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Oxford Data Analysis
     Teaching
           Chick Judd


Additive and Interactive Models
  with Continuous Predictors
Basic Model Comparisons
Yi = β 0 + β1 Ξ 1ι + ... + β κ Ξ κι + ε Αι     ö
                                               Yi = β0 + β1 Ξ 1 ι + ... + βκ Ξ κι

                                                )2
                                  SSE = ∑ (Ψ − Ψ
                                            ι   ι



                             Model Ê vs.Ê
                                    AÊ Model Ê
                                             C

                     Model Ê − ⊇
                            AÊ ΣΣΕ ( Α) ;⊇ παραµ ετερσ
                                         ΠΑ⊇
                     Μοδελ⊇ − ⊇ (Χ ) ;⊇ παραµ ετερσ
                          Χ⊇ ΣΣΕ      ΠΧ⊇

                               ΣΣΕ( Χ ) − ΣΣΕ( Α)     ΣΣΡ
                     PRE =                        =
                                    ΣΣΕ( Χ )        ΣΣΕ( Χ )

                                ΠΡ Ε ( ΠΑ − ΠΧ )     ΣΣΡ ( ΠΑ − ΠΧ )
           FPA− ΠΧ ,⊇ν − ΠΑ =                      =
                              (1 − ΠΡ Ε ) ( ν − ΠΑ) ΣΣΕ ( Α) ( ν − ΠΑ)
Additive Model Comparisons
 A :ÊTime = 24.605 + .167 Αγε − .257 Μιλ
                                       εσ    SSE ( A) = 741.195
Χ1 :⊇ ε = 23.55
     Τιµ                                    ΣΣΕ (Χ1) = 2211.091
Χ 2 :⊇ ε = 14.955 + .217 Αγε
     Τιµ                                    ΣΣΕ (Χ 2 ) = 1714.720
Χ 3 :⊇ ε = 31.911 − .280 Μιλ
     Τιµ                    εσ
                                            ΣΣΕ (Χ 3 ) = 1029.013
Additive Model Comparisons

    Time = 24.605 + .167 Αγε − .257 Μιλ
 A :Ê                                 εσ              SSE ( A) = 741.195
Χ1 :⊇ ε = 23.55
    Τιµ
                                                     ΣΣΕ (Χ1) = 2211.091

              SSR = 2211.091 − 741.195 = 1469.896


                                1469.896
                        PRE =            = .665
                                2211.091


                           .665 2      1469.896 2
              F2 ,77                 =            = 76.35
                       (1 − .665 ) 77 741.195 77
Additive Model Comparisons

 A :ÊTime = 24.605 + .167 Αγε − .257 Μιλ
                                       εσ               SSE ( A) = 741.195
Χ 2 :⊇ ε = 14.955 + .217 Αγε
     Τιµ                                               ΣΣΕ (Χ 2 ) = 1714.720

                  SSR = 1714.720 − 741.195 = 973.525


                                 973.525
                        PRE =            = .567
                                1714.720


                     .567 1      973.525 1                     2
         F1,77                 =           = 101.34 = (10.06 )
                 (1 − .567 ) 77 741.195 77
Additive Model Comparisons

 A :ÊTime = 24.605 + .167 Αγε − .257 Μιλ
                                       εσ               SSE ( A) = 741.195
Χ 3 :⊇ ε = 31.911 − .280 Μιλ
     Τιµ                    εσ
                                                       ΣΣΕ (Χ 3 ) = 1029.013

                 SSR = 1029.013 − 741.195 = 287.818


                                287.818
                       PRE =            = .280
                               1029.013


                     .280 1      287.818 1                    2
         F1,77                 =           = 29.90 = ( 5.47 )
                 (1 − .280 ) 77 741.195 77
Additive Model Comparisons
 A :ÊTime = 24.605 + .167 Αγε − .257 Μιλ
                                       εσ           SSE ( A) = 741.195
Χ1 :⊇ ε = 23.55
     Τιµ                                         ΣΣΕ (Χ1) = 2211.091
Χ 2 :⊇ ε = 14.955 + .217 Αγε
     Τιµ                                         ΣΣΕ (Χ 2 ) = 1714.720
Χ 3 :⊇ ε = 31.911 − .280 Μιλ
     Τιµ                    εσ
                                                 ΣΣΕ (Χ 3 ) = 1029.013

