the
points, the less it is able to explain
17. r 2, is useful because it gives the proportion of the
variance (fluctuation) of one variable that is
predictable from the other variable
It is a measure that allows us to determine how
certain one can be in making predictions from a
certain model/graph
The coefficient of determination is the ratio of the
explained variation to the total variation
The coefficient of determination is such that 0 < r 2 <
1, and denotes the strength of the linear association
between x and y
18. The Coefficient of determination represents the
percent of the data that is the closest to the line of
best fit
For example, if r = 0.922, then r 2 = 0.850
Which means that 85% of the total variation in y
can be explained by the linear relationship between
x and y (as described by the regression equation)
The other 15% of the total variation in y remains
unexplained
19. Spearmans rank coefficient
A method to determine correlation when the data
is not available in numerical form and as an
alternative the method, the method of rank
correlation is used. Thus when the values of the
two variables are converted to their ranks, and
there from the correlation is obtained, the
correlations known as rank correlation.
20. Computation of Rank
Correlation
Spearman’s rank correlation coefficient
ρ can be calculated when
Actual ranks given
Ranks are not given but grades are given but not
repeated
Ranks are not given and grades are given and
repeated