Package 'causl'

Description

Adjust values of copula parameters individually

Usage

adj_vars(
  cop_pars,
  strong = character(0),
  weak = character(0),
  factor = c(5, 0.2)
)

Arguments

Description

Obtain samples from a causal model using the rejection sampling approach of Evans and Didelez (2024). Kept for compatibility of simulations from that paper.

Usage

causalSamp(
  n,
  formulas = list(list(z ~ 1), list(x ~ z), list(y ~ x), list(~1)),
  pars,
  family,
  link = NULL,
  dat = NULL,
  method = "rejection",
  control = list(),
  seed
)

Arguments

Details

Samples from a given causal model using rejection sampling (or, if everything is discrete, direct sampling).

The entries for formula and family should each be a list with four entries, corresponding to the $Z$, $X$, $Y$ and the copula. formula determines the model, so it is crucial that every variable to be simulated is represented there exactly once. Each entry of that list can either be a single formula, or a list of formulae. Each corresponding entry in family should be the same length as the list in formula or of length 1 (in which case it will be repeated for all the variables therein).

We use the following codes for different families of distributions: 0 or 5 = binary; 1 = normal; 2 = t-distribution; 3 = gamma; 4 = beta; 6 = log-normal.

The family variables for the copula are also numeric and taken from VineCopula. Use, for example, 1 for Gaussian, 2 for t, 3 for Clayton, 4 for Gumbel, 5 for Frank, 6 for Joe and 11 for FGM copulas.

pars should be a named list containing: either entries z, x, y and cop, or variable names that correspond to the LHS of formulae in formulas. Each of these should themselves be a list containing beta (a vector of regression parameters) and (possibly) phi, a dispersion parameter. For any discrete variable that is a treatment, you can also specify p, an initial proportion to simulate from (otherwise this defaults to 0.5).

Link functions for the Gaussian, t and Gamma distributions can be the identity, inverse or log functions. Gaussian and t-distributions default to the identity, and Gamma to the log link. For the Bernoulli the logit and probit links are available.

Control parameters are oversamp (default value 10), trace (default value 0, increasing to 1 increases verbosity of output), max_oversamp (default value 1000), warn (which currently does nothing), max_wt which is set to 1, and increases each time the function is recalled. Control parameters also include cop, which gives a keyword for the copula that defaults to "cop".

This function is kept largely for the replication of simulations from Evans and Didelez (2024).

Value

A data frame containing the simulated data.

References

Evans, R.J. and Didelez, V. Parameterizing and simulating from causal models (with discussion). Journal of the Royal Statistical Society, Series B, 2024.

Examples

pars <- list(z=list(beta=0, phi=1),
             x=list(beta=c(0,0.5), phi=1),
             y=list(beta=c(0,0.5), phi=0.5),
             cop=list(beta=1))
causalSamp(100, family = rep(1,4), pars = pars)

Description

Copula family functions

Usage

get_copula(family_index, link)

gaussian_causl_cop(link)

t_causl_cop(link)

emp_causl_cop(link)

sim_cop(causl_copula, beta_matrix, other_pars, model_matrix)

Arguments

Details

get_copula returns the causl_copula that corresponds to the particular integer given. So far, 1 for Gaussian and 2 for t copulas are implemented.

The copula_fam functions return a list that contains, for each valid family:

• name: the name of the family;

• ddist: function to evaluate the density;

• rdist: function to obtain samples from copula;

• pdist: copula function

• pars: character vector of the parameter names;

• default: list of default values;

• link: the chosen link function.

Functions

• get_copula(): get copula family function

• gaussian_causl_cop(): Gaussian copula family

• t_causl_cop(): t copula family

• emp_causl_cop(): empirical copula family

• sim_cop(): simulate from copula family

Description

This defines a causl_model object, that can be used either for simulation or inference.

Usage

causl_model(
  formulas,
  family,
  pars,
  link,
  dat = NULL,
  method = "inversion",
  kwd = "cop",
  control = list()
)

Arguments

Details

The components formulas and family must both be specified, and have matching lengths. If pars is specified, then the model can be used for simulation and inference, if not then only for inference. link is optional, and if not specified default links will be used.

