Inference by gyvwpsjkko

VIEWS: 7 PAGES: 15

									            Inference
Revealing, informative plots often provoke the
   question “Is what we see really there?”
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     What is statistical
      significance?

• Closely related to inference
• How likely are these results to be true for
  the population we’re interested in?
• Or, what’s the probability we see a strong
  effect from chance alone?
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                                                        value                                                                                                                                   value

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                                                             ●
                                                             ●       ●
                                                                     ●           ● ●
                                                                                 ● ●                ●       ●●          ●   ●       b   ●        ●       ● ●          ●●   ●       ●    ●               ●    ●
                                                                                                                                                                                                             ●                  ●
                                                                                                                                                                                                                                ●           ● ●
                                                                                                                                                                                                                                             ●         ●           ●   ●




a   ●                ● ●     ●●        ●●
                                        ●   ●            ●   ●●              ●       ●
                                                                                     ●       ● ●●       ●           ●           ●   a                ●           ●●        ●●      ●            ●●
                                                                                                                                                                                                ●           ● ●           ●
                                                                                                                                                                                                                        ● ●     ●
                                                                                                                                                                                                                                ●       ●         ●        ●   ●           ●




        −2                   −1                     0                                    1                                  2               −2                   −1                         0                                       1                                  2
                                                        value                                                                                                                                   value

b                ●       ●             ●●
                                        ●   ●            ●   ●       ●       ● ●     ●
                                                                                     ●       ● ● ●              ●   ●●      ●   ●   b                    ●       ●    ●●   ●●      ●                ●       ●●
                                                                                                                                                                                                             ●          ● ●     ●●
                                                                                                                                                                                                                                ●       ●        ● ●       ●       ●




a   ●        ●       ●       ●●   ●●   ●    ●   ●            ●   ●
                                                             ● ● ●               ● ●
                                                                                   ●          ●         ●   ●                       a   ●        ● ●         ●   ●         ●
                                                                                                                                                                           ●       ●    ●           ●
                                                                                                                                                                                                    ●
                                                                                                                                                                                                    ●       ●               ●    ●          ●●         ●       ●       ●   ●




        −2                   −1                     0                                    1                                  2               −2                   −1                         0                                       1                                  2
                                                        value                                                                                                                                   value
     How does it work?




                                               Ra Re jec
                                                 nd pe t
                                                   om at
                                                     Re


                                                     ise
•   If the variables are independent, then the
    structure will remain the same when the
    dependence between ordered pairs is broken.
•   Take one column of data and permute the
    values, make a plot, repeat many times.
•   Is the real data plot distinguishable from the
    permuted data plots? If so, the structure in
    the plot of data is real.
                                                 PC2
                                               PC3                                                                   PC6PC4                                                      PC2
                             !                                                                                          PC1
                                                                                                                         PC2                                              !
                                                                                                                                                                          PC1
                                                                                                                                                                         PC4

                                                                                                                                                                                                 Example:
                                                                                                                                                               !        !PC6
                     !       !        PC6                                                             !                                                                ! !               !
                           !                                                                                                                                                !                !
                                                                                        !       !
                   !       !    PC5 PC4                                                                                 PC5                                                     PC3
                 !!                                                             !                                                                     ! !                  !      PC5
                       ! !
                       ! !
                                                                                                !                                                    !!
                                                                                                !                !                                      !



                                                                                                                                                                                                 inference for
                              !                                                                      !                                                                     !
             !     !!        !                                                               !         !
                                                                                                      ! !                               !                                  !!
              !                 !                                                   !       !                                                                                !
                  ! !                                                       !                            !
                                                                                                         !
                 ! !      !                                                                 !!                                                                         !         !
                                                                                        !                                                   !
               !           !                                                                         !           !                                         !                         !
               !                                                                    !                                          !                           !
                                                                                                     !                                                    !        !        !



                                                                                                                                                                                                 classification
                                                                !       !                                                                            !
                                   !                                                        !
                                                                                            !                !!                                                                          !
                       !     !!                                             !               !                                  !                                      !!
                    ! !  !                                          !        !          !                      !                                      !            ! !!!                     !
               !                                                                                     !         !                                                      !
                     !               !
                 !
                  !! !                                                                                                                                         ! !
                                 !                                                          !
                          !                                                     !            !                                                                                   !




