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  1. Overview of Statistical Tests Available BIOL457/657 5 February 2014

  2. Deciding Which Test to Use • Is DV meristic, mensural, or categorical? • Is IV continuous or discrete in its variation? • If IV is discrete, are two groups being compared, or more than two? • Are there multiple IVs?

  3. Textbook Rear Inside Cover

  4. No-difference H0 or no-correlation H0?Is DV meristic/mensural or categorical?

  5. II. Normal or non-normal data?

  6. III. Sample data compared to what?

  7. IV. One comparison or more than one?

  8. V. Paired or unpaired data?

  9. VI. Testing for cause-and-effect [XY]?

  10. VII.Eureka

  11. Comparing Two Sets of DatawithMeristic or Mensural Data

  12. Student’s t-Test • Parametric test • Compare two independent samples • One version for equal variances • Less powerful version for unequal variances • H0: There is no difference in [DV] between two groups or treatments [IV]

  13. HANDOUTIverson, 2002Razorback Musk Turtles of OK and AR

  14. Paired t-test • Parametric • Data paired rather than drawn from independent populations • Assumes equal variances

  15. HANDOUTLong et al., 1998Hemlocks on Tip-up Mounds

  16. Mann-Whitney U-test • aka Rank-Sum Test • Nonparametric comparison of two samples of meristic or mensural DV

  17. HANDOUTGodfrey and Bryant, 2000European Robin Energy Expenditure

  18. Comparing Two (or More) Sets of DatawithCategorical Data

  19. z-test • aka Binomial Test of Proportions • Two samples • DV occurs in two categories for each sample • H0: There is no difference in binomial outcomes [DV] between two samples [IV]

  20. HANDOUTHoikkala and Aspi, 1993Drosophila Mating

  21. Chi-square (χ2) Tests • χ2 Goodness-of-Fit Test • Sample distribution vs. hypothesized/theoretical distribution • χ2 Test for Independence • Sample distribution vs. sample distribution • Mantel-Haenszelχ2 Test for Independence • Compares distributions of samples with two or more IVs that may influence DV distribution

  22. HANDOUTSpinks et al., 2000Mole Rat Sex Ratios

  23. HANDOUTHarrold, 1982Snails and Starfish Predation

  24. HANDOUTCarney et al., 1996Iris Seeds

  25. Comparing Three or More Sets of DatawithMeristic or Mensural Data

  26. One-way Analysis of Variance (ANOVA) • Parametric • H0: There is no difference in [DV] among three or more groups or treatments [IV] • Rejection of H0 requires post hoc testing to identify which samples differ significantly from which other samples • Fisher’s Least Significant Difference (LSD) test • Tukey’s test • Student-Newman-Keuls test • Bonferroni t-tests • Duncan’s Multiple Range test • Sheffé comparisons

  27. HANDOUTGodfrey and Bryant, 2000European Robin Energy Expenditure Note: For both tests, post hoc Scheffé comparisons show CHILLED was significantly greater than other two treatments, which did not differ from one another.

  28. Kruskal-Wallis Test • Nonparametric equivalent of ANOVA • Rejection of H0 requires nonparametric analog of post hoc testing • Dunn’s Multiple Comparisons test

  29. HANDOUTZaviezo and Mills, 2000Parasitoid Clutch Size

  30. Two-Way (or More) ANOVA • Two IVs analyzed simultaneously • Multiple H0s: • No difference in [DV] between/among groups as established by IV1 • No difference in [DV] between/among groups as established by IV2 • No difference in [DV] attributable to the interaction of IV1 and IV2 • More complex models for three or more IVs

  31. HANDOUTParadise, 2000Artificial Treeholes

  32. Examining the Correlation of Variables with Continuous Variation

  33. Correlation Analysis • H0: There is no correlation of DV with IV • Positive correlation: Direct relationship • Negative correlation: Indirect relationship • No correlation: Variation in DV not related to variation in IV

  34. HANDOUTEllers et al., 1998Parasitoid Body Size and Fitness

  35. Simple Linear Regression • If they are correlated, fit DV and IV data to a best-fitting line y = ax + b DV = a(IV) + b a = slope b = y-intercept

  36. HANDOUTBeukema et al., 2000Polychaete Worm Biomass

  37. Multiple Linear Regression • >1 continuous IV • Multidimensional best-fitting line: DV = a(IV1) + b(IV2) + c(IV3) … + z (ac are slopes in separate dimensions; z is intercept) • With each added IV, partial correlation coefficient assesses whether it significantly improves fit to DV • H0: There is no correlation of DV with IVx, after correction for correlation of DV with other IVs in model

  38. HANDOUTLong et al., 2005Arctic Ground Squirrel Activity

  39. Analysis of Covariance • Two IVs: IVc continuous, IVd discrete • Typically, IVc regarded as a confounding variable, known/suspected to be correlated with DV • H0: No difference in DV between/among treatments or groups (IVd), after correction for correlation of DV with IVc.

  40. HANDOUTCrespi, 1988Thrip Fitness

  41. The General Linear Model (GLM) • Mix of multiple linear regression (continuous IVs) and ANCOVA (allows for discrete IVs) • Number of IVs limited only by sample size • H0 for each IV states that its variation is not significantly related to variation in DV (given relationship of other IVs to DV) • Interaction of IVs can also be examined in additional H0s

  42. Logistic Regression • IV is continuous, but DV is discrete and categorical • H0: Probability of categorization of DV is not correlated with IV

  43. HANDOUTViñuela, 1997Black Kite Nesting

  44. Multivariate Analyses • Variety of techniques that take large data sets with numerous variables and… • …allow relatively simple graphical representations • …identify interrelationships of variables

  45. Phylogenetic Comparative Methods • Used for analyzing data from related species • Necessary because phylogenetic relationships cause violation of the assumption of independence of data points • Primarily used for correlation analysis, but can be adapted for use in other types of analysis

  46. Monte Carlo Randomizations • Uses 100s to 1000s of randomizations of the sample data to create a null distribution for the H0 • Parametric assumption thus not necessary • Used to create highly specific H0s for particular scenarios one may wish to analyze