Day 12: Stochastic Actor-Oriented Models

Is This Actually a SAOM Question?

A SAOM is designed for panel networks in which actors have opportunities to change outgoing ties between observed waves.

Can the substantive process be represented as actors making small tie changes while the network and possibly actor behavior co-evolve?

Longitudinal data alone do not make a SAOM appropriate. The actor-oriented micro-step story must be a defensible approximation.

Open the walkthrough

Did Similarity Produce the Relationship?

Or did the relationship make the actors more similar?

States: Do states ally because their security policies already align, or do alliances bring their policies closer?
Legislators: Do legislators collaborate because they already hold similar positions, or does collaboration bring their positions closer?
Armed groups: Do groups cooperate because their strategies already align, or does cooperation lead their strategies to converge?

Repeated observations help us see what changed first. They do not record every change between waves, so the model shows patterns consistent with selection or influence rather than proving either process.

Read the substantive translation

The Specifications Answer Different Applied Questions

Specification	Main question	What we should be able to say
Network SAOM	Which local tie changes are more attractive to actors?	Reciprocity raises the objective-function score of a specific candidate change
Co-evolution SAOM	Are similarity patterns consistent with selection, influence, or both?	Similar peers are selected more often, while influence is weaker or uncertain
TERGM comparator	Is change better represented as a network transition distribution?	The prior graph predicts the next graph without assigning every intermediate change to an actor

Use the applied model compass

Pre-Flight Checks Come Before Fitting

Data feature	Question to settle before fitting
Actor composition	Who was actually present and eligible during each interval?
Structural zero	Which particular ties were genuinely impossible, not merely absent?
Network change	Is there enough continuity and change to estimate a process?
Measurement	Are missingness, behavior ranges, direction, and tie definitions defensible?

netify verifies actor order, direction, and dimensions across the friendship waves before the RSiena array is created.

A structural zero removes a particular tie from the possible choices. A composition-change object determines when an actor belongs to the modeled population. Joining or leaving is not a control variable.

Run the data checks

The Model Fills the Space Between Waves

Observed:

$Y(t_1)\qquad\longrightarrow\qquad Y(t_2)$

Modeled between them:

$Y^{(0)}\rightarrow Y^{(1)}\rightarrow Y^{(2)}\rightarrow\cdots\rightarrow Y^{(M)}$

At each micro-step, one actor gets an opportunity to change one outgoing tie or make no change.

The micro-steps are a model-based latent path, not events observed in the data.

Walk through a micro-step

Rate and Evaluation Parameters Do Different Jobs

Rate function

How often actors receive opportunities to make a change between waves.

Evaluation function

Which available network state an actor tends to prefer when an opportunity occurs.

More opportunities do not imply a preference for more ties. Opportunity and direction of change are separate parts of the model.

The Evaluation Function Scores Candidate States

$f_i(y)=\sum_k \beta_k s_{ki}(y)$

When actor $i$ has an opportunity, candidate next states receive scores based on statistics such as:

Outdegree
Reciprocity
Transitive triplets
Covariate-related selection

For a proposed tie $i\rightarrow j$ , which statistics change for actor $i$ ?

Estimation Simulates the Missing Path

The panel shows the beginning and end of each interval, not the changes in between. RSiena therefore:

Uses the rate function to generate opportunities.
Uses the evaluation function to assign probabilities to the available one-step changes.
Simulates a complete path to the next observed wave.
Checks whether that simulated path produces the kinds of changes seen in the panel.
Revises the parameters and tries again.

The estimator learns which rate and evaluation parameters make the observed changes typical under the assumed micro-step process.

Walk Through SAOM Estimation

What RSiena Is Solving

Let $S_{\mathrm{obs}}$ be the selected change statistics in the observed panel and $S_{\mathrm{sim}}(\theta)$ the same statistics after simulating paths with parameters $\theta$ .

$E_{\theta}\!\left[S_{\mathrm{sim}}(\theta)\right]-S_{\mathrm{obs}}=0$

Robbins-Monro stochastic approximation moves $\theta$ to reduce the simulated-minus-observed discrepancy. The final phase holds the estimate fixed and uses further simulations for standard errors and convergence checks.

Default SAOM estimation is simulation-based method of moments. It is not maximizing a likelihood. Convergence means the targeted moments match within Monte Carlo error, not that the actor-oriented story has been proven.

Fit the Smallest Model That Answers the Question

Start with:

A density or outdegree effect
Reciprocity when substantively plausible
A carefully chosen closure effect
The focal covariate or behavior mechanism

Do not begin with every effect in the RSiena catalog. Each added statistic changes the actor’s choice surface and the convergence problem.

Fit the first model

Convergence Is About Simulated Moments Matching Targets

RSiena adjusts parameters until statistics simulated under the current model align with the observed change statistics.

Inspect:

Overall maximum convergence ratio
Effect-specific t-ratios for deviations
Stability across repeated runs or continued estimation

A printed coefficient is not interpretable until the estimation has converged well enough for the simulated moment conditions.

Break and repair convergence

What the First Fit Is Describing

Between observed waves, the model represents students as receiving opportunities to reconsider one outgoing friendship nomination. At each opportunity, one nomination can be added, removed, or left unchanged.

