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# TESTING THE STRENGTH OF THE MULTIPLE REGRESSION MODEL - PowerPoint PPT Presentation

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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Presentation Transcript

OF THE

MULTIPLE REGRESSION MODEL

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

• 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

• 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” (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

• 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 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.

• “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

• 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?

DFE = Total DF- DFR

Total DF = n-1

SSR

SSE

Total SS = (yi - )2

MSE = SSE/DFE

F = MSR/MSE

P-value for the F test

• 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?

• 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)

• Reject H0 (Accept HA) if:

t-value for test of 3 = 0

p-value for test of 3 = 0

• 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

• 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.

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

In this model Land is significant.

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

• 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?

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 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

Scatterplot Due to Changes in the x’s?

This is not what is really going on

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

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