1. Spotting pseudoreplication 1.Inspect spatial (temporal) layout of the experiment 2.Examine degrees of freedom in analysis

2. Degrees of freedom (df) Number of independent terms used to estimate the parameter = Total number of datapoints – number of parameters estimated from data

3. Example: Variance If we have 3 data points with a mean value of 10, what’s the df for the variance estimate? Independent term method: Can the first data point be any number? Can the second data point be any number? Can the third data point be any number? Yes, say 8 Yes, say 12 No – as mean is fixed ! Variance is  (y – mean) 2 / (n-1)

4. Example: Variance If we have 3 data points with a mean value of 10, what’s the df for the variance estimate? Independent term method: Therefore 2 independent terms (df = 2)

5. Example: Variance If we have 3 data points with a mean value of 10, what’s the df for the variance estimate? Subtraction method Total number of data points? Number of estimates from the data? df= 3-1 = 2 3 1

6. Example: Linear regression Y = mx + b Therefore 2 parameters estimated simultaneously (df = n-2)

7. Example: Analysis of variance (ANOVA) ABCa1 b1 c1a2 b2 c2a3 b3 c3a4 b4 c4ABCa1 b1 c1a2 b2 c2a3 b3 c3a4 b4 c4 What is n for each level?

8. Example: Analysis of variance (ANOVA) ABCa1 b1 c1a2 b2 c2a3 b3 c3a4 b4 c4ABCa1 b1 c1a2 b2 c2a3 b3 c3a4 b4 c4 n = 4 How many df for each variance estimate? df = 3 df = 3 df = 3

9. Example: Analysis of variance (ANOVA) ABCa1 b1 c1a2 b2 c2a3 b3 c3a4 b4 c4ABCa1 b1 c1a2 b2 c2a3 b3 c3a4 b4 c4 What’s the within-treatment df for an ANOVA? Within-treatment df = 3 + 3 + 3 = 9 df = 3 df = 3 df = 3

10. Example: Analysis of variance (ANOVA) ABCa1 b1 c1a2 b2 c2a3 b3 c3a4 b4 c4ABCa1 b1 c1a2 b2 c2a3 b3 c3a4 b4 c4 If an ANOVA has k levels and n data points per level, what’s a simple formula for within-treatment df? df = k(n-1)

11. Spotting pseudoreplication An experiment has 10 fertilized and 10 unfertilized plots, with 5 plants per plot. The researcher reports df=98 for the ANOVA (within-treatment MS). Is there pseudoreplication?

12. Spotting pseudoreplication An experiment has 10 fertilized and 10 unfertilized plots, with 5 plants per plot. The researcher reports df=98 for the ANOVA. Yes! As k=2, n=10, then df = 2(10-1) = 18

13. Spotting pseudoreplication An experiment has 10 fertilized and 10 unfertilized plots, with 5 plants per plot. The researcher reports df=98 for the ANOVA. What mistake did the researcher make?

14. Spotting pseudoreplication An experiment has 10 fertilized and 10 unfertilized plots, with 5 plants per plot. The researcher reports df=98 for the ANOVA. Assumed n=50: 2(50-1)=98

15. Why is pseudoreplication a problem? Hint: think about what we use df for!

16. How prevalent? Hurlbert (1984): 48% of papers Heffner et al. (1996): 12 to 14% of papers

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

18. Variance Ecological rule # 1: Everything varies …but how much does it vary?

19. Variance S 2 = Σ (x i – x ) 2 n-1 x Sum-of-square cake

20. Variance S 2 = Σ (x i – x ) 2 n-1 x

21. Variance S 2 = Σ (x i – x ) 2 n-1 What is the variance of 4, 3, 3, 2 ? What are the units?

22. Variance variants 1. Standard deviation (s, or SD) = Square root (variance) Advantage: units

23. Variance variants 2. Standard error (S.E.) = s n Advantage: indicates precision

24. How to report We observed 29.7 (+ 5.3) grizzly bears per month (mean + S.E.). A mean (+ SD)of 29.7 (+ 7.4) grizzly bears were seen per month + 1SE or SD - 1SE or SD

25. Distributions Normal Quantitative data Poisson Count (frequency) data

26. Normal distribution 67% of data within 1 SD of mean 95% of data within 2 SD of mean

27. Poisson distribution mean Mostly, nothing happens (lots of zeros)

28. Poisson distribution Frequency data Lots of zero (or minimum value) data Variance increases with the mean

29. 1.Correct for correlation between mean and variance by log-transforming y (but log (0) is undefined!!) 2.Use non-parametric statistics (but low power) 3.Use a “generalized linear model” specifying a Poisson distribution What do you do with Poisson data?

30. Null (Ho): no effect of our experimental treatment, “status quo” Alternative (Ha): there is an effect Hypotheses

31. Whose null hypothesis? Conditions very strict for rejecting Ho, whereas accepting Ho is easy (just a matter of not finding grounds to reject it). A criminal trial? Exotic plant species? WTO?

32. Hypotheses Null (Ho) and alternative (Ha): always mutually exclusive So if Ha is treatment>control…

33. Types of error Type 1 error Type 2 error Reject HoAccept Ho Ho true Ho false

34. Usually ensure only 5% chance of type 1 error (ie. Alpha =0.05) Ability to minimize type 2 error: called power Types of error