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Friday 31 July 2015

Possible Application of Approximate Bayesian Computation for Networks

I've been thinking about possible applications for the network simulation/analysis program that Steve Thompson and I developed as part of my PhD thesis.*

I propose to investigate the effect of response order/timing in respondent-driven sampling on estimates of network parameters.

Here I'm assuming that samples are taken in waves. That is, a collection of seed members of the population are identified and given the initial coupons. If the standard deviation of the response times is small compared to the mean, the network of the population is being sampled in a manner similar to breadth first search (BFS). If the standard deviation of response times is large relative the mean, the network ends up being sampled in a manner closer to that of a depth first search (DFS). Each of these sampling methods has the potential to provide vastly different information about the sample.

Such an investigation would include four parts:

1) Motivating questions: Why would we care about the time it takes members of the population to be entered into the sample? Because perhaps these times could be influenced by different incentive structures. If they can, what is best? If they cannot, we can do a what-if analysis to explore counter-factuals. Does timing matter? What sampling and incentive setups are robust to the effects of response times? Are there statistical methods that can adjust for the effects of response times?

2) Find some real respondent-driven samples, preferably by looking in the literature of PLoS-One and using the data that is included with publications, but possibly by asking other researchers individually.

Look at the time stamp of each observation in each data set, if available, and fit a distribution such as gamma(r, beta) to the time delay between giving a recruitment coupon and that recruitment coupon being used. Compare the parameter estimates that each data set produces to see if there is a significant difference between them and see if there's any obvious reason or pattern behind the changes.

3) Generate a few networked populations and sample from each one many times using the different response-delay time distributions found in Part 2. Are there any significant changes to the network statistics that we can compute from the samples we find? That is, how does the variation of the statistics between resamples under one delay time distribution compare to the variation between delay time distributions?

4) Employ the full ABCN system to get estimates of whatever network parameters we can get for case ij, 1 <= i,j <= M, where M is the number of datasets we find. Case ij would be using the observed sample from the ith dataset, with simulations in the ABCN system using the delay distribution estimated from the jth dataset.

This way, we could compare the variation in the network parameters attributable to the sample that was actually found, and how much was attributable to the difference in time it took for recruitments to be entered into the survey. Also, we effectively will have performed a what-if analysis on the datasets we use - and seeing if the conclusions from the datasets would have been different if the recruited respondents had been responded with different delay structures.

*This network simulation/analysis system takes an observed sample of network and computes a battery of summarizing statistics of the sample. Then it simulates and samples from many networks and computes the same battery from each sample. It estimates the parameters of the original sample by looking at the distribution of parameters from the simulation samples that were found to be similar to the observed sample.

This will all be explained again, and in greater detail when I post my thesis after the defense in... gosh.. a month. Basically it's statistical inference turned upside down, where you generate the parameters and see if they make the sample you want, instead of starting with the sample and estimating a parameter value or distribution. The base method is called Approximate Bayesian Computation.

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