Multivariate Statistics Summary and Comparison of Techniques. Multivariate Techniques

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1 Multivariate Statistics Summary and Comparison of s P The key to multivariate statistics is understanding conceptually the relationship among techniques with regards to: < The kinds of problems each technique is suited for < The objective(s) of each technique < The data structure required for each technique < Sampling considerations for each technique < Underlying mathematical model, or lack thereof, of each technique < Potential for complementary use of techniques 1 Model y1 + y yi y 1 + y y i = x y 1 + y y i = x 1 + x x j s Multivariate ANOVA Multi-Response Permutation Analysis of Similarities Mantel Test Discriminant Analysis Logistic Regression Classification Trees Indicator Species Analysis Canonical Correlation Multivariate Regression Trees 2

2 Objective 3 Extract gradients of maximum variation Establish groups of similar entities Test for & describe differences among groups of entities or predict group membership Extract gradients of variation in dependent variables explainable by independent variables Variance Emphasis Emphasizes variation among individual sampling entities by defining gradients of maximum total sample variance; describes the interentity variance structure. 4

3 Variance Emphasis Emphasizes both differences and similarities among individual sampling entities by clustering entities based on inter-entity resemblance. 5 Variance Emphasis Emphasizes variation among groups of sampling entities; describes the inter-group variance structure. 6

4 Variance Emphasis Emphasizes variation among individual sampling entities by defining gradients of maximum total sample variance explainable by environmental variables 7 Dependence Type Interdependence Interdependence Dependence Dependence 8

5 Data Structure One set; >>2 variables One set; >>2 varibles Two sets; 1 grouping variable, >>2 discriminating variables Two sets; >>2 response variables, >>2 explanatory variables 9 Obs Group X-set Y-set 1 A a 11 a 12 a a 1p b 11 b 12 b b 1m 2 A a 21 a 22 a a 2p b 21 b 22 b b 2m 3 A a 31 a 32 a a 3p b 31 b 32 b b 3m n A a n1 a n2 a n3... a np b n1 b n2 b n3... b nm n+1 C c 11 c 12 c c 1p n+2 C c 21 c 22 c c 2p n+3 C c 31 c 32 c c 3p N C c n1 c n2 c n3... c np (PCA, PCO, CA, DCA, NMDS) 10

6 Obs Group X-set Y-set 1 A a 11 a 12 a a 1p b 11 b 12 b b 1m 2 A a 21 a 22 a a 2p b 21 b 22 b b 2m 3 A a 31 a 32 a a 3p b 31 b 32 b b 3m n A a n1 a n2 a n3... a np b n1 b n2 b n3... b nm n+1 C c 11 c 12 c c 1p n+2 C c 21 c 22 c c 2p n+3 C c 31 c 32 c c 3p N C c n1 c n2 c n3... c np s (MANOVA, MRPP, ANOSIM, Mantel; DA, LR, CART, ISA) 11 Obs Group X-set Y-set 1 A a 11 a 12 a a 1p b 11 b 12 b b 1m 2 A a 21 a 22 a a 2p b 21 b 22 b b 2m 3 A a 31 a 32 a a 3p b 31 b 32 b b 3m n A a n1 a n2 a n3... a np b n1 b n2 b n3... b nm n+1 C c 11 c 12 c c 1p n+2 C c 21 c 22 c c 2p n+3 C c 31 c 32 c c 3p N C c n1 c n2 c n3... c np (RDA, CCA, CAP, COR) 12

7 Sample Characteristics N (from known or unknown # pop's) N (from known or unknown # pop's) N (from known # pop's) or N1, N2,... (from separate pop's) N (from one pop) 13 If the research objective is to: P Describe the major ecological gradients of variation among individual sampling entities, and/or to portray sampling entities along "continuous" gradients of maximum sample variation, then use... P Assume linear relationship to ecological gradients... P Assume unimodal relationship to ecological gradients... P Assume no particular relationship; only monotonic relationship between input and output dissimilarities... Unconstrained Ordination PCA, PCO(MDS) CA(RA), DCA NMDS 14

