Concept

Can we improve on an unbiased estimator using SRS?


We can improve our parameter variances!


How? Break our sampling frame into homogeneous parts.

Concept

  • define strata (singular, stratum)
  • randomly sample within ALL strata
  • try to create homogenous strata
  • ideally a higher sample size within strata with higher variance

Concept

  • auxiliary information
  • depends on ability to select beneficial strata
  • almost always going to improve precision; unlikely to do worse
    • allocate more samples to a stratum to lower the overall variance

What are strata used in fish/wildlife studies?

Estimator

We sample \(y_{ih}\) within strata \(h\) from 1 … \(L\) and units \(i\) from 1 … \(n_h\).

\[ \bar{y}_{h} = \hat{\mu}_{h} = \frac{1}{n_h} \sum_{i=1}^{n_h} y_{hi} \]


\[ \hat{\mu}_{st} = \frac{1}{N} \sum_{h=1}^{L} N_{h}\hat{\mu}_{h} \]

Estimator

\[ \hat{\sigma}^2_{\hat{\mu},h} = \frac{1}{n_h -1} \sum_{i=1}^{n_h}\left(y_{hi}-\hat{\mu}_{h}\right)^2 \]


\[ \hat{\sigma}^2_{\hat{\mu},st} = \sum_{h=1}^{L} \left(\frac{N_h}{N}\right)^2 \frac{N_h-n_h}{N_h}\frac{\hat{\sigma}^2_{\hat{\mu},h}}{n_h} \]

Boreal Toad (Case Study)

Goal: to know the mean number of boreal toad egg masses per pond in RMNP


Goal 2: Use stratification to reduce the sampling variance.

Boreal Toad (Case Study)

  • \(N = 6\) ; \(L = 2\)
  • \(N_{h} = N/L = 3\)
  • \(n_{h} = 2\)
  • \(\mu = 8\)
Pond egg.mass strata
A 2 1
B 6 1
C 8 1
D 10 2
E 10 2
F 12 2

Boreal Toad (Case Study)

How may unique combinations?

First strata

N_h = 3; # num. of strata
n_h = 2 # sample size of units within strata
choose(N_h,n_h)
[1] 3

Second strata

N_h=3; # num. of strata
n_h = 2 # sample size of units within strata
choose(N_h,n_h)
[1] 3

All Combinations

choose(N_h,n_h)*choose(N_h,n_h)
[1] 9

Sample S1.1 S1.2 S2.1 S2.2 Mean.S1 Mean.S2 Var.S1 Var.S2
1 A B D E 4 10 8 0
2 A B D F 4 11 8 2
3 A B E F 4 11 8 2
4 A C D E 5 10 18 0
5 A C D F 5 11 18 2
6 A C E F 5 11 18 2
7 B C D E 7 10 2 0
8 B C D F 7 11 2 2
9 B C E F 7 11 2 2



\[ \hat{\sigma}^2_{\hat{\mu},st} = \sum_{h=1}^{L} \left(\frac{N_h}{N}\right)^2 \frac{N_h-n_h}{N_h}\frac{\hat{\sigma}^2_{h}}{n_h} \]

S1.1 S1.2 S2.1 S2.2 Mean.S1 Mean.S2 Var.S1 Var.S2 Var.mean
A B D E 4 10 8 0 0.33
A B D F 4 11 8 2 0.42
A B E F 4 11 8 2 0.42
A C D E 5 10 18 0 0.75
A C D F 5 11 18 2 0.83
A C E F 5 11 18 2 0.83
B C D E 7 10 2 0 0.08
B C D F 7 11 2 2 0.17
B C E F 7 11 2 2 0.17


Stratifed

E[Sampling Distribution Variance] = 0.44


SRS

E[Sampling Distribution Variance] = 4.26

Boreal Toad (Case Study)

Boreal Toad (Case Study)

Boreal Toad (Case Study)

Boreal Toad (Case Study)

