What is quantitative ecology and what lead to the quantitative trend in ecology?
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
What are the 5 steps in ecological research?
Make observations
Formulate objectives/hypotheses
Collect and analyze data
Draw conclusions
Evaluate hypotheses
What are Krebs’ rules for ecological methodology?
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
Why are preliminary and pilot studies important?
Address simple questions of technique
Determine sample sizes needed for a particular level of precision
What are the different scales of measurement in ecology?
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
Compare and contrast accuracy and precision.
Accuracy - measure of bias
Precision - measure of spread
Why is statistical inference difficult in ecology?
Usually can not sample in random manner
Hidden variables confound our interpretations
Statistics can only deal with random error, not detecting biased data
Compare and contrast absolute abundance with relative abundance
Absolute abundance - exact count of species
Relative abundance - measure of dominance relative to co-ocurring species
What are the common measures of abundance used in ecological studies?
Frequency - % of sampling points or quadrats with species
Density - # of individuals per unit area
Biomass - dominance (cover), production, number
What are the different ways that we measure abundance in animal populations?
Mark-recapture
Removal/resight
Distance
What is the Peterson estimate?
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
What are the assumptions of the Peterson estimate?
Closed population
Equal catchability
Marking doesn't alter catchability
Marks aren't lost between sampling
All marks are reported upon discovery in second sample
What is the Seber modification of the Peterson estimate and why is it used?
Reduces the over-estimation bias of the Peterson estimate
What is the Schnabel method, when is it used?
Extension of the Peterson mark-recapture estimate for population size
Used for closed populations and multiple samples
What are the assumptions of the Schnabel method?
Closed population
Equal catchability
Marking doesn't alter catchability
Marks aren't lost between sampling
All marks are reported upon discovery in second sample
What is the Jolley-Seber method?
Mark-recapture method for open populations
Estimate of population size
When and how is the Jolly-Seber method accomplished?
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
What are the assumptions of the Jolley-Seber method?
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
What affects precision in the Jolly-Seber method?
Capture probability
# of sampling intervals
Survival rates
What removal/resight methods are used for exploited populations?
Change-in-ratio
Eberhardt's removal
Catch-Effort
What are the Change-in-ratio methods?
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
What is Eberhardt’s removal method and when is it used?
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
What are catch-effort methods and when are they used?
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
What is the Leslie estimate and what is the theory behind the method?
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
What is the Moran-Zippen method?
Two sample method of removal
Assumes equal removal effort for both samples
What are resight methods and how do we estimate density in animal populations?
Methods used to "resight" radio-collared/tagged individuals
Maximum likelihood estimation for density
What is the boundary strip method?
Used to estimate density
Adds a "boundary strip" around the trapping area which is half the movement radius of the animals under study
What is the nested grids method and is it biased?
If a large area is sampled you can break the data up into a series of nested grids to estimate density
Large positive bias
What is a trapping web and why use this method?
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
What is a quadrat and when is it used for sampling?
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.
What are the requirements for using quadrats?
Area (or volume) counted is known
Organisms are immobile during counting
What are quadrat size effects?
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
What are quadrat shape effects?
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
What is an isodiametric quadrat?
Quadrat has the same diameter (e.g., circle, square)
These have fewer problems with parallax and moving around
What is simple random sampling?
A subset chosen entirely by chance from the entire population
What is meant by stratified random sampling?
The population is divided into non-overlapping subpopulations
Once these strata are chosen, you sample each stratum separately
What are the benefits of stratification?
Means and confidence intervals can be estimated for each subpopulation
Can strata be given different weights?
Stratum weights can be assigned if subpopulations are unequal
What is meant by systematic sampling?
To regularly or systematically place samples
When and why is systematic sampling used?
Simplicity of application
Sample evenly across whole population or habitat
What is the statistical problem with using a systematic sample and how common is the problem in ecological systems?
