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ANOVA

ANOVA. ESM 206 11 February 2002. 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?. Data. Three groups of fish (about 60 each):

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ANOVA

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  1. ANOVA ESM 206 11 February 2002

  2. 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?

  3. Data • 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

  4. Multiple Comparisons • We have established that independent variable doesn’t 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 • Do a bunch of t-tests… • …but the more tests we do, the more likely we are to find a small P-value

  5. 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

  6. Two-factor ANOVA

  7. Assumptions of ANOVA

  8. Metals & fish • 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

  9. Splus

  10. ANOVA table Df Sum Mean F P of Sq Sq Treatment 1 14.73 14.73 110.04 <<0.0001 Fish 2 4.33 2.16 16.17 <<0.0001 Treatment X Fish 2 1.52 0.76 5.69 0.004 Residuals 175 23.43 0.13

  11. 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

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