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Review of yesterday

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  1. Review of yesterday

  2. Consider • Don’t sum the cars • Don’t reject the actual hypothesis H1(when p<0.05 you reject the Null hypothesis H0) • Use simple variable names not Y/N or Å. • Don’t use function names as variable names, e.g., • length • mean • t

  3. 16 14 12 10 8 6 4 Red ants Black ants Logistic regression 2  2 tables Categoric 1.0 Melica 0.8 0.6 Prob. of choosing Melica 0.4 0.2 0.0 Response variable Luzula 4.5 5.5 6.5 7.5 Ant size Regression Anova Continuous - - Seed size Continuous Categoric Explanatory variable

  4. You now can test a lot! • Most cases with: • 1 response • 1 explanatory • Today: • Summarize • Add some more technical issues • Tomorrow: • 2 explanatory variables

  5. Binomial test

  6. Binomial test binom.test(18,20) Exact binomial test data: 18 and 20 number of successes = 18, number of trials = 20, p-value = 0.0004025 alternative hypothesis: true probability of success is not equal to 0.5 95 percent confidence interval: 0.6830173 0.9876515 sample estimates: probability of success 0.9

  7. 2x2 Fisher Test

  8. 2x2 Fisher Test fisher.test(naildata) Fisher's Exact Test for Count Data data: naildata p-value = 0.5006

  9. Logistic regression

  10. Logistic regression mod.glm<-glm(Y.N~Length,binomial) anova(mod.glm,test=”Chi”) Df Deviance Resid. Df Resid. Dev P(>|Chi|) NULL 46 64.109 Length 1 1.855 45 62.254 0.173

  11. Regression Ash seeds

  12. Regression mod.reg<-lm(falltime~seedlength) summary(mod.reg) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.55923 0.65656 2.375 0.024 * seedlength 0.02933 0.01905 1.540 0.13

  13. Critical values

  14. Risk of by chance only > 5 % 95% 70 60 50 40 Normal probability 30 2,5% 2,5% 20 10 0 Difference

  15. Risk by chance = 6.5 +6.5 = 13 % 70 60 50 40 Normal probability 30 6.5% 6.5% 20 10 0 Difference

  16. Regression mod.reg<-lm(falltime~seedlength) summary(mod.reg) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.55923 0.65656 2.375 0.024 * seedlength 0.02933 0.01905 1.540 0.13

  17. Anova

  18. Anova mod.aov<-aov(twiglength~treeside) summary(mod.aov) Df Sum Sq Mean Sq F value Pr(>F) treeside 1 15.129 15.129 7.5222 0.009244 ** Residuals 38 76.427 2.011

  19. Anova

  20. Anova

  21. Confidence intervals • …shows how sure we are of a group mean. • The confidence interval will contain the ”true” mean in 95 % of the time. • The larger our sample size the more sure (= confident!) we are of our sample mean  the confidence interval decreases • And (of course…), the more variation within groups, the less sure we get  confidence interval increases

  22. Is there any difference? Seed size in plants

  23. p = 0,047

  24. Seed size

  25. p = 0,0047

  26. Flower size in Geranium

  27. p = 0,0037

  28. Flower size in Geranium

  29. p = 0,094

  30. Orchid seeds

  31. p = 0, 40

  32. Bat penises

  33. p < 0,0001

  34. bootstrap • to pull oneself up by one's bootstraps(idiomatic) • To begin an enterprise or recover from a setback without any outside help; to succeed only on one's own effort or abilities.We can't get a loan, so we'll just have to pull ourselves up by our bootstraps.

  35. shrimp-booting...

  36. Bootstrap for tests 120 80 No. boot-samples 60 40 20 0 -5 0 5 10 15 20 25 boot.difference

  37. Break?

  38. Anova mod.aov<-aov(twiglength~treeside) summary(mod.aov) Df Sum Sq Mean Sq F value Pr(>F) treeside 1 15.129 15.129 7.5222 0.009244 ** Residuals 38 76.427 2.011

  39. How does this F-test work? • The idea behind the F-test is to check whether the variation caused by the explanatory variable is larger than the random variation in each data point. • In this case the variation caused by the explanatory variable is the variation among groups. • The random variation is simply the variation that is NOT explained by the explanatory variable.

  40. How does this F-test work? • Under the hood. • A frog example. female male

  41. We measure 4 + 4 frogs 10 15 12 14 8 11 9 7

  42. Frog stripchart

  43. Frog stripchart

  44. Group means

  45. Explained by sex

  46. Random variation = residuals

  47. Is a significant amount explained by sex?

  48. The F-tests uses Sums of squares Sums of squares explained by sex= (group mean – total mean)2 for all points= the squares of the blue lines male - total = -2; female - total = 2 22 + 22 + 22 + 22 + 22 + 22 + 22 + 22 = 32

  49. The F-tests uses Sums of squares Sums of squares for resisduals= (y value - group mean)2 for all points= the squares of the red lines -0.752 + 2.252 + 0.252 + -1.752 -2.752 + 2.252 + -0.752 + 1.252 = 23.5

  50. Get the ”avarage” variation explained by sex and residuals Divide by degress of freedom Sex  # groups -1 [  32/1 = 32] Resdiduals  # frogs – sex df -1[  23.5 / 6 =3.9 ] The F-ratio is then 32/3.9 = 8.2