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Statistics review 1

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?. 3cm. 15cm. Urchin size.

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Statistics review 1

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

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

  3. 3cm 15cm Urchin size Sum-of-square cake Variance

  4. Sum-of-square cake 3cm 15cm Urchin size 3cm 15cm Urchin size

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

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

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

  8. + 1SE or SD - 1SE or SD 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

  9. Distributions Normal • Quantitative data Poisson • Count (frequency) data

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

  11. Poisson distribution mean Mostly, nothing happens (lots of zeros)

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

  13. What do you do with Poisson data? • Correct for correlation between mean and variance by log-transforming y (but log (0) is undefined!!) • Use non-parametric statistics (but low power) • Use a “generalized linear model” specifying a Poisson distribution

  14. Hypotheses • Null (Ho): no effect of our experimental treatment, “status quo” • Alternative (Ha): there is an effect

  15. 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?

  16. Hypotheses Null (Ho) and alternative (Ha): always mutually exclusive So if Ha is treatment>control…

  17. Types of error Reject Ho Accept Ho Ho true Ho false

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

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