                                                                                                                                                                                                 Which of
                                                PC3                                                                                                                                PC6
                                                                                                                                                                                   PC4
                                                                                                                                                                                 PC3PC1
             !                                                                      !                                   PC4
                                              PC4                                                             PC2
                                                                                                              PC1                                                                    PC2
                          ! ! !               PC6                               !                                                                !
                         ! !
                                                          PC1
                                                                                                                       PC5                                 !                             PC5


                                                                                                                                                                                                 these is
                                                PC2
                                               !!                                       !                            PC6
                         !
                         !                                                                                                                           !
                           ! !
                                                                                        !                                                                  !
                                                                                                                                                           !
                                          !                                                                                                      !
                                 !                                                               !                                                   !
                                                                                                                                                     ! !     !                               !
                     !                                                                           !                                            !         !
                                          !                                                                     !                                         !
                                                                                                              !!                                !   ! ! !!
         !                       !                                                               !!                    !


                                                                                                                                                                                                 different?
                     !
                         !
                                           !        !                       !                                                                         ! !
                                                                                                                                                      !        !
                                          !                         !                                                                         !!
                                                                                            !                !
                                                                                                             !                                        !
                                     !         !                            !    !            !                                                  ! !      ! !
                                                                                 !                       !                                  !   !         !
                         !            !                             !           !!                                          !                     !!    !
                                          !        !
                                                   !
                                                           !                     !          !!           ! !!              !                            !        !
             !                                                                                           !                 !
     !                                                                                                                                                                      !
     !                !
                     !!                                                                          !        !                  !
                                                                                                          !                                           !                    !
                     ! !             !    !
                                          !                                                              !
                 !                                              !                           !                                                        !
                                                                                                                                                      !




                                                                                                                      PC1                                                      PC2
                                                                                                                                                                               PC3
                                              PC5 PC1
                                                   PC6                                                        PC3                                    !         !                 PC6
                                              PC4                                                                          PC2                                              PC5!
                         !                                                                                                                       !                                !
                                 !! !                                                                                                                                       PC1
 !                               ! !
         !                                     PC2                                                                                      !                                   ! !  PC4
                     !
                     !     !                        ! !                                                                                          !
                                                                                                                                                                       !
                                                                                                                                                                       !
                     !! !!                                                                                                                      !            !         !
                     !
                        !! !    !                                                                                                  !            !           !
                                                                                                                                                           !!!
                                                                                                                                                                   !   !
             !                !                                                                                                                                    !
                        !
                       ! !                                                                                                          !   !                          ! !   !
                                                          !                                                                                 !!                       !
                          ! !                      ! !!                                                  !                                                         ! ! !!
 !                                                                                                        !
                                                                                                                                                 !                   !
                             !                             !     !                                              ! !!
                                                                                                                  !
                                                                                                             ! ! !!
     !                                                          ! ! !!                                         !                                                   !     !
                                                                                                                                   !
                                     !                                 !                                          ! !
                                                                                                                  !!
                             ! !                                   !! ! ! !
                                                                     ! !                                      !
                                                                                                              !                                  !
                                                                                                                                                                                 !
                                                                            !                                       ! !!
                                          !
                                 !                                   ! !
                                                                   !! !
                                                                       ! !                                                 !
                                     !                               !                                                                      !




Fig. 6.2. Testing the assumption that species labels correspond to three groups
in the data, using a projection pursuit guided tour with the LDA index. The plots
                                                 PC2
                                               PC3                                                                   PC6PC4                                                      PC2
                             !                                                                                          PC1
                                                                                                                         PC2                                              !
                                                                                                                                                                          PC1
                                                                                                                                                                         PC4

                                                                                                                                                                                                 Example:
                                                                                                                                                               !        !PC6
                     !       !        PC6                                                             !                                                                ! !               !
                           !                                                                                                                                                !                !
                                                                                        !       !
                   !       !    PC5 PC4                                                                                 PC5                                                     PC3
                 !!                                                             !                                                                     ! !                  !      PC5
                       ! !
                       ! !
                                                                                                !                                                    !!
                                                                                                !                !                                      !



                                                                                                                                                                                                 inference for
                              !                                                                      !                                                                     !
             !     !!        !                                                               !         !
                                                                                                      ! !                               !                                  !!
              !                 !                                                   !       !                                                                                !
                  ! !                                                       !                            !
                                                                                                         !
                 ! !      !                                                                 !!                                                                         !         !
                                                                                        !                                                   !
               !           !                                                                         !           !                                         !                         !
               !                                                                    !                                          !                           !
                                                                                                     !                                                    !        !        !