Description

Checks existence of beta vectors and then assesses appropriate length

Usage

check_rej(formulas, family, pars, dims, kwd)

check_pars(formulas, family, pars, dummy_dat, LHSs, kwd, dims)

Arguments

Functions

• check_rej(): Checks for rejection sampling

Description

Density of a multivariate copula

Usage

dGaussCop(x, Sigma, log = FALSE, use_cpp = TRUE, N)

dtCop(x, Sigma, df, log = FALSE, use_cpp = TRUE)

dfgmCopula(x, alpha, log = FALSE)

Arguments

Details

Computes the density for data from a Gaussian or t-copula. Currently use_cpp only works for dGaussCop.

Value

numeric vector of densities

Functions

• dGaussCop(): Gaussian copula

• dtCop(): t-Copula density

• dfgmCopula(): bivariate FGM copula

Description

A faster Vectorized conditional copula function where we don't build redudendant copulas if unique(eta) is small

Usage

cVCopula(U, copula, param, par2 = NULL, inverse = FALSE, cdf = FALSE)

Arguments

Details

Should have nrow(U) = length(param).

Description

Density of a Mixed Copula

Usage

dGaussDiscCop(x, m, Sigma, eta, log = FALSE, use_cpp = TRUE)

Arguments

Value

numeric vector of densities

Description

Empirical CDF

Usage

emp_cdf(xo, inv = FALSE, zero = "min")

Arguments

Details

zero can be "min" (for the minimum value of xo), "m1" (for the minimum minus 1) or "mInf" (for -Inf).

Description

Empirical copula

Usage

emp_cop(u, pts = u, smoothing = c("none", "checkerboard"))

Arguments

Description

Extract parameter estimates and standard errors

Usage

ests_ses(fit, beta, merged_formula, kwd)

Arguments

Description

Check properties of family variables

Usage

is_categorical(x)

is_discrete(x)

Arguments

Details

If the specific function does represent any family then NA is returned.

is_categorical returns a logical indicating if the object is the input object represents a categorical or ordinal variable.

is_discrete returns a logical indicating if the object is the input object represents a discrete random variable.

Functions

• is_categorical(): Check if family is categorical

• is_discrete(): Check if family is discrete

Description

Obtain list of family functions from numeric or character representation

Usage

family_list(family, func_return = get_family)

Arguments

Examples

family_list(c(1,3,5))
family_list(c("t","binomial"))

Description

Data frames containing

• val: an integer

• family: a vector giving the associated parametric family for that integer.

The integer val may be used in place of the name of the parametric family when specifying the family object.

Usage

family_vals

familyVals

copula_vals

Format

family_vals is a data.frame with 9 rows and 2 columns

familyVals is the same object as family_vals

copula_vals is a data.frame with 7 rows and 2 columns

Details

familyVals will be removed in version 1.0.0.

Functions

• familyVals: Old name

• copula_vals: Values for copula families

Description

Fit multivariate copula regression model

Usage

fit_causl(
  dat,
  formulas = list(y ~ x, z ~ 1, ~x),
  family = rep(1, length(formulas)),
  link,
  cop_pars,
  use_cpp = TRUE,
  control = list(),
  other_pars = list(),
  method = "optim"
)

fitCausal(
  dat,
  formulas = list(y ~ x, z ~ 1, ~x),
  family = rep(1, length(formulas)),
  link,
  par2,
  sandwich = TRUE,
  use_cpp = TRUE,
  control = list()
)

Arguments

Details

formulas is list of three or more formulae giving predictors of y-margin, z-margin(s) and interaction parameters. Fit is by maximum likelihood.

family takes numeric values of the following forms:

control has the same arguments as the argument in optim, as well as sandwich, a logical indicating if sandwich estimates of standard errors should be computed, newton, a logical which controls whether Newton iterates should be performed at the end, and cop which can edit the restricted variable name for the left-hand side of formulae. Useful for altering are trace (1 shows steps of optimization) and maxit for the number of steps.

cop_pars is currently just a numeric scalar, generally representing the degrees of freedom to use in a t-copula.

The list other_pars should be named with the relevant variables, and each entry should be a named list containing the relevant parameters. As an example, for a t-copula, the required parameter is the degrees of freedom, df.

Warning By default, none of the variables should be called cop, as this is reserved for the copula. The reserved word can be changed using the argument cop within control.

Value

Returns a list of class cop_fit.

