Proportions

Proportions

Strong Interest in fish/wildlife

Examples?

Proportions

\[ y \in \{0,1\} \] \[ 1 = \text{occur} \\0 = \text{not occur} \]

. . .

\[ \hat{p} = \frac{1}{n}\sum_{i=1}^{n}y_i \]

. . .

\[ \hat{\sigma}^2_{p} = \left(\frac{N-n}{N}\right)\frac{\hat{p}(1-\hat{p})}{n-1} \]

Proportion area occupied

How does sample grid effect \(\hat{p}\) and \(\hat{\sigma}^2_{p}\)?

Proportion area occupied

How does sample grid effect \(\hat{p}\) and \(\hat{\sigma}^2_{p}\)?

Proportion area occupied

How does sample grid effect \(\hat{p}\) and \(\hat{\sigma}^2_{p}\)?

Var(p)

. . .

SD(p)

Soooooo useful!

\[ \hat{\sigma}_{p} = \sqrt\frac{\hat{p}(1-\hat{p})}{n-1} \]

. . .

  • how many fish should I tag?
  • how many elk should I track?
  • how many point intercepts (plant survey) should I take?
  • how many plots (camera traps / plant survey) should I sample?

SD(p)

Lets visualize a few standard deviations of p

n, p, sd(p)

My Wolverine ‘Problem’

A Sampling ‘Problem’

  • Sites surveyed by the States were selected via SRS


  • a handful of additional sites were surveyed by other entities (e.g., NGO) using the same field protocol

. . .

Should I include the additional/ancillary surveys as data?

Should I report the cells with occurrences at all?

. . .

In small groups: Discuss and then tell me what to do and explain.

Simulation

What if?

  • N = 600
  • p = 0.5
  • SRS of n = 200
  sites occu
1     1    1
2     2    1
3     3    0
4     4    0
5     5    0
6     6    1

SRS

Sampling Distribution of \(\hat{p}\)

Estimator Bias = -0.00005
Distribution Var = 0.00084

Ancillary Occurences

SRS of n = 200 with 5% ancillary and occupied sites

Estimator Bias = 0.02382
Distribution Var = 0.00073

Ancillary Occurences

SRS of n = 200 with 10% ancillary and occupied sites

Estimator Bias = 0.04536
Distribution Var = 0.00069

My Wolverine ‘Problem’

Should I include the ancillary surveys as data?

Should I report the cells with occurrences at all?

Unequal Probability of Sampling

Instead of adding sites w/o clear sampling probabilities, we make these explicit.

Identify sites that are important and increase their probability of being selected for sampling

Unequal Probability of Sampling

  • 1/2 sites with 3x the probability of being sampled versus other half

  • sample function has an argument prob

. . .

\[ \hat{p} = \frac{1}{n}\sum_{i=1}^{n}y_i \]

. . .

Estimator Bias = 0.13753

Unequal Probability of Sampling

Generalized Unequal Probability Estimator

Thompson Ch. 6.3

\[ \hat{\theta} = \frac{\sum_{i} y_i / \pi_i}{\sum_{i}1 / \pi_i} \]

  • \(\pi_{i}\) = probability of selecting sample \(i\).

. . .

Estimator Bias = -0.03147

“not precisely unbiased” - Thompson