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