Step three: statistical analyses to test biological hypotheses

1. Step three: statistical analyses to test biological hypotheses General protocol continued

2. Biological hypotheses and statistical tests • Hypotheses driven by Biology • Statistics depend on data and hypotheses • NO NEW STATISTICAL TOOLS ARE NEEDED FOR MORPHOMETRICS!! • Explanatory hypotheses: relative position of specimens in data space:relationship among specimens in data space • Confirmatory hypotheses: compare groups, associate shape with other variables, etc.

3. Some hypotheses (shape related) • How do populations and species differ? • Does the observed variation generate a predictable pattern? • Are there additional factors (ecological, evolutionary) correlated with variation? • How does shared evolutionary history affect the observed patterns?

4. Do populations differ? Is there a predictable pattern? Correlated factors? Effect of phylogeny? MANOVA, CVA PCA, UPGMA Regression, 2B-PLS Comparative Method Hypotheses as statistical tests

5. Exploratory data analysis • Investigate data using only Y-matrix of shape variables (PWScores + U1,U2) • Specimens are points in high-dimensional data space • Look for patterns and distributions of points • Generate summary plot of data space (ordination) • Look for relationships of points (clustering)

6. Ordination and dimension reduction • Visualize high dimensional data space as succinctly as possible • Describe variation in original data with new set of variables (typically orthogonal vectors) • Order new variables by variation explained (most – least) • Plot first few dimensions to summarize data • Principal Components Analysis (PCA) one approach (others include: PCoA, MDS, CA, etc.)

7. PCA: what does it do? • Rotates data so that main axis of variation (PC1) is horizontal • Subsequent PC axes are orthogonal to PC1, and are ordered to explain sequentially less variation • The goal is to explain more variation in fewer dimensions

8. PCA: interpretations • Eigenvectors are linear combinations of original variables (interpreted by PC loadings of each variable) • PCA PRESERVES EUCLIDEAN DISTANCES among objects • PCA does NOTHING to the data, except rotate it to axes expressing the most variation; it loses NO INFORMATION (if all PC vectors retained) • If the original variables are uncorrelated, PCA not helpful in reducing dimensionality of data • PCA does not find a particular factor (e.g., group differences, allometry): it identifies the direction of most variation, which may be interpretable as a ‘factor’ (but may not)

9. Example: leatherside chub

10. Clustering • Data are dots in a high-dimensional space (Y-matrix) • Can we connect to dots for groupings, where clusters represent groups of similar specimens? • Cluster methods generate ‘1-dimensional view’ of relationships, based on some criterion • Clustering requires distance (or similarity) between points • MANY different criteria • Clustering is algorithmic, not algebraic (i.e., it is a procedure, or set of rules for connecting data)

11. Clustering: UPGMA

12. Conclusions: exploratory methods • Useful tools for summarizing shape variation • Help you understand your data through visualizing variation (both ordination plots and cluster diagrams) • Help describe relationships among specimens in terms of overall similarity

13. Confirmatory data analysis • Investigate data using shape variables (Y-matrix) and other (independent) variables (X-matrix) • Test for patterns of shape variation • Independent variables determine type of statistical test

14. Types of independent variables • Categorical: variables delineating groups of specimens (e.g., male/female, species, etc.) • Continuous: variables on a continuous scale (e.g., size, moisture, age, etc.) • Different statistical methods for each

15. Categorical: shape differences among groups Continuous: relationship of variables and shape Continuous: association of variables and shape MANOVA Mult. Regression 2B-PLS (2-Block Partial Least squares) Some statistical tests MANOVA and multivariate regression are both GLM statistics (General Linear Models)

16. Group differences: MANOVA • Is there a difference in shape between groups? • Multivariate generalization of ANOVA • Compares variation within groups to variation between groups • Significant MANOVA: Group means are different in shape

17. MANOVA • RW1-RW30 Utah chub • Source Sex Loc Sex X loc IL/SL Size Wilks' Lambda 0.38888016 4.66 30 89 <.0001 Wilks' Lambda 0.00308619 3.26 240 706.35 <.0001 Wilks' Lambda 0.10138762 1.40 180 533.33 0.0020 Wilks' Lambda 0.61907356 1.83 30 89 0.0159 Wilks' Lambda 0.75516916 0.96 30 89 0.5318

