Description Usage Arguments Details Value References See Also Examples

Metropolis Coupled Markov Chain Monte Carlo (MC3) Sampling for the GGUM

1 2 3 4 5 6 7 8 9 | ```
ggumMC3(data, sample_iterations = 10000, burn_iterations = 10000,
sd_tune_iterations = 5000, temp_tune_iterations = 5000,
temp_n_draws = 2500, swap_interval = 1, flip_interval = NA,
n_temps = length(temps), temps = NULL, optimize_temps = TRUE,
temp_multiplier = 0.1, proposal_sds = NULL, theta_init = NULL,
alpha_init = NULL, delta_init = NULL, tau_init = NULL,
theta_prior_params = c(0, 1), alpha_prior_params = c(1.5, 1.5, 0.25,
4), delta_prior_params = c(2, 2, -5, 5), tau_prior_params = c(2, 2,
-6, 6), return_sds = TRUE, return_temps = TRUE)
``` |

`data` |
An integer matrix giving the individuals' responses;
note the item options should be of the form 0, 1, ...
(an example of preparing data for analysis is given in the vignette,
available via |

`sample_iterations` |
An integer vector of length one giving the number of iterations the sampler should complete (default is 10000) |

`burn_iterations` |
An integer vector of length one giving the number of iterations to burn in (default is 10000) |

`sd_tune_iterations` |
An integer vector of length one; the number of iterations to use to tune the proposals before the burn-in period begins (default is 5000). If 0 is given, the proposals are not tuned. |

`temp_tune_iterations` |
An integer vector of length one; if a
temperature schedule is not provided in the |

`temp_n_draws` |
An integer vector of length one; if a temperature
schedule is not provided in the |

`swap_interval` |
The period by which to attempt chain swaps; e.g. if swap_interval = 100, a state swap will be proposed between two adjacent chains every 100 iterations (default is 1) |

`flip_interval` |
(Optional) If given, provides the number of iterations
after which the sign of the thetas and deltas should be changed.
For example, if |

`n_temps` |
The number of chains; should only be given if |

`temps` |
(Optional) A numeric vector giving the temperatures;
if not provided and |

`optimize_temps` |
A logical vector of length one; if TRUE and a
temperature schedule is not provided in the |

`temp_multiplier` |
A numeric vector of length one; if a temperature
schedule is not provided and |

`proposal_sds` |
(Optional) A list of length four where is element is a numeric vector giving standard deviations for the proposals; the first element should be a numeric vector with a standard deviation for the proposal for each respondent's theta parameter (the latent trait), the second a vector with a standard deviation for each item's alpha (discrimination) parameter, the third a vector with a standard deviation for each item's delta (location) parameter, and the fourth a vector with a standard deviation for each item's tau (option threshold) parameters. If not given, the standard deviations are all set to 1.0 before any tuning begins. |

`theta_init` |
(Optional) Either a numeric vector giving an initial value for each respondent's theta parameter, or a numeric matrix giving an initial value for each respondent's theta parameter for each parallel chain; if not given, the initial values are drawn from the prior distribution |

`alpha_init` |
(Optional) Either a numeric vector giving an initial value for each item's alpha parameter, or a numeric matrix giving an initial value for each item's alpha parameter for each parallel chain; if not given, the initial values are drawn from the prior distribution |

`delta_init` |
(Optional) Either a numeric vector giving an initial value for each item's delta parameter, or a numeric matrix giving an initial value for each item's delta parameter for each parallel chain; if not given, the initial values are drawn from the prior distribution |

`tau_init` |
(Optional) Either a list giving an initial value for each item's tau vector, or a list of lists giving an initial value for each item's tau vector for each parallel chain; if not given, the initial values are drawn from the prior distribution |

`theta_prior_params` |
A numeric vector of length two; the mean and standard deviation of theta parameters' prior distribution (where the theta parameters have a normal prior; the default is 0 and 1) |

`alpha_prior_params` |
A numeric vector of length four; the two shape parameters and a and b values for alpha parameters' prior distribution (where the alpha parameters have a four parameter beta prior; the default is 1.5, 1.5, 0.25, and 4) |

`delta_prior_params` |
A numeric vector of length four; the two shape parameters and a and b values for delta parameters' prior distribution (where the delta parameters have a four parameter beta prior; the default is 2, 2, -5, and 5) |

`tau_prior_params` |
A numeric vector of length four; the two shape parameters and a and b values for tau parameters' prior distribution (where the tau parameters have a four parameter beta prior; the default is 2, 2, -6, and 6) |

