PWS 551 Quantitative Ecology.txt

• The application of mathematical and statistical tools to address questions in ecology
• The trend reflects the demand to interpret larger and more complex data sets
• Also in response to criticism from other scientific disciplines for ecologists to use more quantitative approaches

• Make observations
• Formulate objectives/hypotheses
• Collect and analyze data
• Draw conclusions
• Evaluate hypotheses

• 1. Not everything that can be measured should be measured
• 2. Find a problem and state your objectives clearly
• 3. Collect data that will satisfy your objectives and a statistician
• 4. Some ecological questions are impossible to answer at the present time
• 5. Decide on # of significant figures for continuous data prior to starting
• 6. Never report an estimate without measure of possible error
• 7. Be skeptical about the results of significance tests 
• 8. Never confuse statistical with biological significance
• 9. Code all your data and enter it on a computer in some readable format
• 10. Garbage in, garbage out

• Address simple questions of technique
• Determine sample sizes needed for a particular level of precision

• Nominal - attributes like sex or species
• Ranking - ranks attributes in relation to one another, but not a numerical value 
• Interval/ratio - all characteristics of ranking, but know distance between classes; data may be discrete or continuous

• Accuracy - measure of bias
• Precision - measure of spread

• Usually can not sample in random manner
• Hidden variables confound our interpretations
• Statistics can only deal with random error, not detecting biased data

• Absolute abundance - exact count of species 
• Relative abundance - measure of dominance relative to co-ocurring species

• Frequency - % of sampling points or quadrats with species
• Density - # of individuals per unit area
• Biomass - dominance (cover), production, number

• Mark-recapture
• Removal/resight
• Distance

• Simplest mark-recapture method to estimate population size
• Mark individuals once, release them, and then recapture to check for marks
• Tends to overestimate; bias is large for small populations

• Closed population
• Equal catchability
• Marking doesn't alter catchability
• Marks aren't lost between sampling
• All marks are reported upon discovery in second sample

Reduces the over-estimation bias of the Peterson estimate

• Extension of the Peterson mark-recapture estimate for population size
• Used for closed populations and multiple samples

• Closed population
• Equal catchability
• Marking doesn't alter catchability
• Marks aren't lost between sampling
• All marks are reported upon discovery in second sample

• Mark-recapture method for open populations
• Estimate of population size

• 3 or more mark-recapture samples
• Individuals are marked to be specific for that sampling time
• Samples are usually point samples of short duration separated by a long duration from the next sample

• Equal catchability
• Equal survivability for marked individuals
• Individuals do not lose their marks and marks are not missed in sampling
• Sampling time is negligible in relation to the time between samples

• Capture probability
• # of sampling intervals
• Survival rates

• Change-in-ratio 
• Eberhardt's removal
• Catch-Effort

• Estimate of abundance/density from the change in sex ratio during a hunting season
• Assumptions:
• Composed of males/females and adults or young
• A differential change in the numbers of the two types of organisms occurs during the observation period

• If a population size index can be made before and after removal of a known # of individuals, we can use the indices to estimate absolute density
• Easier use of removal data
• Effective when >40% of the population is seen and >20% is removed

• Estimate population size by the decline in catch-per-unit effort over time
• Highly restricted because it only works well if a large fraction of population is removed so that there is a decline in the catch-per-unit-effort

• Catch-per-unit-effort is directly proportional to the existing population size
• Because the population must be declining time to time by an amount equal to the catch, a regression plot (accumulated catch vs. catch-per-unit-effort) should be a straight line
• Doesn't assume equal effort

• Two sample method of removal
• Assumes equal removal effort for both samples

• Methods used to "resight" radio-collared/tagged individuals
• Maximum likelihood estimation for density

• Used to estimate density
• Adds a "boundary strip" around the trapping area which is half the movement radius of the animals under study

• If a large area is sampled you can break the data up into a series of nested grids to estimate density
• Large positive bias

• Used to estimate density
• Type of removal method, because each individual is counted only the first time and recaptures are ignored
• Trap density is lower as you move away from center
• Assumes every animal in the center is caught
• Individuals are not attracted to web from outside area

A measured area, of any shape and size, that is used as a sample area in a survey of plants or of sessile/sedentary animals.