Waves 1 to 2

About 6.52 opportunities per student

Waves 2 to 3

About 5.21 opportunities per student

These rates are not log odds. They describe opportunities in the fitted process, not the number of friendships each student changed. An opportunity can produce no change, and the same tie can be reconsidered more than once.

The First Fit in One Table

Interpret each estimate as a contribution to the score of one proposed micro-change.

Friendship feature	Estimate (SE)	What the sign says
One more outgoing nomination	$-2.78\ (0.16)$	Additional nominations are less attractive
Return a nomination	$2.48\ (0.22)$	Returning a nomination is more attractive
Close one friend-of-friend path	$0.66\ (0.15)$	Closing a path is more attractive
Create one three-cycle	$-0.09\ (0.29)$	Little evidence of a separate cycle pattern
Choose someone with the same baseline smoking	$0.14\ (0.15)$	Little evidence of a smoking-similarity pattern

Start with the sign and describe the proposed change. Say: “Holding the other modeled features constant, this feature makes the proposed change more or less attractive.” Exponentiate only if the relative weight helps.

Returned Friendships Carry the Strongest Signal

Suppose student $i$ is considering nominating student $j$ .

$j$ Does Not Nominate $i$

Adding $i\rightarrow j$ creates another outgoing nomination but does not create reciprocity.

$j$ Already Nominates $i$

Adding $i\rightarrow j$ returns the nomination and creates reciprocity.

$\exp(2.48)\approx 11.9$

Holding the other modeled features constant, a proposed nomination that returns an existing nomination receives about twelve times the choice weight of an otherwise comparable nomination that does not.

Friends of Friends Are More Attractive Candidates

Start with:

$\text{Alice}\rightarrow\text{Ben} \qquad\text{and}\qquad \text{Ben}\rightarrow\text{Cara}.$

If Alice adds a nomination to Cara, the move closes one transitive path:

$\text{Alice}\rightarrow\text{Cara}.$

$\exp(0.66)\approx 1.94$

Holding the other modeled features constant, closing one friend-of-friend path nearly doubles the choice weight of the proposed nomination. Closing three paths contributes $3(0.66)=1.98$ to its score.

What the First Fit Does Not Show

Three-Cycles

Estimate: -0.09, SE: 0.29

No clear evidence of a separate circular pattern such as $i\rightarrow j\rightarrow k\rightarrow i$ after reciprocity and transitivity are included.

Same Baseline Smoking

Estimate: 0.14, SE: 0.15

No clear evidence that students with the same wave-1 smoking status receive more weight as friendship choices after accounting for the network terms.

Friendships tend to be returned, and friends of friends tend to become connected. Baseline smoking similarity adds little in this specification. Because smoking is treated as a fixed wave-1 covariate, this model cannot study whether smoking changes with friendships. That requires the co-evolution model next.

One Proposed Nomination Changes Several Statistics

Before Alice’s decision:

$\text{Alice}\rightarrow\text{Ben}\rightarrow\text{Cara}, \qquad \text{Alice}\rightarrow\text{Elena}\rightarrow\text{Cara}, \qquad \text{Cara}\rightarrow\text{Alice}.$

Adding $\text{Alice}\rightarrow\text{Cara}$ produces:

Feature	Change	Estimate	Contribution to Alice’s score
One more outgoing nomination	$1$	$-2.78$	$-2.78$
Returned nomination	$1$	$2.48$	$+2.48$
Friend-of-friend paths closed	$2$	$0.66$	$+1.32$
Combined change			$+1.02$

A total of 1.02 means 2.77 times the weight of making no change, not a tie probability. The model separates contributions to its score, not Alice’s reason for the nomination.

Work through the combined calculation

Interpret the Same-Smoking Coefficient

The smoking comparison is a 0-to-1 change:

Same-smoking indicator	Meaning
$0$	The two students report different smoking categories
$1$	The two students report the same smoking category

The estimated same-smoking coefficient is $0.14$ :

$\exp(0.14)=1.15.$

At a modeled friendship-change opportunity, a nomination to a student with the same smoking status is about 15% more likely to be chosen than an otherwise identical nomination to a student with a different smoking status.

This is a conditional probability ratio between two possible nominations. The estimate is 0.14 with SE 0.15, so the difference is uncertain.

Walk through the 0-to-1 comparison

One Pattern, Two Possible Histories

At the final wave, Alice and Cara are friends and report similar drinking. That endpoint alone does not tell us how they got there.

Selection

They reported similar drinking first, and the friendship appeared later.

Influence

They were friends first, and their reported drinking became more similar later.

Several waves give us information about sequence, but we still do not observe every friendship and behavior change between interviews. The model supplies a particular account of that missing path.

The Co-Evolution Model Uses Two Equations

Both friendship and drinking now change. Each has its own rates and evaluation equation.

