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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 s2 2 (x x ) i n 1 x 3cm Sum-of-square cake Urchin size 15cm x 3cm Urchin size 15cm Sum-of-square cake 3cm x Urchin size 15cm Variance s 2 (x i x) 2 n 1 What is the mean and variance of 4, 3, 3, 2 ? Mean = 3, Variance = 0.67 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 s.e. n 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 16 67% of data within 1 SD of mean 14 12 10 8 6 4 2 0 mean 95% of data within 2 SD of mean Poisson distribution 18 16 14 12 10 8 6 4 2 0 mean Mostly, nothing happens (lots of zeros) Poisson distribution • Frequency data • Lots of zero (or minimum value) data • Variance increases with the mean 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 Hypotheses • Null (Ho): no effect of our experimental treatment, “status quo” • Alternative (Ha): there is an effect 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 Reject Ho Ho true Ho false Accept Ho Type 1 error Type 2 error Types of error • Usually ensure only 5% chance of type 1 error (ie. Alpha =0.05) • Ability to minimize type 2 error: called power