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TESTING THE STRENGTH OF THE MULTIPLE REGRESSION MODELPowerPoint Presentation

TESTING THE STRENGTH OF THE MULTIPLE REGRESSION MODEL

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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

- 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”

- 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” (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

- 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 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

- “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

- 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?

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?

- 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)

The t-test for a particular factor IN THIS MODEL

- Reject H0 (Accept HA) if:

t-value for test of 3 = 0

p-value for test of 3 = 0

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

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

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

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.

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

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?

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
- Adjusted R2 = 1-MSE/SST

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

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