PWS 551 - Midterm

The application of mathematical and statistical tools to questions in the field of ecology.  The trend of quantitative ecology 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
• -form objectives/hypothesis
• -collect data
• -analyze data
• -make conclusions
• --reject hypothesis

• 1. not everything that can be measured should be measured
• 2. find a problem and state your objectives clearly
• 3. collect data that will achieve your objectives and make a statistician happy
• 4. some ecological questions are impossible to answer at the present time
• 5. with continuous data, save time and money by deciding on the number of significant figures needed in the data before you start the experiment
• 6. never report an ecological estimate without some measure of its possible error
• 7. be skeptical about the results of statistical tests of significance
• 8. never confuse statistical significance with biological significance
• 9. code all your ecological data and enter it on a computer in some readable format
• 10. garbage in, garbage out

• critical in both field and lab experiments
• they help address simple questions of technique
• they help determine the sample sizes needed for a particular level of precision

• Nominal scale: attributes like sex or species. Can only determine if one object is different from another. Common in ecology, only formal property is equivalence
• Ranking scale: ranks things, but not a numerical value, just ranked in relation to one another. 
• Interval or ratio scales: have all characteristics of ranking scale, but know distance between classes. Data may be discrete or continuous

• accuracy - measure of bias
• precision - measure of spread

Just because something is statistically significant, does not mean its biologically significant, and the reverse is true.  Statistics can only deal with random error; biased data is hard to detect.

• Frequency - percentage of sampling points or quadrats where species occur
• Density - number of individuals per unit area
• biomass
• dominance - cover
• production
• numberrelative/absolute - absolute is an exact count of species? relative abundance is another measure of dominance

• Peterson
• closed population
• single marking, single recapture
• based on probability: estimated size of pop at time of marking (N)  = (total number captured in second sample (C )* number marked in first sample (M) ) / number recaptured (R)
• widely used
• produces biased estimates - tends to overestimate, bias is large for small populations, several formulas to reduce bias

• Assumptions: 
• closed population
• all animals have an equal chance of getting caught
• does not alter catchability
• must randomly sample marked and unmarked individuals
• animals do not lose marks b/t sampling
• successive samples must be taken in same life history stage
• sampling must not span periods of increased mortality or emigration
• Seber Modification: 
• unbiased if (M+C) > N and nearly unbiased if there are at least 7 recaptures of marked animals

• extension of petersen method for multiple samples
• assumptions:
• same for petersen’s estimate
• easier to pick up violations of the assumptions

• mark recapture samples are taken on three or more occasions
• individuals are marked to be specific for that sampling time
• can know when individual was last captured
• samples are usually point samples of short duration separated by a long duration from the next sample
• bonus:
• can estimate probability of survival
• can estimate addition rate (dilution rate) and loss rate for the population
• assumptions:
• every individual has same chance of being caught
• every marked individual has same probability of surviving
• individuals do not lose their marks, and marks are not missed in sampling
• sampling time is negligible in relation to the time between samples
• Precision increases when:
• capture probability goes up
• number of sampling times increases
• survival rates are higher

• developed so wildlife managers could get estimates of populations under harvest
• Methods for exploited population techniques:
• change in ratio:
• estimated from field data on the change in sex ratio during a hunting season
• assumptions:
• composed of 1. male and female and 2. adults or young
• the differential change in the numbers of the two types of organisms occurs the observation period

• Animal Abundance:
• Eberhardt’s removal method:
• easier use of removal data
• as with mark-recapture, works best when a high fraction of the population is seen and a high fraction is removed
• percentage seen must be >40%, percentage removed must be >20% to be precise
• catch-effort methods:
• estimate population size by the decline in catch-per-unit effort with time
• highly restricted in its use because it will only work well if a large fraction of population is removed so that there is a decline in the catch - per-unit-effort
• assumptions:
• pop is closed
• equal catchability
• probability of each being caught in a trap is constant throughout the experiment

• Animal Abundance:
• Leslie estimate
• catch-per-unit-effort is directly proportional to the existing population size
• pop must be declining time to time by an amount equal to the catch, accumulated catch graphed against catch-per-unit-effort will be a straight line
• Moran Zippen
• not really sure it fits in in the above samples
• 2 sample method
• assumes equal effort for both samples
• See lab exercise*

