posteriorsize: Estimating hidden population size using RDS data

In sspse: Estimating Hidden Population Size using Respondent Driven Sampling Data

Estimating hidden population size using RDS data

Description

posteriorsize computes the posterior distribution of the population size based on data collected by Respondent Driven Sampling. The approach approximates the RDS via the Sequential Sampling model of Gile (2008). As such, it is referred to as the Sequential Sampling - Population Size Estimate (SS-PSE). It uses the order of selection of the sample to provide information on the distribution of network sizes over the population members.

Usage

posteriorsize(
  s,
  s2 = NULL,
  previous = NULL,
  median.prior.size = NULL,
  interval = 10,
  burnin = 5000,
  maxN = NULL,
  K = FALSE,
  samplesize = 1000,
  quartiles.prior.size = NULL,
  mean.prior.size = NULL,
  mode.prior.size = NULL,
  priorsizedistribution = c("beta", "flat", "nbinom", "pln", "supplied"),
  effective.prior.df = 1,
  sd.prior.size = NULL,
  mode.prior.sample.proportion = NULL,
  alpha = NULL,
  visibilitydistribution = c("cmp", "nbinom", "pln"),
  mean.prior.visibility = NULL,
  sd.prior.visibility = NULL,
  max.sd.prior.visibility = 4,
  df.mean.prior.visibility = 1,
  df.sd.prior.visibility = 3,
  beta_0.mean.prior = -3,
  beta_t.mean.prior = 0,
  beta_u.mean.prior = 0,
  beta_0.sd.prior = 10,
  beta_t.sd.prior = 10,
  beta_u.sd.prior = 10,
  mem.optimism.prior = NULL,
  df.mem.optimism.prior = 5,
  mem.scale.prior = 2,
  df.mem.scale.prior = 10,
  mem.overdispersion = 15,
  visibility = TRUE,
  type.impute = c("median", "distribution", "mode", "mean"),
  Np = 0,
  n = NULL,
  n2 = NULL,
  mu_proposal = 0.1,
  nu_proposal = 0.15,
  beta_0_proposal = 0.2,
  beta_t_proposal = 0.001,
  beta_u_proposal = 0.001,
  memmu_proposal = 0.1,
  memscale_proposal = 0.15,
  burnintheta = 500,
  burninbeta = 50,
  parallel = 1,
  parallel.type = "PSOCK",
  seed = NULL,
  maxbeta = 90,
  supplied = list(maxN = maxN),
  max.coupons = NULL,
  recruit.time = NULL,
  recruit.time2 = NULL,
  include.tree = TRUE,
  unit.scale = FALSE,
  optimism = TRUE,
  reflect.time = FALSE,
  equalize = TRUE,
  verbose = FALSE
)

