1. BIOSTATISTICS Analysis of Variance (ANOVA)

2. Copyright ©2012, Joanna Szyda INTRODUCTION 1.Applicability 2.One way analysis of variance 3.Partitioning the observed variability 4.Hypotheses testing 5.Example 6.Variants of ANOVA

3. APPLICABILITY

4. Copyright ©2012, Joanna Szyda APPLICABILITY 1.Dependent variable (y) continuous 2.Normal distribution 3.Analysis of Variance: factors related to variation among y 4.Identical / proportional number of observations within a group

5. ONE WAY ANOVA

6. Copyright ©2012, Joanna Szyda DATA SET 1.Nitrogen content in reed (% of dry matter) 2.3 locations (A, B, C), experiment in 1996 r. 3.Flowermere, Cambridge ABC 3.063.412.92 2.603.232.88 2.553.933.25 2.423.742.64 2.353.183.28

7. Copyright ©2012, Joanna Szyda DATA SET 3.063.412.92 2.603.232.88 2.553.933.25 2.423.742.64 2.353.183.28 Observed variability Different locations 3.063.412.92 2.603.232.88 2.553.933.25 2.423.742.64 2.353.183.28

8. Copyright ©2012, Joanna Szyda DATA SET 3.063.412.92 2.603.232.88 2.553.933.25 2.423.742.64 2.353.183.28 Observed variability 3.063.412.92 2.603.232.88 2.553.933.25 2.423.742.64 2.353.183.28 Genotypes 3.063.412.92 2.603.232.88 2.553.933.25 2.423.742.64 2.353.183.28 Distance from the shore 3.063.412.92 2.603.232.88 2.553.933.25 2.423.742.64 2.353.183.28 Other

9. PARTITIONING THE OBSERVED VARIABILITY

10. Copyright ©2012, Joanna Szyda PARTITIONING THE OBSERVED VARIABILITY ANALYSIS OF VARIANCE MODEL N content = μ + location + e Analysed effect Does it influence variability ??? Test Effects common for all observations No influence of variability of y Effects not measured May influence variability

11. Copyright ©2012, Joanna Szyda PARTITIONING THE OBSERVED VARIABILITY Total variability Variability within group A Variability within group B Variability within group C Variability between groups

12. Excel: example Copyright ©2012, Joanna Szyda PARTITIONING THE OBSERVED VARIABILITY ONE WAY ANOVA MODEL SOURCE OFSUM OFDEGREES OFMEAN VARIABILITYSQUARESFREEDOMSQUARE Between gr. (location) Within gr. error Total

13. HYPOTHESES TESTING

14. Copyright ©2012, Joanna Szyda HYPOTHESES TESTING 1.Formulate hypotheses H 0 and H 1 H 0 : difference in locations does not affect N content H 1 : difference in locations affects N content H 0 : N A = N B = N C H 1 : N A ≠ N B ≠ N C 2.Set the significance level  MAX = 0.05 3.Choose the statistical test and calculate test value

15. Copyright ©2012, Joanna Szyda HYPOTHESES TESTING mean variability of y explained by different locations mean variability of y not explained by different locations

16. Copyright ©2012, Joanna Szyda HYPOTHESES TESTING 3.Choose the statistical test and calculate test value 4.Determine distribution of the test: 5.Determine  t : 6.Decision:  t <  max H 0 H 1 difference in locations affects N content

17. EXAMPLE

18. Copyright ©2012, Joanna Szyda LITERATURE EXAMPLE Hutchinson et al. (2004) Brain 127: 1403-1414 Influence of physical activity on pain perception related to moderate spinal cord damage Model organism - rat 4 activity patterns: no activity, treadmill, swimming, standing After 7 weeks – spinal cord biochemical activity scoring

19. Copyright ©2012, Joanna Szyda LITERATURE EXAMPLE

20. VARIANTS OF ANOVA

21. Copyright ©2012, Joanna Szyda VARIANTS OF ANOVA More than one effect – e.g.. 2-way ANOVA, 3-way ANOVA, … Considering interdependence between effects nesting interaction More than one dependent variable – analysis of covariance (ANCOVA)

22. Copyright ©2012, Joanna Szyda VARIANTS OF ANOVA – 2 WAY ANOVA WITH INTERACTIONS SOURCE OFSUM OF DEGREES OF MEAN VARIABILITYSQUARES FREEDOM SQUARE Between gr. (effect A) Between gr. (effect B) Interaction Within gr. error Total

23. Copyright ©2012, Joanna Szyda VARIANTS OF ANOVA – 2 WAY HIERARCHICAL ANOVA SOURCE OFSUM OFDEGREES OFMEAN VARIABILITYSQUARESFREEDOMSQUARE Between gr. A Between gr. B within gr A Within gr A(B) error Total

24. Copyright ©2014 Joanna Szyda ANOVA