Sampling I

Population and Sample

Define 1) census, 2) sample, and 3) sampling?

Population and Sample

To use a sample to describe a ‘population’ is called what?

Definitions

Census?
- Complete enumeration of the population (i.e., the whole)
What is a sample?
- The act, process, or technique of selecting a representative part of a population for the purpose of determining parameters or characteristics of the whole population
What is sampling?
- Selection of a part of a population to be used to estimate characteristics of the population
- To make inference about the population from a part, i.e., the sample.

Sampling Types

Convenient sampling - when most convenient units are chosen
Judgment sampling - obtained at the discretion of someone who is familiar with the characteristics of the population
Probability sampling - based on a process that the probability a unit is selected is known

Central paradox of Sampling

Impossible to know from a sample whether it is representative
We rely on the process, not the outcome.

Why Sample?

Large populations are physically prohibitive to census
Fiscally prohibitive to census
Timeliness
Observing might be destructive. For example?

Common Sample Units

plots/quadrats - small area to measure/count plants, seeds, insects, etc.
points - measurements are taken from a set of points established throughout a population
transects - line segments in which observations are taken from
individual organisms - the organism is the sample unit or the organism defines the location of the sample unit

Population and Sample

Trees of sufficient size for nesting by spotted owls
House characteristics that northern flickers most often damage
Survival of rainbow trout in the presence of whirling disease
Surveys, e.g., number of people that
- bird watch / hunt / fish / think wolves should live in CO

Population and Sample

Often a spatial/temporal context to our target population

Population and Sample

Title: Muskrat occurrence in Rhode Island shows little evidence of land use change driving declines

Population and Sample

Title: Land cover attributes affect the distribution of rooting damage by wild pigs (Sus scrofa)

Surveys within 1) The Savannah River Site (green) 2) Newberry County (blue), 3) Hampton County (red)
“We generated 9 random points to act as the center of respective 200-ha survey grids”.
“Properties were primarily composed of mixed agricultural-forested cover”.

Thinking Backwards

People often think backwards from what data they have to identify the Population.

E.g., Nesting shorebirds are censused each summer at a handful of beaches.

Can we use these sites as samples to apply to all beaches in the area?

Design vs. Sampling

Experimental Design:
- Deliberately perturbing a part of a ‘population’ to compare it’s effect to a part that was not perturbed
Sampling Design:
- The process of obtaining a representative sample to characterize a ‘population’ w/o necessarily perturbing it.

Design- vs Model-based Infernce

What are important differences according to Thompson and Hankin et al.?

Benefits or costs of either?

Important Considerations

randomness
fixed
undefinable infinite superpoulation vs. real target population

Design- vs Model-based Infernce

Sample Language

A sample (height in inches):

\(\textbf{y} = [69, 54, 72, 61, 58, 71]\)

A sample unit:

\(y_{2} =54\)

Sample size:

\(n = 6\)

More Language

What is a statistic?

An estimate of a population parameter from a sample

\[\hat{\mu} = \left(\left(\sum_{i=1}^{n}y_{i}\right)\times \frac{1}{n}\right) = 57.7\]

\(n\) is a sample parameter (size of sample)
\(\hat{\mu}\) is an estimate of a population parameter (\(\mu\)) from the estimator (mathematical rule for calculation)
57.7 is a statistic (specific value)

More Language

\[\mu =\left(\sum_{i=1}^{N}y_{i}\right)\times \frac{1}{N} = 54\]

\(\mu\) is a population parameter (measure of central tendency)
\(N\) is a population parameter (size of all possible sample units)
\(54\) is the value of the population parameter

Sampling Error

Sampling Error
- The difference b/w a sample statistic (specific value) and the true value of a population paramter
- 57.7 - 54 = 3.3 (inches) sampling error
- Due solely to incomplete enumeration of the population (chance)
- Protection against this is large sample size

Sampling Variation and Error

Target Population: Weight of all black bears in a region

How would you describe a sampling frame relevant to this target population?

Sampling Variation and Error

Sampling Error

Vector of bear weights

  pop.weights[1:5]

[1] 421.7759 185.2152 321.6069 221.8225 284.4739

# Sample Size
  n = 50

# Sample and estimate mean one time
  sample.1 = mean(
                  sample(pop.weights,
                         n,
                         replace = FALSE
                         )
                  )

# Calculate Sampling Error
  sample.1 - pop.mu.weights

[1] 3.822778

Is this a problem?

Sampling Variation vs Error

Sampling variation is the process and sampling error is an outcome.

The differences between samples (sampling variation) lead to differences between sample statistics and population parameters (sampling error).

Sampling Variation

Calculate many many sample means

# Create function to sample 50 units and take the mean
 sample.mean.fn = function(target,n){
                                     mean(
                                         sample(target,n)
                                          )
                                     }
# Repeat the above function 20000 times
  mu.hat = replicate(20000,
                     sample.mean.fn(pop.weights,n)
                     )

Sampling Variation

Estimator Bias

Expected Bias (of the estimator) = average sample mean - population mean

\[ \text{Bias}(\hat{\mu},\mu) = \bar{\hat{\mu}} - \mu \\ \]

\[ \text{Bias}(\hat{\mu},\mu) = E[\hat\mu] - \mu \\ \]

\[ \text{Bias}(\hat{\mu},\mu) = \left(\int_{x} g(x)p(x|\mu)dx\right) - \mu \]

Expectations are an average over all possible samples of size n.

Is our estimator biased?

Precision

How close are repeated measures to each other?

Probability

We can also summarize the sampling variation into a probability

For n = 50, how likely is it that one sample will be within 10% of the true population parameter?

lower = pop.mu.weights - pop.mu.weights*0.05
upper = pop.mu.weights + pop.mu.weights*0.05

length(which(mu.hat > lower & mu.hat < upper)) / length(mu.hat)

[1] 0.9758

Accuracy

Typically a measure of how close the average sample value is to truth

Sampling Variation

Which sampling distribution would you prefer?

Sampling Bias

Sampling Bias
- Systematic tendency of selecting certain sample units; makes the samples unrepresentative to the target population
- Examples in fish/wildlife??