Arguments

s	either a vector of integers or an rds.data.frame providing network size information. If a rds.data.frame is passed and visibility=TRUE, the default, then the measurement error model is to used, whereby latent visibilities are used in place of the reported network sizes as the size variable. If a vector of integers is passed these are the network sizes in sequential order of recording (and the measurement model is not used).
s2	either a vector of integers or an rds.data.frame providing network size information for a second RDS sample subsequent to the first RDS recorded in s. If a rds.data.frame is passed and visibility=TRUE, the default, then the measurement error model is to used, whereby latent visibilities are used in place of the reported network sizes as the size variable. If a vector of integers is passed these are the network sizes in sequential order of recording (and the measurement model is not used).
previous	character; optionally, the name of the variable in s2 indicating if the corresponding unit was sampled in the first RDS.
median.prior.size	scalar; A hyperparameter being the mode of the prior distribution on the population size.
interval	count; the number of proposals between sampled statistics.
burnin	count; the number of proposals before any MCMC sampling is done. It typically is set to a fairly large number.
maxN	integer; maximum possible population size. By default this is determined from an upper quantile of the prior distribution.
K	count; the maximum visibility for an individual. This is usually calculated as round(stats::quantile(s,0.80)). It applies to network sizes and (latent) visibilities. If logical and FALSE then the K is unbounded but set to compute the visibilities.
samplesize	count; the number of Monte-Carlo samples to draw to compute the posterior. This is the number returned by the Metropolis-Hastings algorithm.The default is 1000.
quartiles.prior.size	vector of length 2; A pair of hyperparameters being the lower and upper quartiles of the prior distribution on the population size. For example, quartiles.prior.size=c(1000,4000) corresponds to a prior where the lower quartile (25%) is 1000 and the upper (75%) is 4000.
mean.prior.size	scalar; A hyperparameter being the mean of the prior distribution on the population size.
mode.prior.size	scalar; A hyperparameter being the mode of the prior distribution on the population size.
priorsizedistribution	character; the type of parametric distribution to use for the prior on population size. The options are beta (for a Beta prior on the sample proportion (i.e. n/N)), flat (uniform), nbinom (Negative-Binomial), and pln (Poisson-log-normal). The default is beta.
effective.prior.df	scalar; A hyperparameter being the effective number of samples worth of information represented in the prior distribution on the population size. By default this is 1, but it can be greater (or less!) to allow for different levels of uncertainty.
sd.prior.size	scalar; A hyperparameter being the standard deviation of the prior distribution on the population size.
mode.prior.sample.proportion	scalar; A hyperparameter being the mode of the prior distribution on the sample proportion n/N.
alpha	scalar; A hyperparameter being the first parameter of the beta prior model for the sample proportion. By default this is NULL, meaning that 1 is chosen. it can be any value at least 1 to allow for different levels of uncertainty.
visibilitydistribution	count; the parametric distribution to use for the individual network sizes (i.e., degrees). The options are cmp, nbinom, and pln. These correspond to the Conway-Maxwell-Poisson, Negative-Binomial, and Poisson-log-normal. The default is cmp.
mean.prior.visibility	scalar; A hyper parameter being the mean visibility for the prior distribution for a randomly chosen person. The prior has this mean.
sd.prior.visibility	scalar; A hyper parameter being the standard deviation of the visibility for a randomly chosen person. The prior has this standard deviation.
max.sd.prior.visibility	scalar; The maximum allowed value of sd.prior.visibility. If the passed or computed value is higher, it is reduced to this value. This is done for numerical stability reasons.
df.mean.prior.visibility	scalar; A hyper parameter being the degrees-of-freedom of the prior for the mean. This gives the equivalent sample size that would contain the same amount of information inherent in the prior.
df.sd.prior.visibility	scalar; A hyper parameter being the degrees-of-freedom of the prior for the standard deviation. This gives the equivalent sample size that would contain the same amount of information inherent in the prior for the standard deviation.
beta_0.mean.prior	scalar; A hyper parameter being the mean of the beta_0 parameter distribution in the model for the number of recruits.
beta_t.mean.prior	scalar; A hyper parameter being the mean of the beta_t parameter distribution in the model for the number of recruits. This corresponds to the time-to-recruit variable.
beta_u.mean.prior	scalar; A hyper parameter being the mean of the beta_u parameter distribution in the model for the number of recruits. This corresponds to the visibility variable.
beta_0.sd.prior	scalar; A hyper parameter being the standard deviation of the beta_0 parameter distribution in the model for the number of recruits.
beta_t.sd.prior	scalar; A hyper parameter being the standard deviation of the beta_t parameter distribution in the model for the number of recruits. This corresponds to the time-to-recruit variable.
beta_u.sd.prior	scalar; A hyper parameter being the standard deviation of the beta_u parameter distribution in the model for the number of recruits. This corresponds to the visibility variable.
mem.optimism.prior	scalar; A hyper parameter being the mean of the distribution of the optimism parameter.
df.mem.optimism.prior	scalar; A hyper parameter being the degrees-of-freedom of the prior for the optimism parameter. This gives the equivalent sample size that would contain the same amount of information inherent in the prior.
mem.scale.prior	scalar; A hyper parameter being the scale of the concentration of baseline negative binomial measurement error model.
df.mem.scale.prior	scalar; A hyper parameter being the degrees-of-freedom of the prior for the standard deviation of the dispersion parameter in the visibility model. This gives the equivalent sample size that would contain the same amount of information inherent in the prior for the standard deviation.
