Aquaculture growth data experimental design, statistical
analyses and detection limits
Helgi Thorarensen, Godfrey
Kubiriza and Albert K. Imsland
How reliable are our
conclusions in growth studies?
• Do we find all differences that truly
exist?
• Do we conclude that there are no
effects where they truly exist?
• Are there likely unsound conclusions in
the scientific literature?
Tank2
Tank1
Tankb
Tank1
Tank2
n fish
n fish
....
n fish
n fish
n fish
Tank2
n fish
....
....
Tankb
Treatment2
n fish
Tank1
n fish
....
Tankb
Treatment1
n fish
Design of growth studies
Treatmenta
Design of growth studies
Treatment2
Treatmenta
Tankb
Tank1
Tank2
Tankb
Tank1
Tank2
Tankb
n fish
n fish
n fish
n fish
n fish
n fish
....
n fish
Tank2
Tank1
Treatment1
n fish
n fish
The responses
.... of all fish in each tank
.... are assumed to be independent
....
estimates of the responses of the population to the treatment
Design of growth studies
Treatment2
Treatmenta
Tankb
Tank1
Tank2
Tankb
Tank1
Tank2
Tankb
n fish
n fish
n fish
n fish
n fish
n fish
....
n fish
Tank2
Tank1
Treatment1
n fish
n fish
The responses
.... of all fish in each tank
.... are assumed to be independent
....
estimates of the responses of the population to the treatment
The responses of individual fish are not independent estimates of the
responses of the population to the treatment
ANOVA
Tankb
Tank1
Tank2
Treatmenta
....
Tankb
Tank2
Tank1
Treatment2
Tankb
Tank2
Tank1
Treatment1
....
.... tank
The ANOVA....is performed on total biomass
or mean size in each
Mixed-model ANOVA
Tankb
Tank1
Tank2
Tankb
Tank1
Tank2
n fish
n fish
n fish
n fish
n fish
.... The tanks are nested....within treatments
....
Tankb
Treatmenta
....
n fish
Treatment2
n fish
Tank2
n fish
n fish
Tank1
Treatment1
Which approach to use
• If the design is balanced
– (same number of fish in each, tank same
number of tanks in each treatment)
Simple and mixed model ANOVAs will give
the same result
• If the desing is not balanced
The mixed model ANOVA should be used
How reliable are our results
from ANOVA?
• The design may not allow the detection
of differences that truely exist
What is the minimum difference we
can detect in growth studies?
• Statistical power:
The probability of detecting a significant
difference where one truely exists
• The accepted statistical power is 80%
Statistical power depends on
1. The difference among treatment groups
2. The variance of the data
a) Among fish within a tank
b) Among tanks receiving identical treatments
3. The number of replicate tanks
4. The number of fish within each tank
5. The number of treatments tested
Statistical power depends on
1. The difference among treatment groups
2. The variance of the data
a) Among fish within a tank
b) Among tanks receiving identical treatments
3. The number of replicate tanks
4. The number of fish within each tank
5. The number of treatments tested
Survey
• We analysed the variance in our own
(32) growth studies on five species
• Used this information to estimate the
minimum difference we are likely to
detect (MDD%) with 80% statistical
power
• Added information from 29 studies on
25 species in one volume of
Aquaculture
Final variation in growth
studies
32 growth studies performed in our own facilities on 5 species
Mean
Range
Coefficient of variation for fish within tanks
30.6%
15-56%
Coefficient of variation for tanks within
treatments
4.5%
2-5%
29 growth studies of 24 species of fish published in 2013 and 2014 in Aquaculture
Coefficient of variation for fish within tanks
Coefficient of variation for tanks within
treatments
Mean
28%
Range
23-36%
5%
0-49%
The minimum detectable difference
with 80% statistical power – average
variance
b=2
b=3
b=4
b=5
b=6
60
MDD (%)
50
40
30
20
There is little gained by
increasing n over 50-100
10
0
100
200
300
400
Sample size (n)
500
600
The minimum detectable difference
with 80% statistical power – high
variance
60
MDD (%)
50
40
30
Larger variance gives larger
MDD%
20
10
0
100
200
300
400
Sample size (n)
500
600
The minimum detectable difference
with 80% statistical power
b=2
b=3
b=2
b=4
bb==53
bb==64
b=5
b=6
60
60
MDD
MDD(%)
(%)
50
50
40
40
30
30
20
20
Low variance gives smaller
MDD
10
10
0
0
100
100
200
300
400
200
300
400
Sample
Samplesize
size(n)
(n)
500
500
600
600
45
14
40
12
35
10
8
Final CV
Final CV
Initial and final variance are
correlated
30
25
20
15
6
4
2
0
10
5
10 15 20 25 30 35 40 45
Initial CV
0
2
4
Initial CV
6
8
10
Recent growth studies
published in aquaculture
Mean
Range
Level of replication
3
2-6
Number of fish within each tank (n)
26
4-100
b=2
b=3
b=4
b=5
b=6
60
MDD (%)
50
40
The average replication level
and n gives MDD% between
20-30%
30
20
10
0
100
200
300
400
Sample size (n)
500
600
Conclusions
• Similar experimental designs are suitable for
different species
• Smaller initial variance gives better statistical
power at the end of study
• The minimum detectable difference with 80%
statistical power in published growth studies
is 25-30%
• There are likely studies in the literature that
have failed to detect differences/effects where
they truly exist