(12) Bivariate Data

7/30/2019 (12) Bivariate Data

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Applied Statistics and Computing Lab

BIVARIATE DATA


Indian School of Business


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Learning goals

Understanding bivariate data

Understanding the idea of correlation

Understanding linear regression

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Bivariate Data

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Applied Statistics and Computing Lab4


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Why study variables together Variation in one variable may or may not affect

the variation in another variable Understanding the relationship

When the value of one variable changes, compare the

other variable for: Direction of movement and

: Magnitude of movement

PredictionIf a new value of one variable is observed, can we

predict the corresponding value of the other variable?

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Statistics for bivariate dataX Y

.. ..

.. ..

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X|Y .. .. Totals

..

..

Totals 1

E(XY)

Data type I ( + + + )

=

Data type II ( + + + )

,

=

,

Data type I:

Data type II: (tabulating relative frequencies;

in case if there are multiple observationswith same values of X and Y)

E(X), E(Y), V(X) and V(Y) are calculated as

per the univariate mean and varianceformulae


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Covariance (denoted by Cov) We understand the variation in a single variable by looking at the

movement of its values from a central tendency

For a bivariate data, we want to look at the combined deviation The sign of such a measure may tell us about the two variables and

how they covary

Hence we can take product of the two sets of deviations

The Covariance calculates just this! It is defined as the expected value of the product of the deviation

of X from its mean, and the deviation of Y from its mean*

A reasonable measure of joint variation10

).(),cov(

)().()(),cov(

))]())(([(),cov(

n

y

n

x

n

xyYX

YEXEXYEYX

YEYXEXEYX

=

=

=

*Aczel A., Sounderpandian J. Complete business statistics


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Covariance (contd.) Covariance is independent of change of origin but

affected by change of scale

Covariance of 2 variables is always lesser than or equalto the product of variances of those two variables

Unit of covariance is obtained by taking a product ofthe units of X and Y

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)],[cov(),cov(

)(and

)(For

YXcdVU

d

bYV

c

aXU

=

=

=

)().(),cov( YVarXVarYX


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Covariance (contd.) cov(Waist circumference, adipose tissue area) = 643.39

Can we compare this with another covariance? For the Body measurement data, consider both the Weight

and the Height of all the individuals

What is the covariance between Height and Weight for boththe genders?

= 27.13Kg. : Cms. and

= 40.38Kg. : Cms.

What information do we obtain by comparing these two

covariance values?

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Standardization If we standardize both the variables, the covariance is independent

of the unit of measurement

Makes the covariances of both categories comparable It would then lie between [-1,1]

The number is closer to 0 => the variables do not covary much

The number closer to 1 or -1 => the variables covary highly

, = 0.43

, = 0.53

The height and weight are moderately related to each other, forboth the genders

We will see that this covariance is the same as the measure westudy next!

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Correlation coefficient Denoted by (called rho)

Defined as the measure of the degree of linear association between the

two variables X and Y* Indicates the strength of and direction in which the two variables would

move, in relation with each other

Calculated as the proportion of the covariance between X and Y, to theproduct of standard deviations of X and Y

=( , )

Correlation coefficient is also termed as the Pearson Product-momentCorrelation Coefficient

, = 0.77 ( , ) = 0.43

( , ) = 0.53

14*Aczel A., Sounderpandian J. Complete business statistics


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Properties of Correlation coefficient Correlation coefficient of two variables is equal to the

covariance of their standardised forms

Lies between -1 and 1 (extremes included)

1 1

It is a dimension-free measure or a measure free of

units Is independent of both, change of origin and change of

scale

=( )

=( )

,

=

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Perfect positive correlation.

If one of X or Y increases,

the other one must increaseas per an exact linear

relation. Similarly if one

decreases, the other

decreases by the same rule.

Perfect negative correlation.


the other must decrease asper an exact liner relation.

Similarly if one decreases,

the other increases by the

same rule.

No linear relationship. Strong negative correlation.

If one of X or Y increases,the other decreases as per a

moderately strong linear

relation. Similarly if one

decreases, the other

increases by the same rule.

Strong negative correlation.


the other decreases as per a

very strong linear relation.


the other increases by the

same rule.

Moderate positive

correlation. If one of X or Y

increases, the other must

increase as per a moderately

strong linear relation.


the other decreases by the

same rule.

Weak positive correlation. If

one of X or Y increases, the

other must increase as per a

weak linear relation.


the other decreases by the

same rule.

No linear relationship.

Visuals from Aczel A., Sounderpandian J. Complete business statistics


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Limitations of correlation coefficient

Correlation

coefficient = 0.911!

