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Multiple covariate distancesampling (MCDS)
• Aim: Model the effect of additional covariates on detection probability, inaddition to distance, while assuming probability of detection at zero distance is 1
• References:
• Marques (F) and Buckland (2004) Covariate models for the detection function. Chapter 3 inBuckland et al. (eds). Advanced Distance Sampling.
• Marques (T) et al. (2007) Improving estimates of bird density using multiple covariate distancesampling. The Auk 127: 1229-1243.
• Section 5.3 of Buckland et al. (2015) Distance Sampling: Methods and Applications
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Contents
•Why additional covariates?
•Multiple covariate models
•Estimating abundance
•MCDS in Distance
2nd Lecture:
•Complications• Clusteredpopulations
• Adjustm entterm s
• S tratification
•MCDS analysis guidelines
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x
g(x)
x
g(x)
In conventional distance sampling(CDS) analysis all factors affectingdetectability, except distance, areignored
In reality, many factors mayaffect detectability
Sources of heterogeneity:
Object : species, sex, cluster size
Effort: observer, habitat, weather
Why additional covariates?
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Examples of heterogeneity 1Effect of time of day on Rufous Fantail birds in Micronesia (point transects). Ramsey et. al. 1987.Biometrics 43:1-11
x
x
g(x)
g(x)
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Examples of heterogeneity 2
Effect of sea state (and other covariates) on sea turtles in the Eastern Tropical Pacific(shipboard line transects). Beavers and Ramsey, 1998, J. Wildl. Manage. 63: 948-957
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Examples of heterogeneity 3
Effect of cluster size on beer cans. Otto and Pollock, 1990, Biometrics 46: 239-245
Gp Size=1 Gp Size=4
Gp Size=2 Gp Size=5
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HTI
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PR
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DISTANCE
DISTANCEDISTANCE
DISTANCE
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Why worry about heterogeneity?
• Pooling robustness works for all but extreme levels of heterogeneity
• Potential bias if density is estimated at a ‘lower level’ than detection function (e.g.density by geographic region, detection function global)
• Could potentially increase precision of detection function estimate
• Interest in sources of heterogeneity in their own right (e.g. group size)
In CDS, we use models that are pooling robust, so why worry about heterogeneity?
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Dealing with heterogeneityStratification
Requires estimating separate detection function parameters for each stratum, so often not possible due to lackof data
Model as covariates in detection function
Allows a more parsimonious approach:
- can model effect of numerical covariates
- can ‘share information’ about detectionfunction shapebetween covariate levels
~ 680
~ 320
~ 140
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cxpaxkm
jsjj /)()(
1
1
g(x) = Pr[animal at distance x is detected]
Key function
jth series adjustment term
Scaling constant to ensureg(0) = 1
Multiple covariate models Recap of CDS models
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CDS models continued
1)(xk
2
2
2
xxk exp)(
Key functions
Hazard rate
Half-normal
Neg. exp.
Uniform
Series adjustments
Cosine cos(jπxs)
Polynomial xsj
Hermite poly. Hj(xs)
xsare scaled distances (see later)
xxk exp)(
bx
xk
exp)( 1
Scale parameter
Shape parameter
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Modelling with covariates –ignoring adjustments terms (for now)
J
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10 exp)(
2
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2 )(exp),(
z
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)(exp),(
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g(x,z) = Pr[animal at distance xand covariates z is detected]
Assume the covariates affect the scaleof the key function, not its shape.So choose keyfunctions with a scale parameter
Let
e.g. Hazard rate
Half normal
kis used here to denote the “key” function
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Modelling with covariates
)exp(.
)exp().exp(
exp)(
)(exp),(
sAA
s
ss
s
xsxg
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Example: Dolphin tuna vessel data
Model: half-normal, with no adjustments
Covariate: cluster size, s
~ 680
~ 320
~ 140
From distance outputÂ1 = 2.331Â2 = 0.00895
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Estimating abundance without covariatesusing Horvitz-Thompson estimator
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A
Lincludedanim alN
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i 22
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Recall that f(x)= pdf of observed x’s
Because g(0)=1 by assumption, then f(0) = 1/μ
So )(ˆˆ 02
fL
nAN
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Estimating abundance with covariates
n
i i
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i i
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i zL
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zLincludedanim alN
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)(ˆ)(ˆP rˆ
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)z,(
)z,(
)z,()|(
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Because g(0,z)=1 by assumption, then )()|( zz 10 f
So
n
iif
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AN
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)|(ˆˆ z
Note similarity to CDS estimator
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MCDS in Distance
In Model Definition, choose MCDS analysis engine
See Chapter 9 of online Users Guide
Covariate type:– Factor covariates classify the data into distinct classes
or levels. Can be numerical or text. One parameterper factor level.
– Non-factor (i.e., continuous) covariates must benumerical (integer or decimal). One parameter percovariate + 1 for the intercept.