Statistics review

1. Statistics review Basic concepts: • Variability measures • Distributions • Hypotheses • Types of error • Common analyses • T-tests • One-way ANOVA • Randomized block ANOVA • Two-way ANOVA

2. The t-test Asks: do two samples come from different populations? YES NO DATA Ho A B

3. The t-test Depends on whether the difference between samples is much greater than difference within sample. A B Between >> within… A B

4. The t-test Depends on whether the difference between samples is much greater than difference within sample. A B Between < within… A B

5. s2 + s2 n1 n2 The t-test T-statistic= Difference between means Standard error within each sample

6. s2 + s2 n1 n2 The t-test How many degrees of freedom? (n1-1) + (n2-1) Why does this seem familiar?

7. T-tables Careful! This table built for one-tailed tests. Only common stats table where to do a two-tailed test (A doesn’t equal B) requires you to divide the alpha by 2

8. T-tables Two samples, each n=3, with t-statistic of 2.50: significantly different?

9. T-tables Two samples, each n=3, with t-statistic of 2.50: significantly different? No!

10. If you have two samples with similar n and S.E., why do you know instantly that they are not significantly different if their error bars overlap?

11. If you have two samples with similar n and S.E., why do you know instantly that they are not significantly different if their error bars overlap? } Careful! Doesn’t work the other way around!! • the difference in means < 2 x S.E., i.e. t-statistic < 2 • and, for any df, t must be > 1.96 to be significant!

12. One-way ANOVA General form of the t-test, can have more than 2 samples Ho: All samples the same… Ha: At least one sample different

13. One-way ANOVA General form of the t-test, can have more than 2 samples A B C DATA A B C Ho Ha A B C A C B

14. A B C One-way ANOVA Just like t-test, compares differences between samples to differences within samples Difference between means Standard error within sample T-test statistic (t) MS between groups MS within group ANOVA statistic (F)

15. Mean squares: • MS= Sum of squares • df

16. Everyone gets a lot of cake (high MS) when: Lots of cake (high SS) Few forks (low df) • MS= Sum of squares • df

17. Mean squares: • MS= Sum of squares • df Analogous to variance

18. Variance: Sum of squared differences • S2= Σ (xi – x )2 • n-1 df

19. ANOVA tables SST = SSX SSE

20. ANOVA tables SSE SSX = MSE MSX = df (X) df (E)

21. ANOVA tables } } SSE SSX = MSE MSX = df (X) df (E)

22. Do three species of palms differ in growth rate? We have 5 observations per species. Complete the table!

23. Hint: For the total df, remember that we calculate total SS as if there are no groups (total variance)…

24. Note: treatment df always k-1 Is it significant? At alpha = 0.05, F2,12 = 3.89

25. 2. Randomized block Poor patch Medium patch Good patch BLOCK B BLOCK C BLOCK A

26. Pro: Can remove between-block SS from error SS…may increase power of test Error Error Block Treatment Treatment

27. Con: Blocks use up error degrees of freedom Error Error Block Treatment Treatment

28. Do the benefits outweigh the costs? Does MS error go down? F = Treatment SS/treatment df Error SS/error df Error Error Block Treatment Treatment

29. Two-way ANOVA Just like one-way ANOVA, except subdivides the treatment SS into: • Treatment 1 • Treatment 2 • Interaction 1&2

30. Two-way ANOVA Suppose we wanted to know if moss grows thicker on north or south side of trees, and we look at 10 aspen and 10 fir trees: • Aspect (2 levels, so 1 df) • Tree species (2 levels, so 1 df) • Aspect x species interaction (1df x 1df = 1df) • Error? k(n-1) = 4 (10-1) = 36

32. 5 3 2 Interactions Combination of treatments gives non-additive effect Additive effect: Alder Fir South North

33. Interactions Combination of treatments gives non-additive effect Anything not parallel! South South North North

34. Careful! If you log-transformed your variables, the absence of interaction is a multiplicative effect: log (a) + log (b) = log (ab) y Log (y) South South North North