Statistics review 1 Basic concepts: Variability measures Distributions Hypotheses Types of error Common analyses T-tests One-way ANOVA Two-way ANOVA Randomized.

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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

Variance Ecological rule # 1: Everything varies …but how much does it vary?

Variance Sum-of-square cake Urchin size 3cm15cm

Urchin size 3cm15cm Urchin size 3cm15cm Sum-of-square cake

Variance What is the mean and variance of 4, 3, 3, 2 ? What are the units? Mean = 3, Variance = 0.67

Variance variants 1. Standard deviation (s, or SD) = Square root (variance) Advantage: units

Variance variants 2. Standard error (S.E.) Advantage: indicates precision

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 + 1SE or SD - 1SE or SD

Distributions Normal Quantitative data Poisson Count (frequency) data

Normal distribution 67% of data within 1 SD of mean 95% of data within 2 SD of mean

Poisson distribution 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). Preliminary study? A criminal trial? Chance of a disease epidemic?

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 HoAccept 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

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