Systematic sampling is not random and may incorporate periodic variation
Periodic variation rarely seems to occur in ecological systems
In our lab exercise which method of sample placement best estimated density?
Stratified > regular > random
How do we determine the appropriate size and shape of quadrat to use?
Wiegert's method
Hendricks' method
N_{min }method
Nested quadrats method
Want normal distribution and high precision
What criteria should be met in determining adequate quadrat size and shape?
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
What is Wiegert’s method?
Determines quadrat size and shape
Easy and fast
Select quadrat size and shape that minimizes cost and variance (time = money)
What is Hendricks’ method?
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 (2m^{2} is twice as costly as 1m^{2})
What is the N_{min} method?
Determines adequacy and efficiency of sampling
What is the nested quadrats method?
Determines quadrat size and shape
Used to define a species-area curve for plant communities
What are quadrat size and shape effects?
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)
Take home message on quadrat size and shape
No single quadrat size or shape can be universally used
Do a pilot study to gather means, variances, and costs
When can you ignore recommendations about size and shape of quadrats?
If you want to compare your data with older data gathered with a specific quadrat
If sampling several habitats, species, across seasons
How do we determine where to place quadrats?
There is no evidence that random placement of sample units gives better results than stratified or regular placement
How do we determine sampling strategy?
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
What is sampling design?
Method of placement and # of samples must be decided before sampling
What is meant by adaptive sampling?
Take advantage of spatial pattern in the population to obtain more precise measures of abundance
What is an adaptive cluster sample?
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
What is meant by multi-stage sampling?
Used to describe any design where there are > 2 levels of sample selection
What is cover?
% of area (ground) covered by vegetation or other material
What are the four basic types of cover obtained in ecological sampling?
Basal cover
Canopy cover
Foliar cover
Ground cover
Why do we measure cover?
Cover is an indicator of ecological processes
Cover also serves as a management indicator for monitoring
How does cover measure dominance?
Cover is an ecological indicator of which species are dominating the site
What are the common methods for assessing cover?
Simple quadrats (1m^{2} square)
Line intercept
Point quadrats
Loop methods
Ocular estimate method
What is the line-intercept method?
Assessment of the distance (and sometimes width) along a line that is intercepted by a plant species or type
What are point methods (to assess cover)?
Examination of many points on a site to estimate the proportion of "hits" of a species
What are ocular methods (to assess cover)?
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
From our lab exercise of estimating plant cover what are factors to consider prior to sampling?
Time efficiency
Vegetation type
How to calculate variance and SD
Sample consistency
In our lab of estimating plant cover which method best estimated cover?
Ocular > 1-point > line-intercept > true mean > 10-point
What is the difference between correlation and regression?
Correlation - variables simply vary together
Regression - variables vary together but there is evidence of a causal relationship
What is a contingency table?
Display format used to analyze the relationship between two or more categorical variables
What is a chi-square analysis?
Used to determine if there is a difference between expected and observed frequencies in one or more categories
Non-parametric
What is the chi-square formula?
How do you calculate expected frequencies for a chi-square analysis of a contingency table?
What are the types of t-tests?
Equal variance (parametric)
Unequal variance (non-parametric)
What are ANOVAs?
Analysis of variance that compares two or more means
Parametric
What are plotless sampling procedures and when are they used?
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
How do plotless sampling techniques work?
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
What is line transect sampling?
Method of estimating density
An observer travels along a randomly positioned line, recording the distances and angles from the line to each organism detected
What are the assumptions of line transect sampling?
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
What is the detection function of line-transect sampling?
Detectability will fall off with distance from the center line of the transect
What are the estimators for density using line transect sampling?
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)
What are the two basic approaches to distance methods?
Select random organism and measure distance to nearest neighbor
Select random point and measure the distance from the point to the nearest neighbor
If an area has moderate to high density of sessile organisms and they are randomly dispersed, are distance methods or quadrats better?
