1. Blocks and pseudoreplication

2. This lecture will cover:
Blocks
Experimental units (replicates)
Pseudoreplication
Degrees of freedom

3. Good options for increasing sample size:
More replicates
More blocks
False options for increasing sample size:
More “repeated measurements”
Pseudoreplication

4. Ecological rule #1: the world is not uniform!
Good patch
Medium patch
Poor patch

5. 3 options in assigning treatments: Randomly assign Systematic
Randomized block
Good patch
Medium patch
Poor patch

6. 1. Randomly assign Statistically robust Pros? Cons?
Good patch
Medium patch
Poor patch
Pros?
Cons?
Statistically robust
With small n, chance of all in a bad patch

7. 1. Randomly assign
Good patch
Medium patch
Poor patch
What’s the chance of total spatial segregation of treatments?
Pros?
Cons?

8. 2. Systematic No clumping possible Pros? Cons?
Good patch
Medium patch
Poor patch
Pros?
Cons?
No clumping possible
Violates random assumption of statistics…but is this so bad?

9. 3. Randomized block BLOCK A BLOCK B BLOCK C Good patch Medium patch
Poor patch
BLOCK A
BLOCK B
BLOCK C

10. 3. Randomized block Note:
BLOCK A
BLOCK B
BLOCK C
Note:
Do not have to know if patches differ in quality
Must have all treatment combinations represented in each block
If WANT to test treatment x block interaction, need replication within blocks

11. How to analyze a blocked design in JMP (Method 1)
Basic stats> Oneway.
Add response variable, treatment (“grouping”) and block.
Click OK

12. How to analyze a blocked design in JMP (Method 2)
Open fit model tab. Enter y-variable.
Add treatment, block and –if desired- treatment x block to “effects”.
Click on block in effects box and change attributes to random.
4. Change Method option to EMS (not REML)

13. Good options for increasing sample size:
More replicates
More blocks
False options for increasing sample size:
More “repeated measurements”
Pseudoreplication

14. Experimental unit
Scale at which independent applications of the same treatment occur
Also called “replicate”, represented by “n” in statistics

15. Experimental unit
Example: Effect of fertilization on caterpillar growth

16. What is our per treatment sample size? What is our treatment n?
Experimental unit ?
+ F
- F
+ F
- F
What is our per treatment sample size? What is our treatment n?
n=2

17. Experimental unit ?
+ F
- F
n=1

18. Pseudoreplication Misidentifying the scale of the experimental unit;
Assuming there are more experimental units (replicates, “n”) than there actually are

19. Why is this a pseudoreplicated design?
+ F
- F

20. Example 1.
Hypothesis: Insect abundance is higher in shallow lakes

21. Example 1.
Experiment:
Sample insect abundance every 100 m along the shoreline of a shallow and a deep lake

22. Example 2.
What’s the problem ?
Spatial autocorrelation

23. Example 2.
Hypothesis: Two species of plants have different growth rates

24. Example 2.
Experiment:
Mark 10 individuals of sp. A and 10 of sp. B in a field.
Follow growth rate
over time
If the researcher declares n=10, could this still be pseudoreplicated?

25. Example 2.

26. Example 2.
time

27. Temporal pseudoreplication:
Multiple measurements on SAME individual, treated as independent data points
time
time

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

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

30. 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?
Yes, say 8
Can the second data point be any number?
Yes, say 12
Can the third data point be any number?
No – as mean is fixed !
Variance is  (y – mean)2 / (n-1)

31. 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)

32. 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? 3
Number of estimates from the data? 1
df= 3-1 = 2

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

34. Example: Analysis of variance (ANOVA)
A B C
a1 b1 c1
a2 b2 c2
a3 b3 c3
a4 b4 c4
What is n for each level?

35. Example: Analysis of variance (ANOVA)
A B C
a1 b1 c1
a2 b2 c2
a3 b3 c3
a4 b4 c4
df = 3 df = 3 df = 3
n = 4
How many df for each variance estimate?

36. Example: Analysis of variance (ANOVA)
A B C
a1 b1 c1
a2 b2 c2
a3 b3 c3
a4 b4 c4
df = 3 df = 3 df = 3
What’s the within-treatment df for an ANOVA?
Within-treatment df = = 9

37. Example: Analysis of variance (ANOVA)
A B C
a1 b1 c1
a2 b2 c2
a3 b3 c3
a4 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)

38. 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?

39. 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

40. 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?

41. 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

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

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