Multivariate Analysis and Cluster Analysis Notes

Multivariate Analysis: Distance and Dissimilarity Measures

Introduction to Distance and Dissimilarity Measures

• Multivariate analysis focuses on distance and dissimilarity measures.
• Principal components analysis (PCA) indirectly uses Euclidean distance measures.
• These measures are crucial for analyzing data and forming groupings, which leads to cluster analysis.
• Distance matrices can also be called dissimilarity matrices; sometimes similarity matrices are used.

Overview of the Next Few Lectures

• Concentrate on distance/dissimilarity matrices and similarity matrices.
• Discuss transformations and standardizations of data.
• Cover a range of cluster analysis methods, focusing on hierarchical and non-hierarchical clustering.
• Emphasis on identifying and describing patterns, similar to PCA.

Key Concepts and Hypothesis Testing

• Focus on describing patterns rather than hypothesis testing in PCA and clustering.
• Hypothesis testing will be covered in more detail next week.

Resources and Practice

• Recommended readings: Quinn & Keough (both old and new editions).
• Practice examples and interpret data using available online resources.
• Utilize datasets from both old and new editions for lectures and labs.
• Examples in the lab build upon the background provided in the lecture.

Why Use Distance/Dissimilarity Matrices?

• Analyzing multivariate data, like invertebrate species at different sites, necessitates considering interrelationships among variables.
• Avoid issues with Type I errors and account for species interactions.
• Simplify data into a more manageable format.

Applications of Distance and Dissimilarity Matrices

• Clustering: Form groups from distance/dissimilarity matrices to classify data.
• Ordination: Rearrange data in multi-dimensional space to produce a map (non-metric multi-dimensional scaling).
• Statistical Testing: Use Anosim/permanova to statistically test differences.
• Applicable to both natural experiments and manipulative experiments.

Defining Distance and Dissimilarity

• Evaluate species composition or morphology of organisms.
• Determine how different or alike samples are using distance and dissimilarity measures.
• Dissimilarity is often used for non-metric data, while distance is used for metric data.
• Matrices are often visualized in two or three-dimensional space.

Similarity vs. Dissimilarity Matrices

• Dissimilarity matrices are bounded by 0 and 100: 0 means identical, 100 means nothing in common.
• Similarity matrices: 100 means identical, 0 means nothing in common.
• Confusing similarity and dissimilarity can lead to incorrect results.

Mutual Absences

• Mutual absences: when a species is not present in both samples being compared.
• Mutual absences can cause statistical and biological problems.
• Linking sites based on mutual absences may not be biologically meaningful in ecological studies.
• Example: Antarctica and the tropics both lacking emus doesn't make them similar.
• In habitat data, mutual zeros may be important (e.g., 0% litter cover).

Simple Example and Dissimilarity Matrix

• Link sites together based on species composition.
• A dissimilarity matrix indicates how close sites are in multivariate space.
• Lower numbers indicate closer proximity in multivariate space.
• Example numbers: Sites 2 and 4 are most similar; sites 1 and 3 are most dissimilar.

Calculating Distances and Dissimilarities

• Use metric measurements for continuous or ratio data.
• Use non-metric measurements for ordinal or nominal count data.

Euclidean Distance

• Metric distance used in principal components analysis.
• Based on Pythagoras' theorem, actually Euclidean theory.
• d = \sqrt{\sum (xi - yi)^2}
• Where xi and yi are the values of the ith variable for the two points being compared.
• Link samples with exact same values, bounded by zero, with no upper limit.

Metric vs. Non-Metric and Bray-Curtis

• In two-dimensional space, Euclidean distance can be derived geometrically.
  • Non-metric distance measurements cannot derive the distance.
• Metric uses absolute derived distances and non-metric data uses ranks.

Bray Curtis

• A commonly used non-metric measurement in Ecology.
• Derived by botanists to avoid linking samples based on joint absences.
• Also known as percentage dissimilarity.
• Ignores joint absences, and determinants are the variables with high values.
• Suited for species abundance data.
• BC{ij}=\frac{\sum |x{ij}-x{ik}|}{\sum (x{ij}+x_{ik})}
• Where:
  • BC_{ij} is the Bray-Curtis dissimilarity between samples i and j
  • x_{ij} is the abundance of species k in sample i
  • x_{ik} is the abundance of species k in sample j

Bray Curtis Example

• Calculate the absolute differences of species A-E between site 1 and site 2.
• Calculates the addition of them and if you add zero to that it doesn't affect the number.

Implementation in R

• Using the Vegan package, calculate Bray-Curtis similarities/dissimilarity matrix.

Choosing Metric vs. Non-Metric

• Principal components uses Euclidean, so you can use the principal components in your linear regressions or your ANOVAs because it has the same sort of properties.
• Presence of zeros in species data often means you don't want things linked.
• In measurement data, joint absences may be important.

Standardization and Transformation

• Transform in univariate stats to make things more normal or to satisfy homogeneity of variance.
• In multivariate data analysis, transformations are used for different reasons.
• Non-metric analysis doesn't care about normality.

Transformations

• Transformations can down-weight very common species by changing the scale of measurement.
• Examples include square root, log (adding one to avoid log of zero), fourth-root, and presence/absence transformations.
• log(x+1)
• Where x is the original value, add one because you can't log zero because a log of zero is negative infinity.

Examples of Transformations

• Square root transformation makes the differences between numbers smaller.
• Fourth-root transformation further reduces the emphasis on common species.
• Presence/absence transformation converts all values to 1 (present) or 0 (absent).

Transformation Considerations

• Consider the context of the data and analysis.
• Do not blindly apply transformations without justification.
• Determine if you want dominant variables to dominate the analysis.
• In exam questions, consider if the transformation is to increase linearity or to address the dominance of certain species.

Standardization

• Standardize to make each species equally important or to make each sample equally important.
• Useful for comparing samples of different sizes or with different sampling efforts.
• Express values as proportions or relative to the maximum value.

Calculating Proportions

• Divide each species number by the total.
• Useful when comparing the relative importance of species or samples.

Important Considerations

• Standardizations can change the interpretation of results.
• Compare raw data analysis to standardized data analysis.
• Transformation is usually better than standardization.

Summary

• Understand dissimilarity and distance measures.
• Know the difference between metric (Euclidean) and non-metric (Bray-Curtis) measures.
• Understand the role of transformations in both metric and non-metric analyses.
• Apply distance and dissimilarity matrices in clustering and ordination.

Clustering Analysis

Introduction to Cluster Analysis

• Cluster analysis groups samples based on the extent and samples kit.
• Methods use similarity coefficients between samples (Euclidean or Bray-Curtis).
• Can custom groups or map them in two or three-dimensional states.

Main Question in Clustering

• Looks against Descriptive things and not a statistical test.
• Do samples form natural groupings?
• May be used in taxonomy, genetics, ecology, soil science, etc.

Genetic Clusters

• Genetic data often has different assumptions about mutation rates and stuff.
• Caveali Zivosa is a called genetic distance
• Techniques in clustering and ordination.
• Treat diagram based on Genetic distance.