Linear Regression Ed.3

7/30/2019 Linear Regression Ed.3

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SitiHawa/MTE3105/ed.2

Linear Regression


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Think Of

Below is 2 sets of data, set A

(coursework score) and set B(test score) and think of the

conclusion that you can make

between this sets of data.

Set A

83.0

92.0

80.0

83.080.0

80.0

90.0

83.0

80.0

80.094.0

94.0

80.0

80.0

90.0

90.092.0

94.0

Set B

71.0

82.0

71.0

51.071.0

61.0

65.0

60.0

80.0

82.064.0

59.0

47.0

57.0

71.0

78.083.0

80.0


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Think Of

Dependent variable

Independent variable

Dependent

Or

Independent

Plot graph for this data

and explain it to the

class


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Linear Regression

Example : A set of data and the plot of a "best-fit" straight linethrough the data.


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Linear Regression

The relationship between two sets of data (xand y) is linear,when the data is plotted (yvsx) the result is a straight line.

This relationship is having a linear correlation and follows theequation of a straight line, y = mx+b .


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Linear Regression : Concept

Example of linear regression with one dependent and one independent variable.
http://en.wikipedia.org/wiki/Image:Normdist_regression.png


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Scatterplots

A scatterplot is a graphic toolused to display the relationshipbetween two quantitativevariables.

A scatterplot consists of an X axis(the horizontal axis), a Y axis (thevertical axis), and a series of dots.

Each dot on the scatterplotrepresents one observation from adata set. The position of the dot onthe scatterplot represents its X andY values.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.0 2.0 4.0 6.0 8.0

x

y
http://stattrek.com/Help/Glossary.aspx?Target=Quantitative%20variablehttp://stattrek.com/Help/Glossary.aspx?Target=Quantitative%20variable


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Linear Regression : Scatter Diagram

A scatterplot is often employed to identify potential associations between

two variables : explanatory variable and response variable.

Positive association (positive

slope)

Negative association

(negative slope)No association


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Linear Regression

Given a set of data (xi, yi) with n data points, the slope and y-intercept can be determined using the following:

22 )()(

)(

xxn

yxxyn

m

n

xmyb


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Correlation Coefficient

Given a set of data (xi , yi) with n data points, the correlation

coefficient, r can be determined by :

2222 )()()()(

)(

yynxxn

yxxynr

See example using MS Excel
http://localhost/var/www/apps/conversion/tmp/scratch_10/Linear%20Regression%20Eg1.xlsxhttp://localhost/var/www/apps/conversion/tmp/scratch_10/Linear%20Regression%20Eg1.xlsx


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Linear Regression

Correlation coefficient, r, is a measure of the reliability of the linear

relationship between thexand yvalues.

r= 1 indicates an exact linear relationship betweenxand y.

Values ofrclose to 1 indicate excellent linear reliability.

If the correlation coefficient is relatively far away from 1, the predictions

based on the linear relationship, y = mx + b , will be less reliable.


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Correlation Coefficients, r

Correlation coefficients measure thestrength of association between two

variables.

The most common correlation coefficient,called the Pearson product-moment

correlation coefficient, measures the

strength of the linear association betweenvariables.


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Scatterplots and Correlation Coefficients

The scatterplots below show how different patterns of data produce

different degrees of correlation.

Maximum positive correlation (r = 1.0) Strong positive correlation (r = 0.80) Zero correlation (r = 0)

Minimum negative correlation (r = -1.0) Moderate negative correlation (r = -0.43) Strong correlation with outlier (r = 0.71)
http://stattrek.com/Help/Glossary.aspx?Target=Scatterplothttp://stattrek.com/Help/Glossary.aspx?Target=Scatterplot


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Linear Regression : Concept

Simple Linear Regression is the method for finding the "line of best fit" betweenthe dependent variable, y, and the independent variable, x.

Simple: only one independent variable

In general, the goal of linear regression is to find the line that best predicts Yfrom

X.

Linear regression does this by finding the line that minimizes the sum of the

squares of the vertical distances of the points from the line.

The Least Squares Regression Line is the line which minimizes the sum of thesquare or the error of the data points.


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How to Interpret a Correlation Coefficient

The sign and the absolute value of a correlation coefficient describe the

direction and the magnitude of the relationship between two variables.

The value of a correlation coefficient ranges between -1 and 1.

The greater the absolute value of a correlation coefficient,

the stronger the linearrelationship.

The strongest linear relationship is indicated by a correlation coefficient of -1 or 1.

The weakest linear relationship is indicated by a correlation coefficient equal to 0.

A positive correlation means that if one variable gets bigger, the other

variable tends to get bigger.

A negative correlation means that if one variable gets bigger, the other

variable tends to get smaller.
http://stattrek.com/Help/Glossary.aspx?Target=Absolute%20valuehttp://stattrek.com/Help/Glossary.aspx?Target=Absolute%20value


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Interpolation & Extrapolation

Interpolation is making a prediction within the range of

values of the predictor in the sample used to generate themodel.

Extrapolation is making a prediction outside the range of

values of the predictor in the sample used to generate the

model.

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Linear Regression Ed.3