Multiple Regression

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Multiple Regression

• Selecting the Best Equation

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Techniques for Selecting the "Best" Regression
Equation

• The best Regression equation is not necessarily
  the equation that explains most of the variance
  in Y (the highest R2).
• This equation will be the one with all the
  variables included.
• The best equation should also be simple and
  interpretable. (i.e. contain a small no. of
  variables).
• Simple (interpretable) Reliable - opposing
  criteria.
• The best equation is a compromise between these
  two.

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• We will discuss several strategies for selecting
  the best equation
•  
• All Possible Regressions
• Uses R2, s2, Mallows Cp
•   Cp RSSp/s2complete - n-2(p1)
• "Best Subset" Regression
• Uses R2,Ra2, Mallows Cp
• Backward Elimination
• Stepwise Regression

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An Example

• In this example the following four chemicals are
  measured
• X1 amount of tricalcium aluminate, 3 CaO -
  Al2O3
• X2 amount of tricalcium silicate, 3 CaO - SiO2
• X3 amount of tetracalcium alumino ferrite, 4
  CaO - Al2O3 - Fe2O3
• X4 amount of dicalcium silicate, 2 CaO - SiO2
• Y heat evolved in calories per gram of
  cement.

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The data is given below
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I All Possible Regressions

• Suppose we have the p independent variables X1,
  X2, ..., Xp.
• Then there are 2p subsets of variables

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• Variables in Equation Model
• no variables Y b0 e
• X1 Y b0 b1 X1 e
• X2 Y b0 b2 X2 e
• X3 Y b0 b3 X3 e
• X1, X2 Y b0 b1 X1 b2 X2 e
• X1, X3 Y b0 b1 X1 b3 X3 e
• X2, X3 Y b0 b2 X2 b3 X3 e and
• X1, X2, X3 Y b0 b1 X1 b2 X2 b2 X3 e

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• Use of R2
• 1. Assume we carry out 2p runs for each of the
  subsets.
• Divide the Runs into the following sets
• Set 0 No variables
• Set 1 One independent variable.
• ...
• Set p p independent variables.
• 2. Order the runs in each set according to R2.
• 3. Examine the leaders in each run looking for
  consistent patterns
• - take into account correlation between
  independent variables.

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• Example (k4) X1, X2, X3, X4
• Variables in for leading runs 100 R2
• Set 1 X4. 67.5
• Set 2 X1, X2. 97.9
• X1, X4 97.2
• Set 3 X1, X2, X4. 98.234
• Set 4 X1, X2, X3, X4. 98.237
•  
• Examination of the correlation coefficients
  reveals a high correlation between X1, X3 (r13
  -0.824) and between X2, X4 (r24 -0.973).
•  
• Best Equation Y b0 b1 X1 b4 X4 e

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Use of R2
Number of variables required, p, coincides with
where R2 begins to level out
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• Use of the Residual Mean Square (RMS) (s2)
• When all of the variables having a non-zero
  effect have been included in the mode then the
  residual mean square is an estimate of s2.
• If "significant" variables have been left out
  then RMS will be biased upward.

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• No. of Variables
• p RMS s2(p) Average s2(p)
• 1 115.06, 82.39,1176.31, 80.35 113.53
• 2 5.79,122.71,7.48,86.59.17.57 47.00
• 3 5.35, 5.33, 5.65, 8.20 6.13
• 4 5.98 5.98
• - run X1, X2 - run X1, X4 s2-
  approximately 6.

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Use of s2
Number of variables required, p, coincides with
where s2 levels out
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• Use of Mallows Cp
• If the equation with p variables is adequate then
  both s2complete and RSSp/(n-p-1) will be
  estimating s2.
• If "significant" variables have been left out
  then RMS will be biased upward.

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• Then
• Thus if we plot, for each run, Cp vs p and look
  for Cp close to p 1 then we will be able to
  identify models giving a reasonable fit.