Stratification Allocation

Sample Size per Strata

  • uniform distribution; \(n_1 = 2 ; n_2 = 2\)
  • variable distribution; e.g., \(n_1 = 3 ; n_2 = 1\)
    • allocation proportional to size (e.g. area): \(n_h = \frac{nN_h}{N}\)
    • optimal allocation for fixed \(n\): \(n_h = \frac{nN_h\sigma_h}{\sum_{k=1}^L N_k\sigma_k}\)

Stratification Allocation

Allocate most of our samples to the strata with the highest variance

Pond egg.mass strata
A 2 1
B 6 1
C 8 1
D 10 2
E 10 2
F 12 2

How many possible sample combinations are there?

Stratification Allocation

Sample S1.1 S1.2 S1.3 S2.1 Mean.S1 Mean.S2 pop.means
1 A B C D 5.333333 10 7.666667
2 A B C E 5.333333 10 7.666667
3 A B C F 5.333333 12 8.666667
  • removed the variance from stratum 1

Stratification Allocation

Reverse the situation- allocate more samples to the least variable stratum

Pond egg.mass strata
A 2 1
B 6 1
C 8 1
D 10 2
E 10 2
F 12 2
Sample S1.1 S2.1 S2.2 S2.3 Mean.S1 Mean.S2 pop.means
1 A D E F 2 10.66667 6.333333
2 B D E F 6 10.66667 8.333333
3 C D E F 8 10.66667 9.333333
  • compared to population means of 7.6, 7.6, and 8.6
  • Estimator population means is still unbisaed
    • \(E[\hat{\mu}_{st}] = (6.333 + 8.333 + 9.333) / 3 = 8 = \mu\)

SRS Estimator

What if we ignored the stratification and used the SRS sample mean estimator?

S1 S2 S3
2 6 8
10 10 10
10 10 10
12 12 12
  • Sample means are 8.5, 9.5, 10

  • Population mean is 9.3333333

  • \(E[\hat{\mu}_{SRS}] \neq \mu\)

Unequal Sampling Fractions

Pond egg.mass strata
A 2 1
B 6 1
C 8 1
D 10 2
E 10 2
F 12 2
  • Strata 1, \(\text{weight}_{i}\) = \(1/n_h = 1/1 =1\)
  • Strata 2, \(\text{weight}_{i}\) = \(1/n_h = 1/3\)

Unequal Sampling Fractions

\[ \hat{\mu}_{st} = \frac{1}{L}\sum_{h=1}^L \sum_{i=1}^{n_h} y_{hi}\times \text{weight}_h \]

Strata S1 S2 S3 Weight
1 2 6 8 1.0000000
2 10 10 10 0.3333333
2 10 10 10 0.3333333
2 12 12 12 0.3333333

Sample 1

\[ \hat{\mu}_{st} = \frac{\left(2\times1\right) + \left(10\times1/3 + 10\times1/3 + 12\times1/3 \right)}{2} \]

Pseudo Random Sampling

  • In practice random sampling can be expensive; can feel wasteful
  • Stratification improves the ‘targetedness’ of sampling
  • Sometimes the samples per strata are random but then the selection of individuals or specific subplots within sampled strata are left to the observers to choose. (Quota Sampling)

Quota Sampling

Observer Freedom

  • selection bias is present
  • selection procedure is ill-defined; Standard Error’s have no valid estimator
  • “But quota sampling always achieves the same sample size [within strata] as random sampling”

Quota Sampling

  • The increased flexibility comes at a reduction of sample credibility
  • Selection bias
  • Non-response bias
  • No valid measures of uncertainty

Summary

  • Units need a probability of being sampled but don’t need to be the same
  • Stratification requires accounting for within strata sample size (i.e., the weighting)
  • Stratification allocations allow us a means to reduce the population sampling distribution variance
  • Sampling an entire strata is perfectly fine
  • Construct strata so their averages are different as possible and their variances are as small as possible (i.e., approx. homogenous)
  • Need at least \(n_h \geq 2\) to estimate variability in a stratum