Metal toxicity in fish

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Metal toxicity in fish

• Clark Fork River (Montana) contaminated with
  mixture of toxic metals
• Can trout develop resistance to metals toxicity
  by being exposed to low concentrations of metals?

• Three groups of fish (about 60 each)
• Hatchery brown trout
• Hatchery rainbow trout
• Clark Fork River brown trout
• Two treatments (half of each group)
• Control keep fish in clean water for 3 weeks
• Treatment keep fish in weakly contaminated water
  for 3 weeks
• All fish survived this stage
• Measurement
• Expose fish to highest Clark Fork River
  concentrations
• Measure time to death in hours

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Single factor ANOVA
The ith observation in group j
The error deviation of the observation from the
j group mean
Overall mean across all groups (grand mean)
Deviation of the jth group mean (level from
the grand mean
Sum of as is zero
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Single factor ANOVA Effect of fish source on
toxicity resistance
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Two factor ANOVA
Deviation of kth level of factor 2 from the
grand mean
Grand mean
Error
Deviation of jth level of factor 1 from the
grand mean
Interaction deviation from j group mean of
factor 1 associated with being in kth group of
factor 2
ith obs in group j of factor 1 and group k of
factor 2
Sum of as is zero
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Hypothesis testing for two-factor ANOVA
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Assumptions of ANOVA Metals fish

• Samples random and independent
• Within-group variation normally distributed
• Homoscedasticity (groups have same variances)

• The factors (independent variables) are
• Species/source of fish
• Acclimation treatment yes/no
• The variances are very different among groups
• Range from 20 to 1200
• Log transform greatly reduces this

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Fish Metals
Df Sum Sq Mean Sq
F value Pr(gtF) Fish 2 4.5812
2.2906 17.1087 1.635e-07 Metals 1
14.4823 14.4823 108.1684 lt 2.2e-16
FishMetals 2 1.5249 0.7624 5.6946
0.004018 Residuals 175 23.4301 0.1339

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Fish Metals
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Multiple Comparisons

• We have established that independent variable
  doesnt have the same mean across all levels of
  the independent variable
• but which means are different?
• With n levels, there are n(n-1)/2 comparisons
• Even more if there are multiple factors
• Do a bunch of t-tests
• but the more tests we do, the more likely we are
  to find a small P-value

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Bonferroni correction

• Comparison-wise error rate Probability of type-I
  error in a given test
• Family-wise error rate Probability of at least
  one type I error in all the tests
• Also called experiment-wise error
• To obtain family-wise error rate a across m
  tests, set comparison-wise rate to

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Multiple comparisons

• Control vs. treatment, for each fish (3)
• Control fish vs. one another (3)
• Treatment fish vs. one another (3)
• Treatment effect per fish, vs. one another (3)
• For experiment-wise a of 0.05, need to set
  test-wise a to 0.004