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Exploratory Data Analysis and Multivariate Strategies Andrew Mead (School of Life Sciences)

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1 Exploratory Data Analysis and Multivariate Strategies Andrew Mead (School of Life Sciences)

2 Multi-… approaches in statistics  Multiple comparison tests  Multiple testing adjustments  Methods for adjusting the significance levels when doing a large number of tests (comparisons between treatments) within a single analyses  Multiple regression analysis  Statistical model building with more than one explanatory variable  Multi-factor analysis of variance  Analysis of designed experiments with more than one explanatory factor  Multivariate Analysis  Methods to summarise and explore the relationships among multiple response variables, and/or to assess differences among “treatments” based on multiple response variables 2

3 Multivariate data  Responses by a number of individuals to a range of different (related) questions in a (social studies) survey  Counts of different species in a range of locations in an ecological survey  Measurements of a range of traits of individual people, animals, plants, products, …  Different medical measurements made on a group of patients  Measurements of gene expression / protein expression, metabolite expression in biological samples  Counts of sequence consensus matches from microbial samples  Data on the “distances” between a number of objects 3

4 Multivariate questions  Identifying groups of objects with similar (and different) responses  e.g. UK landscape areas with similar percentage composition of different land covers (woodland, arable, urban, …)?  Identifying the particular measurement that contribute to the variability among a set of objects  e.g. weed species that are present under different long-term herbicide strategies?  Identifying the particular measurements that contribute to differences between groups of objects  e.g. which genes discriminate between patients with and without some form of cancer?  Identifying the particular measurements that explain variation in some over-arching response variable  e.g. which traits of predatory insects influence the predation rate of aphids? 4

5 Exploratory data analysis  Summary statistics / graphical summaries  Variability for each variable/measurement  Groups of observations  both pre-determined – to find potential differences  and to be identified – based on each individual variable  Scatter plots / correlations  Associations between pairs of variables 5

6 Univariate analysis  For each individual variable:  Hypothesis tests  Choice depends on the question to be answered  Analysis of variance  For variables measured in designed experiments  Regression analysis  To build statistical models to describe how one response variable depends on one (or more) explanatory variables  Generalised Linear Models (GLMs)  For data where standard assumptions do not hold!  Time Series Analysis  … 6

7 Multivariate analysis  For a set of “correlated” variables:  Assess relationships between variables  Consider the effects of “treatments” on these relationships  Consider how a “response” depends on these relationships  Multivariate methods concerned with “data reduction”  Summarise the correlations between variables  Produce a smaller set of (uncorrelated) variables containing the important information  For a set of “related” objects  Identify groups of similar objects  Identify differences between groups of similar objects  And what makes the objects similar! 7

8 Simple graphical summaries 1  For compositional data  e.g. numbers of onion bulbs in different marketable size grades  Present as a stacked bar- chart  For raw data  For percentage of the total 8

9 Simple graphical summaries 2  More general data  e.g. different measurements on a set of plants  Scatter plots for each pair of variables  Present in a matrix  Calculate linear correlation coefficient for each pair of variables 9

10 Two forms of data matrix  The DATA matrix  p variables for each of n samples (observations)  Presented in a rectangular matrix  n rows and p columns  The ASSOCIATION matrix  Distance, similarity or dissimilarity  Between every pair of variables or every pair of samples  Symmetric square matrix  n-by-n - between samples  p-by-p - between variables  just show lower triangle  Turns multivariate data into univariate data? p variables n samples p variables Lower Triangle 10

11 Analysing Association Data 11  Start with associations  Distances between locations on a map  Psychometric (sensory) similarities between products  Construct associations from data  Depends on the types of data  Binary (presence/absence) data  Simple matching coefficient; Jaccard coefficient; …  Continuous data  Euclidean distance; Manhattan distance; …  Similarities or Disimilarities/Distances