                 SS         df         MS       F          PRE
    Model     1469.90             2   734.95   76.35          .66
    Miles      973.33             1   973.33   101.34         .57
    Age        287.92             1   287.92   29.90          .28
    Error       741.19           77     9.63
    Total       221.09           79    27.99
Confidence Intervals and
  the Basics of Power
                 Φ ;1 ,ν − ΠΑ ΜΣΕ
                  α
          b±
                 ( ν − 1) ( σ ) τολ
                                2
                             Ξ

                                    ΜΣΕ
          = β ± τα ,ν − ΠΑ
                              ( ν − 1) ( σΞ ) τολ
                                          2




                        4 ( 9.626 )
        .167 ±
   Age :Ê                                  = .167 ± .061
                   79 (133.73 ) .974


                             4 ( 9.626 )
          .257 ±
   Miles :Ê                                  = .257 ± .051
                     79 (191.13 ) .974
Standardized Coefficients
                                σΞ
                    βΨΞ   = βΨΞ
                                σΨ



     A :ÊTime = 24.605 + .167 Αγε − .257 Μιλ
                                           εσ
    Χ1 :⊇ ε = 23.55
         Τιµ
    Χ 2 :⊇ ε = 14.955 + .217 Αγε
         Τιµ
    Χ 3 :⊇ ε = 31.911 − .280 Μιλ
         Τιµ                   εσ



     A :ÊTime = .365 Αγε − .671 Μιλ
                                  εσ
    Χ1 :⊇ ε = 0
         Τιµ
    Χ 2 :⊇ ε = .474 Αγε
         Τιµ
    Χ 3 :⊇ ε = −.730 Μιλ
         Τιµ             εσ
Additive Model Interpretation
Additive Model Interpretation
Additive Model Interpretation
Generic Additive Model Versus One
    Allowing Different Slopes
            ö
            Y = β0 + β1 Ξ + β2 Ζ
            
            Ψ = ( β0 + β1 Ξ ) + ( β2 ) Ζ
            
            Ψ = (β + β Ζ ) + (β ) Ξ
                    0    2         1



        ö
        Y = ( β0 + β1 Ξ ) + ( β2 + β3 Ξ ) Ζ
        
        Ψ = ( β0 + β2 Ζ ) + ( β1 + β3 Ζ ) Ξ


         ö
         Y = β0 + β1 Ξ + β2 Ζ + β3 ΞΖ
Age and Miles Interactive Model

Model Ê :Time = 18.899 + .309 Αγε − .069 Μιλ − .005 ( Αγε∗ Μιλ )
       A                                   εσ                εσ
                         ΣΣΕ ( Α) = 697.625

          Model Ê :Time = 24.605 + .167 Αγε − .257 Μιλ
                C                                    εσ
                          ΣΣΕ ( Α) = 741.195
                  SSR = 741.195 − 697.625 = 43.57

         43.57
PRE =           = .059
        742.195
                                   .059 1       43.57 1                    2
                         F1,76               =           = 4.75 = ( 2.18 )
                               (1 − .059 ) 76 697.625 76
These parameter estimates aren’t interesting!




                                Rang
                                    e of D
                                          ata
Interactive Model: Uncentered and
          Centered Predictors
  Time = 18.899 + .309 Αγε − .069 Μιλ − .005 ( Αγε∗ Μιλ )
                                    εσ                 εσ
                        ΣΣΕ = 697.625

Time = 23.552 + .166 ΑγεΧ − .258 Μιλ Χ − .005 ( ΑγεΧ ∗ Μιλ Χ )
                                   εσ                    εσ
                         ΣΣΕ = 697.625

  Time = (18.889 + .309 Αγε ) + ( −.069 − .005 Αγε ) Μιλ
                                                       εσ
  Τιµ ε = ( 23.552 + .166 ΑγεΧ ) + ( −.258 − .005 ΑγεΧ ) Μιλ Χ
                                                           εσ


    Time = (18.889 − .069 Μιλ ) + ( .309 − .005 Μιλ ) Αγε
                            εσ                    εσ
    Τιµ ε = ( 23.552 − .258 Μιλ Χ ) + ( .166 − .005 Μιλ Χ ) Αγε
                              εσ                      εσ
“Main Effects” in Interactive Models?

  Time = 18.899 + .309 Αγε − .069 Μιλ − .005 ( Αγε∗ Μιλ )
                                    εσ                 εσ

       Certainly Not!