                                                                                                                                                                                                 classification
                                                                !       !                                                                            !
                                   !                                                        !
                                                                                            !                !!                                                                          !
                       !     !!                                             !               !                                  !                                      !!
                    ! !  !                                          !        !          !                      !                                      !            ! !!!                     !
               !                                                                                     !         !                                                      !
                     !               !
                 !
                  !! !                                                                                                                                         ! !
                                 !                                                          !
                          !                                                     !            !                                                                                   !




                                                                                                                                                                                                 Which of
                                                PC3                                                                                                                                PC6
                                                                                                                                                                                   PC4
                                                                                                                                                                                 PC3PC1
             !                                                                      !                                   PC4
                                              PC4                                                             PC2
                                                                                                              PC1                                                                    PC2
                          ! ! !               PC6                               !                                                                !
                         ! !
                                                          PC1
                                                                                                                       PC5                                 !                             PC5


                                                                                                                                                                                                 these is
                                                PC2
                                               !!                                       !                            PC6
                         !
                         !                                                                                                                           !
                           ! !
                                                                                        !                                                                  !
                                                                                                                                                           !
                                          !                                                                                                      !
                                 !                                                               !                                                   !
                                                                                                                                                     ! !     !                               !
                     !                                                                           !                                            !         !
                                          !                                                                     !                                         !
                                                                                                              !!                                !   ! ! !!
         !                       !                                                               !!                    !


                                                                                                                                                                                                 different?
                     !
                         !
                                           !        !                       !                                                                         ! !
                                                                                                                                                      !        !
                                          !                         !                                                                         !!
                                                                                            !                !
                                                                                                             !                                        !
                                     !         !                            !    !            !                                                  ! !      ! !
                                                                                 !                       !                                  !   !         !
                         !            !                             !           !!                                          !                     !!    !
                                          !        !
                                                   !
                                                           !                     !          !!           ! !!              !                            !        !
             !                                                                                           !                 !
     !                                                                                                                                                                      !
     !                !
                     !!                                                                          !        !                  !
                                                                                                          !                                           !                    !
                     ! !             !    !
                                          !                                                              !
                 !                                              !                           !                                                        !
                                                                                                                                                      !




                                                                                                                      PC1                                                      PC2
                                                                                                                                                                               PC3
                                              PC5 PC1
                                                   PC6                                                        PC3                                    !         !                 PC6
                                              PC4                                                                          PC2                                              PC5!
                         !                                                                                                                       !                                !
                                 !! !                                                                                                                                       PC1
 !                               ! !
         !                                     PC2                                                                                      !                                   ! !  PC4
                     !
                     !     !                        ! !                                                                                          !
                                                                                                                                                                       !
                                                                                                                                                                       !
                     !! !!                                                                                                                      !            !         !
                     !
                        !! !    !                                                                                                  !            !           !
                                                                                                                                                           !!!
                                                                                                                                                                   !   !
             !                !                                                                                                                                    !
                        !
                       ! !                                                                                                          !   !                          ! !   !
                                                          !                                                                                 !!                       !
                          ! !                      ! !!                                                  !                                                         ! ! !!
 !                                                                                                        !
                                                                                                                                                 !                   !
                             !                             !     !                                              ! !!
                                                                                                                  !
                                                                                                             ! ! !!
     !                                                          ! ! !!                                         !                                                   !     !
                                                                                                                                   !
                                     !                                 !                                          ! !
                                                                                                                  !!
                             ! !                                   !! ! ! !
                                                                     ! !                                      !
                                                                                                              !                                  !
                                                                                                                                                                                 !
                                                                            !                                       ! !!
                                          !
                                 !                                   ! !
                                                                   !! !
                                                                       ! !                                                 !
                                     !                               !                                                                      !




Fig. 6.2. Testing the assumption that species labels correspond to three groups
in the data, using a projection pursuit guided tour with the LDA index. The plots
  V22                                                                                                                        V6
                                                                                                                       V27                              !


                                                                                                                                                                                     Which one of
                                                                     V41
                                                                      V6    !
              V45
V41                                                                                                                    V49                     !
         !                                                                                                                     !                !
  V33 V25
    !
 V18 V42
  !!    !      !
                       !                                       V43
                                                             V46 !                                                       V30V25 !
                                                                                                                           V1
                                                                                                                                                     !            !!         !