12 Finding groups of similar objects 12  Hierarchical Cluster Analysis  Aim: to arrange the objects into homogenous groups  Output:  Dendrogram showing how objects are joined together  Levels of similarity/distance at which groups are formed or divided  Primarily a descriptive technique  Interpretation includes identification of “how many groups?”  Agglomerative methods  Start with individual objects, group the two most similar together, re- calculate similarity between new group and other objects, and continue until all objects in one group  Different rules (algorithms) for re-calculating similarities, resulting in different dendrograms

13 Simple example  Relative intensity of fluorescence spectrum at four different wavelengths  Calculate distances using the “Euclidean” metric  standardised by the mean absolute deviation  Illustrate HCA using Single Link (Nearest Neighbour) algorithm  Distance to new group is the minimum of the distances to the objects being grouped CompoundWavelength (nm) A B C D E F G H I J K L

14 Step 1 A0.000 B C D E F G H I J K L ABCDEFGHIJKL  Identify minimum distance  between E and F  Join these objects into a group  Re-calculate all distances to this group  And repeat! 14

15 Dendrograms 15

16 Non-Hierarchical Clustering  Aim: to divide units into a number of mutually exclusive groups  Optimize some suitable criterion directly from the data matrix  Does not analyse the similarity matrix  Criteria include  Maximise the total Euclidean distance between groups  Minimise the determinant of the within-group variance-covariance matrix, pooled over groups  Repeat for different numbers of groups  Usually start with a large number of groups, and gradually reduce the number  Grouping is not hierarchical, i.e. best 3-group solution may not be best 2-group solution with one group divided into 2 sub-groups  Need a rule to determine the “right” number of groups  Also known as K-means clustering 16

17 Analysing Association Data  Multidimensional Scaling (MDS) and Principal Co-ordinate Analysis (PCO)  Analyse the same matrix of similarities or distances to produce a multidimensional picture of the relationships between units  Generates an “ordination” or configuration for a set of objects  Matches the inter-point distances to the dissimilarities or distances  PCO works with similarities  Produces an analytical solution (metric scaling)  Matches configuration distances to the observed dissimilarities based on the sum of squared differences  MDS works with distances or dissimilarities  Produces an iterated solution (non-metric scaling)  Matches the configuration distances to the observed dissimilarities based on rank orders (monotonic regression) 17

18 MDS Output for fluorescence data 18

19 Exploring patterns  Principal Component Analysis (PCA)  Aim: to identify the (combinations of) variables that explain the variability within a data set  Primarily a descriptive technique  Usually for quantitative variables  Starts with DATA matrix (p variables by n units)  Transforms original set of correlated variables into new set of orthogonal (independent) variables  Linear combinations of original variables  First principal component accounts for as much of the variability in the data as possible  Second principal component accounts for as much of the remaining variability as possible, and is orthogonal to the first  etc. 19

20 Matrix algebra  PCA best described in terms of matrix algebra  In common with almost all multivariate analysis methods  PCA is an eigenvalue decomposition of the matrix of associations between the variables  Produces two matrices  A diagonal matrix containing the eigenvalues  A rectangular (n-by-p) matrix containing the eigenvectors  Three possible matrices of associations can be used  Constructed from the original data matrix  Sum of Squares and Products Matrix (SSPM)  Variance-Covariance Matrix  Correlation Matrix  Different results from the PCA applied to each 20

21 PCA Output  Roots (eigenvalues)  How much of the variation is explained by each component  Expressed as a percentage of total  Indicates how many components are necessary  Loadings (eigenvectors)  How each original variable contributes to each principal component  Shows which variables are important  Scores  Values of each observation on each principal component 21

22 Fluorescence example  Relative intensity of fluorescence spectrum at four different wavelengths CompoundWavelength (nm) A B C D E F G H I J K L

