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Lecture 2: Replication and pseudoreplication

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Lecture 2:Replication and pseudoreplication

- Experimental units (replicates)
- Pseudoreplication
- Degrees of freedom

Scale at which independent applications of the same treatment occur

Also called “replicate”, represented by “n” in statistics

Example: Effect of fertilization on caterpillar growth

+ F

+ F

- F

- F

n=2

+ F

- F

n=1

Misidentifying the scale of the experimental unit;

Assuming there are more experimental units (replicates, “n”) than there actually are

+ F

- F

Example 1.

Hypothesis: Insect abundance is higher in shallow lakes

Example 1.

Experiment:

Sample insect abundance every 100 m along the shoreline of a shallow and a deep lake

Example 2.

What’s the problem ?

Spatial autocorrelation

Example 2.

Hypothesis: Two species of plants have different growth rates

- 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?

Example 2.

Example 2.

time

Temporal pseudoreplication:

Multiple measurements on SAME individual, treated as independent data points

time

time

- Inspect spatial (temporal) layout of the experiment
- Examine degrees of freedom in analysis

Number of independent terms used to estimate the parameter

= Total number of datapoints –

number of parameters estimated from data

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)

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)

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

Example: Linear regression

Y = mx + b

Therefore 2 parameters estimated simultaneously

(df = n-2)

Example: Analysis of variance (ANOVA)

ABC

a1 b1 c1

a2 b2 c2

a3 b3 c3

a4 b4 c4

What is n for each level?

Example: Analysis of variance (ANOVA)

ABC

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?

Example: Analysis of variance (ANOVA)

ABC

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 = 3 + 3 + 3 = 9

Example: Analysis of variance (ANOVA)

ABC

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)

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?

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

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?

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

Hint: think about what we use df for!

Hurlbert (1984): 48% of papers

Heffner et al. (1996): 12 to 14% of papers