Correlation and regression

Correlation(Pearson & spearman)
&Linear Regression

Correlation
 Semantically, Correlation means Co-together and
Relation.
 Statistical correlation is a statistical technique which
tells us if two variables are related.

PEARSON CORRELATION
 measures the degree of linear association between
two interval scaled variables analysis of the
relationship between two quantitative outcomes,
e.g., height and weight

Assumption under Pearson’s
Correlation Coefficient
 Assumption 1: The correlation coefficient r assumes
that the two variables measured
form a bivariate normal distribution population.
 Assumption 2: The correlation coefficient r measures
only linear associations: how nearly the data
falls on a straight line. It is not a good summary of
the association if the scatterplot has a nonlinear
(curved) pattern.

Assumptions Contd.
 Assumption 3: The correlation coefficient r is not a
good summary of association if the data are
heteroscedastic.(when random variables have the
same finite variance. It is also known as homogenity
of variance)
 Assumption 4: The correlation coefficient r is not a
good summary of association if the data have
outliers.

Strength of Relationship
 r lies between -1 and 1. Values near 0 means no (linear)
correlation and values near ± 1 means very strong
correlation.


Interpretation of the value of r

Coefficient of Determination
 Pearson's r can be squared , r 2, to derive a
coefficient of determination.
 Coefficient of determination - the portion of
variability in one of the variables that can be
accounted for by variability in the second
variable

Example
 Pearson's r can be squared , r 2
 If r=0.5, then r2=0.25 If r=0.7 then r2=0.49
 Thus while r=0.5 versus 0.7 might not look so
different in terms of strength, r2 tells us that r=0.7
accounts for about twice the variability relative to
r=0.5

Spearman’s Rank Correlation
 Spearman's rank correlation coefficient
or Spearman's rho, is a measure of statistical
dependence between two variables.

Interpretation
 The sign of the Spearman correlation indicates the
direction of association between X (the independent
variable) and Y (the dependent variable). If Y tends to
increase when X increases, the Spearman correlation
coefficient is positive. If Y tends to decrease when X
increases, the Spearman correlation coefficient is
negative. A Spearman correlation of zero indicates
that there is no tendency for Y to either increase or
decrease when X increases

Repeated Ranks
 If there is more
than one item with
the same value ,
then they are
given a common
rank which is
average of their
respective ranks
as shown in the
table.

 The raw data in the
table below is used to
calculate the
correlation between
the IQ of an with the
number of hours
spent in front
of TV per week.
Example

Regression
 One variable is a direct cause of the other or if the
value of one variable is changed, then as a direct
consequence, the other variable also change or if the
main purpose of the analysis is prediction of one
variable from the other


Regression
 Regression: the dependence of dependent variable Y
on the independent variable X.
 Relationship is summarized by a regression equation.
y = a + bx
 A=intercept at y axis
 B=regression coefficient

The Least Squares Method
 The line of regression is the line which gives the best
estimate to the value of one variable for any specific
value of the other variable. Thus the line of regression
is the line of “best fit” and is Obtained by the principle
of least squares.
 This principle consists in minimizing the sum of the
squares of the deviations of the actual values of y
from their estimate values given by the line of best fit

Example
 Fit a least square line to the following data

Correlation and regression

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Correlation and regression