Tutorial 4: Selection and influence, network-behavior co-evolution

Goal. Friends resemble each other. Is that because people select friends who are like them, or because friends influence each other so that they become alike? An SAOM can estimate both processes in one model, with the network and a behavior as two outcomes that change together.

Data. The Glasgow data (see Tutorial 1), with the friendship networks glasgow1-glasgow3 and the drinking behavior alcohol1-alcohol3. Script and log: example.do, example.log.

The behavior

The alcohol variables count how often a pupil drinks, in five ordered categories from “non” to “more” (coded 1 to 5). nwsaom needs a behavior with a small number of ordered integer values, one variable per wave.

. tabulate alcohol1

            alcohol1 |      Freq.     Percent        Cum.
---------------------+-----------------------------------
                 non |          5       10.00       10.00
once or twice a year |         16       32.00       42.00
        once a month |         12       24.00       66.00
         once a week |         14       28.00       94.00
                more |          3        6.00      100.00
---------------------+-----------------------------------
               Total |         50      100.00

nwsaom describe accepts the behavior as well and adds the information that a SAOM works with: the mean in each wave and how many actors increase, decrease or keep their value in each period.

. nwsaom describe, waves(glasgow1 glasgow2 glasgow3) behavior(alcohol1 alcohol2 alcohol3)
...
wave            mean          sd   missing
1              2.880       1.118         0
2              3.100       1.199         0
3              3.360       0.985         0
----------------------------------------------------------
period          up      down      same
1->2            16         6        28
2->3            18         9        23

Drinking increases on average from wave to wave, and in each period a good part of the pupils (22 and 27 of 50) change their value, so there is behavior change to explain.

A model with selection and influence

The behavior is added with behavior(), which takes the variables of all waves, in order. Its effects are then listed:

linear and quadratic: the baseline tendencies of the behavior itself (the linear shape effect is required, as outdegree is for the network). A negative quadratic effect pulls actors towards the middle of the scale.
avalt: influence. Actors move towards the average drinking level of their friends.
behsim: selection. Actors prefer to name as friends others whose current drinking is similar to their own.

. nwsaom, waves(glasgow1 glasgow2 glasgow3) outdegree reciprocity gwesp(.69) behsim behavior(alcohol1 alcohol2 alcohol3) linear quadratic avalt seed(1)
-------------------------------------------------------------------------------------------------------------------------
SAOM co-evolution (Method of Moments), waves: glasgow1 glasgow2 glasgow3
Actors: 50                             Periods: 2
Behavior: alcohol1 alcohol2 alcohol3
-------------------------------------------------------------------------------------------------------------------------
-------------------------------------------------------------------------------
glasgow1_to~3 | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
--------------+----------------------------------------------------------------
    outdegree |  -2.940128    .167233   -17.58   0.000    -3.267899   -2.612357
  reciprocity |   2.464193   .1974097    12.48   0.000     2.077277    2.851109
    gwesp_.69 |   1.613435   .1633764     9.88   0.000     1.293224    1.933647
       behsim |   1.550423   .6206369     2.50   0.012     .3339968    2.766849
   beh_linear |   .3974465   .1944195     2.04   0.041     .0163913    .7785016
beh_quadratic |  -.5829936   .3035246    -1.92   0.055    -1.177891    .0119037
    beh_avalt |   1.280615   .7760478     1.65   0.099    -.2404103    2.801641
-------------------------------------------------------------------------------
Rate parameters (estimated, one per inter-wave period):

             |   period1    period2 
-------------+---------------------
network_rate |    6.5156     5.1150 
          se |    1.0511     0.7958 
behavior_r~e |    1.3450     1.7957 
          se |    0.3293     0.4649 
Overall maximum convergence ratio: 0.160 (criterion: below 0.25; met)
Convergence t-ratios (criterion: all below 0.1 in absolute value): not met for rate_beh_period1 (0.127)
Convergence is good only if both criteria are met. Refit starting from these estimates with theta0(); see nwsaom.

The network part of the table reads as before: gwesp_.69 is the clustering effect (the gwesp family of effects is used here instead of transtrip). There are now two rates, one for the network and one for the behavior. The model converged (overall ratio 0.160, below 0.25), with one t-ratio slightly above 0.1, which is reported separately. A restart as in Tutorial 2 would be a sensible last step before reporting it.