• Area (or volume) counted is known
• Organisms are immobile during counting

• Small quadrats require larger sample sizes for a specified level of adequacy than larger quadrats
• Small quadrats are more expensive than large based on number of objects sampled per unit time
• Coefficient of variation values are largest for small quadrats and sparse populations on average

• Rectangle quadrats are more accurate for aggregated populations because they typically include portions of aggregated and unoccupied patches in a single sample
• Circles are poor because they generally land completely within or outside of a patch
• Each quadrat shape tends to over estimate density parameters

• Quadrat has the same diameter (e.g., circle, square)
• These have fewer problems with parallax and moving around

A subset chosen entirely by chance from the entire population

• The population is divided into non-overlapping subpopulations 
• Once these strata are chosen, you sample each stratum separately

Means and confidence intervals can be estimated for each subpopulation

Stratum weights can be assigned if subpopulations are unequal

To regularly or systematically place samples

• Simplicity of application
• Sample evenly across whole population or habitat

• Systematic sampling is not random and may incorporate periodic variation
• Periodic variation rarely seems to occur in ecological systems

Stratified > regular > random

• Wiegert's method
• Hendricks' method
• N_min method
• Nested quadrats method
• Want normal distribution and high precision

• No species should occur (or be equally abundant) in all quadrats
• All important species should be in the sample at least occasionally
• Observe all parts of the quadrat with minimum personal movement
• Time for specified reliability should be minimized

• Determines quadrat size and shape
• Easy and fast
• Select quadrat size and shape that minimizes cost and variance (time = money)

• Determines quadrat size and shape
• More accurate, strict assumptions, but slower
• Variance decreases with larger quadrat size
• Time to read a quadrat is proportional to size (2m² is twice as costly as 1m²)

Determines adequacy and efficiency of sampling

• Determines quadrat size and shape
• Used to define a species-area curve for plant communities

• Rectangular quadrats are the best in heterogeneous habitats
• Circular quadrats work well with permanently marked points in sequential sampling (not in grasslands)
• Size and shape effects are not about biased abundance estimates as much as they are about efficiency
• Do pilot studies to determine optimal size and shape for your particular study (statistically, ecologically, and logistically)

• No single quadrat size or shape can be universally used
• Do a pilot study to gather means, variances, and costs

• If you want to compare your data with older data gathered with a specific quadrat
• If sampling several habitats, species, across seasons

There is no evidence that random placement of sample units gives better results than stratified or regular placement

• Clearly specify the statistical population
• Decide your sampling units and your experimental units
• Select a sample and adopt a variety of sampling plans as necessary

Method of placement and # of samples must be decided before sampling

Take advantage of spatial pattern in the population to obtain more precise measures of abundance

Begins with normal random sampling but when an organism of interest is detected, additional quadrats in the vicinity of the original quadrat are added to the sample

Used to describe any design where there are > 2 levels of sample selection

% of area (ground) covered by vegetation or other material

• Basal cover
• Canopy cover
• Foliar cover
• Ground cover

• Cover is an indicator of ecological processes
• Cover also serves as a management indicator for monitoring

Cover is an ecological indicator of which species are dominating the site

• Simple quadrats (1m² square)
• Line intercept
• Point quadrats
• Loop methods
• Ocular estimate method

Assessment of the distance (and sometimes width) along a line that is intercepted by a plant species or type

Examination of many points on a site to estimate the proportion of "hits" of a species

Uses concept that even if it is not possible to accurately estimate precise cover for a given quadrat, it may be easy to estimate broad cover classes

• Time efficiency
• Vegetation type
• How to calculate variance and SD
• Sample consistency

Ocular > 1-point > line-intercept > true mean > 10-point

• Correlation - variables simply vary together  
• Regression - variables vary together but there is evidence of a causal relationship

Display format used to analyze the relationship between two or more categorical variables

• Used to determine if there is a difference between expected and observed frequencies in one or more categories
• Non-parametric

• Equal variance (parametric)
• Unequal variance (non-parametric)

• Analysis of variance that compares two or more means
• Parametric

• Another way of estimating abundance and density
• An alternative to mark/recapture and quadrats
• They are useful for plants and animals that do not move much or can be located before they move

• Based on the reciprocal relationship between density and nearness of individuals to one another
• Only necessary to know the distance between regularly spaced individuals to calculate density

• Method of estimating density
• An observer travels along a randomly positioned line, recording the distances and angles from the line to each organism detected

• Animals directly on the transect line will never be missed
• Animals do not move before being detected and are never counted twice
• Distances and angles are measured exactly
• Sightings of individual animals are independent events

Detectability will fall off with distance from the center line of the transect

D = n/2La

• D = density of animals/unit area
• n = # animals seen on transect
• L = total length of transect
• a = half the effective strip width (a constant)

• Select random organism and measure distance to nearest neighbor
• Select random point and measure the distance from the point to the nearest neighbor