mem.overdispersion	scalar; A parameter being the overdispersion of the negative binomial distribution that is the baseline for the measurement error model.
visibility	logical; Indicate if the measurement error model is to be used, whereby latent visibilities are used in place of the reported network sizes as the unit size variable. If TRUE then a rds.data.frame need to be passed to provide the RDS information needed for the measurement error model.
type.impute	The type of imputation to use for the summary visibilities (returned in the component visibilities. The imputes are based on the posterior draws of the visibilities. It can be of type distribution, mode,median, or mean with median the default, being the posterior median of the visibility for that person.
Np	integer; The overall visibility distribution is a mixture of the Np rates for 1:Np and a parametric visibility distribution model truncated below Np. Thus the model fits the proportions of the population with visibility 1:Np each with a separate parameter. This should adjust for an lack-of-fit of the parametric visibility distribution model at lower visibilities, although it also changes the model away from the parametric visibility distribution model.
n	integer; the number of people in the sample. This is usually computed from s automatically and not usually specified by the user.
n2	integer; If s2 is specified, this is the number of people in the second sample. This is usually computed from s automatically and not usually specified by the user.
mu_proposal	scalar; The standard deviation of the proposal distribution for the mean visibility.
nu_proposal	scalar; The standard deviation of the proposal distribution for the CMP scale parameter that determines the standard deviation of the visibility.
beta_0_proposal	scalar; The standard deviation of the proposal distribution for the beta_0 parameter of the recruit model.
beta_t_proposal	scalar; The standard deviation of the proposal distribution for the beta_t parameter of the recruit model. This corresponds to the visibility variable.
beta_u_proposal	scalar; The standard deviation of the proposal distribution for the beta_u parameter of the recruit model. This corresponds to the time-to-recruit variable.
memmu_proposal	scalar; The standard deviation of the proposal distribution for the log of the optimism parameter (that is, gamma).
memscale_proposal	scalar; The standard deviation of the proposal distribution for the log of the s.d. in the optimism model.
burnintheta	count; the number of proposals in the Metropolis-Hastings sub-step for the visibility distribution parameters (\theta) before any MCMC sampling is done. It typically is set to a modestly large number.
burninbeta	count; the number of proposals in the Metropolis-Hastings sub-step for the visibility distribution parameters (\beta) before any MCMC sampling is done. It typically is set to a modestly large number.
parallel	count; the number of parallel processes to run for the Monte-Carlo sample. This uses MPI or PSOCK. The default is 1, that is not to use parallel processing.
parallel.type	The type of parallel processing to use. The options are "PSOCK" or "MPI". This requires the corresponding type to be installed. The default is "PSOCK".
seed	integer; random number integer seed. Defaults to NULL to use whatever the state of the random number generator is at the time of the call.
maxbeta	scalar; The maximum allowed value of the beta parameter. If the implied or computed value is higher, it is reduced to this value. This is done for numerical stability reasons.
supplied	list; If supplied, is a list with components maxN and sample. In this case supplied is a matrix with a column named N being a sample from a prior distribution for the population size. The value maxN specifies the maximum value of the population size, a priori.
max.coupons	The number of recruitment coupons distributed to each enrolled subject (i.e. the maximum number of recruitees for any subject). By default it is taken by the attribute or data, else the maximum recorded number of coupons.
recruit.time	vector; An optional value for the data/time that the person was interviewed. It needs to resolve as a numeric vector with number of elements the number of rows of the data with non-missing values of the network variable. If it is a character name of a variable in the data then that variable is used. If it is NULL then the sequence number of the recruit in the data is used. If it is NA then the recruitment is not used in the model. Otherwise, the recruitment time is used in the model to better predict the visibility of the person.
recruit.time2	vector; An optional value for the data/time that the person in the second RDS survey was interviewed. It needs to resolve as a numeric vector with number of elements the number of rows of the data with non-missing values of the network variable. If it is a character name of a variable in the data then that variable is used. If it is NULL, the default, then the sequence number of the recruit in the data is used. If it is NA then the recruitment is not used in the model. Otherwise, the recruitment time is used in the model to better predict the visibility of the person.
include.tree	logical; If TRUE, augment the reported network size by the number of recruits and one for the recruiter (if any). This reflects a more accurate value for the visibility, but is not the self-reported degree. In particular, it typically produces a positive visibility (compared to a possibility zero self-reported degree).
unit.scale	numeric; If not NULL it sets the numeric value of the scale parameter of the distribution of the unit sizes. For the negative binomial, it is the multiplier on the variance of the negative binomial compared to a Poisson (via the Poisson-Gamma mixture representation). Sometimes the scale is unnaturally large (e.g. 40) so this give the option of fixing it (rather than using the MLE of it). The model is fit with the parameter fixed at this passed value.
optimism	logical; If TRUE then add a term to the model allowing the (proportional) inflation of the self-reported degrees relative to the unit sizes.
reflect.time	logical; If TRUE then the recruit.time is the time before the end of the study (instead of the time since the survey started or chronological time).
equalize	logical; If TRUE and the capture-recapture model is used, adjusts for gross differences in the reported network sizes between the two samples.
verbose	logical; if this is TRUE, the program will print out additional information, including goodness of fit statistics.