Functions

• fitCausal(): old name

Description

Fit using maximum likelihood estimation

Usage

fit_optim(
  dat,
  full_form,
  family,
  fam_cop,
  link,
  mm,
  cop_pars,
  LHSs,
  other_pars,
  control,
  use_cpp
)

Arguments

Description

Tools for manipulating formulas

Usage

lhs(formulas, surv = FALSE)

lhs(formulas) <- value

rhs_vars(formulas)

tidy_formulas(formulas, kwd, prefix = "V")

Arguments

Details

lhs returns a character vector containing left-hand sides of a list of formulae. If surv=TRUE then two responses are returned in the event of the left-hand side being a valid Surv object. ⁠lhs<-⁠ allows one to assign the left-hand sides of variables in the obvious way.

tidy_formulas ensures that all formulae in a list have a left hand side, by giving them names of the form Vn where n is some positive integer. The prefix V can be changed using the argument prefix.

rhs_vars extracts all the variables used on the right-hand sides of a list of formulas.

Functions

• lhs(): Obtain left-hand sides from list of formulas

• lhs(formulas) <- value: Assign left-hand sides to list of formulas

• rhs_vars(): Extract variables from right-hand sides

• tidy_formulas(): Tidy up formulae

Description

Attempts to ensure that values after passing through the standard link function used for Gaussian copulas will have the specified value. For other copulas this will not target the correct range, but it can still be used by considering how the relevant link functions work for the Gaussian and other copula.

Usage

gen_cop_pars(formulas, data, range = c(-1, 1), ...)

Arguments

Value

A list suitable for the cop entry of the pars argument of rfrugalParam

Description

Create a dummy dataset for the purpose of checking coefficient numbers

Usage

gen_dummy_dat(family, pars, dat, LHSs, dims)

Arguments

Description

Return causl_fam function from integer index

Usage

get_family(val)

gaussian_causl_fam(link)

t_causl_fam(link)

Gamma_causl_fam(link)

binomial_causl_fam(link)

beta_causl_fam(link)

categorical_causl_fam(link)

ordinal_causl_fam(link)

Arguments

Details

The functions gaussian_causl_fam() etc. represent the functions that are returned by get_family().

A few function of this form can be defined by the user, and it should return the following:

• name: the name of the relevant family;

• ddist: a function returning the density of the distributions;

• qdist: a function returning the quantiles from probabilities;

• rdist: a function to sample values from the distribution;

• pdist: a cumulative distribution function;

• pars: a list of the names of the parameters used;

• default: a function that returns a list of the default values for an observation and each of the parameters;

• link: the specified link function.

The function should also give the output the class "causl_family", so that it is interpreted appropriately. Note that ddist should have a log argument, to allow the log-likelihood to be evaluated.

The only parameterization of the categorical family currently implemented is the multivariate logistic parameterization. For a random variable $X$ with $K$ states, dependence on a vector $\boldsymbol{Z}$ uses:

$\log \dfrac{P(X=k)}{P(X=1)} = \beta_{k}^T \boldsymbol{Z},$

and the $\beta_k$ vectors are stored as $(\beta_2,\dots,\beta_K)$.

The ordinal family is parameterized using a variation of the ordinal logistic regression model. This takes the logits of entries in the cumulative distribution function and treats the covariates variables linearly on that scale. Suppose $\boldsymbol{Z}$ is the vector of covariates and there are $K$ levels, then

$\log \dfrac{P(X \leq k)}{P(X > k)} = \beta_k^T \boldsymbol{Z}.$

As in the categorical case, the vectors $\beta_k$ are stored as $(\beta_1,\ldots,\beta_{K-1})$.

Functions

• gaussian_causl_fam(): Gaussian distribution family

• t_causl_fam(): Student's t distribution family

• Gamma_causl_fam(): Gamma distribution family

• binomial_causl_fam(): binomial distribution family

• beta_causl_fam(): beta distribution family

• categorical_causl_fam(): multinomial/categorical distribution family

• ordinal_causl_fam(): ordinal categorical distribution family

See Also

family_vals

Description

Get maximum weight for each segment of a distribution

Usage

get_max_weights(pars, forms_X, fam_X, qden, fam_Z, LHS_Z, ranges, link, ...)

