In regression analysis we analyze the . relationship. . between . two or more. variables.. The relationship between two or more variables could be . linear or non linear. .. This week . first talk . ID: 241572
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Regression Analysis
In regression analysis we analyze the
relationship
between
two or more
variables.
The relationship between two or more variables could be
linear or non linear
.
This week
first talk
about
the simplest case.
Simple
Linear Regression : Linear Regression Between Two Variables
How we could
use available data
to investigate such a relationship
?
H
ow
could we
use this relationship to forecast future
.
While our interest it to investigate the relationship between demand (y) and time (x). But the concept is general, for example,
advertising
could be the
independent variable
and sales to be the
dependent variable
.
Slide2Simple Linear Relationship
Linear relationship between two variables is stated as
y = b0 + b1 xThis is the general equation for a line b0 : Intersection with y axisb1 : The slopex : The independent variabley : The dependent variable
b
1
> 0
b
1
< 0
b
1
= 0
Slide3
Scatter Diagram
Slide4Graphical  Judgmental Solution
b
1
b
0
1
Slide5Graphical  Judgmental Solution
Slide6Graphical  Judgmental Solution
Slide7SSE : Pictorial Representation
y
10
 y
10
^
Slide8SST : Pictorial Representation
y
10

y
y
i
Slide9SSE , SST and SSR
SST : A measure of how well the
observations cluster around ySSE : A measure of how well the observations cluster around ŷ If x did not play any role in vale of y then we shouldSST = SSEIf x plays the full role in vale of y then SSE = 0SST = SSE + SSRSSR : Sum of the squares due to regressionSSR is explained portion of SSTSSE is unexplained portion of SST
Slide10Coefficient of Determination for Goodness of Fit
SSE = SST  SSR
The largest value for SSE is
SSE = SST
SSE = SST =======> SSR = 0
SSR/SST = 0 =====> the worst fit
SSR/SST = 1 =====> the best fit
Slide11Coefficient of Determination for Pizza example
In the Pizza example,
SST = 15730
SSE = 1530
SSR = 15730  1530 = 14200
r
2
=
SSR/SST : Coefficient of Determination
1
r
2
0
r
2
= 14200/15730 = .9027
In other words, 90% of variations in y can be explained by the regression line.
Slide12Coefficient of Determinationr 2 = SSR/SST = 100/114 = .88 The regression relationship is very strong since 88% of the variation in number of cars sold can be explained by the linear relationship between the number of TV ads and the number of cars sold.
Example : Read Auto Sales
Slide13Correlation Coefficient = Sign of b1 times Square Root of the Coefficient of Determination)
The Correlation Coefficient
Correlation coefficient is a measure of the strength of a linear association between two variables. It has a value between 1 and +1
R
xy
= +1
: two variables are perfectly related through a line with positive slope.
R
xy
= 1
: two variables are perfectly related through a line with negative slope.
R
xy
= 0
: two variables are not linearly related.
Slide14Coefficient of Determination and Correlation Coefficient are both measures of associations between variables.Correlation Coefficient for linear relationship between two variables.Coefficient of Determination for linear and nonlinear relationships between two and more variables.
Correlation Coefficient and Coefficient of Determination
Slide15Next Slides