overdispersed {networkscaleup}

Fit Overdispersed model to ARD (Gibbs-Metropolis)

Description

This function fits the ARD using the Overdispersed model using the original Gibbs-Metropolis Algorithm provided in Zheng, Salganik, and Gelman (2006). The population size estimates and degrees are scaled using a post-hoc procedure. For the Stan implementation, see overdispersedStan.

Usage

overdispersed(
  ard,
  known_sizes = NULL,
  known_ind = NULL,
  G1_ind = NULL,
  G2_ind = NULL,
  B2_ind = NULL,
  N = NULL,
  warmup = 1000,
  iter = 1500,
  refresh = NULL,
  thin = 1,
  verbose = FALSE,
  alpha_tune = 0.4,
  beta_tune = 0.2,
  omega_tune = 0.2,
  init = "MLE"
)

Arguments

ard

The 'n_i x n_k' matrix of non-negative ARD integer responses, where the '(i,k)th' element corresponds to the number of people that respondent 'i' knows in subpopulation 'k'.

known_sizes

The known subpopulation sizes corresponding to a subset of the columns of ard.

known_ind

The indices that correspond to the columns of ard with known_sizes. By default, the function assumes the first n_known columns, where n_known corresponds to the number of known_sizes.

G1_ind

A vector of indices denoting the columns of 'ard' that correspond to the primary scaling groups, i.e. the collection of rare girls' names in Zheng, Salganik, and Gelman (2006). By default, all known_sizes are used. If G2_ind and B2_ind are not provided, 'C = C_1', so only G1_ind are used. If G1_ind is not provided, no scaling is performed.

G2_ind

A vector of indices denoting the columns of 'ard' that correspond to the subpopulations that belong to the first secondary scaling groups, i.e. the collection of somewhat popular girls' names.

B2_ind

A vector of indices denoting the columns of 'ard' that correspond to the subpopulations that belong to the second secondary scaling groups, i.e. the collection of somewhat popular boys' names.

N

The known total population size.

warmup

A positive integer specifying the number of warmup samples.

iter

A positive integer specifying the total number of samples (including warmup).

refresh

An integer specifying how often the progress of the sampling should be reported. By default, resorts to every 10 verbose = FALSE.

thin

A positive integer specifying the interval for saving posterior samples. Default value is 1 (i.e. no thinning).

verbose

Logical value, specifying whether sampling progress should be reported.

alpha_tune

A positive numeric indicating the standard deviation used as the jumping scale in the Metropolis step for alpha. Defaults to 0.4, which has worked well for other ARD datasets.

beta_tune

A positive numeric indicating the standard deviation used as the jumping scale in the Metropolis step for beta Defaults to 0.2, which has worked well for other ARD datasets.

omega_tune

A positive numeric indicating the standard deviation used as the jumping scale in the Metropolis step for omega Defaults to 0.2, which has worked well for other ARD datasets.

init

A named list with names corresponding to the first-level model parameters, name 'alpha', 'beta', and 'omega'. By default the 'alpha' and 'beta' parameters are initialized at the values corresponding to the Killworth MLE estimates (for the missing 'beta'), with all 'omega' set to 20. Alternatively, init = 'random' simulates 'alpha' and 'beta' from a normal random variable with mean 0 and standard deviation 1. By default, init = 'MLE' initializes values at the Killworth et al. (1998b) MLE estimates for the degrees and sizes and simulates the other parameters.

Details

This function fits the overdispersed NSUM model using the Metropolis-Gibbs sampler provided in Zheng et al. (2006).

Value

A named list with the estimated posterior samples. The estimated parameters are named as follows, with additional descriptions as needed:

alphas

Log degree, if scaled, else raw alpha parameters

betas

Log prevalence, if scaled, else raw beta parameters

inv_omegas

Inverse of overdispersion parameters

sigma_alpha

Standard deviation of alphas

mu_beta

Mean of betas

sigma_beta

Standard deviation of betas

omegas

Overdispersion parameters

If scaled, the following additional parameters are included:

mu_alpha

Mean of log degrees

degrees

Degree estimates

sizes

Subpopulation size estimates

References

Zheng, T., Salganik, M. J., and Gelman, A. (2006). How many people do you know in prison, Journal of the American Statistical Association, 101:474, 409–423

Examples

# Analyze an example ard data set using Zheng et al. (2006) models
# Note that in practice, both warmup and iter should be much higher
data(example_data)

ard = example_data$ard
subpop_sizes = example_data$subpop_sizes
known_ind = c(1, 2, 4)
N = example_data$N

overdisp.est = overdispersed(ard,
known_sizes = subpop_sizes[known_ind],
known_ind = known_ind,
G1_ind = 1,
G2_ind = 2,
B2_ind = 4,
N = N,
warmup = 50,
iter = 100)

# Compare size estimates
data.frame(true = subpop_sizes,
basic = colMeans(overdisp.est$sizes))

# Compare degree estimates
plot(example_data$degrees, colMeans(overdisp.est$degrees))

# Look at overdispersion parameter
colMeans(overdisp.est$omegas)