killworth.start: Killworth Starting Values for MCMC
In NSUM: Network Scale Up Method

Description Usage Arguments Details Value Author(s) References See Also Examples

This function uses the Killworth estimates to calculate reasonable starting values for the MCMC estimation.

1	killworth.start(dat, known, N)

`dat`	a matrix of non-negagtive integers, the `(i,k)`-th entry represents the number of people that the `i`-th individual knows from the `k`-th subpopulation.
`known`	a vector of positive numbers, the sizes of known subpopulations. All additional columns of `dat` are treated as unknown.
`N`	a positive number, the (known) total population size.

The function killworth.start allows for the estimation reasonable starting values for many of the parameters in the MCMC function nsum.mcmc based on Killworth's network scale-up model. These are the default starting values where applicable. For simple subpopulation size estimation using Killworth's model, see the function killworth.

A list with four components:

`NK.start`	a vector of positive numbers with length equal to the total number of unknown subpopulations, the starting values for the sizes of the unknown subpopulations\.
`d.start`	a vector of positive numbers with length equal to the number of individuals, the starting values for the network degrees.
`mu.start`	a real number, the starting value for the location parameter for the log-normal distribution of network degrees.
`sigma.start`	a positive number, the starting value for the scale parameter for the log-normal distribution of network degrees.

Rachael Maltiel and Aaron J. Baraff

Maintainer: Aaron J. Baraff <ajbaraff at uw.edu>

Killworth, P., Johnsen, E., McCarty, C., Shelley, G., and Bernard, H. (1998a), "A Social Network Approach to Estimating Seroprevalence in the United States," Social Networks, 20, 23-50.

Killworth, P., McCarty, C., Bernard, H., Shelley, G., and Johnsen, E. (1998b), "Estimation of Seroprevalence, Rape, and Homelessness in the United States using a Social Network Approach," Evaluation Review, 22, 289-308.

Maltiel, R., Raftery, A. E., McCormick, T. H., and Baraff, A. J., "Estimating Population Size Using the Network Scale Up Method." CSSS Working Paper 129. Retrieved from https://www.csss.washington.edu/Papers/2013/wp129.pdf

killworth.start, nsum.mcmc

## load data
data(McCarty)

## simulate from model with barrier effects
sim.bar <- with(McCarty, nsum.simulate(100, known, unknown, N, model="barrier",
                                       mu, sigma, rho))

## estimate Killworth starting values
dat.bar <- sim.bar$y
start <- with(McCarty, killworth.start(dat.bar, known, N))

## estimate unknown population size from MCMC
mcmc <- with(McCarty, nsum.mcmc(dat.bar, known, N, model="barrier", iterations=100,
                                burnin=50, NK.start=start$NK.start, d.start=start$d.start, 
                                mu.start=start$mu.start, sigma.start=start$sigma.start))