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

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

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

5. T-statistic= Difference between means Standard error within each sample The t-test s 2 + s 2 n 1 n 2

6. How many degrees of freedom? The t-test s 2 + s 2 n 1 n 2 (n 1 -1) + (n 2 -1) Why does this seem familiar?

7. T-tables v0.100.050.025 13.0786.31412.706 21.8862.9204.303 31.6382.3533.182 41.5332.1322.776 infinity1.2821.6451.960 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 v0.100.050.025 13.0786.31412.706 21.8862.9204.303 31.6382.3533.182 41.5332.1322.776 infinity1.2821.6451.960 Two samples, each n=3, with t-statistic of 2.50: significantly different?

9. T-tables v0.100.050.025 13.0786.31412.706 21.8862.9204.303 31.6382.3533.182 41.5332.1322.776 infinity1.2821.6451.960 Two samples, each n=3, with t-statistic of 2.50: significantly different? No!

10. v0.100.050.025 13.0786.31412.706 21.8862.9204.303 31.6382.3533.182 41.5332.1322.776 infinity1.2821.6451.960 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. v0.100.050.025 13.0786.31412.706 21.8862.9204.303 31.6382.3533.182 41.5332.1322.776 infinity1.2821.645 1.960 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? } 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! Careful! Doesn’t work the other way around!!

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

13. General form of the t-test, can have more than 2 samples One-way ANOVA ABC ABC ABC ABC DATA HoHa

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

15. MS= Sum of squares df Mean squares:

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

18. Variance: S 2 = Σ (x i – x ) 2 n-1 Sum of squared differences df

19. ANOVA tables dfSSMSFp Treatment (between groups) df (X)SSX Error (within groups) df (E)SSE Totaldf (T)SST SST = SSXSSE

20. ANOVA tables dfSSMSFp Treatment (between groups) df (X)SSX df (X) Error (within groups) df (E)SSE df (E) Totaldf (T)SST SSX SSE df (X)df (E) MSX = = MSE

21. ANOVA tables dfSSMSFp Treatment (between groups) df (X)SSX df (X) MSX MSE Look up ! Error (within groups) df (E)SSE df (E) Totaldf (T)SST } } SSX SSE df (X)df (E) MSX = = MSE

22. Do three species of palms differ in growth rate? We have 5 observations per species. Complete the table! dfSSMSFp Treatment (between groups) 69 Error (within groups) k(n-1) Total104

23. Hint: For the total df, remember that we calculate total SS as if there are no groups (total variance)… dfSSMSFp Treatment (between groups) 69 Error (within groups) k(n-1) Total104

24. Note: treatment df always k-1 Is it significant? At alpha = 0.05, F 2,12 = 3.89 dfSSMSFp Treatment (between groups) 26934.511.8? Error (within groups) 12352.92 Total14104

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

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

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

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

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

30. 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? Two-way ANOVA k(n-1) = 4 (10-1) = 36

31. vdfSSMSF Aspect1SS(Aspect)MS(Aspect)MS(As) MSE Species1SS(Species)MS(Species)MS(Sp) MSE Aspect x Species 1SS(Int)MS(Int) MSE Error (within groups) 36SSEMSE Total39SST

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

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

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