Gaussian Observation Model

Gaussian Observation Model

• Used for normally-distributed continuous data like temperature (which ranges from zero to infinity)
• Can also be used for positive continuous data like length and weight if the values are large enough that the probability of $y_i<0$ is negligible
  • If the values are very small, and your model has some probability of a zero or negative weight/length, then this is a problem

Parameter range

• $y_i\in (-\mathrm{\infty },\mathrm{\infty })$
• ${\mu }_i\in (-\mathrm{\infty },\mathrm{\infty })$
• ${\sigma }^2\in (0,\mathrm{\infty })$

Log-Gaussian

Log-Gaussian observation model

• Gaussian Observation Model with a log transform (link function): $\log y_i\sim N({\mu }_i,{\sigma }^2)$
• works better for continuous postitive data that is close to zero, since it prevents the model from predicting any zero vales in the resposnse
  • For example distance or biomass data which is often close to zero

Parameter range

Same as Gaussian, but $y_i\in (0,\mathrm{\infty })$ (must be positive)

Poisson observation model

Poisson Observation Model

• Used for discrete count data with a density ($\lambda$)
• Since it uses density, this works for different transect lenghts (representing sampling effort)
• Use a $\log ({\lambda }_i)$ as a link function back to "real space"
• Examples of data - counts of individuals, numbers of incidents, species richness

Parameter Range

• $y_i\in \{0,1,2,\dots \}$ (postitive integers)
• ${\lambda }_i\in (0,\mathrm{\infty })$

Negative-Binomial

Negative-Binomial Observation Model

• Basically a combination of Poisson Observation Model with Random Effects. The random effect here is called the "overdispersion parameter" $r$
• It's used when you have data where the variance exceeds the mean - e.g. data with lots of zeroes that bring down the mean
  • The overdispersion parameter accounts for this

Binomial observation model

Binomial observation model

• Modeling the number of successes based on the sample size and probability of success
• Used for:
  • number of successes in $n_i$ experiments
  • Number of sampled animals that test positive for a disease
• Uses a logit function as link
• Note that offset isn't easily applicable to Binomial or Bernouilli models (e.g. sampling effort) - best thing to do is to add a covariate in your GLM for this

Parameter range

• $n_i\in \{0,1,2,\dots \}$
• $y_i\in \{0,1,2,\dots ,n_i\}$
  • The number of successes can't exceed the number of trials
• ${\theta }_i\in (0,1)$

Bernoulli observation model

Bernoulli observation model

• Like a binomial model where $n=1$
• Used for binary data ($y_i\in \{0,1\}$) based on a probability ($\theta$) of success - for example:
  • the probability of presence/absense of species in sampling sites
  • The probability of success/failure in a single experiment
• Uses a logit function as a link:

Parameter range

• $y_i\in \{0,1\}$
• ${\theta }_i\in (0,1)$

Hurdle Observation Models

Hurdle Observation Models

• A family of models that really use two observation models:
  • One model if $y_i=0$
    • $p_H(y_i|{\theta }_i)=1-{\theta }_i$
  • A different model if $y_i>0$
    • Depends on the second model used - can be any other kindof model
• This is another wat to deal with zero-inflated data
• For example, if you are measuring biomass when there is a chance that biomass will be zero, you can use a Hurdle log-Gaussian Model: