Statistical Power

1
Statistical Power
2
Ho Treatments A and B the same HA Treatments
A and B different
3
Critical value at alpha0.05
Points on this side, only 5 chance from
distribution A.
Frequency
A
Area 5
A could be control treatment B could be
manipulated treatment
4
If null hypothesis true, A and B are identical
Probability that any value of B is significantly
different than A 5
A
B
Probability that any value of B will be not
significantly different from A 95
5
What you say
Decide NOT significantly different (do not reject Ho) Decide significantly different (reject Ho)
Ho true (same) Type 1 error
Ho false (different) Type 2 error
Reality
6
If null hypothesis true, A and B are identical
Probability that any value of B is significantly
different than A 5 likelihood of type 1
error
A
B
Probability that any value of B will be not
significantly different from A 95
7
If null hypothesis false, two distributions are
different
Probability that any value of B is significantly
different than A 1- beta power
A
B
Probability that any value of B will be not
significantly different from A beta
likelihood of type 2 error
8
Effect size
A
B
Effect size difference in means SD
9
1. Power increases as effect size increases
Power
Effect size
A
B
Beta likelihood of type 2 error
10
2. Power increases as alpha increases
Power
A
B
Beta likelihood of type 2 error
11
3. Power increases as sample size increases
Low n
A
B
12
3. Power increases as sample size increases
High n
A
B
13
Alpha
Effect size
Power
Sample size
14
Types of power analysis A priori Useful for
setting up a large experiment with some pilot
data Posteriori Useful for deciding how
powerful your conclusion is (definitely? Or
possibly). In manuscript writing, peer reviews,
etc.
15
Example Fox hunting in the UK (posteriori)
16

• Hunt banned (one year only) in 2001 because of
  foot-and-mouth disease.
• Can examine whether the fox population increased
  in areas where it used to be hunted (in this
  year).
• Baker et al. found no effect (p0.474,
  alpha0.05, n157), but Aebischer et al. raised
  questions about power.

Baker et al. 2002. Nature 419 34 Aebischer et
al. 2003. Nature 423 400
17
157 plots where the fox population monitored.
Alpha 0.05 Effect size if hunting affected
fox populations 13
18
157 plots where the fox population monitored.
Alpha 0.05 Effect size if hunting affected
fox populations 13 Power 0.95 !
19
Class exercise Means and SD of parasite load
(pgt0.05) Daphnia magna 5.9 1 (n
3) Daphnia pulex 4.9 1 (n 3) (1) Did the
researcher have enough power (gt0.80)? (2)
Suggest a better sample size. (3) Why is n3
rarely adequate as a sample size? Hint see
table.
20

• Good options for increasing sample size
• More replicates
• More blocks
• False options for increasing sample size
• More repeated measurements
• Pseudoreplication