Introduction to distance sampling

• Line transects
• Simple estimates of abundance
• Why is detectability important?
• What is a detection function?
• First look at fitting models in R

• Take a sample in some fixed areas
• Find density/abundance in covered area
• Multiply up to get abundance

• Take a sample in some fixed areas
  • Sample is representative
• Find density/abundance in covered area
  • Estimator is “good”
• Multiply up to get abundance
  • Sample is representative

• We've assumed certain detection so far
• This rarely happens in the field
• Distance to the object is important
  • (Other things too, more on that later)
  • Detectability should decrease with increasing distance

Credit Scott and Mary Flanders

• Plots: what if an animal is just outside the box?

• Strips: what if an animal is just outside the strip?

• Line transects: record everything (within reason), then discard later

  • Decide strip width (truncation distance) later

• Detection function: \( \mathbb{P}(\text{ detection } \vert \text{ at distance } x) \)
• Integrate out the conditioning \( \Rightarrow \mathbb{P}(\text{ detection }) = \hat{p} \)
• “Inflate” \( n \) by \( \hat{p} \) to estimate abundance

• Surveyed 5 lines (each \( 1*0.025 \) units)
  • Total covered area \( a=5*1*0.02 = \) 0.2
• Probability of detection \( \hat{p} = \int_0^w \frac{g(x)}{w}dx= \) 0.5981
• Saw \( n= \) 60 animals
• Inflate to \( n/\hat{p}= \) 100.31
• Estimated density \( \hat{D}=\frac{n/\hat{p}}{a}= \) 502
• Total area \( A=1 \)
• Estimated abundance \( \hat{N}=\hat{D}A= \) 502

• Efficient survey design
• Relax the assumption of perfect detection
• Exchange assumptions for data
• More information = better inference

1. Animals are distributed independent of lines
2. On the line, detection is certain
3. Distances are recorded correctly
4. Animals don't move before detection

Credit Cork Whale Watch

• Model \( \mathbb{P}\left( \text{detection } \vert \text{ animal at distance } x \right) \)
• (Hence the integration)
• Many different forms, depending on the data
• All share some characteristics

• Have a “shoulder”
  • we see things nearby easily
• Monotonic decreasing
  • never increasing with increasing distance
• “Model robust”
  • lots of forms/flexible models
• “Pooling robust”
  • individual heterogeneity averages out
• “Efficient”
  • models don't need lots of parameters

• There are many options
• A restricted set we'll cover in this course…
  • Half-normal
  • Hazard-rate
  • adjustments to the above

• These models are flexible
• What about adding more flexibilty by “adjusting” them
• Options:
  • Cosine series
  • Polynomials
  • Hermite polynomials
• Add extra flexibility

1. Pick some formulations, fit models
2. Check assumptions are violated
3. Goodness of fit
4. Select models
5. Estimate \( \hat{N} \) (and uncertainty!)

• Need to have data setup a certain way
  • a data.frame with one row per observation
  • at least 2 columns, named “object” and “distance”

   distance object size SeaState
1  246.0173      1    2      3.0
2 1632.3934      2    2      2.5
3 2368.9941      3    1      3.0
4  244.6977      4    1      3.5
5 2081.3468      5    1      4.0
6 1149.2632      6    1      2.4

• Using the package Distance
• Function ds() does most of the work

library(Distance)
df_hn <- ds(distdata, truncation=6000, adjustment = NULL)
df_hr <- ds(distdata, truncation=6000, key="hr", adjustment = NULL)

summary(df_hn)

Summary for distance analysis 
Number of observations :  132 
Distance range         :  0  -  6000 

Model : Half-normal key function 
AIC   : 2252.06 

Detection function parameters
Scale Coefficients:  
            estimate         se
(Intercept) 7.900732 0.07884776

                       Estimate          SE         CV
Average p             0.5490484  0.03662569 0.06670757
N in covered region 240.4159539 21.32287580 0.08869160

• More efficient sampling
  • No census
• Collect additional information
  • Distances
• Estimate detection
• Use \( \mathbb{P}(\text{detection}) \) to correct counts

• Model checking and selection
• Estimating abundance in R
• Stratification
• What else affects detectability?