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Y X

0.6 2.01

0.2 2

0.2 2

0.2 2

0.1 2

0.1 2

0.1 2

0.05 2

0.05 2

0 2

X Y

-3 9

-2 4

-1 1

0 0

1 1

2 4

3 9

Correlation coefficient= 0

Yet, there exists aperfect quadraticrelation between Xand Y


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Correlation and causality A huge Roger Federer fan!

Watches several Fedearer - Nadal matches live

on television

Has recorded that Federer loses approximately80% of the matches, that this fan watches live

Does he cause Federer to lose, by watching

the match?

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Other measures Rank correlation

To measure the degree of correlation between two ordinal variables or

rankings

: Company rankings given by two different publications

: Ranks of universities published on two websites

Consider two groups of women. They are grouped based on whether they use a

particular brand of shampoo (say Shampoo A) or not. For each of the groups,responses are collated to indicate which of the five characteristics about their

shampoo are most important to them.

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Characteristics Group 1 rankings Group 2 rankings D=(rank 2 rank 1)

Characteristic 1 1 5 4



Characteristic 4 5 1 -4

Characteristic 5 4 2 -2


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Other measures (contd.) Spearmans Rank correlation coefficient ():

= 1

6

( 1)

Where, d= difference between 2 ranks of each object

n= Number of objects

This rank correlation is also equal to the Pearson product-moment

correlation applied to the ranks organised in an ascending order

Lies in the interval [-1,1]

Higher the positive correlation coefficient, greater the degree of

agreement between two ranks

Higher the negative correlation coefficient (closer to -1), greater the

degree of disagreement between two ranks

A correlation coefficient of 0 indicates that there is absolutely no similarity

in the two ranks given to the same object

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Other measures (contd.) Kendalls Tau ():

= ( )

12 ( 1)

For n objects with ranks , ; for each i=1,2,,n, a pair of observations ( , )

and , is said to be,

concordantif the ranks of both elements agree i.e. both ( > ) and

> OR both ( < ) and <

discordantif( > ) and ( < ) OR ( < ) and ( > ), the pair is

said to be discordant

Neither concordant nor discordant if( = ) or =

Lies in the interval [-1,1]

If the agreement between two rankings is perfect, coefficient = 1

If the disagreement between two rankings is perfect, coefficient = -1

If the rankings are independent, the coefficient would be close to 0

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Linear Regression Suppose now, the variation in one variable (X) influences the

variation in the other variable (Y)

Is the adipose tissue area is influenced by waist circumference? Are ice-cream sales affected by the temperature in the city?

The variable X i.e. the variable that influences, is also referred to as

the predictor variable or the independent variable or theexplanatory variable

The variable Y i.e. the variable that is being influenced, is alsoreferred to as the outcome variable or the dependent variable orthe explanatory variable

Can we draw one line such that the equation of that line explainsthe relation between X and Y?

Which line describes the relationship in a reasonable way?

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Linear regression (contd.)

24Visuals from Aczel A., Sounderpandian J. Complete business statistics

This line minimizes the sum of squared vertical distances


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Linear regression (contd.) Simple linear regression model:

=

+

+

where, Y=Outcome variable

X=Predictor variable

=Random component in the model

=( )( )

( )

= -

If we can safely assume linear relationship between Xand Y, this model predicts average value by which Y willchange for one unit change in X

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The model is estimated using Method of least squares

This method tries to minimize the sum of squared errors

There are other methods of estimation

Visuals from Aczel A., Sounderpandian J. Complete business statistics


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Linear regression (contd.) Goodness of the model depends on the strength

of linear relationship between X and Y The error could comprise of factors other than X,

that may affect Y

The coefficient of determination or is a

measure of the strength of linearity in the

relationship

It indicates the proportion of variation in Y, that is

explained by X

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Linear regression (contd.) Fitting a linear regression for the Waist circumference-Adipose tissue data gives following output in R:

We get the following regression equation: = 71.26 + 0.2( )

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Coefficients: Estimate Std. Error t value Pr(>|t|)

(Intercept) 71.26327 1.88565 37.79


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R-codesFunction R-code

Dotplot install.packages(TeachingDemos)

library(TeachingDemos)

dots(variable name)

Scatter plot plot(variable1 name,variable2 name)

Covariance cov(variable1 name,variable2 name)

Correlation cor(variable1 name,variable2 name)Spearmans rank correlation cor(variable1 name,variable2 name,

method=spearman)

Kendalls tau cor(variable1 name,variable2 name,

method=kendall)Linear regression lm(response variable ~ explanatory

variable)

Regression line abline(response variable ~ explanatory

variable)

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Thank you

Documents

(12) Bivariate Data