1. Lecture 2: Replication and pseudoreplication

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

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

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

5. Experimental unit ?
+ F
+ F
- F
- F
n=2

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

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

8. When is this a pseudoreplicated design?
+ F
- F

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

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

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

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

13. 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?

14. Example 2.

15. Example 2.
time

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

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

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

19. 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)

20. 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)

21. 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

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

23. 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?

24. 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?

25. 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

26. 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)

27. 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?

28. 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

29. 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?

30. 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

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

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