                                                                                                                                                                                     these is the real
          !            !                                                                                                         !! !
      !
                                                             ! !
                                                               !
                                                                        !
                                                                                                                            !      !                !!
               !                                                                                            !             !      !    !
      !            !
           !                                                  ! !
                                                                    !
                                                                      !
                                                                                                                               !    !                           !! ! !!          !
          !            !                                      !! ! !                                                         !                              !       !
                       ! !                                  !      ! !                   !    !                          !
                                                            !    !                                                !


                                                                                                                                                                                     data?
               !       !!                                           !               !! ! !
                                                                                         !                                                !                          !   !   !
                                                                                        !   ! !
                                                                                           ! !                                !                             !
                        !                                                             !!                                                  !
                                                                                                                                          !
                       ! !                                                                                                                                       !
                  !  ! !                                     !              !   !                   !
               !     !!!                                                                                                                           !     !
                !          !
                        !                                                                    !                                             !
                   !
                      ! ! !                                                                 !
                                !     !
                       !                                                        !




                                                                 V35
                                                                   V8                                                        V25
                                                                                                                              V21
                                                                                                                               V42
V39
V19                                        !               V32                                                    !                                                          !
                                                                                                                                                   !!
                                                                                                        !              V39                !
                                                                                            !                    !!!                                 !                           !
              V48                                          V41                                                                    !                 !
                                                                                                                                                    !
 V9                                 ! !
                                                                  V17                       !                                 V11          !   !     !
  V4                        !       !  !                         V16                                               !           !!
                                                                                                                                !         !    ! !!
                            !           !                                                       !                                                !    !
                       !    !        !!
                                                                                                            !            ! ! !               ! !
                       !              !                                                                        !                         !! ! ! ! !
                                                                                                                                         ! !    !     !
                                    ! !!                                                            !                        !        !!
                       !!                                                                   !               !!
                                                                                                             !
                                                                                                             !
                                                                                                                          !
                                                                                                                           ! !
                          !
                   ! ! !                                                    ! !
                                                                                                            !!                                                       !
                  !   !                                                 ! ! !
                           !                                     !      !                               !
              !!                                                              !                     !
               !    !                                                            !                          !
                    ! !                                                      !! !                   !
                !     !!                                              !
                                                                                 !
                   !                                                      !
                 !
                 !    !                                                   !
                                                                          !
                                                                          !
      !                                                                    !
  !                    !
                                                                                        !




  V5 V25                                                     V22                                                        V27 V15
                                                                                                                            V39
                                 !                                V7 !                                                                         !        !
                                               !                                                                                                                  !
                                                                  V25                                                                                   !
                                                                                                                                                        !
                                      !                                                                                            V2          !
V44                                                               V8 !!                                                V40                          !            !!!
                        !         !                             !     !!            !                                             V6                            !
   V50                          ! !                            V28
                                                             V41       !
                   !! !          ! !                                                                                                                      !
                                                                 !      ! !                                                           !                !    !
                     !                                                   ! !
                      !          !!    ! !                         ! !!
                                                                     !! !                                                                      !        !!
                                !                                           !                                                                          !
                       !!                                               ! !                                                                                          !   !
                                                                                                                                                    !                        !
                            ! !
                            !! !                                      ! !!                                                                              !
                                                                                                                                                                !!
                                   !                        ! !      !! ! ! !
                                                                           !
                                                                          ! !
                            !              !                          !
                                 !                                                                                                                               ! !!            !
                                 !                                                                                                                                  !
                             !                                          !
                                                                        ! !                                                                                      !
                              !            !       !                     !                                                                                         !   !
                                                       !                   !        !                                                                               !!
                                 !                                                                                                                               !
                                !                                                                                                                             !
                                  !                                                                                                                          !!
                    !        ! !                                        !                                                                                   !!




Fig. 6.3. Testing the assumption that species labels correspond to three groups in
simulated data, using a projection pursuit guided tour with the LDA index. The
  V22                                                                                                                        V6
                                                                                                                       V27                              !


                                                                                                                                                                                     Which one of
                                                                     V41
                                                                      V6    !
              V45
V41                                                                                                                    V49                     !
         !                                                                                                                     !                !
  V33 V25
    !
 V18 V42
  !!    !      !
                       !                                       V43
                                                             V46 !                                                       V30V25 !
                                                                                                                           V1
                                                                                                                                                     !            !!         !


