Multivariate methods

Remember you should

1

A pharmaceutical company has a drug that may help an illness that causes fever (temperature in degrees Celsius), blood pressures, and “aches” (scored on an index). Data is collected for several patients. To determine if the drug actually helps, test for differences in multivariate means for the fever, pressure and aches column, against the grouping variable treatment.

illness <- read.csv("https://docs.google.com/spreadsheets/d/e/2PACX-1vRlcjpU0XHfXF1WId1C5ZYX0YdY53KI9Nv91_tNCMj4z4iTjr-XMW1L_Ln8j3ahk5GUPZy4kGzSlA96/pub?gid=1322236994&single=true&output=csv",
                    stringsAsFactors = T)

2

Darlingtonia californica is a partly carnivorous pitcher plant that grows in fens and along seeps and streams in the mountains of Oregon and California. Its pitchers are tubular l eaves with a round hood and a mouth at the base of the hood (see figure below). A “fishtail” appendage hangs from the mouth. Wasps and other prey are attracted to nectar secreted by extrafloral nectaries along the hood, mouth, and fishtail. Plants absorb nutrients excreted by a food web of bacteria, protozoa, mites, and fly larvae that break down the prey.

Measurements of 87 plants from four sites were made by Ellison and Farnsworth (2005, The cost of carnivory for Darlingtonia californica (Sarraceniaceae): evidence from relationships among leaf traits. Am. J. Botany 92: 1085-1093). Their measurements are available using

pitcher <- read.csv("https://docs.google.com/spreadsheets/d/e/2PACX-1vQZf2mS4NmfBUUsn7lY2RTpuVjuWvRYN4MdLNt2XdS4WepolrxvWCKBI5diKBMWPLhdbEGwP-hfWOnz/pub?gid=1427497144&single=true&output=csv",
                    stringsAsFactors = T)

I obtained them from the web page (http://harvardforest.fas.harvard.edu/personnel/web/aellison/publications/primer/primer.html) of A. M. Ellison for the book by Gotelli and Ellison (2004, A primer of ecological statistics. Sinauer, Sunderland, Mass.). To simplify, outliers have been removed. Most plant traits in the file are illustrated in the image below, and trait labels are fairly self-explanatory. Keel width measures the span of the pitcher tube. “Wing” traits refer to the lengths of the fishtail appendage.

Photograph of a Darlingtonia californica pitcher with morphological measurements indicated (lower diameter at ground level not shown). Note the translucent hood and the pronounced fishtail appendage attached to the proximal side of the mouth
  • Use a MANOVA to consider differences in plant traits (do not follow-up with almost 20 ANOVA’s! Just consider why PCA might be useful with large datasets!
  • Use principal component analysis to investigate variation among individual plants in their dimensions. Along the way, make sure you
    • construct screeplots
    • determine how many principal components to retain (and why)
    • Use biplots and/or loadings to see if you can understand/interpret the first few principal components

3

Using the same plant dataset, use linear discriminant analysis to classify the various sites.

4

Using the same plant dataset, use cluster analysis to determine how many clusters are supported by the data.

5

The data for this exercise are rodent species abundance from 28 sites in California (Bolger et al. 1997, Response of rodents to habitat fragmentation in coastal Southern California, Ecological Applications 7: 552–563).

This data comes from the (website)[http://www.zoology.unimelb.edu.au/qkstats/data.htm) of Quinn and Keough (2002, Experimental Design and Data Analysis for Biologists, Cambridge Univ. Press, Cambridge, UK). Data is available via

rodents <- read.csv("https://docs.google.com/spreadsheets/d/e/2PACX-1vTLRwuI1cQ61RZOVJFwi0jhO85fonqR7oZHzy_9A5fVwxuZQ2A6iBnlLG2Z-33rwNnycqNUUh1_XuMU/pub?gid=1403553505&single=true&output=csv", 
                    stringsAsFactors = T)

The 9 species are indicated by variable (column) names. Genus abbreviations are: Rt (Rattus), Rs (Reithrodontomys), Mus (Mus), Pm (Peromyscus), Pg (Perognathus), N (Neotoma) and M (Microtus). Rattus and Mus are invasive species, whereas the others are native.

  • Analyze the dat using correspondence analysis
    • interprent any results (loadings!)