Lab Week 5 Lab Due Tuesday, September 30

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In this lab, you will explore methods for describing distances between populations and samples using provided R code, particularly focusing on Section 5.3 of your text. You will then write your own functions to calculate distances based on proportions (Section 5.4, equations 5.5 and 5.6) and presence-absence data (Section 5.5). The assignment involves generating and analyzing distance matrices, creating custom functions for similarity and dissimilarity indices, and applying these to biological data such as genetic and geographic distances. Additionally, you will conduct a Mantel test to assess correlations between genetic and geographic distances in butterfly populations. Throughout, you will interpret the results, compare different distance measures, and explore their implications for ecological and evolutionary studies.

Paper For Above instruction

The exploration of biological distances and similarity indices is central to understanding relationships among populations in ecology and genetics. The assignment guides students through practical applications in R, emphasizing the calculation of distances based on population means, proportions, and presence-absence data, along with statistical tests to interpret ecological patterns.

Calculating Population Distances Using R

Initially, students will utilize existing R code for the distance matrix function (distmatrix()) to compute distances between skull data populations, comparing Euclidean and Mahalanobis metrics. This exercise emphasizes matrix algebra relationships, where the Euclidean distance depends on direct point-to-point differences, while the Mahalanobis distance adjusts for correlations among variables. By executing the code and analyzing the results, students will gain insight into how these measures reflect underlying data structure. Notably, the contrasting distance matrices reveal the influence of variable covariances on perceived similarity among populations, with potential implications for biological interpretations.

Two commands in the R code are likely new: as.dist() and dist(). The dist() command computes a dissimilarity matrix or vector based on specified methods such as Euclidean distance. It simplifies the process of calculating pairwise distances among observations. The as.dist() function converts a matrix object into a 'dist' object, which is suitable for plotting and analysis. Understanding these commands enhances skills in matrix manipulation and distance computation in R, essential for ecological data analysis.

Distance Measures Based on Proportions

Moving into compositional data, students will construct a function that calculates dissimilarity and similarity indices based on species proportions across colonies or populations. Using equations 5.5 and 5.6, the function should take two proportion vectors with elements summing to 1 and output the dissimilarity and similarity measures. These indices include the dissimilarity index (commonly called Bray-Curtis or other) and similarity indices such as the Sørensen or Jaccard metrics. Graphical representation of scenarios with no overlap, complete overlap, and partial overlap provides intuitive understanding of how these indices quantify biological similarity.

Efficiency and accuracy in coding are vital; therefore, commenting the R function clarifies its logic. Displaying example calculations for each scenario demonstrates how index values change with the degree of class overlap among populations, facilitating interpretation of ecological distances in real-world data.

Presence-Absence Data and Ecological Similarity

Further complexity is introduced through presence-absence data, where each site-survey involves recording whether species are present or absent. The task involves creating a function to compute four similarity indices: Simple Matching, Ochiai, Dice-Sorensen, and Jaccard. These indices, normalized between 0 (no similarity) and 1 (complete similarity), are essential for biodiversity and community ecology studies. The function inputs are counts of presences and absences, which should be carefully specified and related to ecological interpretations.

By applying the function to three scenarios—no overlap (species present in one site only), complete overlap (species shared across sites), and partial overlap—the differences in similarity values reveal the sensitivity of each index. Such analyses support conclusions about community structure and species turnover.

Genetic and Geographic Distances via Mantel Test

The final part involves analyzing a butterfly dataset to examine whether genetic similarity correlates with geographic distance among populations. Using the ade4 package, students will compute a genetic distance matrix based on a prior measure, such as that from Problem 2. A Mantel test with permutations assesses the correlation, with scatter plots visualizing the relationship. Repeating the test with alternative distance measures (e.g., based on different equations) investigates the robustness of the correlation.

The nested loops fill a genetic distance matrix by applying the custom distance function, and the Mantel test determines the significance of the correlation between genetic and geographic distances. Interpreting these results reveals whether spatial proximity influences genetic similarity, a fundamental question in landscape genetics and conservation biology.

Overall, this comprehensive exercise enhances understanding of how various distance and similarity metrics quantify ecological and evolutionary relationships, and how statistical tests validate or refute observed patterns in biological data.

References

• Legendre, P., & Legendre, L. (2012). Numerical Ecology (3rd ed.). Elsevier.
• R Core Team. (2023). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.r-project.org/
• Clarke, K. R., & Warwick, R. M. (2001). Change in Marine Communities: An Approach to Statistical Analysis and Interpretation. PRIMER-E.
• Oksanen, J., et al. (2023). vegan: Community Ecology Package. R package version 2.6-0. https://CRAN.R-project.org/package=vegan
• Borcard, D., Gillet, F., & Legendre, P. (2018). Numerical Ecology with R. Springer.
• Legendre, P., & Gallagher, E. D. (2001). Ecologically meaningful transformations for ordination of species data. Ecology, 82(7), 2192–2207.
• Mantel, N. (1967). The detection of disease clustering and a generalized regression approach. Cancer Research, 27(2), 209–220.
• Stanisz, A. (2007). Measurement in Psychiatry: A Practical Guide. CRC Press.
• Jackson, D. A. (1993). Stopping Rules in Principal Components Analysis: A Comparison of Heuristic and Statistical Approaches. Ecology, 74(8), 2204-2214.
• Thorpe, P., & co-authors. (2015). Use of the Mantel test for genetic distance analysis. Methods in Ecology and Evolution, 6(7), 720–731.