Tutorial 1: A first relational event model on a phone-call log

Goal. Declare a sequence of timed events, fit a relational event model (REM) with nwrem, and understand what the coefficients say.

Data. The phone calls among 84 university students that the R package goldfish ships as the calls data of its Social Evolution example (data from a study of MIT students, distributed with goldfish): 439 calls over about six weeks in 2008, each with the caller, the person called and the time (in seconds) of the call. The two data files (events, actors) are part of this section. You need the core nwcommands and the nwrem package (see Install); the commands were run with nwrem 1.0.1 and the core 3.0.0. The complete script and log are in example.do and example.log. For a general introduction to event models, see Intro to Longitudinal Network Models.

What a REM does, in four sentences

A relational event is one act from a sender to a receiver at a known moment (a call, an e-mail, a hand-shake). A REM (Butts 2008) looks at the events one after the other and asks at each moment: among all the ordered pairs of actors that could have interacted next, why was it this pair? Effects such as “the sender has been very active so far” or “the sender has called this receiver before” raise or lower the chance of a pair, and nwrem estimates by how much. Only the order of the events enters the likelihood (it is a partial likelihood, as in a Cox model), so there is no intercept to estimate.

Look at the events and declare them

. import delimited "../data/calls_events.csv", clear
. list in 1/5

     +--------------------------------+
     | sender   receiver    eventtime |
     |--------------------------------|
  1. |     72         50   1220733470 |
  2. |     43         51   1221102974 |
  3. |     43         51   1221784293 |
  4. |     43         22   1221785882 |
  5. |     43         55   1221787264 |
     +--------------------------------+

. count
  439

. display "span in days: " (eventtime[_N] - eventtime[1]) / 86400
span in days: 42.125949

The log names 54 of the 84 students. The other 30 neither call nor are called, but they are possible callers and receivers, so they belong to the set of pairs the model compares. A list of events creates only the actors it mentions, so we add the missing ones as isolates with nwaddnodes (the lines that build the list miss are in example.do):

. nwset sender receiver, directed eventtime(eventtime) name(calls)
. nwaddnodes calls, nodenames(`miss')

The option eventtime() says that each row is an event with a time and not a lasting tie. There are now 84 x 83 = 6,972 ordered pairs that could have happened at any moment.

Model 1: activity

nodsnd is the sender’s normalized out-degree: the share of all earlier events in which this actor was the sender. nidrec is the receiver’s normalized in-degree: the share of earlier events in which this actor was the receiver.

. nwrem calls, nodsnd nidrec
------------------------------------------------------------
Relational event model (ordinal partial likelihood, MLE)
Network: calls                         Actors: 84
Events: 439                            Log likelihood: -3.12e+03
------------------------------------------------------------
------------------------------------------------------------------------------
       calls | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
      nodsnd |   10.57478   .3530761    29.95   0.000     9.882761    11.26679
      nidrec |   11.32606   .3391393    33.40   0.000     10.66136    11.99077
------------------------------------------------------------------------------

How to read it. The effects are shares between 0 and 1, so the coefficients are large. A sender whose share of earlier calls is 0.1 higher has exp(1.06), about 2.9 times the odds of making the next call. A receiver whose share of earlier receipts is 0.1 higher is about exp(1.13), 3.1 times as likely to be the one called. People who have been busy stay busy, and people who have been called get called.

Model 2: history of the pair

frpsndsnd (“inertia”) is the share of the sender’s own earlier calls that went to this receiver. frrecsnd (“reciprocity of receipt”) is the share of the sender’s earlier receipts that came from this receiver.

. nwrem calls, frpsndsnd frrecsnd
...
Events: 439                            Log likelihood: -2.27e+03
...
   frpsndsnd |   6.254331   .1453025    43.04   0.000     5.969544    6.539119
    frrecsnd |   2.034655   .1400245    14.53   0.000     1.760212    2.309098

. display "events: " e(N) ", actors: " e(nodes) ", log likelihood: " %9.2f e(ll)
events: 439, actors: 84, log likelihood:  -2274.05
. display "effect of inertia on the odds: x" %6.1f exp(_b[frpsndsnd] * 0.1) " per 10 percentage points"
effect of inertia on the odds: x   1.9 per 10 percentage points

If another 10 percentage points of the sender’s calls had gone to a given person, the odds that the next call goes to that person again are multiplied by 1.9. Calling back someone who called you earlier also increases the odds (2.03 on the log scale for a sender whose receipts all came from this person). This model fits far better than the activity model: the log likelihood rises from about -3,120 to -2,274 (the log likelihood is printed in rounded form in the header; e(ll) holds the exact value). How people behave toward each other explains the calls much better than how active they are in general.

Both together

. nwrem calls, nodsnd nidrec frpsndsnd frrecsnd
...
Events: 439                            Log likelihood: -2.12e+03
...
      nodsnd |   5.392451   .4376142    12.32   0.000     4.534743     6.25016
      nidrec |   .9485414   .5809882     1.63   0.103    -.1901745    2.087257
   frpsndsnd |   5.782841   .1612638    35.86   0.000      5.46677    6.098913
    frrecsnd |   1.640027   .1538415    10.66   0.000     1.338504    1.941551

Once the history of the pair is in the model, the activity effects shrink: nodsnd halves (5.4) and nidrec falls to 0.95 and is no longer distinguishable from zero (the interval runs from -0.19 to 2.09). Much of what looked like “popular people get called” in Model 1 was people calling those they always call. This is the usual reason for fitting several effects together and not one at a time.

A word on precision

The degree-type effects are running shares that move closer to one another (towards 1/84) as the log grows. Therefore their estimates can stay noisy, and several of them in one model can be nearly collinear, so that different starting values give visibly different individual coefficients with about the same log likelihood. If it matters, check the stability of a coefficient under the other specifications, as in Tutorial 3.

Things to try

Fit nwrem calls, frpsndsnd alone, and then nwrem calls, frrecsnd alone.
Replace the shares by recency: nwrem calls, rsndsnd rrecsnd asks whether the most recent partners are the ones returned to.
Add seed(1) to any call; it only decides which optimizer restart is tried first and never changes a converged result.

Tutorial 2 adds actor and pair covariates and the recency effects. Back to the overview of nwrem.