Statistics 483

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Chapter 5

• Multivariate Regression
• Inferences and model comparison

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Example Fuel Consumption
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Comparing Two Regression Models

• Full Model
• Reduced Model
• Is the full model significantly better than the
  reduced model in explaining the variation in y?
• E.g.

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F-test for comparing Two Regression Models
To test H0 ?L1 ?L2 ?k 0 versus Ha
At least one of the ?L1 , ?L2, , ?k is not
equal to 0
Test Statistic
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F-test for comparing Two Regression Models

• P-value P(F(df1,df2) gt Fratio)
• Reject H0 in favor of Ha if p-value lt a
• Fdf1,df2 is based on (k-L) numerator and
• n-(k1) denominator degrees of freedom

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Example Fuel Consumption
Full Model
Reduced Model
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Example Fuel Consumption
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Prediction
Prediction Equation
is the point prediction of an individual value of
the dependent variable when the values of the
independent variables are x01, x02, , x0k.
b1, b2, , bk are the least squares point
estimates of the parameters ?1, ? 2, , ?
k. x01, x02, , x0k are specified values of the
independent predictor variables x1, x2, , xk.
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Example Fuel Consumption
Data see Fuel2
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Using The Regression Equation to Make Predictions

• Predict the amount of Fuel consumption used for a
  home (i) if the average temperature is Xi1 400
  and the Chill index is Xi2 10.
• E(Y) 13.157 - 0.090 (Xi1) 0.076 (Xi2)
• 13.157 - 0.090 (40) 0.076 (10)
• 10.317

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Confidence Intervals for mean of y
Prediction
If the regression assumptions hold,
100(1 - a) prediction interval for the mean
value of y
Calculation of the standard error (s.e.) requires
matrix algebra
ta/2 is based on n-(k1) degrees of freedom
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Prediction of Mean Values in JMP

• Type in x values in the dataset ? Leave y value
  empty ? select Analyze ? select Fit Model ? fuel
  into Y ? put temp and chill into Add ? select Run
  Model ? click red triangle or click right mouse
  button ? select Save Columns ? select Predicted
  Values OR select Mean Confidence Interval

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Fitting Curves to Data
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Idea of Fitting Curves

• Instead of fitting y and x, fit y and g(x)
• Here, g(x) means a transformation of x
• E.g, if x and y are related in a curvilinear
  fashion, then perhaps x2 and y have a linear
  relationship

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Example Telemarketing

• A telemarketing division sells the companys
  service
• - 20 employees
• Data on the number of months of employment (x)
• and the number of calls placed per day (y)

Data Telemarketing
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Example continued
As the number of months on the job increases,
the number of calls also increases. But the rate
of increase begins to slow over time
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Example Telemarketing
The fitted model calls -0.14 2.31 months -
0.04 months2
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Example Telemarketing
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Example MPG Versus HP
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MPG versus HP

• What kind of relationship do we have?
• MPG?0?1(1/HP)e

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The Quadratic Regression Model

• Relationship Between the Response Variable and
  the Explanatory Variable is a Quadratic
  Polynomial Function
• Useful When Scatter Diagram Indicates Non-linear
  Relationship
• Quadratic Model
• The Second Explanatory Variable is the Square
  of the First Variable

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Quadratic Regression Model
Quadratic model may be considered when a scatter
diagram takes on the following shapes
Y
Y
Y
Y
X1
X1
X1
X1
?2 gt 0
?2 gt 0
?2 lt 0
?2 lt 0
?2 the coefficient of the quadratic term
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Testing for Significance Quadratic Model

• Testing for Overall Relationship
• Similar to test for linear model
• F test statistic

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Testing for Significance Quadratic Model

• Testing the Quadratic Effect
• Compare quadratic model
• with the linear model
• Hypotheses
• (No quadratic term)
• (Quadratic term is
  needed)

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Heating Oil Example
Determine if a quadratic model is needed for
estimating heating oil used for a single family
home in the month of January based on average
temperature and amount of insulation in inches.
Data Heating
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Heating Oil Example
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Heating Oil Example t Test for Quadratic Model

• Testing the Quadratic Effect
• Model with quadratic insulation term
• Model without quadratic insulation term
• Hypotheses
• (No quadratic term in
  insulation)
• (Quadratic term is needed
  in insulation)

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Quadratic Regression in JMP

• Analyze ? Fit Model ? put Oil in Y ? put Temp
  Insul in Add ? click on Insul in both left
  and right boards ? click on Cross ? Run Model

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Example Solution
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Example Solution
Is quadratic term in insulation needed on monthly
consumption of heating oil? Test at ? 0.05.
H0 ?3 0 H1 ?3 ? 0 df 11
Test Statistic
P-value0.1249 Decision Do not reject H0 at ?
0.05
Conclusion There is not sufficient evidence for
the need to include quadratic effect of
insulation on oil consumption.
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A second-order Regression Model
Model
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Telemarketing Data 1st order Non Constant
Variance
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Tests for Lack of Fit
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F test for testing nonlinearity
F test Pure error Lack of-fit Lack of
fit / Pure error
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Telemarketing example, 1st order
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• F ratio Lack of fit / Pure error
• 4.038 / 0.50
• 8.077
• Df112, df26
• P-value
• P(F(12,6) gt 8.077) 0.008

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Corrections for non-constant Variance

• Polynomial regression
• Transformation of X
• Transformation on X and Y
• E.g., the natural logarithm of y, ln(y)
• ln(y) is appropriate when the residual plot shows
  increasing error variance
• Square root transformation is another possibility

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1. Polynomial regression

• Check beta related to the quadratic term
• Check if the residual plot with the new linear
  model has been improved

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Telemarketing example 2nd order
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