18. MANOVA: post hoc tests • Pairwise comparisons using Generalized Mahalanobis Distance (D2 or D) • Convert D2→T2 → F to test • For experiment-wise error rate, adjust using Bonferroni: α exp = α / # comparisons

19. Discriminant analysis: CVA & DFA • ‘Combination’ of MANOVA and PCA • Tests for group differences (MANOVA) • PCA of among-group variation relative to within-group variation • Suggests which groups differ on which variables • Can ‘classify’ specimens to groups • Special case: 2 groups= discriminant function analysis (DFA)

20. DFA/CVA: post-hoc tests • For DFA/CVA, compare difference among groups using Generalized Mahalanobis Distance (D2) • Mahalanobis D2 is logical choice because CVA/DFA is MANOVA, and the PCA is relative to within-group variability (i.e., VCV ‘standardized’) • Convert D2→T2 → F to perform statistical test • Experiment-wise error rate adjusted as before (i.e., adjusted α)

21. Continuous variation: regression • Is there a relationship between shape and some other variable? • Multivariate regression of shape on continuous variable • Significant regression implies shape changes as a function of other variable (e.g., size)

22. Example of shape on size in mountain sucker Multivariate tests of significance: Statistic Value Fs df1 df2 Prob Wilks' Lambda: 0.34356565 22.822 36 430.0 3.580E-078 Pillai's trace: 0.65643435 22.822 36 430.0 3.580E-078 Hotelling-Lawley trace: 1.91065190 22.822 36 430.0 3.580E-078 Roy's maximum root: 1.91065190 22.822 36 430.0 3.580E-078 Test that kth root and those that follow are zero: k U Fs df1 df2 Prob 1 0.34356565 22.822 36 430.0 3.580E-078

23. Continuous variation: association 2B-PLS • Is there an association between shape and some other set of variables (not causal)? • Find pairs of linear combinations for X & Y that maximize the covariation between data sets • Linear combinations are constrained to be orthogonal within each set (like PC axes) but NOT between data sets • Calculations less complicated for 2B-PLS (because fewer mathematical constraints) • Analogous to ‘multivariate correlation’ • 2B-PLS is called SINGULAR WARPS when shape is one or more of the data sets. Bookstein et al., 2003: J. of Hum. Evol.)

24. Resampling methods • Methods that take many samples from original data set in some specified way and evaluate the significance of the original based on these samples • Resampling approaches are nonparametric, because they do not depend of theoretical distributions for significance testing (they generate a distribution from the data) • Are very flexible, and can allow for complicated designs • Very useful in morphometrics, and can be used for: • Testing standard designs • Testing non-standard designs • Testing when sample sizes small relative to # of variables

25. Randomization (permutation) • Proposed by Fisher (1935) for assessing significance of 2-sample comparison (Fisher’s exact test) • Fisher’s exact test: a total enumeration of possible pairings of data • Randomization can be used to determine most any test statistic • Protocol • Calculate observed statistic (e.g., T-statistic): Eobs • Reorder data set (i.e. randomly shuffle data) and recalculate statistic Erand • Repeat many times to generate distribution of statistic • Percentage of Erand more extreme than Eobs is significance level

26. Randomization: comments • Randomization EXTREMELY useful and flexible technique • How and what to resample depends upon data and hypothesis • Regression and correlation: shuffle Y vs. X • Group comparison (e.g., ANOVA): shuffle Y on groups • Some tests (e.g., t-test) may depend on direction (1-tailed vs. 2-tailed) • Also useful when no theoretical distribution exists for statistic, or when design is ‘non-standard’ • This is frequently the case in E&E studies

27. Step four: Graphical depiction of results • Strength of landmark-based TPS approach • Can view deformation of TPS grid among groups or with continuous variable

28. Superimposition

29. Effect of relative intestinal length: measure of trophic level Long IL/SL 3.0 Short IL/SL 0.72

30. Effect of gradient on shape in mountain sucker Low High