Time = 23.552 + .166 ΑγεΧ − .258 Μιλ Χ − .005 ( ΑγεΧ ∗ Μιλ Χ )
                                   εσ                    εσ

       Maybe, but not really
           Time = 24.605 + .167 Αγε − .257 Μιλ
                                             εσ

       Maybe, but not ideal


 The nature of an interaction: “It depends”
Paradoxical Effects of Centering only One
                Predictor

  Time = 18.899 + .309 Αγε − .069 Μιλ − .005 ( Αγε∗ Μιλ )
                                    εσ                 εσ

 Time = 23.552 + .166 ΑγεΧ − .258 Μιλ Χ − .005 ( ΑγεΧ ∗ Μιλ Χ )
                                    εσ                    εσ

  Time = 31.132 + .309 ΑγεΧ − .258 Μιλ − .005 ( ΑγεΧ ∗ Μιλ )
                                        εσ                εσ
  Time = ( 31.132 − .258 Μιλ ) + ( .309 − .005 Μιλ ) ΑγεΧ
                           εσ                    εσ
   Τιµ ε = ( 31.132 + .309Τιµ εΧ ) + ( −.258 − .005 ΑγεΧ ) Μιλ
                                                             εσ

  Time = 16.848 + .166 Αγε − .069 Μιλ Χ − .005 ( Αγε∗ Μιλ Χ )
                                       εσ                   εσ
   Time = (16.848 + .166 Αγε ) + ( −.069 − .005 Αγε ) Μιλ Χ
                                                        εσ
   Τιµ ε = (16.848 − .069 Μιλ Χ ) + ( .166 − .005 Μιλ Χ ) Αγε
                            εσ                      εσ
Informative Effects of Centering around
       other Interesting Values



 Time = 34.319 + .308 Αγε 50 − .307 Μιλ − .005 ( Αγε 50 ∗ Μιλ )
                                      εσ                    εσ


   Time = ( 34.319 + .308 Αγε 50 ) + ( −.307 − .005 Αγε 50 ) Μιλεσ
Surprising Invariance of Slope of Product,
        Given changes in tolerance

     Time = 18.899 + .309 Αγε − .069 Μιλ − .005 ( Αγε∗ Μιλ )
                                       εσ                εσ
                                           4 ( 9.179 )
           CI age* miles = −.005 ±
                                       79 ( 380387 ) .064
                      = −.005 ± .0043

    Time = 23.552 + .166 ΑγεΧ − .258 Μιλ Χ − .005 ( ΑγεΧ ∗ Μιλ Χ )
                                       εσ                    εσ

                                            4 ( 9.179 )
             CI age* miles = −.005 ±
                                         79 ( 24276 )1.00
                         = −.005 ± .0043

