# Regression Analysis by jlranm

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```									Regression Analysis
Chapter 10
Regression and Correlation
Techniques that are used to establish whether there is a
mathematical relationship between two or more variables, so that
the behavior of one variable can be used to predict the behavior of
others. Applicable to “Variables” data only.
• “Regression” provides a functional relationship (Y=f(x))
between the variables; the function represents the “average”
relationship.
• “Correlation” tells us the direction and the strength of the
relationship.

analysis starts with a Scatter Plot Y vs X
TheThe analysis starts with a ScatterPlot ofof Y vs X.
2
Simple Linear Regression
What is it?
Determines if Y
depends on X and
provides a math
equation for the
y
relationship
(continuous data)

Examples:
x
Process conditions

and product properties
Does Y depend on X?

budget
Which line is correct?

3
Simple Linear Regression

rise
m = slope =
run
Y

b = Y intercept

rise
= the Y value
at point that
the line
intersects Y                                run
axis.
b

0                                           X
A simple linear relationship can be described mathematically by
Y = mX + b

4
Simple Linear Regression
(6 - 3)       1
rise    =                    =
slope =
run           (10 - 4)           2

Y

5

rise
run
intercept = 1

0                                                       X
0        5                10

Y = 0.5X + 1
Simple regression example
   An agent for a residential real estate
company in a large city would like to
predict the monthly rental cost for
apartments based on the size of the
apartment as defined by square
footage. A sample of 25 apartments
in a particular residential
neighborhood was selected to gather
the information

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Size    Rent
850     950
1450     1600
1085     1200
1232     1500
718     950
1485     1700
1136     1650

The data on size
726     935
700     875
956    1150

and rent for the
1100     1400
1285     1650
1985     2300

25 apartments
1369     1800
1175     1400
1225     1450

will be analyzed
1245     1100
1259     1700
1150     1200

in EXCEL.
896    1150
1361     1600
1040     1650
755    1200
1000      800
1200     1750
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Scatter plot
2500
2300
2100
1900
1700
Rent

1500
1300
1100
900
700
500
500       700      900      1100     1300     1500     1700     1900    2100
Size

Scatter plot suggests that there is a ‘linear’ relationship between Rent and Size
8
Interpreting EXCEL output
SUMMARY OUTPUT

Regression Statistics
Multiple R                 0.85
R Square                   0.72
Standard Error           194.60
Observations                25

ANOVA
df            SS               MS           F      Significance F
Regression                 1        2268776.545      2268776.545 59.91376452 7.51833E-08
Residual                  23        870949.4547       37867.3676
Total                     24          3139726

Coefficients   Standard Error      t Stat      P-value     Lower 95%     Upper 95%
Intercept             177.121          161.004          1.100     0.282669853    -155.942      510.184
Size                   1.065            0.138           7.740     7.51833E-08      0.780        1.350

Regression Equation
Rent = 177.121+1.065*Size                                                                 9
Interpretation of the
regression coefficient
   What does the coefficient of Size
mean?

Rent goes up by \$1.065

10
Using regression for
prediction
   Predict monthly rent when
apartment size is 1000 square feet:

Regression Equation:
Rent = 177.121+1.065*Size
Thus, when Size=1000
Rent=177.121+1.065*1000=\$1242 (rounded)

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Using regression for
prediction – Caution!
   Regression equation is valid only over the range
over which it was estimated!
   We should interpolate

   Do not use the equation in predicting Y when X
values are not within the range of data used to
develop the equation.
   Extrapolation can be risky

   Thus, we should not use the equation to predict
rent for an apartment whose size is 500 square
feet, since this value is not in the range of size
values used to create the regression equation.

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Why extrapolation is risky
Extrapolated relationship

True
Relationship

In this figure, we fit our
regression model using
Sample
Data
sample data – but the linear
2.5            4.0
relation implicit in our
regression model does not
hold outside our sample! By
extrapolating, we are making
erroneous estimates!

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Correlation (r)
    “Correlation coefficient”, r, is a measure
of the strength and the direction of the
relationship between two variables.
Values of r range from +1 (very strong
direct relationship), through “0” (no
relationship), to –1 (very strong inverse
relationship). It measures the degree of
scatter of the points around the “Least
Squares” regression line

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Coefficient of correlation
from EXCEL
SUMMARY OUTPUT

Regression Statistics
Multiple R                 0.85
R Square                   0.72
Standard Error           194.60
Observations                25

ANOVA
df            SS               MS           F      Significance F
Regression                 1        2268776.545      2268776.545 59.91376452 7.51833E-08
Residual                  23        870949.4547       37867.3676
Total                     24          3139726

Coefficients   Standard Error      t Stat      P-value     Lower 95%     Upper 95%
Intercept             177.121          161.004          1.100     0.282669853    -155.942      510.184
Size                   1.065            0.138           7.740     7.51833E-08      0.780        1.350
The sign of r is the same as that of the coefficient of X (Size) in the regression
equation (in our case the sign is positive). Also, if you look at the scatter plot,
you will note that the sign should be positive.