8 If the research objective is to: P Establish artificial classes or groups of similar entities where pre-specified, welldefined groups do not already exist, and/or to portray sampling entities in "discrete" groups, then use... P Assign entities to a specified number of groups to maximize within-group similarity or form composite clusters... P Assign entities to groups and display relationships among groups as they form Non-hierarchical Hierarchical If the research objective is to: P Establish artificial classes or groups of entities with similar species composition and abundance where pre-specified, well-defined groups do not already exist, based on measured environmental variables, and/or to portray sampling entities in "discrete" groups representing species assemblages with distinct environmental affinities, then use... Constrained (MRT) 16

9 If the research objective is to: P Differentiate among pre-specified, well-defined classes or groups of sampling entities, and to: P Test for significant differences among groups... < Parametric test... < Nonparametric tests... MANOVA / DA MRPP, ANOSIM, Mantel 17 If the research objective is to: P Differentiate among pre-specified, well-defined classes or groups of sampling entities, and to: P Describe the major ecological differences among groups... < Assume a linear discrimination function... < Assume a logistic discrimination function... < Do not assume any particular function... < Identify indicators for each group... DA LR (MLR) CART (UCT) ISA 18

10 If the research objective is to: P Differentiate among pre-specified, well-defined classes or groups of sampling entities, and to: P Predict group membership of future observations... < Linear classification function... < Logistic classification function... < Decision tree classifier... < Other nonparametric classifiers... DA (LDF) LR (MLR) CART (UCT) Kernel K nearest-neighbor 19 If the research objective is to: P Explain the variation in a continuous dependent variable using two or more continuous independent variables, and/or to develop a model for predicting the value of the dependent variable from the values of the independent variables, then use... Alternatives: Multiple Linear Regression CART (URT) 20

11 If the research objective is to: P Explain the variation in a dichotomous dependent (grouping) variable using two or more continuous and/or categorical independent variables, and/or to develop a model for predicting the group membership of a sampling entity from the values of the independent variables, then use... Alternatives: Multiple Logistic Regression DA CART (UCT) 21 If the research objective is to: P Describe the major ecological patterns in one set of (response) variables explainable by another set of (explanatory) variables, then use... P Assume linear response function of response variables (species) along linear gradients defined by the explanatory variables (environment)... P Assume unimodal response function of response variables (species) along linear gradients defined by the explanatory variables (environment)... P Do not assume any response function... Constrained Ordination or MRT RDA, CAP CCA, DCCA MRT 22

12 If the research objective is to: P Describe the major ecological relationships between two sets of variables expressed as distance matrices; i.e., dissimilarities between samples, then use... P Describe the major ecological relationships between two sets of variables expressed as distance matrices after accounting for a third set of variables (i.e., Y~X Z), then use... Mantel Test Partial Mantel Test 23 Dependence s Independent Variables 24

13 Dependence s CT = Contingency tables SLR = Simple logistic regression MLR = Multiple logistic regression SRA = Simple linear regression MRA = Multiple linear regression T-test = T-test ANOVA = Analysis of variance UCT = Univar. classification trees URT = Univar. regression trees T 2 -test MANOVA = Multivariate analysis of variance DA = Discriminant analysis ISA RDA = Redundancy analysis CCA = Can. correspond. analysis COR = Canonical corr. analysis MRT = Hotelling s T 2 = Indicator species analysis CAP = Can. prin. coord. analysis = Multivar. regression trees 25 Dependence s Independent Variables CT CT CT SLR SLR MLR UCT CT CT CT UCT DA DA SLR MLR UCT UCT DA DA CT DA CT DA CT RDA MRT CAP MRT COR CCA COR SRA SRA MRA SRA MRA T-test ANOVA ANOVA URT URT T 2 -test Manova Manova RDA DA DA MRT CAP MRT ISA ISA COR CCA COR RDA CAP CCA RDA CAP CCA 26

14 Advantages of Multivariate Statistics P Reflect more accurately the true multidimensional, multivariate nature of natural systems. P Provide a way to handle large data sets with large numbers of variables. P Provide a way of summarizing redundancy in large data sets. P Provide rules for combining variables in an "optimal" way. 27 Advantages of Multivariate Statistics P Provide a solution to the multiple comparison problem by controlling experimentwise error rate. P Provide a means of detecting and quantifying truly multivariate patterns that arise out of the correlational structure of the variable set. P Provide a means of exploring complex data sets for patterns and relationships from which hypotheses can be generated and subsequently tested experimentally. 28