# Inference by gyvwpsjkko

VIEWS: 7 PAGES: 15

• pg 1
```									            Inference
Revealing, informative plots often provoke the
question “Is what we see really there?”
b                ●       ●            ● ●   ●   ●       ●    ●
●●       ●    ●
●●   ●       ●●       ● ●       ●       ●
group

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

−2                       −1                               0                                   1       2
value
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?
b            ●   ●       ●   ●            ●●         ●      ●    ●
●       ●    ● ●
● ●        ●        ●   ● ●●        ●           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

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

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

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

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

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

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

p-value = 0.08
value                                                                                                                           value

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

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

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

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

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

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

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

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

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

p-value < 0.01
value                                                                                                                                   value

b            ● ●                  ●●   ●    ●   ●            ●
●
●       ●
●           ● ●
● ●                ●       ●●          ●   ●       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.

```
To top