• Identify features of data that drive analyses

• Think critically about what the data could tell you

• Start thinking about the distributional forms for data

Question \(\rightarrow\) Data \(\rightarrow\) Model \(\rightarrow\) Inference \(\rightarrow\) Prediction

Question \(\rightarrow\) Data \(\rightarrow\) Model \(\rightarrow\) Inference \(\rightarrow\) Prediction

Ideas from the audience?

Sex?

Fecundity?

Growth?

Survival?

Movement?

Thoughts from you all?

Abundance?

Survival?

Spatial distribution?

Movement/migration?

Question \(\rightarrow\) Data \(\rightarrow\) Model \(\rightarrow\) Inference \(\rightarrow\) Prediction

Ideas, anyone?

1 Detection \(\rightarrow\) presence/absence

2+ Detections \(\rightarrow\) survival, movement

1 Detection \(\rightarrow\) presence/absence

2+ Detections \(\rightarrow\) survival, movement

1 Measurement \(\rightarrow\) fecundity, age, size

2+ Measurements \(\rightarrow\) growth

Detections \(\rightarrow\) presence/absence

Counts \(\rightarrow\) density or survival/movement

Can you see where this is going?

Nonexhaustive counts

Exhaustive counts

(Non)exhaustive surveys

Depletions

(Non)exhaustive surveys

Depletions

Capture/Tag/Recapture

Anyone have thoughts here?

Sex

Age

Fecundity

Counts/Census

Individual survival

How about this?

Size (length, mass)

Biomass

Survival (population)

Survival (0.78)

Composition (~55% age-4)

Density (0.14 per ha)

Survival (7 of 9 survived \(\approx\) 0.78)

Composition (4 age-3, 18 age-4, 11 age-5 \(\rightarrow\) ~55% age-4)

Density (3 animals in 21 ha plot \(\approx\) 0.14 per ha)

Which of these give you more confidence?

A) 3 / 9 \(\approx\) 0.33

B) 300 / 900 \(\approx\) 0.33

Binary (0,1) \(\rightarrow\) Bernoulli

Binary (0,1) \(\rightarrow\) Bernoulli

Classic “coin toss”

set.seed(514)
## flip a coin 1000 times
x <- rbinom(n = 1000, size = 1, prob = 0.5)
## count tails (0) and heads (1)
table(x)

## x
##   0   1 
## 511 489

Count (non-negative integers) \(\rightarrow\) Poisson or Negative-Binomial

Count (non-negative integers) \(\rightarrow\) Poisson or Negative-Binomial

Number of times birds were resighted

set.seed(514)
## 37 banded birds; number of resightings
x <- rpois(n = 37, lambda = 1)
## count up resightings
table(x)

## x
##  0  1  2  3  4  5 
##  7 18  5  5  1  1

Composition (D categories) \(\rightarrow\) Binomial (D = 2) or Multinomial (D > 2)

Composition (D categories) \(\rightarrow\) Binomial (D = 2) or Multinomial (D > 2)

Surviving offspring

set.seed(514)
## 1357 tagged smolts; probability of survival is 0.05
x <- rbinom(n = 1357, size = 1, prob = 0.05)
## number of mortalities (0) and survivors (1)
table(x)  # p ~= 0.0593

## x
##    0    1 
## 1281   76

Real numbers (-\(\infty\),\(\infty\)) \(\rightarrow\) Normal

Real numbers (-\(\infty\),\(\infty\)) \(\rightarrow\) Normal

Common “bell-shaped curve”

## normal distribution (mean = 0, sd = 1)
curve(dnorm(x, mean = 0, sd = sqrt(1)), from = -4, to = 4,
      main = "", xlab = "", ylab = "Density", bty = "n",
      cex.lab = 1.5, cex.axis = 1.2, lwd = 2, col = "#32006e")

Density (non-negative real values) \(\rightarrow\) log-Normal or Gamma

Density (non-negative real values) \(\rightarrow\) log-Normal or Gamma

## log-normal distribution (mean = 2, sd = 0.7)
curve(dlnorm(x, meanlog = 2, sdlog = 0.7), from = 0, to = 20,
      main = "", xlab = "", ylab = "Density", bty = "n",
      cex.lab = 1.5, cex.axis = 1.2, lwd = 2,  col = "#32006e")

Proportion (simplex\(^D\)) \(\rightarrow\) Beta (D = 2) or Dirichlet (D > 2)

## beta distribution (mean = 2/3)
curve(dbeta(x, shape1 = 6, shape2 = 3), from = 0, to = 1,
      main = "", xlab = "", ylab = "Density", bty = "n",
      cex.lab = 1.5, cex.axis = 1.2, lwd = 2,  col = "#32006e")