Arguments

Description

Get density of treatments

Usage

get_X_density(dat, eta, phi, qden, family, link, other_par, log = FALSE)

Arguments

Value

a numeric vector of weights

Description

Get univariate densities and uniform order statistics

Usage

glm_dens(x, eta, phi, other_pars, family = 1, link)

univarDens(x, eta, phi, other_pars, family = 1, link)

Arguments

Details

fam follows the usual numeric pattern: 1=normal, 2=t-distribution and 3=Gamma with a log-link.

Value

A list with entries being the numeric vectors u (the quantiles of the input values) and ld (the log density of each observation).

Functions

• univarDens(): old name

Description

Simulate values from some generalized linear models

Usage

glm_sim(family, eta, phi, other_pars, link, quantiles = TRUE)

Arguments

Value

a numeric vector of weights

Description

Insert/remove extra list level for causl_family objects

Usage

insert_lev(x, target_class = "causl_family")

rmv_lev(x, target_class = "causl_family")

Arguments

Functions

• rmv_lev(): remove a level of listing

Description

Obtain link from causl_ objects

Usage

link(x, ...)

## S3 method for class 'causl_family'
link(x, ...)

## S3 method for class 'causl_copula'
link(x, ...)

Arguments

Methods (by class)

• link(causl_family): method for causl_family obect

• link(causl_copula): method for causl_copula object

Description

Helper function to apply link function

Usage

link_apply(eta, link, family_nm, inverse = TRUE)

Arguments

Value

inner product eta transformed by inverse of link function

Description

family and vars should have the same structure if vars is specified.

Usage

link_setup(
  link,
  family,
  vars,
  sources = links_list,
  fam_list = list(family_vals)
)

Arguments

Description

List of link functions for copuulas

Usage

links_cop(family)

Arguments

Details

This function returns the default link function for each possible copula. These are given in the table:

Description

This is a named list whose entries are character vectors of allowed link functions.

Usage

links_list

linksList

Format

An object of class list of length 8.

linksList is the old name for links_list

Functions

• linksList: Old name

Description

Log-likelihood for frugal parameterization

Usage

ll_frugal(pars, dat, formulas, family, link, kwd = "cop")

Arguments

Description

Take collection of formulae and create one formula with all variables on the right-hand side of any of the originals.

Usage

merge_formulas(formulas)

Arguments

Description

Generic for modify

Usage

modify(x, ...)

## Default S3 method:
modify(x, ...)

Arguments

Methods (by class)

• modify(default): default method

See Also

modify.causl_model()

Description

Change one or more components of a causl_model object.

Usage

## S3 method for class 'causl_model'
modify(x, over = FALSE, formulas, family, pars, link, dat, method, kwd, ...)

Arguments

Examples

cm <- causl_model(formulas = list(Z ~ 1, X~ Z, Y ~ 1, ~ 1),
                  family = list(3, 1, 1, 1),
                  pars = list(Z = list(beta = 1, phi = 1),
                              X = list(beta = c(0.5, 1)),
                              Y = list(beta = 0, phi = 1),
                              cop = list(beta = 1)))
modify(cm, pars = list(cop = list(beta = 1.5)))
cm$pars

Description

Negative log-likelihood

Usage

nll2(
  theta,
  dat,
  mm,
  beta,
  phi,
  inCop,
  fam_cop = 1,
  family,
  link,
  cop_pars = NULL,
  use_cpp = TRUE,
  other_pars = list()
)

Arguments

Details

The number of columns of beta should be the number of columns in dat plus the number required to parameterize the copula. The first few columns and the entries in phi are assumed to be in the order of those in dat. If the $i$th family for a variable does not require a dispersion parameter then the value of phi[i] is ignored.

Description

Sets up copula quantities only

Usage

pair_copula_setup(formulas, family, pars, LHSs, quans, ord)

Arguments

Description

Get parameter masks for regression parameters

Usage

par_masks(formulas, family = rep(1, nv), full_form)

Arguments

Description

Display output from causl_model

Usage

## S3 method for class 'causl_model'
print(x, ...)

Arguments

Description

Ultimately should also work for ordinal and categorical cases

Usage

process_discrete_dens(dat, family, LHSs)

Arguments

Description

New function to sort out links and families simultaneously

Usage

process_family_link(family, link, dims, func_return = get_family)

Arguments

Description

Process formulas, families and parameters

Usage

process_inputs(
  formulas,
  family,
  pars,
  link,
  dat,
  method = "inversion",
  control = list()
)

process_formulas(formulas, method, len = 4)

process_family(family, dims, link = link, func_return = get_family)

Arguments

Details

Function that processes and checks the validity of the main arguments used for simulating data. The control argument only requires values useful for obtaining the empirical (conditional) quantiles of pre-specified data.

