Testing the strength of the multiple regression model
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TESTING THE STRENGTH OF THE MULTIPLE REGRESSION MODEL. Test 1: Are Any of the x’s Useful in Predicting y?. We are asking: Can we conclude at least one of the ’s (other than  0 )  0? H 0 :  1 =  2 =  3 =  4 = 0 H A : At least one of these ’s  0  = .05.

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Testing the strength of the multiple regression model

TESTING THE STRENGTH

OF THE

MULTIPLE REGRESSION MODEL


Test 1 are any of the x s useful in predicting y
Test 1: Are Any of the x’s Useful in Predicting y?

We are asking: Can we conclude at least one of the ’s (other than 0)  0?

H0: 1 = 2 = 3 = 4 = 0

HA: At least one of these ’s 0

 = .05


Idea of the test
Idea of the Test

  • Measure the overall “average variability” due to changes in the x’s

  • Measure the overall “average variability” that is due to randomness (error)

  • If the overall “average variability” due to changes in the x’s IS A LOT LARGER than “average variability” due to error, we conclude at least  is non-zero, i.e. at least one factor (x) is useful in predicting y


Total variability
“Total Variability”

  • Just like with simple linear regression we have total sum of squares due to regression SSR , and total sum of squares due to error, SSE, which are printed on the EXCEL output.

    • The formulas are a more complicated (they involve matrix operations)


Average variability
“Average Variability”

  • “Average variability” (Mean variability) for a group is defined as the Total Variability divided by the degrees of freedom associated with that group:

  • Mean Squares Due to Regression

    MSR = SSR/DFR

  • Mean Squares Due to Error

    MSE = SSE/DFE


Degrees of freedom
Degrees of Freedom

  • Total number of degrees of freedom DF(Total) always = n-1

  • Degrees of freedom for regression (DFR) = the number of factors in the regression (i.e. the number of x’s in the linear regression)

  • Degrees of freedom for error (DFE) = difference between the two = DF(Total) -DFR


The f statistic
The F-Statistic

  • The F-statistic is defined as the ratio of two measures of variability. Here,

  • Recall we are saying if MSR is “large” compared to MSE, at least one β ≠ 0.

  • Thus if F is “large”, we draw the conclusion is that HA is true, i.e. at least one β ≠ 0.


The f test
The F-test

  • “Large” compared to what?

  • F-tables give critical values for given values of 

  • TEST: REJECT H0 (Accept HA) if:

    F = MSR/MSE > F,DFR,DFE


Results
RESULTS

  • If we do not get a large F statistic

    • We cannot conclude that any of the variables in this model are significant in predicting y.

  • If we do get a large F statistic

    • We can conclude at least one of the variables is significant for predicting y .

    • NATURAL QUESTION --

      • WHICH ONES?


Testing the strength of the multiple regression model

DFR = #x’s

DFE = Total DF- DFR

Total DF = n-1

SSR

SSE

Total SS = (yi - )2


Testing the strength of the multiple regression model

MSR = SSR/DFR

MSE = SSE/DFE

F = MSR/MSE

P-value for the F test


Results1
Results

  • We see that the F statistic is 20.89762

  • This would be compared to F.05,3,34

    • From the F.05 Table, the value of F.05,3,34 is not given.

    • But F.05,3,30 = 2.92 and F.05,3,40 = 2.84.

    • And 20.89762 > either of these numbers.

    • The actual value of F.05,3,34 can be calculated by Excel by FINV(.05,3,34) = 2.882601

  • USE SIGNIFICANCE F

    • This is the p-value for the F-Test

    • Significance F = 7.46 x 10-8 = .0000000746 < .05

    • Can conclude that at least one x is useful in predicting y


Test 2 which variables are significant in this model
Test 2: Which Variables Are Significant IN THIS MODEL?

  • The question we are asking is, “taking all the other factors (x’s) into consideration, does a change in a particular x (x3, say) value significantly affect y.

  • This is another hypothesis test (a t-test).

  • To test if the age of the house is significant:

    H0: 3 = 0 (x3 is not significant in this model)

    HA: 3  0 (x3 is significant in this model)



Testing the strength of the multiple regression model

t-value for test of 3 = 0

p-value for test of 3 = 0


Reading printout for the t test
Reading Printout for the t-test

  • Simply look at the p-value

    • p-value for 3 = 0 is .02194 < .05

      • Thus the age of the house is significant in this model

  • The other variables

    • p-value for 1 = 0 is .0000839 < .05

      • Thus square feet is significant in this model

    • p-value for 2 = 0 is .15503 > .05

      • Thus the land (acres) is not significant in this model


Does a poor t value imply the variable is not useful in predicting y
Does A Poor t-value Imply the Variable is not Useful in Predicting y?

  • NO

  • It says the variable is not significant IN THIS MODEL when we consider all the other factors.

  • In this model – land is not significant when included with square footage and age.

  • But if we would have run this model without square footage we would have gotten the output on the next slide.


Testing the strength of the multiple regression model

p-value for land is .00000717. Predicting y?

In this model Land is significant.


Testing the strength of the multiple regression model
Can it even happen that F says at least one variable is significant, but none of the t’s indicate a useful variable?

  • YES

    EXAMPLES IN WHICH THIS MIGHT HAPPEN:

    • Miles per gallon vs. horsepower and engine size

    • Salary vs. GPA and GPA in major

    • Income vs. age and experience

    • HOUSE PRICE vs. SQUARE FOOTAGE OF HOUSE AND LAND

  • There is a relation between the x’s –

    • Multicollinearity


Approaches that could be used when multicollinearity is detected
Approaches That Could Be Used When Multicollinearity Is Detected

  • Eliminate some variables and run again

  • Stepwise regression

    This is discussed in a future module.


Test 3 what proportion of the overall variability in y is due to changes in the x s
Test 3 --What Proportion of the Overall Variability in y Is Due to Changes in the x’s?

R2

  • R2 = .442197

  • Overall 44% of the total variation in sales price is explained by changes in square footage, land, and age of the house.


What is adjusted r 2
What is Adjusted R Due to Changes in the x’s?2?

  • Adjusted R2 adjusts R2 to take into account degrees of freedom.

  • By assuming a higher order equation for y, we can force the curve to fit this one set of data points in the model – eliminating much of the variability (See next slide).

  • But this is not what is going on!

    R2 might be higher – but adjusted R2 might be much lower

  • Adjusted R2 takes this into account

  • Adjusted R2 = 1-MSE/SST


Testing the strength of the multiple regression model

Scatterplot Due to Changes in the x’s?

This is not what is really going on


Review
Review Due to Changes in the x’s?

  • Are any of the x’s useful in predicting y IN THIS MODEL

    • Look at p-value for F-test – Significance F

    • F = MSR/MSE would be compared to F,DFR,DFE

  • Which variables are significant in this model?

    • Look at p-values for the individual t-tests

  • What proportion of the total variance in y can be explained by changes in the x’s?

    • R2

    • Adjusted R2 takes into account the reduced degrees of freedom for the error term by including more terms in the model


Testing the strength of the multiple regression model

4 Places to Look on Excel Printout Due to Changes in the x’s?

4- R2

What proportion of y can be

explained by changes in x?

2- Significance F

Are any variables useful?

3- p-values for t-tests

Which variables are significant

in this model?

1-regression equation