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Class1

  • 1. Oxford Data Analysis Teaching Chick Judd Additive and Interactive Models with Continuous Predictors
  • 2. Basic Model Comparisons Yi = β 0 + β1 Ξ 1ι + ... + β κ Ξ κι + ε Αι ö Yi = β0 + β1 Ξ 1 ι + ... + βκ Ξ κι  )2 SSE = ∑ (Ψ − Ψ ι ι Model Ê vs.Ê AÊ Model Ê C Model Ê − ⊇ AÊ ΣΣΕ ( Α) ;⊇ παραµ ετερσ ΠΑ⊇ Μοδελ⊇ − ⊇ (Χ ) ;⊇ παραµ ετερσ Χ⊇ ΣΣΕ ΠΧ⊇ ΣΣΕ( Χ ) − ΣΣΕ( Α) ΣΣΡ PRE = = ΣΣΕ( Χ ) ΣΣΕ( Χ ) ΠΡ Ε ( ΠΑ − ΠΧ ) ΣΣΡ ( ΠΑ − ΠΧ ) FPA− ΠΧ ,⊇ν − ΠΑ = = (1 − ΠΡ Ε ) ( ν − ΠΑ) ΣΣΕ ( Α) ( ν − ΠΑ)
  • 3. Additive Model Comparisons A :ÊTime = 24.605 + .167 Αγε − .257 Μιλ εσ SSE ( A) = 741.195 Χ1 :⊇ ε = 23.55 Τιµ ΣΣΕ (Χ1) = 2211.091 Χ 2 :⊇ ε = 14.955 + .217 Αγε Τιµ ΣΣΕ (Χ 2 ) = 1714.720 Χ 3 :⊇ ε = 31.911 − .280 Μιλ Τιµ εσ ΣΣΕ (Χ 3 ) = 1029.013
  • 4. Additive Model Comparisons Time = 24.605 + .167 Αγε − .257 Μιλ A :Ê εσ SSE ( A) = 741.195 Χ1 :⊇ ε = 23.55 Τιµ ΣΣΕ (Χ1) = 2211.091 SSR = 2211.091 − 741.195 = 1469.896 1469.896 PRE = = .665 2211.091 .665 2 1469.896 2 F2 ,77 = = 76.35 (1 − .665 ) 77 741.195 77
  • 5. Additive Model Comparisons A :ÊTime = 24.605 + .167 Αγε − .257 Μιλ εσ SSE ( A) = 741.195 Χ 2 :⊇ ε = 14.955 + .217 Αγε Τιµ ΣΣΕ (Χ 2 ) = 1714.720 SSR = 1714.720 − 741.195 = 973.525 973.525 PRE = = .567 1714.720 .567 1 973.525 1 2 F1,77 = = 101.34 = (10.06 ) (1 − .567 ) 77 741.195 77
  • 6. Additive Model Comparisons A :ÊTime = 24.605 + .167 Αγε − .257 Μιλ εσ SSE ( A) = 741.195 Χ 3 :⊇ ε = 31.911 − .280 Μιλ Τιµ εσ ΣΣΕ (Χ 3 ) = 1029.013 SSR = 1029.013 − 741.195 = 287.818 287.818 PRE = = .280 1029.013 .280 1 287.818 1 2 F1,77 = = 29.90 = ( 5.47 ) (1 − .280 ) 77 741.195 77
  • 7. Additive Model Comparisons A :ÊTime = 24.605 + .167 Αγε − .257 Μιλ εσ SSE ( A) = 741.195 Χ1 :⊇ ε = 23.55 Τιµ ΣΣΕ (Χ1) = 2211.091 Χ 2 :⊇ ε = 14.955 + .217 Αγε Τιµ ΣΣΕ (Χ 2 ) = 1714.720 Χ 3 :⊇ ε = 31.911 − .280 Μιλ Τιµ εσ ΣΣΕ (Χ 3 ) = 1029.013 SS df MS F PRE Model 1469.90 2 734.95 76.35 .66 Miles 973.33 1 973.33 101.34 .57 Age 287.92 1 287.92 29.90 .28 Error 741.19 77 9.63 Total 221.09 79 27.99
  • 8. Confidence Intervals and the Basics of Power Φ ;1 ,ν − ΠΑ ΜΣΕ α b± ( ν − 1) ( σ ) τολ 2 Ξ ΜΣΕ = β ± τα ,ν − ΠΑ ( ν − 1) ( σΞ ) τολ 2 4 ( 9.626 ) .167 ± Age :Ê = .167 ± .061 79 (133.73 ) .974 4 ( 9.626 ) .257 ± Miles :Ê = .257 ± .051 79 (191.13 ) .974
  • 9. Standardized Coefficients σΞ βΨΞ = βΨΞ σΨ A :ÊTime = 24.605 + .167 Αγε − .257 Μιλ εσ Χ1 :⊇ ε = 23.55 Τιµ Χ 2 :⊇ ε = 14.955 + .217 Αγε Τιµ Χ 3 :⊇ ε = 31.911 − .280 Μιλ Τιµ εσ A :ÊTime = .365 Αγε − .671 Μιλ εσ Χ1 :⊇ ε = 0 Τιµ Χ 2 :⊇ ε = .474 Αγε Τιµ Χ 3 :⊇ ε = −.730 Μιλ Τιµ εσ