Value

posteriorsize returns a list consisting of the following elements:

pop	vector; The final posterior draw for the degrees of the population. The first n are the sample in sequence and the reminder are non-sequenced.
K	count; the maximum visibility for an individual. This is usually calculated as twice the maximum observed degree.
n	count; the sample size.
samplesize	count; the number of Monte-Carlo samples to draw to compute the posterior. This is the number returned by the Metropolis-Hastings algorithm.The default is 1000.
burnin	count; the number of proposals before any MCMC sampling is done. It typically is set to a fairly large number.
interval	count; the number of proposals between sampled statistics.
mu	scalar; The hyper parameter mean.prior.visibility being the mean visibility for the prior distribution for a randomly chosen person. The prior has this mean.
sigma	scalar; The hyper parameter sigma being the standard deviation of the visibility for a randomly chosen person. The prior has this standard deviation.
df.mean.prior.visibility	scalar; A hyper parameter being the degrees-of-freedom of the prior for the mean. This gives the equivalent sample size that would contain the same amount of information inherent in the prior.
df.sd.prior.visibility	scalar; A hyper parameter being the degrees-of-freedom of the prior for the standard deviation. This gives the equivalent sample size that would contain the same amount of information inherent in the prior for the standard deviation.
Np	integer; The overall visibility distribution is a mixture of the 1:Np rates and a parametric visibility distribution model truncated below Np. Thus the model fits the proportions of the population with visibility 1:Np each with a separate parameter. This should adjust for an lack-of-fit of the parametric visibility distribution model at lower visibilities, although it also changes the model away from the parametric visibility distribution model.
mu_proposal	scalar; The standard deviation of the proposal distribution for the mean visibility.
nu_proposal	scalar; The standard deviation of the proposal distribution for the CMP scale parameter of the visibility distribution.
N	vector of length 5; summary statistics for the posterior population size. MAP maximum aposteriori value of N Mean AP mean aposteriori value of N Median AP median aposteriori value of N P025 the 2.5th percentile of the (posterior) distribution for the N. That is, the lower point on a 95% probability interval. P975 the 97.5th percentile of the (posterior) distribution for the N. That is, the upper point on a 95% probability interval.
maxN	integer; maximum possible population size. By default this is determined from an upper quantile of the prior distribution.
sample	matrix of dimension samplesize\times 10 matrix of summary statistics from the posterior. This is also an object of class mcmc so it can be plotted and summarized via the mcmc.diagnostics function in the ergm package (and also the coda package). The statistics are: N population size. mu scalar; The mean visibility for the prior distribution for a randomly chosen person. The prior has this mean. sigma scalar; The standard deviation of the visibility for a randomly chosen person. The prior has this standard deviation. visibility1 scalar; the number of nodes of visibility 1 in the population (it is assumed all nodes have visibility 1 or more). lambda scalar; This is only present for the cmp model. It is the \lambda parameter in the standard parameterization of the Conway-Maxwell-Poisson model for the visibility distribution. nu scalar; This is only present for the cmp model. It is the \nu parameter in the standard parameterization of the Conway-Maxwell-Poisson model for the visibility distribution.
vsample	matrix of dimension samplesize\times n matrix of posterior draws from the unit size distribution for those in the survey. The sample for the ith person is the ith column.
lpriorm	vector; the vector of (log) prior probabilities on each value of m=N-n - that is, the number of unobserved members of the population. The values are n:(length(lpriorm)-1+n).
burnintheta	count; the number of proposals in the Metropolis-Hastings sub-step for the visibility distribution parameters (\theta) before any MCMC sampling is done. It typically is set to a modestly large number.
verbose	logical; if this is TRUE, the program printed out additional information, including goodness of fit statistics.
predictive.visibility.count	vector; a vector of length the maximum visibility (K) (by default K=2*max(sample visibility)). The kth entry is the posterior predictive number persons with visibility k. That is, it is the posterior predictive distribution of the number of people with each visibility in the population.
predictive.visibility	vector; a vector of length the maximum visibility (K) (by default K=2*max(sample visibility)). The kth entry is the posterior predictive proportion of persons with visibility k. That is, it is the posterior predictive distribution of the proportion of people with each visibility in the population.
MAP	vector of length 6 of MAP estimates corresponding to the output sample. These are: N population size. mu scalar; The mean visibility for the prior distribution for a randomly chosen person. The prior has this mean. sigma scalar; The standard deviation of the visibility for a randomly chosen person. The prior has this standard deviation. visibility1 scalar; the number of nodes of visibility 1 in the population (it is assumed all nodes have visibility 1 or more). lambda scalar; This is only present for the cmp model. It is the \lambda parameter in the standard parameterization of the Conway-Maxwell-Poisson model for the visibility distribution. nu scalar; This is only present for the cmp model. It is the \nu parameter in the standard parameterization of the Conway-Maxwell-Poisson model for the visibility distribution.
mode.prior.sample.proportion	scalar; A hyperparameter being the mode of the prior distribution on the sample proportion n/N.
median.prior.size	scalar; A hyperparameter being the mode of the prior distribution on the population size.
mode.prior.size	scalar; A hyperparameter being the mode of the prior distribution on the population size.
mean.prior.size	scalar; A hyperparameter being the mean of the prior distribution on the population size.
quartiles.prior.size	vector of length 2; A pair of hyperparameters being the lower and upper quartiles of the prior distribution on the population size.
visibilitydistribution	count; the parametric distribution to use for the individual network sizes (i.e., visibilities). The options are cmp, nbinom, and pln. These correspond to the Conway-Maxwell-Poisson, Negative-Binomial, and Poisson-log-normal. The default is cmp.
priorsizedistribution	character; the type of parametric distribution to use for the prior on population size. The options are beta (for a Beta prior on the sample proportion (i.e. n/N), nbinom (Negative-Binomial), pln (Poisson-log-normal), flat (uniform), and continuous (the continuous version of the Beta prior on the sample proportion. The default is beta.

