Two Variable Relationships Positive Association

1
Two Variable Relationships(Positive Association)
Y
X
(a) Linear
2
Two Variable Relationships(Negative Association)
Y
X
(b) Linear
3
Two Variable Relationships
Y
X
(c) Curvilinear
4
Two Variable Relationships
Y
X
(d) Curvilinear
5
Two Variable Relationships(No association)
Y
X
(e) No Relationship
6
Correlation

• The correlation coefficient is a quantitative
  measure of the strength of the linear
  relationship between two variables. The
  correlation ranges from 1.0 to - 1.0. A
  correlation of ? 1.0 indicates a perfect linear
  relationship, whereas a correlation of 0
  indicates no linear relationship.

7
Correlation

• SAMPLE CORRELATION COEFFICIENT
• where
• r Sample correlation coefficient
• n Sample size
• x Value of the independent variable
• y Value of the dependent variable

8
Correlation

• SAMPLE CORRELATION COEFFICIENT
• or the algebraic equivalent

9
Correlation(sales in midwest)
10
Correlation
11
Correlation
Correlation between Years and Sales
Software Correlation Output
12
Correlation

• TEST STATISTIC FOR CORRELATION
• where
• t Number of standard deviations r is from 0
• r Simple correlation coefficient
• n Sample size

13
Correlation Significance Test
Rejection Region ? /2 0.025
Rejection Region ? /2 0.025
Since t4.752 gt 2.048, reject H0, there is a
significant linear relationship
14
Correlation

• Spurious correlation occurs when there is a
  correlation between two otherwise unrelated
  variables.

15

• SUM OF RESIDUALS

SUM OF SQUARED RESIDUALS
16

• TOTAL SUM OF SQUARES
• where
• SST Total sum of squares
• n Sample size
• y Values of the dependent variable
• Average value of the dependent variable

17
Measures of VariationThe Sum of Squares

• SST Total Sum of Squares
• measures the variation of the Yi values around
  their mean Y

_

• SSR Regression Sum of Squares
• explained variation attributable to the
  relationship between X and Y

• SSE Error Sum of Squares
• variation attributable to factors other than the
  relationship between X and Y. This is the measure
  you minimize to get the coefficient estimates.

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Measures of Variation The Sum of Squares
Y
Ù
SSE å(Yi - Yi )2
Yi b0 b1Xi
Ù
_
SST å(Yi - Y)2
_
Ù
SSR å(Yi - Y)2
_
Y
X
Xi
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Two data points (x1,y1) and (x2,y2) of a certain
sample are shown.
y2
y1
x1
x2
Total variation in y
Variation explained by the regression line)
Unexplained variation (error)
20
Measures of Variation The Sum of Squares
Example
SSR
SSE
SST
21

• SUM OF SQUARES ERROR (RESIDUALS)
• where
• SSE Sum of squares error
• n Sample size
• y Values of the dependent variable
• Estimated value for the average of y for
  the given x value

22

• SUM OF SQUARES REGRESSION
• where
• SSR Sum of squares regression
• Average value of the dependent variable
• y Values of the dependent variable
• Estimated value for the average of y for the
  given x value

23

• SUMS OF SQUARES

24

• The coefficient of determination is the portion
  of the total variation in the dependent variable
  that is explained by its relationship with the
  independent variable. The coefficient of
  determination is also called R-squared and is
  denoted as R2.

25

• COEFFICIENT OF DETERMINATION (R2)

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Midwest Sales Example

• COEFFICIENT OF DETERMINATION (R2)

69.31 of the variation in the sales data for
this sample can be explained by the linear
relationship between sales and years of
experience.
27
Simple Linear Regression(only!)

• COEFFICIENT OF DETERMINATION SINGLE INDEPENDENT
  VARIABLE CASE
• where
• R2 Coefficient of determination
• r Simple correlation coefficient

28
Coefficients of Determination (R2) and
Correlation (r)
R2 1,
Y
r 1
Y
R2 1,
r -1

Y

b0

b1
X
i
i

Y

b0

b1
X
i
i
X
X
R2 .8,
R2 0,
r 0.9
r 0
Y
Y


Y

b0

b1
X
Y

b0

b1
X
i
i
i
i
X
X
29

• STANDARD DEVIATION OF THE REGRESSION SLOPE
  COEFFICIENT (POPULATION)
• Standard deviation of the regression slope
  (Called the standard error of the slope)
• Population standard error of the estimate

30

• ESTIMATOR FOR THE STANDARD ERROR OF THE ESTIMATE
• where
• SSE Sum of squares error
• n Sample size
• k number of independent variables in the
  model

31

• ESTIMATOR FOR THE STANDARD DEVIATION OF THE
  REGRESSION SLOPE
• where
• Estimate of the standard error of the least
  squares slope
• Sample standard error of the estimate

32

• TEST STATISTIC FOR TEST OF SIGNIFICANCE OF THE
  REGRESSION SLOPE
• where
• b1 Sample regression slope coefficient
• ?1 Hypothesized slope
• sb1 Estimator of the standard error of the
  slope

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Significance Test of Regression Slope (model
utility)
Rejection Region ? /2 0.025
Rejection Region ? /2 0.025
Since t4.753 gt 2.048, reject H0 conclude that
the true slope is not zero
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Simple Regression Steps

• Develop a scatter plot of y and x. You are
  looking for a linear relationship between the two
  variables.
• Calculate the least squares regression line for
  the sample data.
• Calculate the correlation coefficient and the
  simple coefficient of determination, R2.
• Conduct one of the significance tests.

35

• CONFIDENCE INTERVAL FOR THE SLOPE
• or equivalently
• where
• sb1 Std. error of the regression slope
• s Standard error of the estimate

36

• CONFIDENCE INTERVAL FOR
• Point estimate of the dependent variable
• t Critical value with n -
  2 d.f.
• s Standard error of the
  estimate
• n Sample size
• xp Specific value of the independent
  variable
• Mean of independent variable observations

37
Prediction Interval

• PREDICTION INTERVAL FOR

38
Residual Analysis

• Before using a regression model for description
  or prediction, you should do a check to see if
  the assumptions concerning the normal
  distribution and constant variance of the error
  terms have been satisfied. One way to do this is
  through the use of residual plots.