Reading the selection and influence effects.

behsim = 1.55 (z = 2.5, p = 0.012): pupils prefer friends with a similar drinking level. This is evidence for selection.
beh_avalt = 1.28 (z = 1.65, p = 0.099): the sign says that pupils move towards the drinking of their friends, but with this effect the evidence for influence is weaker than for selection.
beh_linear is positive and beh_quadratic negative: the general drift is towards more drinking, with a tendency against the extremes.

A different influence effect

avalt is one way to formalise influence; avsim (“average similarity”) is another; the actor moves in the direction that makes the behavior more similar to that of the friends, rather than towards their average. Swapping one for the other is a robustness check that you should always make:

. nwsaom, waves(glasgow1 glasgow2 glasgow3) outdegree reciprocity gwesp(.69) behsim behavior(alcohol1 alcohol2 alcohol3) linear quadratic avsim seed(1)
...
       behsim |   1.498898   .5391366     2.78   0.005     .4422095    2.555586
   beh_linear |   .3675215   .1538492     2.39   0.017     .0659827    .6690604
beh_quadratic |  -.0726351   .1049503    -0.69   0.489    -.2783339    .1330636
    beh_avsim |   4.005875   1.941244     2.06   0.039     .2011071    7.810643
...
Overall maximum convergence ratio: 0.159 (criterion: below 0.25; met)
Convergence t-ratios (criterion: all below 0.1 in absolute value): met

(The network part and the rates are left out.) Selection is again estimated as about 1.5, and now the influence effect is statistically significant (4.01, z = 2.1, p = 0.039). So the data give some support for influence, but how strongly depends on the specification of the effect. In a real study you would report both, say which was your main specification before you looked, and not pick the more favorable one. Note also that the quadratic effect is now close to zero.

Selection through ego and alter effects

Selection on similarity is not the only possibility. behegox asks whether heavier drinkers send more friendship ties and behaltx whether they receive more:

. nwsaom, waves(glasgow1 glasgow2 glasgow3) outdegree reciprocity gwesp(.69) behsim behegox behaltx behavior(alcohol1 alcohol2 alcohol3) linear quadratic avalt seed(1)
...
       behsim |    1.55185   .7147432     2.17   0.030      .150979    2.952721
      behegox |   .0718023   .1271858     0.56   0.572    -.1774774    .3210819
      behaltx |  -.0519744   .1046156    -0.50   0.619    -.2570172    .1530685
...
Overall maximum convergence ratio: 0.235 (criterion: below 0.25; met)
Convergence t-ratios (criterion: all below 0.1 in absolute value): not met for rate_period2 (-0.115)
Convergence is good only if both criteria are met. Refit starting from these estimates with theta0(); see nwsaom.

Neither is distinguishable from zero, and the similarity effect is unchanged. The overall ratio of this run is 0.235, close to the limit, so for a real analysis this model would be restarted.

Goodness of fit with a behavior

After a co-evolution fit estat gof adds a fourth statistic, the distribution of the behavior, and it simulates network and behavior together (here after the avsim model, the last one fitted):

. estat gof, nsim(100) seed(7)

Goodness of fit: Mahalanobis distance test, 100 simulations at the estimates (2 periods pooled)
------------------------------------------------------------
Outdegree     | Mahalanobis dist. =   13.530   p =  0.140 (no evidence against fit)
Indegree      | Mahalanobis dist. =   11.373   p =  0.340 (no evidence against fit)
Geodesic      | Mahalanobis dist. =   29.953   p =  0.000 (evidence AGAINST fit)
Behavior      | Mahalanobis dist. =    0.981   p =  0.910 (no evidence against fit)
------------------------------------------------------------

The behavior distribution is reproduced very well. The geodesic distances are not, as in the thin model of Tutorial 3; the same remedies (more clustering and degree effects) apply.

Things to try

Use smoke1 smoke2 smoke3 as the behavior (smoking has fewer categories) and ask the same selection/influence question for smoking.
Add egox(alcohol1) altx(alcohol1) as constant covariates and compare with behegox / behaltx: the covariate versions use the fixed first-wave value, the behavior versions the current value, which changes during the simulation.
Fit the network-only model without the behavior and see how the network effects change.
Look up the other behavior effects (avsim split into endowment and creation, beheff() for effects of network position on the behavior, several behaviors at once) in help nwsaom under “Behavior co-evolution effects”.

Where next

Tutorial 5 shows how to test whether an effect should be in the model without fitting it, and how to see how much each effect matters.