23 PCA output  Analysis based on the variance- covariance matrix  Variances of original variables are similar (~25% each)  PC1 accounts for almost 73% of total variability  First two PCs account for nearly 89%  Obtain PC Scores for each compound by multiplying observed values by coefficients (loadings)  View groupings against PCs Eigenvalue Proportion Cumulative VariablePC1PC2PC3PC

24 Example PCA plot A K G D I H J L E B F C 24

25 Biplots 25  Graphical approach to present results from PCA (and a number of other multivariate methods)  Plots objects as points  As for example PCA plot  Plots variables as vectors  Supports interpretation of analysis  Shows those objects that are similar  Identifies variables that are highly correlated  Identifies variables that are particular associated with groups of objects

26 Correspondence Analysis  Analogous to Principal Components Analysis  Appropriate for categorical variables rather than continuous variables  Also known as “reciprocal averaging”  Finds an ordination (ordering) of each categorical variable that maximises the correlation between the two categorical variables  Used in the analysis of ecological community data  e.g. counts of the numbers of different species in different environments – identifies the species associated with particular environments  Extension to Canonical Correspondence Analysis  Incorporates the influence of one or more explanatory variables (such as environmental variables) in finding the ordination  Enables sites to be ranked along each environmental variable, taking account of correlations between species and environmental variables 26

27 Factor Analysis 27  Similar approach to PCA, predominantly used in the social sciences  Principle: that correlations between variables can be explained by a number of common factors, plus a number of specific factors (one for each original variable)  Focused on explaining the covariance between variables  While PCA is focused on explaining the maximum amount of variance in the data  Observed variables are assumed to be linear combinations of hypothetical underlying (and un-observable) factors  Creates an underlying causal model  Original application to the derivation of factors underlying intelligence  Can be used in a hypothesis testing mode (confirmatory factor analysis)

28 Canonical Variate Analysis (CVA)  Similar to PCA in working on the data matrix  Works on within-group SSPM and between-group SSPM  Finds combinations of the original variables to maximise the ratio of between-group variance to within-group variance  Groups are separated as much as possible whilst keeping each group as compact as possible  Combinations can be used to discriminate between the groups  For g groups – at most g-1 combinations of variables to discriminate between them  Need at least g-1 original variables  For a new observation, use “discriminant” functions to identify which group it is most likely to belong to  Discriminant Analysis 28

29 CVA Output  Output  Latent roots (eigenvalues)  How much variation is explained by each component  Expressed as a percentage of total  Root greater than 1 indicates that there is discrimination between groups on that canonical variate  Explicit test for dimensionality  Latent vectors (loadings)  Contributions of each original variable to new canonical variates  Canonical Variate Means  Mean values for each group on the canonical variates  Adjustment terms so that the centroid of group means is at the origin  Produce plots showing each group mean with a 95% confidence interval  Construct confidence intervals for the “population” 29

30 Example – Fisher “iris” data  Measurements of sepal length, sepal width, petal length and petal width for 50 plants of each of three iris species  Plot shows separation between the three species  Loadings (coefficients) indicate which variables are used to separate the groups 30

31 Multivariate ANOVA and Regression  Generalisation of univariate Analysis of Variance  Analyse multiple variables of data from a designed experiment  Assess for the effects of different factors on the whole set of variables  Takes account of the covariance between variables  Interpretation based on matrices of variances and covariances  Similar approach for multivariate regression  Relates a set of correlated response variables to one or more explanatory variables  PC Regression  PLS Regression 31

32 Procrustes Rotation  Provides a way of comparing two (or more) multidimensional configurations of a set of units  Procrustes = Greek inn-keeper who fitted his guests to one size of bed by chopping bits off or stretching bits!  Takes one configuration and fits the second configuration to it  Combinations of rotation, reflection and scaling each axis  Measure how much manipulation is needed to make the configurations similar  Measure how similar it is possible to make them  Generalise to more than two configurations 32

33 Exploratory Data Analysis and Multivariate Strategies Andrew Mead (School of Life Sciences)


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