For causl we use the get_family() function to process character based arguments, but we allow for other functions to be used in packages that build on this one.

Functions

• process_formulas(): Process input for family variables

• process_family(): Process input for family variables

Description

Obtain quantiles for prespecified variables

Usage

process_prespecified(
  dat,
  prespec,
  cond = TRUE,
  nlevs = 5,
  cor_thresh = 0.25,
  tol = sqrt(.Machine$double.eps)
)

Arguments

Details

Currently takes the rank of each entry, and subtracts 1/2 and normalizes by the number of entries. If there are $k$ ties they are randomly sorted with a uniform random variable in the symmetric interval around the rank of width $k/n$.

nlevs is used to classify discrete vs continuous variables. If a variable has more than this number of distinct values, it is considered to be continuous. cor_thresh is used to determine when the correlation is small enough that it need not be modelled. tol is for classifying which ranks are sufficiently close together to add noise as a group.

Description

Get weights for rejection sampling

Usage

rejectionWeights(dat, mms, family, pars, qden, link)

Arguments

Value

a numeric vector of weights

Description

Rescale quantiles to conditional copula

Usage

rescale_cop(U, X, family = 1, pars, df, inverse = TRUE, cdf = FALSE)

rescaleCop(U, X, beta, family = 1, df)

Arguments

Details

The variable to be transformed must be in the final column of U, with variables being conditioned upon in the earlier columns.

family can be 1 for Gaussian, 2 for t, 3 for Clayton, 4 for Gumbel, 5 for Frank, 6 for Joe and 11 for FGM copulas. pars should contain either regression parameters (labelled as beta) or a constant rank correlation tau; df must be specified if family = 2. U should have the same length as X has rows, and X should have the same number of columns as the length of pars$beta.

Value

vector of rescaled quantiles

Functions

• rescaleCop(): Old name, now deprecated

Description

Rescale quantiles to arbitrary random variable.

Usage

rescale_var(U, X, pars, family = 1, link)

rescaleVar(U, X, pars, family = 1, link)

Arguments

Details

family can be 1, 2, 3, 4 or 5 for Gaussian, t-distributed, Gamma distributed, beta distributed or discrete respectively, and 11 for ordinal variables. pars should be a list with entries beta and phi, as well as possibly df, trunc and nlevel if the family is set to 2 or 5. U should have the same length as X has rows, and X should have the same number of columns as the length of pars$beta.

Value

vector of rescaled variables

Functions

• rescaleVar(): Old name, now deprecated

Description

Obtain samples from a causal model parameterized as in Evans and Didelez (2024).

Usage

rfrugal(n, causl_model, control = list())

rfrugalParam(
  n,
  formulas = list(list(z ~ 1), list(x ~ z), list(y ~ x), list(~1)),
  family = c(1, 1, 1, 1),
  pars,
  link = NULL,
  dat = NULL,
  method = "inversion",
  control = list(),
  ...
)

Arguments

Details

Samples from a given causal model under the frugal parameterization.

The entries for formula and family should each be a list with four entries, corresponding to the $Z$, $X$, $Y$ and the copula. formula determines the model, so it is crucial that every variable to be simulated is represented there exactly once. Each entry of that list can either be a single formula, or a list of formulae. Each corresponding entry in family should be the same length as the list in formula or of length 1 (in which case it will be repeated for all the variables therein).

We use the following codes for different families of distributions:

The family variables for the copula are also numeric and taken from VineCopula. Use, for example, 1 for Gaussian, 2 for t, 3 for Clayton, 4 for Gumbel, 5 for Frank, 6 for Joe and 11 for FGM copulas.

pars should be a named list containing variable names that correspond to the LHS of formulae in formulas. Each of these should themselves be a list containing beta (a vector of regression parameters) and (possibly) phi, a dispersion parameter. For any discrete variable that is a treatment, you can also specify p, an initial proportion to simulate from (otherwise this defaults to 0.5).

Link functions for the Gaussian, t and Gamma distributions can be the identity, inverse or log functions. Gaussian and t-distributions default to the identity, and Gamma to the log link. For the Bernoulli the logit, probit, and log links are available.

