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A primer on statistics

A primer on statistics. Just what you need to get from data to conclusions…don’t freak out. Summary. Review on Hypotheses T-test example Common terms Correlation example. Hypothesis Generation. Tautological (vacuously true)

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A primer on statistics

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  1. A primer on statistics Just what you need to get from data to conclusions…don’t freak out

  2. Summary • Review on Hypotheses • T-test example • Common terms • Correlation example

  3. Hypothesis Generation • Tautological (vacuously true) • “We hypothesized occupied habitat would differ from habitat selected at random • Descriptive • “Our objectives were to examine and home range sizes of gray partridge in South Dakota • Not-testable • My totem protects me from evil spirits • Null hypothesis* • statistical hypothesis • a priori false • Not-testable • Gause’s principle of Competitive Exclusion • STILL have scientific value • Models- on principles of nature • can be advanced as deductions and used in hypothesis testing • Poincare- assumption, something taken as true, or imagined for argument’s sake • Hutchinson Niche Theory- the niche is an imaginary phenomenon- provides a construct • Research/ Alternative* Non-scientific Scientific

  4. Research Hypothesis • Whether (If)? • existential, the existence of pattern or phenomenon • How? • how do migratory animals navigate? • Why? • ask for purpose reason or cause • winter hare populations declined because of winter food shortages • What is the cause? • A caused B only if A occurred, B occurred and B would not have occurred without A (can’t test) • What’s the cause of the cause and so on…what caused winter food shortages • Could have several causes • Correlation does not imply causation which requires

  5. Question dictates your stats • Are you looking for differences? • Between two groups? • Two areas? • Do a T-test • compares the means of two groups • whether LCR fish surface breath less than UCR fish • The t test compares 1 variable between two groups. • Are you looking for patterns or similarities? • Among variables? • Is there a relationship between them? • Test for correlations • how two variables vary together • Is the density of grazers in a stream reach correlated with the types of algae present • (e.g. flat head mayfly- stenonema)

  6. T-test ExampleResearch Hypothesis: Construction

  7. T-test ExampleThe Null Hypothesis: Strawman (Ho) • Widely applied in scientific research • NOT a true hypothesis • Used for statistical analysis • Set-up to be rejected or accepted • Accepted= Research hypothesis is false • Rejected= Research hypothesis is supported • Null hypothesis* • statistical hypothesis • a priori false

  8. Experimental example • Low oxygen tanks • Measure time till surface breathing begins • Measure # of surface breaths • Sample Size- # Aquariums, # fish in each? • Fish Species- Mix? All the same? • Oxygen curve- Speed high to low? How low? • Control tank • Other considerations? (time, money) UCR Fish LCR Fish

  9. P-value • Are we finished? • Null- LCR fish and UCR fish will exhibit the same surface breathing • Research- LCR fish will exhibit less surface breathing, and begin surface breathing later • Accept or reject the null?

  10. NO! Is the DIFFERENCE significant? • Compare means • 5.28 (LCR) • 1.88 (UCR) • Are they significantly different? • Unpaired t test results • Statistical significance: • P value < 0.0001 • Cut-off < 0.05 • t = 6.565 • Reject the Null of no difference • Supports our hypothesis • Compare means • 4.15 (LCR) • 4.25 (UCR) • Are they significantly different? • Unpaired t test results • Statistical significance: • P value < 0.0771 • Cut-off < 0.05 • t = 1.182 • Accept the Null of no difference • Reject our hypothesis Time to first surface breath http://www.graphpad.com/quickcalcs/ttest1.cfm?Format=C # surface breath sin 20 minutes

  11. Results & Conclusion • Hypothesis- Overtime, the fish community in the LCR has evolved an enhanced ability to survive in oxygen-poor habitat • Prediction 1: LCR fish will exhibit less surface breathing than UCR fish • Incorrect prediction- reject our hypothesis • Write up as: LCR fish did not surface breath less often than UCR fish (p<0.0771). • Prediction 2: LCR fish will begin surface breathing later than UCR fish • Correct prediction • Write up as: LCR fish began surface breathing later than UCR fish (p<0.0001). • Conclusion- Partial support • Why no evidence of fewer surface breaths? • Design Flaw? • Physiological constraint?

  12. Often Misunderstood Terms • P-value • Theoretical probability of obtaining a larger test statistic (more extreme) given the null hypothesis is true, if the experiment were repeated at infinitum • Ignore what is insignificant and study further what is significant • Chance would only produce the significant effect “once in 20 trials” is arbitrary but convenient • Type l error-a true null hypothesis is rejected-finding significance where there is none • probability of a type I error= alpha level • at alpha=0.05, the probability of a type 1 error = 1-0.95=5% • Type II error-false null hypothesis is rejected- failing to find significance where it exists • probability of type II error= beta level • power= the probability of not committing a type ll error  

  13. Correlation ExampleResearch Hypothesis: Construction

  14. Correlation ExampleThe Null Hypothesis: Strawman (Ho) • Widely applied in scientific research • NOT a true hypothesis • Used for statistical analysis • Set-up to be rejected or accepted • Accepted= Research hypothesis is false • Rejected= Research hypothesis is supported • Null hypothesis* • statistical hypothesis • a priori false

  15. P-value • Are we finished? • Null- There will be NO relationship between the density of algae in a stream reach and the density grazers supported • Research- Stream reaches with more algae will support a higher level of grazers • Accept or reject the null?

  16. Pearson Correlation • Is the density of algae type 1 (x1) related to density of grazers (y)? • Is the density of algae type 2 (x2) related to density of grazers (y)? • http://www.wessa.net/corr.wasp

  17. Enter your data Designate your x variable (independent) Designate your y variable (dependent) Leave blank to include all observations Leave standard size

  18. Algae type 1 vs. Grazer density • Correlation coefficient (r) • 0-0.4 (low), 0.4-0.7 (moderate), 0.7-1 (high • P-value< 0.05 • Report as- The grazer density among our reaches was highly correlated with the density of algae type 1 (r=0.9, p<0.001, df=8)

  19. Algae type 2 vs. Grazer density • Report as: The grazer density among our reaches was not related to the density of algae type 2 (r=-0.22, p<0.26, df=8) • CONCLUSION- The density of algae type 1 is related to the density of grazers

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