• resight methods are used for radio-collared/tagged individuals, uses maximum likelihood estimation
• boundary strip method
• boundary strip is ½ the movement radius of the animals under study
• 1. calculate the average home range size
• 2. compute the ratio of grid size to area of average home range
• 3. computer simulation with elliptical
• home ranges allowed to overlap
• nested grid method
• large positive bias
• trapping web
• type of removal - recaptures are ignored
• trap density is lower as you move away from center
• assume everyone in the center is caught
• assumes that distances from center of web to each trap are measured accurately
• individuals are not attracted to web from outside area

• Quadrat - a measured area, of any shape and size, that is used as a sample area in a biological survey, particularly a survey of plants or of sessile and sedentary animals.
• Requirements for using quadrats:
• 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.  Smaller quadrats are more expensive than large based on number of objects sampled per unit time.  Coefficient of variation statistics are directly related to sample size needed for specified levels of adequacy, the C.V. values are largest for small quadrats and sparse populations on average.

• Rectangle quadrats are more accurate for aggregated populations because they have a greater chance of including portions of aggregated and unoccupied patches in a single sample.  The circle estimated density the worst because it had the greatest chance of falling completely within a patch or completely out of a patch.  
• Iso-diametric quadrats (same diameter: circle or square), have fewer problems with parallax and moving around.
• All shapes over-estimated density because the edge has low density and we excluded much of the edge because the samples that run over the edge are excluded.

A simple random sample is a subset chosen entirely by chance from the entire population.  Each individual sample must have the same probability of being chosen.  Simple random sampling is different from random sampling and should be completely unbiased.  Replacement sampling (two samples are independent; the first sample does not affect what we get on the second sample) is usually more accurate than sampling without replacement.

• In stratified random sampling, the population is divided into subpopulations that do not overlap. Once these strata are chosen, you sample each stratum separately.   Benefits of stratified random sampling: Means and confidence intervals can be estimated for each subpopulation
• Often results in a gain in precision of parameters of entire populations
• Stratum weights can be assigned if subpopulations are unequal.  Often stratum weights are proportions and must add up to 1.  Boundaries between strata can be determined by the cumulative square root of frequency of quadrats method.

• To regularly or systematically place (or take)  samples
• Used for simplicity of application
• Used to sample evenly across whole population or habitat
• The problem with systematic sampling is that it is not random and may incorporate periodic variation
• Periodic variation rarely seems to occur in ecological systems

• How to determine appropriate size:
• Need to minimize edge effects in small quadrat sizes
• Want to get values with normal distribution
• Cost: Minimum number of quadrats required for adequate sample
• Want a quadrat size that gives highest precision (Lowest SE, narrowest CI).
• How to determine appropriate shape:
• Edge effects: circle<square<rectangle
• Long, thin quadrats are better than circular or square ones in heterogeneous habitats
• Circular quadrats work better with permanently marked points in sequential sampling
• Quadrat size and shape effects are not about biased abundance estimates as much as they are about efficiency.
• Simplest approach: Go to the literature and use the same size and shape that everyone else uses
• Better approach: Do pilot studies to determine optimal size and shape for your particular study (Statistically, ecologically, and logistically).

• No species should occur (or be equally abundant) in all quadrats
• All important species should be included in the sample at least occasionally
• Be able to observe all parts of the quadrat with minimum personal movement
• Time required to achieve a specified reliability should be minimized

(easy and fast) Pick quadrat size and shape that minimizes relative cost and relative variance (time=money)

• (not as quick, more accurate, strict assumptions)
• Assumptions:
• Variance decreases with larger quadrat size
• Log of variance will fall linearly with the log of the quadrat size
• slope of the line is between 0 and -1 (if not, this method cannot be used)
• Time to read a quadrat is proportional to size (2m2 is twice as costly as 1m2)

• Used to define a species-area curve for plant communities
• Then can be used to define quadrat size
• Can be any shape

• No single quadrat size or shape can be universally used
• Best to do a pilot study to gather means, variances, and costs
• Pilot studies should be part of every experimental design
• When to ignore recommendations: If you want to compare your data with older data gathered with a specific quadrat, if sampling several habitats, species, across seasons.