Statistics review 1

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Statistics review 1

• Basic concepts
• Variability measures
• Distributions
• Hypotheses
• Types of error

• Common analyses
• T-tests
• One-way ANOVA
• Two-way ANOVA
• Randomized block

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Variance

• Ecological rule 1 Everything varies
• but how much does it vary?

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Variance
4
3cm
15cm
Urchin size
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Variance
What is the mean and variance of 4, 3, 3, 2 ?
Mean 3, Variance 0.67
What are the units?
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Variance variants

• 1. Standard deviation (s, or SD)
• Square root (variance)
• Advantage units

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Variance variants

• 2. Standard error (S.E.)
• Advantage indicates precision

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How to report
Tourist boats observed 29.7 ( 5.3) shark attacks
on seals (mean S.E.) A mean ( SD) of 29.7 (
7.4) shark attacks were seen per month
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Distributions

• Normal
• Quantitative data
• Poisson
• Count (frequency) data

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Normal distribution
67 of data within 1 SD of mean
95 of data within 2 SD of mean
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Poisson distribution
mean
Mostly, nothing happens (lots of zeros)
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Poisson distribution

• Frequency data
• Lots of zero (or minimum value) data
• Variance increases with the mean

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What do you do with Poisson data?

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

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Hypotheses

• Null (Ho) no effect of our experimental
  treatment, status quo
• Alternative (Ha) there is an effect

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Whose null hypothesis?

• Conditions very strict for rejecting Ho, whereas
  accepting Ho is easy (just a matter of not
  finding grounds to reject it).
• Preliminary study?
• A criminal trial?
• Chance of a disease epidemic?

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Hypotheses

• Null (Ho) and alternative (Ha)
• always mutually exclusive
• So if Ha is treatmentgtcontrol

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Types of error
Reject Ho Accept Ho
Type 1 error
Type 2 error
Ho true
Ho false
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Types of error

• Usually ensure only 5 chance of type 1 error
  (ie. Alpha 0.05)
• Ability to minimize type 2 error called power