A variety of sampling methods are implemented. The inversion method with pair-copulas is the default (method="inversion"), but we cam also use a multivariate copula (method="inversion_mv") or even rejection sampling (method="rejection"). Note that the inveresion_mv method simulates the entire copula, so it cannot depend upon intermediate variables.

The only control parameters are cop: which gives a keyword for the copula that defaults to "cop"; quiet which defaults to FALSE but will reduce output if set to TRUE; and (if rejection sampling is selected) careful: this logical enables one to implement the full rejection sampling method, which means we do get exact samples (note this method is generally very slow, especially if we have an outlying value, so the default is FALSE).

Value

A data frame containing the simulated data.

Functions

• rfrugalParam(): old function for simulation

Examples

pars <- list(z=list(beta=0, phi=1),
             x=list(beta=c(0,0.5), phi=1),
             y=list(beta=c(0,0.5), phi=0.5),
             cop=list(beta=1))
rfrugalParam(100, pars = pars)

Description

Sample from multivariate copulas

Usage

rGaussCop(n, Sigma)

rtCop(n, Sigma, df)

rfgmCopula(n, d = 2, alpha)

Arguments

Details

Quicker than rCopula.

Note that rfgmCopula only works for $d = 2$.

Value

A vector of the simulated random variables.

Functions

• rGaussCop(): Gaussian copula

• rtCop(): t-copula

• rfgmCopula(): FGM-copula

Description

Compute robust standard errors

Usage

sandwich_errors(func, pars, method = "simple", trace = 0L, other_args)

Arguments

Description

Simulate copula values

Usage

sim_copula(dat, family, par, df, model_matrix)

sim_CopVal(dat, family, par, df, model_matrix)

Arguments

Details

Returns data frame containing columns y and z1, ..., zk.

The family variables are numeric and taken from VineCopula. Use, for example, 1 for Gaussian, 2 for t, 3 for Clayton, 4 for Gumbel, 5 for Frank, 6 for Joe and 11 for FGM copulas.

Value

A data frame of the same dimension as dat containing the simulated values.

Functions

• sim_CopVal(): Old name, now deprecated

Description

Simulate for single time-step

Usage

sim_inversion(out, proc_inputs)

sim_multi(out, proc_inputs)

sim_rejection(out, proc_inputs, careful)

Arguments

Details

sim_inversion and sim_rejection correspond to performing the sampling by inversion or using rejection sampling.

sim_multi first simulates from the copula then transforms to the correct margins in the correct causal ordering

Functions

• sim_multi(): simulation with multivariate copula

• sim_rejection(): Rejection sampling code

Description

Each entry formulas, family, pars, link is a list with two entries, the first referring to the variable being simulated and the second to the pair-copulas being used.

If the quantile is smaller than tol or bigger than 1 - tol the associated value is moved to tol and 1 - tol respectively.

Usage

sim_variable(
  n,
  formulas,
  family,
  pars,
  link,
  dat,
  quantiles,
  tol = 10 * .Machine$double.eps
)

Arguments

Value

The data frame dat with an additional column given by the left-hand side of formula[[1]].

Description

Simulate from vine copula

Usage

sim_vinecop(dat, family, par, df = NULL, model_matrix, link)

Arguments

Value

A data frame of the same dimensions as dat.

Description

Simulate initial X values

Usage

sim_X(n, fam_x, theta, offset, sim = TRUE)

Arguments

Details

Returns a list that includes a data frame containing a column x, as well as the density that was used to generate it. Possible families are Gaussian (=1), t (=2), Exponential (=3), beta (=4) Bernoulli/categorical (=5) and log-normal (=6).

For the exponential distribution, theta is the mean. Beta can take one or two parameters, and if there is only one it is just repeated.

The offset parameter alters the median for the normal and t-distributions, or the median of the logarithm in the case of a log-normal.

Value

A list with two entries: x a vector of the simulated values, and qden, which contains a function that evaluates to the density of the distribution used to generate those values.

Description

Transform categorical or ordinal parameters into probabilities

Usage

theta_to_p_ord(theta)

theta_to_p_cat(theta)

Arguments

Details

Returns the probabilities implied by given log-linear parameters.

Functions

• theta_to_p_ord(): for ordinal variables

Description

Obtain variable ordering from formulas

Usage

var_order(formulas, dims, inc_cop = TRUE, method)

Arguments