                                                                                                                                                                                     these is the real
          !            !                                                                                                         !! !
      !
                                                             ! !
                                                               !
                                                                        !
                                                                                                                            !      !                !!
               !                                                                                            !             !      !    !
      !            !
           !                                                  ! !
                                                                    !
                                                                      !
                                                                                                                               !    !                           !! ! !!          !
          !            !                                      !! ! !                                                         !                              !       !
                       ! !                                  !      ! !                   !    !                          !
                                                            !    !                                                !


                                                                                                                                                                                     data?
               !       !!                                           !               !! ! !
                                                                                         !                                                !                          !   !   !
                                                                                        !   ! !
                                                                                           ! !                                !                             !
                        !                                                             !!                                                  !
                                                                                                                                          !
                       ! !                                                                                                                                       !
                  !  ! !                                     !              !   !                   !
               !     !!!                                                                                                                           !     !
                !          !
                        !                                                                    !                                             !
                   !
                      ! ! !                                                                 !
                                !     !
                       !                                                        !




V39
V19                                        !               V32
                                                                 V35
                                                                   V8                                             !
                                                                                                                             V25
                                                                                                                              V21
                                                                                                                               V42
                                                                                                                                                    !
                                                                                                                                                                             !
                                                                                                                                                                                     Can’t tell?
                                                                                                                                                                                     Real data is
                                                                                                                                          !        !
                                                                                            !           !        !!!   V39
              V48                                                                                                                 !                  !                           !
 V9                                                        V41    V17                       !                                                       !
                                                                                                                                                    !
  V4                        !       ! !                          V16                                                          V11
                                                                                                                               !!         !!   !     !
                                    !  !                                                                           !            !              ! !!
                            !           !                                                       !                                                !    !
                            !                                                                               !            ! ! !               ! !


                                                                                                                                                                                     50D with no
                       !             !!
                       !              !                                                                        !                         !! ! ! ! !
                                                                                                                                         ! !    !     !
                                    ! !!                                                            !                        !        !!
                       !!                                                                   !               !!
                                                                                                             !
                                                                                                             !
                                                                                                                          !
                                                                                                                           ! !
                          !
                   ! ! !                                                    ! !
                                                                                                            !!                                                       !
                                                                        ! ! !


                                                                                                                                                                                     real clusters.
                  !   !                                          !      !                               !
              !!           !
                    !                                                         !                     !       !
               !                                                                 !
                    ! !                                                      !! !                   !
                !     !!                                              !
                                                                                 !
                   !                                                      !



                                                                                                                                                                                     Apparent
                 !
                 !    !                                                   !
                                                                          !
                                                                          !
      !                                                                    !
  !                    !
                                                                                        !




  V5 V25                         !
                                               !
                                                             V22
                                                                  V7 !                                                  V27 V15
                                                                                                                            V39                !        !         !
                                                                                                                                                                                     clusters occur
                                                                                                                                                                                     because we
                                                                  V25                                                                                   !
                                                                                                                                                        !
                                      !                                                                                            V2          !
V44                                                               V8 !!                                                V40                          !            !!!
                        !         !                             !     !!            !                                             V6                            !
   V50                          ! !                            V28
                                                             V41       !
                   !! !          ! !                                                                                                                      !
                                                                 !      ! !                                                           !                !    !
                     !                                                   ! !


                                                                                                                                                                                     have large P
                      !          !!    ! !                         ! !!
                                                                     !! !                                                                      !        !!
                                !                                           !                                                                          !
                       !!                                               ! !                                                                                          !   !
                                                                                                                                                    !                        !
                            ! !
                            !! !                                      ! !!                                                                              !
                                                                                                                                                                !!
                                                                     !! ! ! !


                                                                                                                                                                                     small n.
                            !      !
                                           !                ! !            !
                                                                          ! !
                                 !                                    !                                                                                          ! !!
                                 !                                                                                                                                  !            !
                             !                                          !
                                                                        ! !                                                                                      !
                              !            !       !                     !                                                                                         !   !
                                                       !                   !        !                                                                               !!
                                 !                                                                                                                               !
                                !                                                                                                                             !
                                  !                                                                                                                          !!
                    !        ! !                                        !                                                                                   !!




Fig. 6.3. Testing the assumption that species labels correspond to three groups in
simulated data, using a projection pursuit guided tour with the LDA index. The
Ways to test for significance

• Independence between variables can be tested
  using permutation methods, where the
  appropriate columns of data values are shuffled
  to break the association in the ordered tuples.
• Distributional forms can tested by simulating
  samples from the distribution using parameters
  estimated from the sample.

								
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