Quadrats - less biased and do not overestimate density

Distance methods - less costly in terms of money and time

• Both this and quarter method determine the average distance between organisms
• Measures distance between the object nearest to the sampling point and a second object that lies outside an exclusion angle of 180 degrees

• Distance from an organism to the one closest to it
• Sometimes incorporates multiple neighbors

• Plotless distance method to estimate density
• Quick, easy, and objective
• A series of random points is selected, often along a transect, with the constraint that points should not be so close that the same individual is measured at two successive points
• The area around each random point is divided into four quadrants and the distance to the nearest tree is measured in each quadrant
• Thus four point-to-organism distances are generated at each random point to the first-, second-, third-, and fourth-nearest neighbors
• Overestimates density

• Plotless distance method to estimate density
• For tree basal area is very accurate and quick
• Use sighting scope
• Turn 360 degrees and only count objects that are greater than the cross-bar
• Issues with parallax especially in grasslands

• Useful for large areas
• Set out 2n sampling points in study area 
• Select half of the 2n points at random and measure distance to nearest organism
• Around other half of points set out plots that include about 5 individuals each
• Select n of these at random
• Measure distance from selected organism to its nearest neighbor

• Most robust estimator of density
• Measure distance from random points to nearest organism and then to its nearest neighbor
• Only used in randomly distributed populations

• All distance methods are sensitive to spatial pattern (biased in clumped populations)
• Estimators based on point to organism are better in clumped
• Estimators based on organism to nearest neighbor are not so good in clumped

• Aggregated - some constraint on the population, environmental heterogeneity, gregarious behavior, reproductive behaviors
• Random - environmental homogeneity or non-selective behavioral patterns
• Uniform - negative interactions and competition

• Poisson - random pattern
• Negative binomial - aggregated

Associations of populations of many species that occupy the same geographical area

• Organismic (holistic)
• Individualistic
• Intermediate

Proposed the open community view and individualistic view

• Transition area between two biomes
• It is where two communities meet and integrate

• Closed have obvious ecotones between species distribution
• Open is overlapping

• Associations between two different species
• Test using a Chi-square

• Delineation of communities can be difficult unless the transition between adjacent communities is abrupt
• —Often arbitrary
• Require statistical approaches such as ordination and cluster analysis

• Ordination
• Cluster analysis

• Binary coefficients
• Distance coefficients
• Correlation coefficients

• Used for presence/absence data
• Jaccard index
• Ochiai index
• Sorensens’ similarity index
• Dice index
• Baroni-Urbani and Buser coefficient

• Measures of dissimilarity rather than similarity
• When a distance coefficient = 0, communities are identical
• Require some measure of species abundance in the community
• Euclidean distance, Bray-Curtis measure, Canberra metric

Simplest similarity measure that deals with presence/absence data

> 20 binary similarity measures

• Binary coefficient of similarity in presence/absence data
• Deficiency - no significance test

Binary coefficient of similarity in presence/absence data

• Binary coefficient of similarity in presence/absence data
• Weights species composition matches between 2 samples more heavily than mismatches
• Deficiency - no significance test

• Simplest binary coefficient of similarity in presence/absence data
• Makes use of negative matches as well as positive matches

• More complex binary coefficient of similarity
• Makes use of negative matches

In general, the Sorenson’s coefficient yields the highest similarity whereas the Jaccard’s coefficient results in the lowest similarity

• Measure of community dissimilarity
• Weighs abundant species more heavily than rare species

• Both are particularly poor in diverse communities with large sample sizes
• Both are best used in situations with low species diversity and small sample size

• Pearsons product moment - used for parametric data
• Spearman’s rank - used for non-parametric data

Percent similarity between two communities

• 
• Similarity index— for quantitative data
• Relatively unbiased
• —Easy to compute

• Similarity index for counts of individuals (not for other abundance estimates based on biomass, productivity, or cover)
• Also a dispersion coefficient

Similarity index for proportional data (biomass, cover, or productivity)

• Similarity index
• Least sensitive to sample size and gives the best representation of similarity
• Can be calculated from raw data (numbers) or from relative abundances (proportions or percentages)

Graphical way of portraying results from a large matrix of interspecific association (or similarity) indices

• Also called nearest neighbor method
• Simple to compute
• One inaccurate sample may compromise the entire process
• Tends to produce long, strung out clusters

• Farthest neighbor clustering method
• Opposite of single linkage
• Tends to produce compact clusters

Consists of the Weighted Pair Group Method of Averaging and the Unweighted Pair Group Method of Averaging (UPGMA)

• Weighted average procedure is more conservative than the unweighted method because it weighs the new addition to the cluster just as heavily as everything else
• Whereas the UPGMA assigns equal weights to each individual component within clusters