Statistical Power
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Transcript of Statistical Power
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Statistical Power
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Ho : Treatments A and B the same
HA: Treatments A and B different
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Points on this side, only 5% chance from distribution A.
Area = 5%
Critical value at alpha=0.05F
req
uen
cy
A
A could be control treatmentB could be manipulated treatment
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AB
If null hypothesis true, A and B are identical
Probability that any value of B will be not significantly different from A = 95%
Probability that any value of B is significantly different than A = 5%
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AB
If null hypothesis true, A and B are identical
Probability that any value of B will be not significantly different from A = 95%
Probability that any value of B is significantly different than A = 5% = likelihood of type 1 error
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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
What you say:R
eali
ty
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AB
If null hypothesis false, two distributions are different
Probability that any value of B will be not significantly different from A = beta = likelihood of type 2 error
Probability that any value of B is significantly different than A = 1- beta = power
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AB
Effect size
Effect size = difference in meansSD
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AB
1. Power increases as effect size increases
Beta = likelihood of type 2 error
Power
Effect size
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AB
2. Power increases as alpha increases
Beta = likelihood of type 2 error
Power
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AB
3. Power increases as sample size increases
Low n
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AB
3. Power increases as sample size increases
High n
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Power
Effect size Alpha
Sample size
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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.
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Example : Fox hunting in the UK(posteriori)
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• 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 (p=0.474, alpha=0.05, n=157), but Aebischer et al. raised questions about power.
Baker et al. 2002. Nature 419: 34Aebischer et al. 2003. Nature 423: 400
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157 plots where the fox population monitored. Alpha = 0.05
Effect size if hunting affected fox populations: 13%
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157 plots where the fox population monitored. Alpha = 0.05
Effect size if hunting affected fox populations: 13%
Power = 0.95 !
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Class exercise:
Means and SD of parasite load (p>0.05):
Daphnia magna 5.9 ± 2 (n = 3)
Daphnia pulex 4.9 ± 2 (n = 3)
(1) Did the researcher have “enough” power (>0.80)?
(2) Suggest a better sample size.
(3) Why is n=3 rarely adequate as a sample size?
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How many samples?
PCBs in salmon from Burrard inlet and Alaska
In an initial survey (3 individuals each), we find the following information (mean, standard deviation)
Burrard – 120.5 ± 75.9 ppbAlaska – 75.2 ± 71.9 ppb
The two error bars overlap, but that’s still a big difference and we only took 3 samples
The difference could be “hidden” the sizes of the errors
This would be reduced by increased samples, but how many should we take?
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How many samples?
Our difference between (q) is ~40, therefore if our confidence limits (SE) were <20ppb, we should have adifference between populations,
Burrard – 120.5 ± 75.9 ppbAlaska – 75.2 ± 71.9 ppb
How many samples do we therefore need??
€
q = 2ts
n
€
40 = 2∗1.9675.9
n
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€
q = 2ts
n
€
40 = 2∗1.9675.9
n
Re-arrange the equation…
€
n =2*1.96* 75.9
40
⎛
⎝ ⎜
⎞
⎠ ⎟2
= 55.4
So we should take 56 samples to be reasonably sure of a significant difference
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Don’t get silly..