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• Run Cp p 1
• no variables 443.2 1
•  
• 1,2,3,4 202.5, 142.5, 315.2, 138.7 2
•  
• 12,13,14 2.7, 198.1, 5.5 3
• 23,24,34 62.4, 138.2, 22.4
•  
• 123,124,134,234 3.0, 3.0, 3.5, 7.5 4
•  
• 1234 5.0 5

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Use of Cp
Cp
p
Number of variables required, p, coincides with
where Cp becomes close to p 1
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II "Best Subset" Regression

• Similar to all possible regressions.
• If p, the number of variables, is large then the
  number of runs , 2p, performed could be extremely
  large.
• In this algorithm the user supplies the value K
  and the algorithm identifies the best K subsets
  of X1, X2, ..., Xp for predicting Y.

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III Backward Elimination

• In this procedure the complete regression
  equation is determined containing all the
  variables - X1, X2, ..., Xp.

• Then variables are checked one at a time and the
  least significant is dropped from the model at
  each stage.

• The procedure is terminated when all of the
  variables remaining in the equation provide a
  significant contribution to the prediction of the
  dependent variable Y.

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• The precise algorithm proceeds as follows

• Fit a regression equation containing all
  variables in the equation.

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• 2. A partial F-test is computed for each of the
  independent variables still in the equation.
•  

The Partial F statistic  
where RSS1 the residual sum of squares with
all variables that are presently in the equation,
RSS2 the residual sum of squares with on of
the variables removed, and MSE1 the Mean
Square for Error with all variables that are
presently in the equation.
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• 3. The lowest partial F value is compared with Fa
  for some pre-specified a .

If FLowest ? Fa then remove that variable and
return to step 2.
If FLowest gt Fa then accept the equation as it
stands.
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• Example (k4) (same example as before) X1,
  X2, X3, X4

1. X1, X2, X3, X4 in the equation.
The lowest partial F 0.018 (X3) is compared
with Fa(1,8) 3.46 for a 0.01.
Remove X3.
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• 2. X1, X2, X4 in the equation.

The lowest partial F 1.86 (X4) is compared with
Fa(1,9) 3.36 for a 0.01.
Remove X4.
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3. X1, X2 in the equation.

• Partial F for both variables X1 and X2 exceed
  Fa(1,10) 3.36 for a 0.01.

Equation is accepted as it stands.
Y 52.58 1.47 X1 0.66 X2
Note F to Remove partial F.
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IV Stepwise Regression

• In this procedure the regression equation is
  determined containing no variables in the model.

• Variables are then checked one at a time using
  the partial correlation coefficient as a measure
  of importance in predicting the dependent
  variable Y.

• At each stage the variable with the highest
  significant partial correlation coefficient is
  added to the model.

• Once this has been done the partial F statistic
  is computed for all variables now in the model is
  computed to check if any of the variables
  previously added can now be deleted.

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• This procedure is continued until no further
  variables can be added or deleted from the model.
  

• The partial correlation coefficient for a given
  variable is the correlation between the given
  variable and the response when the present
  independent variables in the equation are held
  fixed.

• It is also the correlation between the given
  variable and the residuals computed from fitting
  an equation with the present independent
  variables in the equation.

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• Example (k4) (same example as before) X1,
  X2, X3, X4

1. With no variables in the equation.
The correlation of each independent variable with
the dependent variable Y is computed.
The highest significant correlation ( r
-0.821) is with variable X4.
Thus the decision is made to include X4.
Regress Y with X4
-significant thus we keep X4.
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• Compute partial correlation coefficients of Y
  with all other independent variables given X4 in
  the equation.

The highest partial correlation is with the
variable X1. ( rY1.42 0.915).
Thus the decision is made to include X1.
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Regress Y with X1, X4. R2 0.972 , F 176.63
.  
Check to see if variables in the equation can be
eliminated  
For X1 the partial F value 108.22 (F0.10(1,8)
3.46) Retain X1.
For X4 the partial F value 154.295 (F0.10(1,8)
3.46) Retain X4.
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• Compute partial correlation coefficients of Y
  with all other independent variables given X4
  and X1 in the equation.

The highest partial correlation is with the
variable X2. ( rY2.142 0.358). Thus the
decision is made to include X2.
Regress Y with X1, X2,X4. R2 0.982 .
Check to see if variables in the equation can be
eliminated
Lowest partial F value 1.863 for X4 (F0.10(1,9)
3.36) Remove X4 leaving X1 and X2 .
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Examples

• Using Statistical Packages