TESTING THE STRENGTH

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• TESTING THE STRENGTH
• OF THE
• MULTIPLE REGRESSION MODEL

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Test 1 Are Any of the xs 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

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Idea of the Test

• Measure the overall average variability due to
  changes in the xs
• Measure the overall average variability that is
  due to randomness (error)
• If the overall average variability due to
  changes in the xs 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

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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)

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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

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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 xs in the linear regression)
• Degrees of freedom for error (DFE) difference
  between the two DF(Total) -DFR

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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 ß ? 0.
• Thus if F is large, we draw the conclusion is
  that HA is true, i.e. at least one ß ? 0.

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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 gt F?,DFR,DFE

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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?

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Results

• We see that the F statistic is 20.89856
• 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 20.89856 gt 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 lt .05
• Can conclude that at least one x is useful in
  predicting y

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Test 2 Which Variables Are Significant IN THIS
MODEL?

• The question we are asking is, taking all the
  other factors (xs) 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)

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The t-test for a particular factor IN THIS MODEL

• Reject H0 (Accept HA) if

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Reading Printout for the t-test

• Simply look at the p-value
• p-value for ?3 0 is .021931 lt .05
• Thus the age of the house is significant in this
  model
• The other variables
• p-value for ?1 0 is .0000839 lt .05
• Thus square feet is significant in this model
• p-value for ?2 0 is .15501 gt .05
• Thus the land (acres) is not significant in this
  model

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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.

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Can it even happen that F says at least one
variable is significant, but none of the ts
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 xs
• Multicollinearity

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Approaches That Could Be Used When
Multicollinearity Is Detected

• Eliminate some variables and run again
• Stepwise regression
• This is discussed in a future module.

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Test 3 --What Proportion of the Overall
Variability in y Is Due to Changes in the xs?

• R2
• R2 .442199
• Overall 44 of the total variation in sales price
  is explained by changes in square footage, land,
  and age of the house.

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What is Adjusted R2?

• 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

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Scatterplot
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Review

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

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4 Places to Look on Excel Printout