1. Assignment # 2

2. Problem Set-up  Clearly describe the link between the biological hypothesis, the statistical hypothesis and the data collected

3. Example This study measured the tumour volume in mice randomly assigned to four groups, three receiving different anti-cancer drugs and one control receiving no drugs, to test if the drugs were effective in reducing tumour size If a drug was effective, then mice that received it should have a mean tumour volume that was significantly LOWER THAN the control (i.e. one- tailed). Efficiency among drugs was tested by determining which, if any, drug had a significantly LOWER tumour volume (i.e. on tailed)

4. Statistical Hypothesis H o : No difference among group means H a : Difference among group means  Significant ANOVA does not indicate that drugs were effective against cancer;  Difference could be smaller control tumour volume  Need to do multiple comparisons with one-tailed hypothesis H o : µ T ≥µ C H a : µ T< µ C

5. Analysis 1. Examine raw data first 2. Present results of ANOVA, then test assumptions 3. Try a transformation before doing non-Parametric 4. Be sure to explicitly identify if plots and tests are of residuals or raw data

6. Power Considerations Power a concern when you do not reject Ho i.e. pair-wise comparisons Not the ANOVA Power was low for C-T, C-X comparisons Perhaps due to the higher variation in tumour size in those groups, especially in the control Consequence of low power: fail to detect true differences between C – T, C – X, accepted a false null (Type II error)

7. Discussion  Many people provide very little discussion of results, and answers to questions.  Many people simply concluded that, “the data were appropriate”. Thanks for the vote of confidence, but we are looking for a little more intellectual zeal than that!

8. Discussion points 1. Logical inconsistency in pair-wise comparisons and ; V > C, but V = T,X and T, X = C V = C ??? 1. That there was difference in mortality rates among groups (many dead mice in V). a. may explain the smaller variation in tumour size among V group and the higher power to detect a difference in C – V b. the usefulness of V in treating cancer if it has a high mortality rate

9. Discussion Points  Differences in starting tumour size: measure tumour shrinkage, as opposed to final tumour volume to reduce some of the variation and increase power  Variation among individuals; why does the drug work in some but not others?  Any other reasonable suggestion

10. Discussion Points  Simply suggesting more samples was not sufficient  Especially for those who didn’t talk about problems with low power!!

11. Writing Style  Try to write in the more formal style demanded in science.  Use paragraphs that explain what you did, interpret the result refer reader to appropriate statistical result, graphic, table that supports your conclusion.  Include statistical output, in appendix if necessary

12. Example  Visual examination of residuals from the initial ANOVA, comparing tumour volumes among groups, showed evidence of non-normality (Figure 1). This result was confirmed by the K-S test on the residuals (K-S: 0.1083, p = 0.0315).

13. Example # 2  Tumour volume was significantly different among groups (ANOVA: F = 7.78, df = 3,70, p <0.0001), with post-hoc comparisons indicating this was due solely to a significant difference between the C and V groups (Figure 2). There was no significant difference between the control and the other two treatments or among the treatments themselves (Figure 2)