Fitting Network Scale-up Models

Overview

This package fits several different network scale-up models (NSUM) to Aggregated Relational Data (ARD). ARD represents survey responses about how many people each respondent knows in different subpopulations through "How many X's do you know?" questions. Specifically, if respondents are asked how many people they know in subpopulations, then ARD is an by matrix, where the element represents how many people respondent reports knowing in subpopulation . NSUM leverages these responses to estimate the unknown size of hard-to-reach populations. See Laga, et al. (2021) for more details.

In this package, we provide functions to estimate the size and accompanying parameters (e.g. degrees) from 4 papers:

• Killworth, P. D., Johnsen, E. C., McCarty, C., Shelley, G. A., and Bernard, H. R. (1998) plug-in MLE
• Killworth, P. D., McCarty, C., Bernard, H. R., Shelley, G. A., and Johnsen, E. C. (1998) MLE
• Zheng, T., Salganik, M. J., and Gelman, A. (2006) overdispersed model
• Laga, I., Bao, L., and Niu, X (2021) uncorrelated, correlated, and covariate models

Requirements

This package requires the following Python libraries:

• numpy >= 1.24
• pandas >= 2.1
• scipy >= 1.11
• cmdstanpy >= 1.1

PIMLE

The plug-in MLE estimator from Killworth, P. D., Johnsen, E. C., McCarty, C., Shelley, G. A., and Bernard, H. R. (1998) is a two-stage estimator that first estimates the degrees for each respondent by maximizing the following likelihood for each respondent:

$$L(d_i;y,{N_k}) = \prod_{k=1}^{L} {d_i \choose y_{ik}} \left(\frac{N_k}{N}\right)^{y_{ik}}\left(1-\frac{N_k}{N}\right)^{d_i-y_{ik}}$$

where is the number of subpopulations with known . For the second stage, the model plugs in the estimated into the equation

$$\frac{y_{ik}}{d_i} = \frac{N_k}{N}$$

and solves for the unknown for each respondent. These values are then averaged to obtain a single estimated of .

To summarize, stage 1 estimate by

and then these estimates are used in stage 2 to estimate the unknown by

The following demonstrates how to use the killworth function to compute PIMLE estimates of unknown subpopulation sizes:

pimle_est = killworth(ard,
                      known_sizes = sizes[[0, 1, 3]],
                      known_ind = [0,1,3],
                      N = N,
                      model = "PIMLE")

Note that the function may provide a warning saying that at least one was 0. This occurs when a respondent does not report knowing anyone in the known subpopulations. This is an issue for the PIMLE since a 0 value is in the denominator for . Thus, we ignore the responses from respondents that correspond to = 0.

MLE

The MLE estimator from Killworth, P. D., McCarty, C., Bernard, H. R., Shelley, G. A., and Johnsen, E. C. (1998) is also a two-stage model with an identical first stage, i.e

However, the second stage estimates by maximizing the Binomial likelihood with respect to , fixing at the estimated . Thus, the estimate for the unknown subpopulation size is given by

The following demonstrates how to use the killworth function to compute MLE estimates of unknown subpopulation sizes:

mle_est = killworth(ard,
                      known_sizes = np.ravel(sizes)[[0,1,3]],
                      known_ind = [0,1,3],
                      N = N,
                      model = "MLE")

Note that there is no warning here since the denominator depends on the summation of .

Bayesian Models

Now we introduce the two Bayesian estimators implemented in this package.

Overdispersed Model

The overdispersed model proposed in Zheng et al. (2006) assumes the following likelihood:

$$y_{ik} \sim \text{Negative-Binomial}(\text{mean}=e^{a_i+b_k},\text{overdispersion}=\omega_k)$$

Please see the original manuscript for more details on the model structure and priors.

This package fits this overdispersed model either via the Gibbs-Metropolis algorithm provided in the original manuscript (overdispersed) or via Stan (overdispersedStan). We suggest using the Stan version since convergence and effective sample sizes are more satisfactory in the Stan implementation, and does not require tuning jumping scales for Metropolis updates.

In order to identity the and as log-degrees and log-prevalences, respectively, the overdispersed model requires scaling the parameters. In order to scale the parameters, the user must supply at least one subpopulation with known size and the column index corresponding to that known size. Additionally, a two secondary groups may be supplied which can adjust for differences in gender or other binary group classifications. More details of the scaling procedure can be found in the original manuscript.

The following demonstrates how to use the overdispersed and overdispersedStan functions to compute estimates of unknown subpopulation sizes using the Gibbs-Metropolis and Stan implementations of the overdispersed model. Note that in practice, both warmup and iter should be set to higher values:

overdisp_gibbs_metrop_est = overdispersed(
                         ard, 
                         known_sizes = sizes[[0,1,3]], 
                         known_ind = [0,1,3], 
                         G1_ind = 0,
                         G2_ind = 1,
                         B2_ind = 3,
                         N=N,
                         warmup = 500,
                         iter = 1000,
                         verbose = True,
                         init = "MLE")

overdisp_stan = overdispersedStan(
                         ard, 
                         known_sizes = sizes[[0,1,3]], 
                         known_ind = [0,1,3],
                         G1_ind = 0,
                         G2_ind = 1,
                         B2_ind = 3,
                         N = N,
                         chains = 2,
                         cores = 2,
                         warmup = 250,
                         iter = 500)

Correlated Models

The correlated model proposed in Laga et al. (2023) assumes the following likelihood

where critically,

$$\textbf{b}_i \sim \mathcal{N}_k(\mu, \Sigma)$$

so that the responses for each respondent are correlated across subpopulations. Again, and need to be scaled, and they can either be scaled using the same procedure as for the overdispersed model (providing indices corresponding to different groups), by using all known subpopulation sizes, or by weighting groups according to their correlation with other groups. More details about these scaling procedures are provided in Laga et al. (2023).

In this package, model parameters are estimated via Stan. Note that while the full model likelihood depends on , and , any combination of these covariates can be provided. Additionally, we can assume that is a diagonal matrix (i.e. no correlation) by setting the argument model = uncorrelated in the correlatedStan function.

The following demonstrates how to use the correlatedStan function to compute estimates of unknown subpopulation sizes using the Stan implementations of the correlated model. Note that in practice, both warmup and iter should be set to higher values:

correlated_cov_stan = correlatedStan(
    ard,
    known_sizes = sizes[[0,1,3]],
    known_ind = [0,1,3],
    model = "correlated",
    scaling = "weighted",
    x = x,
    z_subpop = z_subpop,
    z_global = z_global,
    N = N,
    chains = 2,
    cores = 2,
    warmup = 250,
    iter = 500,
)

correlated_nocov_stan = correlatedStan(
    ard,
    known_sizes = sizes[[0,1,3]],
    known_ind = [0,1,3],
    model = "correlated",
    scaling = "all",
    N = N,
    chains = 2,
    cores = 2,
    warmup = 250,
    iter = 500,
)

uncorrelated_cov_stan = correlatedStan(
    ard,
    known_sizes = sizes[[0,1,3]],
    known_ind = [0,1,3],
    model = "uncorrelated",
    scaling = "all",
    x = x,
    z_subpop = z_subpop,
    z_global = z_global,
    N = N,
    chains = 2,
    cores = 2,
    warmup = 250,
    iter = 500,
)

uncorrelated_x_stan = correlatedStan(
    ard,
    known_sizes = sizes[[0,1,3]],
    known_ind = [0,1,3],
    model = "uncorrelated",
    scaling = "all",
    x = x,
    N = N,
    chains = 2,
    cores = 2,
    warmup = 250,
    iter = 500,
)