  • 13. Generic Additive Model Versus One Allowing Different Slopes ö Y = β0 + β1 Ξ + β2 Ζ  Ψ = ( β0 + β1 Ξ ) + ( β2 ) Ζ  Ψ = (β + β Ζ ) + (β ) Ξ 0 2 1 ö Y = ( β0 + β1 Ξ ) + ( β2 + β3 Ξ ) Ζ  Ψ = ( β0 + β2 Ζ ) + ( β1 + β3 Ζ ) Ξ ö Y = β0 + β1 Ξ + β2 Ζ + β3 ΞΖ
  • 14. Age and Miles Interactive Model Model Ê :Time = 18.899 + .309 Αγε − .069 Μιλ − .005 ( Αγε∗ Μιλ ) A εσ εσ ΣΣΕ ( Α) = 697.625 Model Ê :Time = 24.605 + .167 Αγε − .257 Μιλ C εσ ΣΣΕ ( Α) = 741.195 SSR = 741.195 − 697.625 = 43.57 43.57 PRE = = .059 742.195 .059 1 43.57 1 2 F1,76 = = 4.75 = ( 2.18 ) (1 − .059 ) 76 697.625 76
  • 15.
  • 16.
  • 17.
  • 18.
  • 19.
  • 20. These parameter estimates aren’t interesting! Rang e of D ata
  • 21. Interactive Model: Uncentered and Centered Predictors Time = 18.899 + .309 Αγε − .069 Μιλ − .005 ( Αγε∗ Μιλ ) εσ εσ ΣΣΕ = 697.625 Time = 23.552 + .166 ΑγεΧ − .258 Μιλ Χ − .005 ( ΑγεΧ ∗ Μιλ Χ ) εσ εσ ΣΣΕ = 697.625 Time = (18.889 + .309 Αγε ) + ( −.069 − .005 Αγε ) Μιλ εσ Τιµ ε = ( 23.552 + .166 ΑγεΧ ) + ( −.258 − .005 ΑγεΧ ) Μιλ Χ εσ Time = (18.889 − .069 Μιλ ) + ( .309 − .005 Μιλ ) Αγε εσ εσ Τιµ ε = ( 23.552 − .258 Μιλ Χ ) + ( .166 − .005 Μιλ Χ ) Αγε εσ εσ
  • 22. “Main Effects” in Interactive Models? Time = 18.899 + .309 Αγε − .069 Μιλ − .005 ( Αγε∗ Μιλ ) εσ εσ Certainly Not! Time = 23.552 + .166 ΑγεΧ − .258 Μιλ Χ − .005 ( ΑγεΧ ∗ Μιλ Χ ) εσ εσ Maybe, but not really Time = 24.605 + .167 Αγε − .257 Μιλ εσ Maybe, but not ideal The nature of an interaction: “It depends”
  • 23. Paradoxical Effects of Centering only One Predictor Time = 18.899 + .309 Αγε − .069 Μιλ − .005 ( Αγε∗ Μιλ ) εσ εσ Time = 23.552 + .166 ΑγεΧ − .258 Μιλ Χ − .005 ( ΑγεΧ ∗ Μιλ Χ ) εσ εσ Time = 31.132 + .309 ΑγεΧ − .258 Μιλ − .005 ( ΑγεΧ ∗ Μιλ ) εσ εσ Time = ( 31.132 − .258 Μιλ ) + ( .309 − .005 Μιλ ) ΑγεΧ εσ εσ Τιµ ε = ( 31.132 + .309Τιµ εΧ ) + ( −.258 − .005 ΑγεΧ ) Μιλ εσ Time = 16.848 + .166 Αγε − .069 Μιλ Χ − .005 ( Αγε∗ Μιλ Χ ) εσ εσ Time = (16.848 + .166 Αγε ) + ( −.069 − .005 Αγε ) Μιλ Χ εσ Τιµ ε = (16.848 − .069 Μιλ Χ ) + ( .166 − .005 Μιλ Χ ) Αγε εσ εσ
  • 24. Informative Effects of Centering around other Interesting Values Time = 34.319 + .308 Αγε 50 − .307 Μιλ − .005 ( Αγε 50 ∗ Μιλ ) εσ εσ Time = ( 34.319 + .308 Αγε 50 ) + ( −.307 − .005 Αγε 50 ) Μιλεσ
  • 25. Surprising Invariance of Slope of Product, Given changes in tolerance Time = 18.899 + .309 Αγε − .069 Μιλ − .005 ( Αγε∗ Μιλ ) εσ εσ 4 ( 9.179 ) CI age* miles = −.005 ± 79 ( 380387 ) .064 = −.005 ± .0043 Time = 23.552 + .166 ΑγεΧ − .258 Μιλ Χ − .005 ( ΑγεΧ ∗ Μιλ Χ ) εσ εσ 4 ( 9.179 ) CI age* miles = −.005 ± 79 ( 24276 )1.00 = −.005 ± .0043