Análisis de regresión y correlación: distribuciones estadísticas multivariadas, Diapositivas de Estadística Empresarial

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Curso 2015-

Estadística Empresarial I

Chapter 4Bivariate Data Analysis

CONTENTS^ 1. Multivariate statistical distributions2. Graphical presentations3. Momentum4. Correlation5. Regression6. Use of regression and correlation

1. Multivariate statistical distributionsMultidimensional Variables (II)Examples of bivariate variables:^ 

Maximum and minimum temperatures registered each day during themonth of february this month. Age and height of your classmates. Monthly mean of the Madrid Stock market general index during the last10 months and the average business volume on same months. Working situation and sex of a group of persons.

Student

Age

Height

1. Multivariate statistical distributionsMultidimensional Variables (III)

Student

Age

Height

Y

X^

y^1

y^2

…^

yj^

…^

yk^

n^ i·

x^1

n^11

n^12

…^

n1j^

…^

n1k^

n1·

x^2

n^21

n^22

…^

n2j^

…^

n2k^

n2·

…^

…^

…^

…^

…^

…^

…^

xi^

ni^

ni^

…^

nij^

…^

nik^

ni·

…^

…^

…^

…^

…^

…^

…^

xh^

nh^

nh^

…^

nhj^

…^

nhk^

nh·

n^ ·j^

n·^

n·^

…^

n·j^

…^

n·k^

N

1. Multivariate statistical distributionsJoint distribution (II) – Correlation Table

Altura

1. Multivariate statistical distributionsJoint distribution (III) – Correlation table Edad

Altura Edad

[1,58-1,64]

(1,64-1,70]

(1,70-1,76]

(1,76-1,82]

(1,82-1,88]

Altura 18 19 20 21 22 23 24 Edad

[1,58-1,64]

(1,64-1,70]

(1,70-1,76]

(1,76-1,82]

(1,82-1,88]

Altura Edad

[1,58-1,64]

(1,64-1,70]

(1,70-1,76]

(1,76-1,82]

(1,82-1,88]

Height Age

[1,58-1,64]

(1,64-1,70]

(1,70-1,76]

(1,76-1,82]

(1,82-1,88]

Total

Total

1. Multivariate statistical distributionsMarginal distributions (I)On a bivariate frequency distribution we are often interested in the separatestudy of each variable, considering them as two separate unidimensionaldistributions, X and Y respectively.To produce we find what is called the

marginal frequency distributions

that is, how many times each value of the variable appears, regardless ofwhich value of the other variable goes together with it.Therefore, for the i-

esim

value of X, its marginal frequency is:

And for the j-

esim

value of Y, its marginal frequency is :

k  ^1 j

ij ·i

n n^

n n n 1 n f f^

·i k^1 j

ij

k ij^1 j ·i



 ^

 ^

 

h ^1 i

ij j·

n n^

n n n 1 n f f^

j· h^1 i

ij

h ij^1 i j·



 ^

 ^

 

Y

X^

y^1

…^

yj^

…^

yk^

n^ i·^

f^ i·

x^1

n^11

…^

n1j^

…^

n1k^

n1·^

f1·

…^

…^

…^

…^

…^

…^

…^

xi^

ni^

…^

nij^

…^

nik^

ni·^

fi·

…^

…^

…^

…^

…^

…^

…^

xh^

nh^

…^

nhj^

…^

nhk^

nh·^

fh·

n^ ·j^

n·^

…^

n·j^

…^

n·k^

N^

f^ ·j^

f·^

…^

f·j^

…^

f·k^

X^

n^ i·^

f^ i·

x^1

n1·^

f1·

…^

…^

xi^

ni·^

fi·

…^

…^

xh^

nh·^

fh· N^

Y

X^

y^1

…^

yj^

…^

yk^

n^ i·^

f^ i·

x^1

n^11

…^

n1j^

…^

n1k^

n1·^

f1·

…^

…^

…^

…^

…^

…^

…^

xi^

ni^

…^

nij^

…^

nik^

ni·^

fi·

…^

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xh^

nh^

…^

nhj^

…^

nhk^

nh·^

fh·

n^ ·j^

n·^

…^

n·j^

…^

n·k^

N^

f^ ·j^

f·^

…^

f·j^

…^

f·k^

Y^

y^1

…^

yj^

…^

yk

n^ ·j^

n·^

…^

n·j^

…^

n·k^

N

f^ ·j^

f·^

…^

f·j^

…^

f·k^

1. Multivariate statistical distributionsMarginal distributions (II)

Y

X^

y^1

…^

yj^

…^

yk^

n^ i·^

f^ i·

x^1

n^11

…^

n1j^

…^

n1k^

n1·^

f1·

…^

…^

…^

…^

…^

…^

…^

xi^

ni^

…^

nij^

…^

nik^

ni·^

fi·

…^

…^

…^

…^

…^

…^

…^

xh^

nh^

…^

nhj^

…^

nhk^

nh·^

fh·

n^ ·j^

n·^

…^

n·j^

…^

n·k^

N^

f^ ·j^

f·^

…^

f·j^

…^

f·k^

X|Y=y

j^

n^ i|j

x^1

n1j

…^

xi^

nij

…^

xh^

nhj n·j

Y

X^

y^1

…^

yj^

…^

yk^

n^ i·^

f^ i·

x^1

n^11

…^

n1j^

…^

n1k^

n1·^

f1·

…^

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xi^

ni^

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nij^

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nik^

ni·^

fi·

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xh^

nh^

…^

nhj^

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nhk^

nh·^

fh·

n^ ·j^

n·^

…^

n·j^

…^

n·k^

N^

f^ ·j^

f·^

…^

f·j^

…^

f·k^

Y|X=x

i^

y^1

…^

yj^

…^

yk

xi^

ni^

…^

nij^

…^

nik^

ni·

1. Multivariate statistical distributionsConditional distributions (II)

1. Multivariate statistical distributionsStatistical dependence and independence (I)When there is no relation between the values reached by two variables, wesay they are independent. Conversely, when there is a perfect relation, i.e.there is a function linking them, we say there is a functional relation, whichmeans their relation can be expressed as

y = f(x)

Nevertheless, there are many situations in which the relation between twovariables is somewhere in between full independence and full functionaldependence. That is what we call

statistical dependence

: to some extent

the behavior of one variable has an influence in the behavior of the other.As the functional dependence does not admit a grading (it exists or itdoesn’t), statistical dependence can be graded, since it can be stronger orweaker,

consider

it

is

closer

to

functional

dependence

or

plain

independence.

1. Multivariate statistical distributionsStatistical dependence and independence (III)Two variables

X and Y

are said to be

statistically independent

when

each joint relative frequency equals the product of the associated relativemarginal frequencies, that is:Or, in absolute frequencies:In this situation, relative conditional frequencies equal their correspondentmarginal relative frequencies. This tells us (independence) that conditioninghas no effects:

j,i n· n n n n^ n

j· ·i ij^





j,i f· f f^

j· ·i ij^

  ·i ·i j· ·i j·

ij j· j|i

f n n n·n n n

n n f^

 

 ^

j· j· j· ·i ·i

ij ·i i|j

f n n n·n n n

n n f^

 

 

1. Multivariate statistical distributionsStatistical dependence and independence (III)Considering statistical independence its necessary to face three differentproblems:^ 

Recognizing the presence of statistical relation. Studying the degree of dependence or intensity of the relation betweenvariables:

correlation theory

Finding the shape of such statistical dependence relation, finding thebest possible functional model:

regression models

2. Graphical presentationsScatter graph or dispersion diagram

2. Graphical presentationsScatter graph or dispersion diagram