`return_sds` |
A logical vector of length one; if TRUE, the proposal standard deviations are stored in an attribute of the returned object named "proposal_sds." The default is TRUE. |

`return_temps` |
A logical vector of length one; if TRUE, the temperatures of the parallel chains are stored in an attribute of the returned object named "temps." The default is TRUE. |

`ggumMC3`

provides `R`

implementation of the MC3 algorithm
from Duck-Mayr and Montgomery (2019).
Some details are provided in this help file, but please see the vignette
(via `vignette("bggum")`

) for a full in-depth practical guide to
Bayesian estimation of GGUM parameters.

Our sampler creates random initial values for the parameters of the model,
according to their prior distributions.
N parallel chains are run, each at a different inverse "temperature";
the first "cold" chain has an inverse temperature of 1, and each subsequent
chain has increasingly lower values (still greater than zero,
i.e. fractional values).
At each iteration, for each chain, new parameter values are proposed
from a normal distribution with a mean of the current parameter value,
and the proposal is accepted probabilistically using a
Metropolis-Hastings acceptance ratio.
The purpose of the chains' "temperatures" is to increase the probability of
accepting proposals for chains other than the "cold" chain recorded for
inference; the acceptance probability in the Metropolis-Hastings
update steps for parameter values are raised to the power of the chain's
inverse temperature.
After every `swap_interval`

th iteration of the sampler, a proposal is
made to swap states between adjacent chains as a Metropolis step.
For details, please read the vignette via `vignette("bggum")`

,
or see Duck-Mayr and Montgomery (2019);
see also Gill (2008) and Geyer (1991).

Before burn-in, the standard deviation of the proposal densities can be
tuned to ensure that the acceptance rate is neither too high nor too low
(we keep the acceptance rate between 0.2 and 0.25).
This is done if proposal standard deviations are not provided as an argument
and `sd_tune_iterations`

is greater than 0.

The temperature schedule can also be tuned using an implementation of the
temperature tuning algorithm in Atchadé, Roberts, and Rosenthal (2011).
This is done if a temperature schedule is not provided as an argument and
`optimize_temps = TRUE`

.
If a temperature schedule is not provided and `optimize_temps = FALSE`

,
each temperature T_t for t > 1 is given by
1 / (1 + `temp_multiplier`

* (t-1)), and T_1 = 1.

A numeric matrix giving the parameter values at each iteration
for the cold chain.
The matrix will additionally have classes "ggum"
(so that `summary.ggum`

can be called on the result)
and "mcmc" with an "mcpar" attribute
(so that functions from the `coda`

package can be used, e.g.
to assess convergence).
If `return_sds`

is `TRUE`

, the result also has an attribute
"proposal_sds", which will be a list of length four giving the standard
deviations of the proposal densities for the theta, alpha, delta, and
tau parameters respectively.
If `return_temps`

is `TRUE`

, the result also has an attribute
"temps", which will be a numeric vector giving the parallel chains'
inverse temperatures.

Atchadé, Yves F., Gareth O. Roberts, and Jeffrey S. Rosenthal.
2011. “Towards Optimal Scaling of Metropolis-Coupled Markov Chain
Monte Carlo.” *Statistics and Computing* 21(4): 555–68.

Duck-Mayr, JBrandon, and Jacob Montgomery. 2019. “Ends Against the Middle: Scaling Votes When Ideological Opposites Behave the Same for Antithetical Reasons.” http://jbduckmayr.com/papers/ggum.pdf.

Geyer, Charles J. 1991. “Markov Chain Monte Carlo Maximum Likelihood.” In Computing Science and Statistics. Proceedings of the 23rd Symposium on the Interface, edited by E. M. Keramides, 156–63. Fairfax Station, VA: Interface Foundation.

Gill, Jeff. 2008. *Bayesian Methods: A Social and Behavioral
Sciences Approach*. 2d ed. Boca Raton, FL: Taylor & Francis.

`ggumProbability`

, `ggumMCMC`

,
`tune_temperatures`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 | ```
## NOTE: This is a toy example just to demonstrate the function, which uses
## a small dataset and an unreasonably low number of sampling interations.
## For a longer practical guide on Bayesian estimation of GGUM parameters,
## please see the vignette ( via vignette("bggum") ).
## We'll simulate data to use for this example:
set.seed(123)
sim_data <- ggum_simulation(100, 10, 2)
## Now we can generate posterior draws:
## (for the purposes of example, we use 100 iterations,
## though in practice you would use much more)
draws <- ggumMC3(data = sim_data$response_matrix, n_temps = 2,
sd_tune_iterations = 100, temp_tune_iterations = 100,
temp_n_draws = 50,
burn_iterations = 100, sample_iterations = 100)
``` |

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