Quadrats - less biased and do not overestimate density
If an area has low density of sessile organisms and they are randomly dispersed, are distance methods or quadrats better?
Distance methods - less costly in terms of money and time
What is random pairs?
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
What is nearest neighbor?
Distance from an organism to the one closest to it
Sometimes incorporates multiple neighbors
What is the point-quarter method?
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
What is the Bitterlich method?
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
What is the Byth and Riply procedure and why is it used?
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
What is the T-Square method?
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
Compare all distance methods
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
What are the three spatial patterns in nature and what gives rise to these patterns?
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
What are the two statistical distributions and their spatial patterns?
Poisson - random pattern
Negative binomial - aggregated
What is a community?
Associations of populations of many species that occupy the same geographical area
What are the historic community concepts?
Organismic (holistic)
Individualistic
Intermediate
Who are Clements and Gleason?
Proposed the open community view and individualistic view
What is meant by ecotone?
Transition area between two biomes
It is where two communities meet and integrate
What is meant by open vs. closed communities?
Closed have obvious ecotones between species distribution
Open is overlapping
What is meant by interspecific associations and how do we test for this?
Associations between two different species
Test using a Chi-square
How are communities classified?
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
How do we determine where one community ends and another begins?
Ordination
Cluster analysis
What are measurements of community similarity?
Binary coefficients
Distance coefficients
Correlation coefficients
What are binary coefficients?
Used for presence/absence data
Jaccard index
Ochiai index
Sorensens’ similarity index
Dice index
Baroni-Urbani and Buser coefficient
What are distance coefficients?
Measures of dissimilarity rather than similarity
When a distance coefficient = 0, communities are identical
Require some measure of species abundance in the community
What do binary coefficients of similarity deal with?
Simplest similarity measure that deals with presence/absence data
How many binary coefficients of similarity are available in the literature?
> 20 binary similarity measures
What is the Jaccard index?
Binary coefficient of similarity in presence/absence data
Deficiency - no significance test
What is the Ochiai index?
Binary coefficient of similarity in presence/absence data
What is the Sorenson’s index?
Binary coefficient of similarity in presence/absence data
Weights species composition matches between 2 samples more heavily than mismatches
Deficiency - no significance test
What is the simple matching coefficient?
Simplest binary coefficient of similarity in presence/absence data
Makes use of negative matches as well as positive matches
What is the Baroni-Urbani and Buser coefficient?
More complex binary coefficient of similarity
Makes use of negative matches
How do binary coefficients of similarity compare?
In general, the Sorenson’s coefficient yields the highest similarity whereas the Jaccard’s coefficient results in the lowest similarity
What is the Bray-Curtis measure?
Measure of community dissimilarity
Weighs abundant species more heavily than rare species
Compare Bray-Curtis and Canberra metric
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
What are the two types of correlation coefficients and when is each used?
Pearsons product moment - used for parametric data
Spearman’s rank - used for non-parametric data
What is percentage similarity?
Percent similarity between two communities
What is Ruzicka’s index?
Similarity index for quantitative data
Relatively unbiased
Easy to compute
What is Morisita’s index?
Similarity index for counts of individuals (not for other abundance estimates based on biomass, productivity, or cover)
Also a dispersion coefficient
What is the simplified Morisita’s index?
Similarity index for proportional data (biomass, cover, or productivity)
What is Horn’s index?
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)
Describe the multivariate technique called cluster analysis?
Graphical way of portraying results from a large matrix of interspecific association (or similarity) indices
What is simple linkage clustering?
Also called nearest neighbor method
Simple to compute
One inaccurate sample may compromise the entire process
Tends to produce long, strung out clusters
What is complete linkage clustering?
Farthest neighbor clustering method
Opposite of single linkage
Tends to produce compact clusters
What is average linkage clustering?
Consists of the Weighted Pair Group Method of Averaging and the Unweighted Pair Group Method of Averaging (UPGMA)
Compare and contrast the weighted and unweighted pair-group methods 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