Calculate equilibrium trait values (theta) for a fitted coevfit object

Calculate equilibrium trait values \(\theta\) for one or more traits given a set of intervention values for other traits from a fitted coevfit object.

Usage

coev_calculate_theta(object, intervention_values = NULL)

Arguments

object

An object of class coevfit

intervention_values

Either NULL (the default) or a named list of variables and associated intervention values for calculating equilibrium trait values. If NULL, calculates the equilibrium states when all parameters are free to vary. Otherwise, all coevolving variables must be declared separately in the named list without repetition. If the intervention value for a particular variable is set to NA, this variable is treated as a free variable. If the intervention value for a particular variable is specified, the variable is held constant at this trait value in the calculation. At least one variable must be declared as a free variable and at least one variable must be held constant (e.g., list(var1 = NA, var2 = 0)).

Value

Posterior samples in matrix format

Details

The equilibrium trait values for freely evolving traits \(\mathbf{\eta}\) are calculated using the following formula:

$$\mathbf{\theta} = -\mathbf{A}^{-1}\mathbf{b}$$

If we hold some variables constant at some value(s) (denoted \(\eta_{h}\)) and let others evolve freely (denoted \(\eta_{f}\)), we can partition the parameters as follows:

• \(\mathbf{A}_{ff}\): selection coefficients to/from the free variables

• \(\mathbf{A}_{fh}\): selection coefficients to the free variables from the held variables

• \(\mathbf{b}_{f}\): continuous time intercepts for the free variables

We can then calculate the equilibrium values for the free variables:

$$\boldsymbol{\theta_f} = -\mathbf{A}_{ff}^{-1} \left( \mathbf{A}_{fh} \mathbf{\eta}_h + \mathbf{b}_f \right)$$

With the overall equilibrium vector being a mix of the free equilibria and the held values:

$$\boldsymbol{\theta | \mathbf{\eta}_h} = \begin{bmatrix} \boldsymbol{\theta}_f \\\mathbf{\eta}_h\end{bmatrix}$$

References

Ringen, E., Martin, J. S., & Jaeggi, A. (2021). Novel phylogenetic methods reveal that resource-use intensification drives the evolution of "complex" societies. EcoEvoRXiv. doi:10.32942/osf.io/wfp95

Sheehan, O., Watts, J., Gray, R. D., Bulbulia, J., Claessens, S., Ringen, E. J., & Atkinson, Q. D. (2023). Coevolution of religious and political authority in Austronesian societies. Nature Human Behaviour, 7(1), 38-45. 10.1038/s41562-022-01471-y

See also

coev_calculate_delta_theta, coev_plot_delta_theta

Author

Scott Claessens scott.claessens@gmail.com, Erik Ringen erikjacob.ringen@uzh.ch

Examples

if (FALSE) { # \dontrun{
# fit dynamic coevolutionary model
fit <- coev_fit(
  data = authority$data,
  variables = list(
    political_authority = "ordered_logistic",
    religious_authority = "ordered_logistic"
  ),
  id = "language",
  tree = authority$phylogeny,
  # additional arguments for cmdstanr::sample()
  chains = 4,
  parallel_chains = 4,
  seed = 1
  )

# calculate theta with no interventions
coev_calculate_theta(
  object = fit
 )

# calculate theta given intervention values
coev_calculate_theta(
  object = fit,
  intervention_values = list(
    political_authority = NA,
    religious_authority = 0
    )
  )
} # }