Spotting pseudoreplication
1. Inspect spatial (temporal) layout of the experiment
2. Examine degrees of freedom in analysis
Degrees of freedom (df)
Number of independent terms used to estimate the parameter
= Total number of datapoints – number of parameters estimated from data
Example: VarianceIf 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)
Example: VarianceIf 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)
Example: VarianceIf 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
Example: Linear regression
Y = mx + b
Therefore 2 parameters estimated simultaneously
(df = n-2)
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?
Example: Analysis of variance (ANOVA)
A B C a1 b1 c1
a2 b2 c2
a3 b3 c3
a4 b4 c4
n = 4
How many df for each variance estimate?
df = 3 df = 3 df = 3
Example: Analysis of variance (ANOVA)
A B C a1 b1 c1
a2 b2 c2
a3 b3 c3
a4 b4 c4
What’s the within-treatment df for an ANOVA?
Within-treatment df = 3 + 3 + 3 = 9
df = 3 df = 3 df = 3
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)
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?
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
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?
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
Why is pseudoreplicationa problem?
Hint: think about what we use df for!
How prevalent?
Hurlbert (1984): 48% of papers
Heffner et al. (1996): 12 to 14% of papers
Statistics review
Basic concepts:
• Variability measures
• Distributions
• Hypotheses
• Types of error
Common analyses
• T-tests
• One-way ANOVA
• Two-way ANOVA
• Randomized block
Variance
Ecological rule # 1: Everything varies
…but how much does it vary?
Variance
S2= Σ (xi – x )2
n-1
x
Sum-of-squarecake
Variance
S2= Σ (xi – x )2
n-1
x
Variance
S2= Σ (xi – x )2
n-1
What is the variance of 4, 3, 3, 2 ?
What are the units?
Variance variants
1. Standard deviation (s, or SD)
= Square root (variance)
Advantage: units
Variance variants
2. Standard error (S.E.)
= s
n
Advantage: indicates precision
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
Distributions
Normal• Quantitative data
Poisson• Count
(frequency) data
Normal distribution
0
2
4
6
8
10
12
14
16
mean
67% of data within 1 SD of mean
95% of data within 2 SD of mean
Poisson distribution
0
2
4
6
8
10
12
14
16
18
mean
Mostly, nothing happens (lots of zeros)
Poisson distribution
• Frequency data
• Lots of zero (or minimum value) data
• Variance increases with the mean
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?
• Null (Ho): no effect of our experimental treatment, “status quo”
• Alternative (Ha): there is an effect
Hypotheses
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?
Hypotheses
Null (Ho) and alternative (Ha):
always mutually exclusive
So if Ha is treatment>control…
Types of error
Type 1 error
Type 2 error
Reject Ho Accept Ho
Ho true
Ho false
• Usually ensure only 5% chance of type 1 error (ie. Alpha =0.05)
• Ability to minimize type 2 error: called power
Types of error