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Homework

Learn about assumptions in statistics, how to detect deviations from normality, and what to do when these assumptions are violated. Explore methods for comparing variances and means in different groups.

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Homework

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  1. Homework • Chapter 11: 13 • Chapter 12: 1, 2, 14, 16

  2. Assumptions in Statistics

  3. The wrong way to make a comparison of two groups “Group 1 is significantly different from a constant, but Group 2 is not. Therefore Group 1 and Group 2 are different from each other.”

  4. A more extreme case...

  5. Interpreting Confidence Intervals Not different Unknown Different

  6. Assumptions • Random sample(s) • Populations are normally distributed • Populations have equal variances • 2-sample t only

  7. Assumptions • Random sample(s) • Populations are normally distributed • Populations have equal variances • 2-sample t only What do we do when these are violated?

  8. Assumptions in Statistics • Are any of the assumptions violated? • If so, what are we going to do about it?

  9. Detecting deviations from normality • Histograms • Quantile plots • Shapiro-Wilk test

  10. Detecting deviations from normality: by histogram Frequency Biomass ratio

  11. Detecting deviations from normality: by quantile plot Normal Quantile Normal data

  12. Detecting deviations from normality: by quantile plot Normal Quantile Biomass ratio

  13. Detecting deviations from normality: by quantile plot Normal Quantile Normal distribution = straight line Non-normal = non-straight line Biomass ratio

  14. Detecting differences from normality: Shapiro-Wilk test Shapiro-Wilk Test is used to test statistically whether a set of data comes from a normal distribition Ho: The data come from a normal distribution Ha: The data come from some other distribution

  15. What to do when the distribution is not normal • If the sample sizes are large, sometimes the standard tests work OK anyway • Transformations • Non-parametric tests • Randomization and resampling

  16. The normal approximation • Means of large samples are normally distributed • So, the parametric tests on large samples work relatively well, even for non-normal data. • Rule of thumb- if n > ~50, the normal approximations may work

  17. Data transformations A data transformation changes each data point by some simple mathematical formula

  18. Log-transformation ln = “natural log”, base e log = “log”, base 10 EITHER WORK Y Y' = ln[Y]

  19. Carry out the test on the transformed data!

  20. Variance and mean increase together --> try the log-transform

  21. Other transformations Arcsine Square-root Square Reciprocal Antilog

  22. Example: Confidence interval with log-transformed data Data: 5 12 1024 12398 ln data: 1.61 2.48 6.93 9.43 ln[Y] ln[Y] ln[Y]

  23. Valid transformations... • Require the same transformation be applied to each individual • Must be backwards convertible to the original value, without ambiguity • Have one-to-one correspondence to original values X = ln[Y] Y = eX

  24. Choosing transformations • Must transform each individual in the same way • You CAN try different transformations until you find one that makes the data fit the assumptions • You CANNOT keep trying transformations until P <0.05!!!

  25. Assumptions • Random sample(s) • Populations are normally distributed • Populations have equal variances • 2-sample t only Do the populations have equal variances? If so, what should We do about it?

  26. Comparing the variance of two groups One possible method: the F test

  27. The test statistic F Put the larger s2 on top in the numerator.

  28. F... • F has two different degrees of freedom, one for the numerator and one for the denominator. (Both are df = ni -1.) The numerator df is listed first, then the denominator df. • The F test is very sensitive to its assumption that both distributions are normal.

  29. Example: Variation in insect genitalia

  30. Example: Variation in insect genitalia

  31. Degrees of freedom For a 2-tailed test, we compare to Fa/2,df1,df2 from Table A3.4

  32. Why a/2 for the critical value? By putting the larger s2 in the numerator, we are forcing F to be greater than 1. By the null hypothesis there is a 50:50 chance of either s2 being greater, so we want the higher tail to include just a/2.

  33. Critical value for F

  34. Conclusion The F= 107.4 from the data is greater than F(0.025), 6,8 =4.7, so we can reject the null hypothesis that the variances of the two groups are equal. The variance in insect genitalia is much greater for polygamous species than monogamous species.

  35. F-test Null hypothesis The two populations have the same variance 21 22 Sample Null distribution F with n1-1, n2-1 df Test statistic compare How unusual is this test statistic? P > 0.05 P < 0.05 Reject Ho Fail to reject Ho

  36. What if we have unequal variances? • Welch’s t-test would work • If sample sizes are equal and large, then even a ten-fold difference in variance is approximately OK

  37. Comparing means when variances are not equal Welch’s t test compared the means of two normally distributed populations that have unequal variances

  38. Burrowing owls and dung traps

  39. Dung beetles

  40. Experimental design • 20 randomly chosen burrowing owl nests • Randomly divided into two groups of 10 nests • One group was given extra dung; the other not • Measured the number of dung beetles on the owls’ diets

  41. Number of beetles caught • Dung added: • No dung added:

  42. Hypotheses H0: Owls catch the same number of dung beetles with or without extra dung (m1 = m2) HA: Owls do not catch the same number of dung beetles with or without extra dung (m1m2)

  43. Welch’s t Round down df to nearest integer

  44. Owls and dung beetles

  45. Degrees of freedom Which we round down to df= 10

  46. Reaching a conclusion t0.05(2), 10= 2.23 t=4.01 > 2.23 So we can reject the null hypothesis with P<0.05. Extra dung near burrowing owl nests increases the number of dung beetles eaten.

  47. Assumptions • Random sample(s) • Populations are normally distributed • Populations have equal variances • 2-sample t only What if you don’t want to make so many assumptions?

  48. Welch’s t-test Null hypothesis The two populations have the same mean 12 Sample Null distribution t with df from formula Test statistic compare How unusual is this test statistic? P > 0.05 P < 0.05 Reject Ho Fail to reject Ho

  49. Non-parametric methods • Assume less about the underlying distributions • Also called "distribution-free" • "Parametric" methods assume a distribution or a parameter

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