stat-11, regression.pdf

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Jahangirnagar University Islam, M.T. http://sites.google.com/site/kjatbd/

Regression Analysis

Md. Tarikul Islam Jahangirnagar University, Bangladesh


Recap-1/2

q So far we have seen

Recapping o Research, types of research, and research methodology

In core of all there is DATA

Basics of statistics o Data, types of data o Place of data in statistics with the definition and

characteristics of statistics

Data collection and sampling o What data to collect? From where? How? o Sampling methods

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Recap-2/2

q So far we have seen

Data presentation o Classification, tabulation, and graphs

Central Tendency o Mean, median, mode, harmonic mean, and geometric mean

Measures of variation o Range, average deviation, and standard deviation

Correlation o Scatter diagram, Pearsons r, and rank correlation


The Course: Topics No Topics

01 Basics of statistics and recap!

02 Collection of data

03 Presentation of data

04 Measures of central tendency

05 Measures of variation

06 Skewness, moments, and kurtosis

07 Correlation analysis

08 Regression analysis

09 Forecasting and time series analysis

10 Probability

11 Sampling

Content

s are su

bject to

change

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Find out a and b

15 = 5a + 25b . (I) 88 = 25a + 151b ..(ii)

a = ? b = ?


Regression

q Regression

o Regression is the technique of predicting the behavior of one variable based on the another one (s); condition to that both variables are correlated

Estimating? Predicting the probable/most probable values for

one variable depending on another Value of one variable is known while that of another

one is unknown One variable is independent and based on that we

calculate the value of the dependent variable

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Correlation vs Regression

q Correlation vs Regression

o Correlation calculates the degree of relationship while regression calculates the nature of relationship

o Regression better shows the cause and effect relationship than correlation


Variables

q Variables

o Again two variables but in real life can be more than two; two variables is called simple regression

o Variables Dependent and independent

X on Y and Y on X as examples

o If both variables are independent?

o Now as we have two variable we would have two regression lines too!

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Regression: Formulae

q Y on X o Y = a + bX

Normal Equations o Y = Na + bX o XY = a X + b X2

q X on Y o X = a + bY

Normal Equations o X = Na + bY o XY = a Y + b Y2

a and b are called parameters and they actually set the position of the line completely!


a and b in graph

q a and b in graph

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Regression: Formulae Explanation

q Formulae Explanation

o a and b can be calculated with the method of least squares With the given two equations in the previous slide Called normal equations

o What is least square method? A method assuming that the best-fit curve of a given

type is the curve that has the minimal sum of the deviations squared (least square error) from a given set of data


What are the relevant items?

q Relevant items

o X o Y o XY o X2 o Y2

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Problem solving: 1/3

q Given data

X Y

1 2

2 5

3 3

4 8

5 7



q We need to fill the shaded areas

X Y X2 Y2 XY

1 2

2 5

3 3

4 8

5 7

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q Lets fill in

X Y X2 Y2 XY

1 2 1 4 2

2 5 4 25 10

3 3 9 9 9

4 8 16 64 32

5 7 25 49 35

X = 15 Y = 25 X2 = 55 Y2 = 151 XY = 88


Regression solution

q With the formula for X on Y

o X on Y X = a + bY

o Normal Equations X = Na + bY XY = a Y + b Y2

15 = 5a + 25b . (I) 88 = 25a + 151b ..(ii) From (I) and (ii): a = 0.5, b = 0.5

So the regression is

X = 0.5 + 0.5 Y

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Use of the regression line

q Use of the regression line

o If the regression line is Y = 10 + 15X

o Then Explain the regression line Calculate the value of Y if X changes by 30 units


Deviation from means

q Deviation from means

o Rather than working with X and Y directly we can work with the deviations from the means of X and Y

o Formula gets a bit change too How?

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Regression: Formulae-New

q Y on X o Y = a + bX changed to

Normal Equations o y = Na + bx o xy = a x + b x2

q X on Y o X = a + bY

Normal Equations o x = Na + by o xy = a y + b y2

Y !Y!

!=!byx(X!X)

!

!and!!!!byx

!=!xy"x2"

!!!!!where,!x!=!X!X,!

!!y =!Y !Y!

X!X!

!=!bxy(Y !Y)

!

!and!!!!bxy

!=!xy"y2"

!!!!!where,!x!=!X!X,!

!!y =!Y !Y!


steps in calculation

q steps in calculation

First mean of two series (X and Y),

Then o

o x2 and y2

o xy

X!

!!!and!!Y!

!!

x!=!X!X!

!!!and!y!=!Y !!Y!

!!

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Practice: given data

X Y 40 2.5

70 6

50 4

60 5

80 4

50 2.5

90 5.5

40 3

60 4.5

60 3


Practice: filling in..

X Y x x2 y y2 xy

40 2.5

70 6

50 4

60 5

80 4

50 2.5

90 5.5

40 3

60 4.5

60 3

600 40

X!

Y!

= 600 10 = 60 = 40 10 = 4

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Practice: filling in..

X Y x x2 y y2 xy

40 2.5 -20 400 -1.5 2.25 30

70 6 +10 100 +2 4 20

50 4 -10 100 0 0 0

60 5 0 0 +1 1 0

80 4 +20 400 0 0 0

50 2.5 -10 100 -1.5 2.25 15

90 5.5 +30 900 +1.5 2.25 45

40 3 -20 400 -1 1 20

60 4.5 0 0 +0.5 .25 0

60 3 0 0 -1 1 0

600 40 0 2400 0 14 130


Practice: calculations

q Practice: calculations

byx

!=!xy!x2!

!=!1302400

!=!0.054

Y !Y!

= !byx(X!X

!

)!=!Y !4!=!0.054!(X!60)!=!Y!=!0.76+0.054X

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Practice @ home

q Do the math

o There are lots of math to do in this chapter o Go through them at home o Get any problem? Speak up in the class beforehand!


qThank You!

qAny Question?!

stat-11, regression.pdf

Documents

sampling o

examples o

o sampling methods

recapping o research

types of data o place

normal equations o x

bx normal equations

na bx o xy