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Analysis of Stratified Surveys
Section 3.7 of Buckland et al. (2001)
Section 2.3 of Buckland et al. (2015)
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Stratification• Why stratify?
• Stratification by:
• Geographic area• Survey• Species / cluster size
• Limitations of Distance
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Stratification is used to:• reduce variance and improve precision• and for producing estimates in regions of interest
Stratify by:• AREA or GEOGRAPHIC REGION
- the study region is partitioned into smaller regions
• SURVEY- used when different surveys cover the same geographic area
• POPULATION/SPECIES/CLUSTER SIZE- same geographic region with different ‘sub-stocks’ in it
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Area/Geographic stratification
Estimate density in each sub-region
321 AAAA
A1
A2
A3
321 DDD ˆ,ˆ,ˆ
333
222
111
DAN
DAN
DAN
ˆˆ
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Total size of study region
Abundance in each sub-region is given by
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ii
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NNNN
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Total abundance is Overall (Global in Distance) density is
Note form of equation
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Example:SCANS III (2016)Small Cetaceans inEuropean Atlantic watersand the North Sea
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Example of stratified data
Region.Label Area Sample.Label Effort distance size1 Ideal 84734 1 86.75 0.10 12 Ideal 84734 1 86.75 0.22 13 Ideal 84734 1 86.75 0.16 3: : : : : : :45 Ideal 84734 13 75.63 0.03 146 Marginal 630582 14 83.33 0.49 147 Marginal 630582 14 83.33 1.94 148 Marginal 630582 14 83.33 1.10 1: : : : : : :99 Marginal 630582 25 107.72 NA 1
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Example: Full geographic stratification
D
Pa1 Pa2 Pa3
E1(s) E2(s) E3(s)
(n/L)1 (n/L)2 (n/L)3
ideal.dat <- whales[whales$Region.Label==“Ideal”, ]whales.ideal <- ds(data=ideal.dat, key=“hr”)
Select strata and fit ds to each strata
E.g. (many ways to perform selection)
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Example: Pa pooledD
Pa123
E1(s) E2(s) E3(s)
(n/L)1 (n/L)2 (n/L)3
Data contains different strata in Region.LabelThe ds function performs this stratification by default
whale.pool <- ds(whales, key="hr”)
E.g. Part of output from summary(whale.pool)
Summary of clusters …
Abundance:Label Estimate se cv lcl ucl df
1 Marginal 12181.313 4638.5533 0.3807926 5499.920 26979.371 12.967422 Ideal 3653.313 910.0737 0.2491091 2181.589 6117.879 17.961423 Total 15834.626 4834.1865 0.3052921 8388.389 29890.769 15.25427
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Doing the same thing withdht2
The same stratification (i.e. pooled Pa) can be achieved with:
dht2(model=whale.pool, flatfile=whales,
strat_formula=~Region.Label,
stratification=“geographical”)
The dht2 function can be used for more complicated stratification (as we will see later)
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Pooled vs Stratified Pa StratifiedIdeal habitat n=39
Marginal habitat n=49
Pooled n=88
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It is a Model Selection Problem
Criterion for stratification of Pa:Fit separate Pa for each strata if
stratastratumpooled
AICAIC
Pooled Stratum 1 Stratum 2 StratumSum
Log likelihood loge(L) -180.490 -72.699 -104.676 -177.375
No. parameters (q) 2 2 2 4
AIC 364.980 149.398 213.352 362.75
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Non-geographic stratification -- Stratification by survey
Survey 1
Survey 2
ii
i DL
L
DLL
LD
LL
LD
ˆ
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2
1
2
21
21
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1
Let Li be effort for survey i
Global density is given by
This is the same form as before, but weightingfactor now depends on effort
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Stratification by survey
Need to use the dht2 function:
dht2(model=whale.pool, flatfile=whales,
strat_formula=~Region.Label,
stratification=“replicate”)
This represents different surveys
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Stratification by species
21 spsp DDD ˆˆˆ
Species 1
Species 2
dht2(model=whales.pool,flatfile=whales,strat_formula=~species,stratification=“object”)
Need this columnin data
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When to usedht2 for density or abundance estimates
• Use density or abundance estimates from ds when:
- no stratification
- simple geographical stratification (i.e. specified in Region.Label)
- no multipliers/cue counts
- Typical encounter rate variance estimatorsRather than the estimators used to produce better variance estimates undersystematic surveys (from precision lecture)
• Call dht2 after detection function fitting with ds in other situations
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Limitations in Distance
• Distance cannot currently do multilevel stratificationin one command
• Two runs are necessary– Estimate Pa, E[s] and n/L by stratum
– Combine strata 1 and 2 to estimate Pa12
• Care must be taken when calculating CVs because thedensity estimates for stratum 1 and 2 have anestimated Pa in common
D
Pa12 Pa3
E1[s] E2[s] E3[s]
(n/L)1 (n/L)2 (n/L)3
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Alternatives to stratification in Distance
• Small sample sizes can lead to low precision in stratum-specific estimates
• An alternative approach to reducing bias due to heterogeneity is Multiple CovariatesDistance Sampling (MCDS)
• Covariates, other than distance, are incorporated into the scale parameter of thedetection function
• MCDS can be used to fit the detection function at multiple levels e.g. stratum-specificdensity estimates can be obtained even if you don't have enough data to fit separatedetection functions for each stratum
• MCDS methods are covered in the next lecture.
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Analysis of Populationsin Clusters
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Mean cluster size estimation
No Size Bias
• Mean of observed sizes does not change with distance
Size Bias
• Smaller clusters less detectable at larger distances
• Mean observed cluster size increases with distance
Distance (x)
Clu
ster
size
(s)
Distance (x)
Clu
ster
size
(s)
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If size bias is present, will be positively biased:ssE )(ˆ
Observed mean
True mean
Distance (x)
Clu
ster
size
(s)
Effect of size bias on sample mean
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Remedy to size bias
• Recognize that detection of cluster• Depends upon cluster size
• Model the dependence in the detection function• using covariates in the detection function
• Details in the next lecture