Details on priors

The best way to specify the prior is via the hyperparameter mode.prior.size which specifies the mode of the prior distribution on the population size. You can alternatively specify the hyperparameter median.prior.size which specifies the median of the prior distribution on the population size, or mean.prior.sample proportion which specifies the mean of the prior distribution on the proportion of the population size in the sample or mode.prior.sample proportion which specifies the mode of the prior distribution on the proportion of the population size in the sample. Finally, you can specify quartiles.prior.size as a vector of length 2 being the pair of lower and upper quartiles of the prior distribution on the population size.

References

Gile, Krista J. (2008) Inference from Partially-Observed Network Data, Ph.D. Thesis, Department of Statistics, University of Washington.

Gile, Krista J. and Handcock, Mark S. (2010) Respondent-Driven Sampling: An Assessment of Current Methodology, Sociological Methodology 40, 285-327.

Gile, Krista J. and Handcock, Mark S. (2014) sspse: Estimating Hidden Population Size using Respondent Driven Sampling Data R package, Los Angeles, CA. Version 0.5, https://hpmrg.org/sspse/.

Handcock MS (2003). degreenet: Models for Skewed Count Distributions Relevant to Networks. Statnet Project, Seattle, WA. Version 1.2, https://statnet.org/.

Handcock, Mark S., Gile, Krista J. and Mar, Corinne M. (2014) Estimating Hidden Population Size using Respondent-Driven Sampling Data, Electronic Journal of Statistics, 8, 1, 1491-1521

Handcock, Mark S., Gile, Krista J. and Mar, Corinne M. (2015) Estimating the Size of Populations at High Risk for HIV using Respondent-Driven Sampling Data, Biometrics.

See Also

network, statnet, degreenet

Examples

data(fauxmadrona)
# Here interval=1 so that it will run faster. It should be higher in a 
# real application.
fit <- posteriorsize(fauxmadrona, median.prior.size=1000,
                                 burnin=20, interval=1, samplesize=100)
summary(fit)

sspse documentation built on Aug. 8, 2023, 9:07 a.m.