R=0.85 suggests a fairly ‘strong’ correlation between size and rent.                           15
Coefficient of determination
(r2)
   “Coefficient of Determination”, r-squared,
(sometimes R- squared), defines the
amount of the variation in Y that is
attributable to variation in X

16
Getting r2 from EXCEL
SUMMARY OUTPUT

Regression Statistics
Multiple R                 0.85
R Square                   0.72
Standard Error           194.60
Observations                25

ANOVA
df            SS               MS           F      Significance F
Regression                 1        2268776.545      2268776.545 59.91376452 7.51833E-08
Residual                  23        870949.4547       37867.3676
Total                     24          3139726

Coefficients   Standard Error      t Stat      P-value     Lower 95%     Upper 95%
Intercept             177.121          161.004          1.100     0.282669853    -155.942      510.184
Size                   1.065            0.138           7.740     7.51833E-08      0.780        1.350

It is important to remember that r-squared is always positive. It is the square of
the coefficient of correlation r. In our case, r2=0.72 suggests that 72% of
variation in Rent is explained by the variation in Size. The higher the value of r2,
the better is the simple regression model.                                                          17
Standard error (SE)
   Standard error measures the
variability or scatter of the observed
values around the regression line.
2100
1900
1700
Rent (\$)

1500
1300
1100
900
700
500
500   1000         1500           2000   2500
Size (square feet)

18
Getting the standard error
(SE) from EXCEL
SUMMARY OUTPUT

Regression Statistics
Multiple R                 0.85
R Square                   0.72
Standard Error           194.60
Observations                25

ANOVA
df            SS               MS           F      Significance F
Regression                 1        2268776.545      2268776.545 59.91376452 7.51833E-08
Residual                  23        870949.4547       37867.3676
Total                     24          3139726

Coefficients   Standard Error      t Stat      P-value     Lower 95%     Upper 95%
Intercept             177.121          161.004          1.100     0.282669853    -155.942      510.184
Size                   1.065            0.138           7.740     7.51833E-08      0.780        1.350

In our example, the standard error associated with estimating rent is \$194.60.                  19
Is the simple regression
model statistically valid?
    It is important to test whether the
regression model developed from
sample data is statistically valid.
    For simple regression, we can use
2 approaches to test whether the
coefficient of X is equal to zero
1.   using t-test
2.   using ANOVA

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Is the coefficient of X equal
to zero?
   In both cases, the hypothesis we
test is:

H 0 : Slope  0
H1 : Slope  0

What could we say about the linear relationship
between X and Y if the slope were zero?

21
Using coefficient information
for testing if slope=0
SUMMARY OUTPUT

Regression Statistics
Multiple R                 0.85
P-value
R Square                   0.72                                                   7.52E-08
Standard Error           194.60                                                   =7.52*10-8
Observations                25                                                    =0.0000000752
ANOVA
df            SS               MS           F      Significance F
Regression                 1        2268776.545      2268776.545 59.91376452 7.51833E-08
Residual                  23        870949.4547       37867.3676
Total                     24          3139726

Coefficients   Standard Error      t Stat      P-value     Lower 95%     Upper 95%
Intercept             177.121          161.004          1.100     0.282669853    -155.942      510.184
Size                   1.065            0.138           7.740     7.51833E-08      0.780        1.350

t-stat=7.740 and P-value=7.52E-08. P-value is very small. If it is smaller than
our a level, then, we reject null; not otherwise. If a=0.05, we would reject null
and conclude that slope is not zero. Same result holds at a=0.01 because the P-
value is smaller than 0.01. Thus, at 0.05 (or 0.01) level, we conclude that the
slope is NOT zero implying that our model is statistically valid.                                   22
Using ANOVA for testing if
slope=0 in EXCEL
SUMMARY OUTPUT

Regression Statistics
Multiple R                 0.85
R Square                   0.72
Standard Error           194.60
Observations                25

ANOVA
df            SS               MS           F      Significance F
Regression                 1        2268776.545      2268776.545 59.91376452 7.51833E-08
Residual                  23        870949.4547       37867.3676
Total                     24          3139726

Coefficients   Standard Error      t Stat      P-value     Lower 95%     Upper 95%
Intercept             177.121          161.004          1.100     0.282669853    -155.942      510.184
Size                   1.065            0.138           7.740     7.51833E-08      0.780        1.350

F=59.91376 and P-value=7.51833E-08. P-value is again very small. If it is
smaller than our a level, then, we reject null; not otherwise. Thus, at 0.05 (or
0.01) level, slope is NOT zero implying that our model is statistically valid. This
is the same conclusion we reached using the t-test.                                               23
Confidence interval for the
slope of Size
SUMMARY OUTPUT

Regression Statistics
Multiple R                 0.85
R Square                   0.72
Standard Error           194.60
Observations                25

ANOVA
df            SS               MS           F      Significance F
Regression                 1        2268776.545      2268776.545 59.91376452 7.51833E-08
Residual                  23        870949.4547       37867.3676
Total                     24          3139726

Coefficients   Standard Error      t Stat      P-value     Lower 95%     Upper 95%
Intercept             177.121          161.004          1.100     0.282669853    -155.942      510.184
Size                   1.065            0.138           7.740     7.51833E-08      0.780        1.350

The 95% CI tells us that for every 1 square feet increase
in apartment Size, Rent will increase by \$0.78 to \$1.35.
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Summary
   Simple regression is a statistical tool that attempts to fit
a straight line relationship between X (independent
variable) and Y (dependent variable)

   The scatter plot gives us a visual clue about the nature of
the relationship between X and Y

   EXCEL, or other statistical software is used to ‘fit’ the
model; a good model will be statistically valid, and will
have a reasonably high R-squared value

   A good model is then used to make predictions; when
making predictions, be sure to confine them within the
domain of X’s used to fit the model (i.e. interpolate); we
should avoid extrapolation

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