Equation	Other included terms	Focal term and estimate (SE)
Friendship	Outdegree, reciprocity, transitive triplets, three-cycles, drinking ego, and drinking alter	Similarity in reported drinking (1 to 5): $1.44\ (0.61)$
Drinking	Baseline movement across the 1-to-5 scale and toward the middle or extremes	Movement toward friends’ average drinking: $1.36\ (0.95)$

The two baseline drinking terms are called linear shape and quadratic shape in RSiena. They describe where drinking tends to move even apart from friends. They are not time trends.

Friendship opportunity: Change one outgoing nomination or make no change. Drinking opportunity: Move one category up or down or make no change. Separate period-specific rates govern how often the opportunities arise.

The fitted friendship process favors students with more similar drinking as possible friends. The estimated movement toward friends’ drinking is positive, but too uncertain to distinguish from no influence.

Build the co-evolution model

Interpret the Drinking-Similarity Coefficient

Reported drinking is coded 1 = none, 2 = once or twice a year, 3 = monthly, 4 = weekly, and 5 = more than weekly. RSiena turns the gap into a similarity score:

Difference in reported drinking	Similarity score
Same category	1.00
One category apart	0.75
Two categories apart	0.50

Moving one category closer raises similarity by $0.25$ . With the estimated coefficient of $1.44$ :

$0.25 \times 1.44 = 0.36,\qquad \exp(0.36)=1.43.$

At a modeled friendship-change opportunity, a student who is one drinking category closer is about 43% more likely to be chosen than an otherwise comparable student.

Walk through the coefficient step by step

Put the Influence Result on the Drinking Scale

Compare the complete joint model with its average-friend influence coefficient at 1.36 against the same model with only that coefficient set to 0. The friendship equation, selection term, drinking shape terms, rates, and starting observations remain fixed.

For students who start with at least one friend and differ from their friends’ average, a smaller next-wave distance from that starting friend average means closer.

Influence coefficient set to 0	Influence coefficient set to 1.36	Change from including the term
0.88 categories	0.82 categories	−0.06 categories

Within this specification, including the fitted influence term reduces the simulated distance to current friends’ starting average by about 0.06 response categories.

The influence estimate is 1.36 with SE 0.95. This small point-estimate contrast is not clear evidence of influence.

Walk through the influence calculation

Separate Equations Are Not Automatic Proof

Possible problem	What it could look like here
An omitted common cause	A new peer group or family change affects both friendship and drinking
Unknown ordering within a wave interval	A friendship and drinking change both occur between interviews
Measurement error	A student omits a friendship nomination or reports drinking imprecisely
A poor timing model	The meaningful changes happen much faster or more slowly than the interview spacing

The two equations let us represent selection and influence separately. A causal claim still requires a defensible design, timing assumptions, measurement, and model specification.

Goodness-of-Fit Asks What the Model Missed

After fitting, simulate friendship and drinking histories from the model and compare features that were not directly forced to match.

Check	What a miss would point toward	Result
Number of nominations received	Missing differences in popularity	$p=.589$
Distribution of drinking	A poor model of behavior change	$p=.912$
Types of three-person configurations	Missing closure, cycles, or subgroup structure	$p=.007$

The model reproduces the broad popularity and drinking distributions, but not the local three-person patterns. Convergence told us estimation finished. This comparison tells us the fitted model still leaves important network structure unexplained.

Run the goodness-of-fit checks

A Failed Check Suggests a Revision, Not a Fishing Expedition

We tried adding degree-based popularity because unequal popularity can create local clustering.

Specification	Overall convergence	Triad-census GOF
Original co-evolution model	$0.163$	$p=.007$
Added popularity effect	$0.266$	$p=.004$

The revised fit did not converge adequately and did not improve the triad check. We cannot call it a successful repair. The next step is to obtain convergence, rerun the check, and consider a theoretically motivated alternative. It is not to keep adding effects until one p-value exceeds .05.

A Score Test Temporarily Pins the New Effect at Zero

We want to know whether indegree popularity belongs in the model. The score test includes that statistic in the calculations but imposes

$\beta_{\text{popularity}}=0.$

fix = TRUE keeps the coefficient at zero.
initialValue = 0 states the restricted value.
test = TRUE asks whether the model strains against that restriction.

RSiena compares the observed popularity-related change with what the restricted model simulates. A large mismatch means the coefficient appears to want to move away from zero.

Test without fitting

What the Score Test Tells Us Here

For the indegree-popularity effect:

$\chi^2=5.27,\qquad p=.022.$

The observed friendship changes contain more popularity-related structure than the restricted model reproduces. Zero does not look adequate for this effect, so a freely estimated popularity term is worth investigating.

It does not tell us:

The coefficient’s direction or size
Its standard error after estimation
How other coefficients will change

Whether the expanded model will converge
Whether goodness-of-fit will improve

Use this as a screen. Then fit the expanded model and check convergence, estimates, and goodness-of-fit.

Time Heterogeneity Challenges One-Process Stories

Are the same evaluation parameters plausible between every pair of waves?

Time-varying effects may be needed when:

Institutions or opportunities change
The observation intervals differ
A shock changes the network process
Behavior scales or compositions shift

A single average parameter can conceal substantively different transition regimes.

SAOM and TERGM Encode Different Temporal Stories

SAOM

Actor-oriented micro-steps, rate and evaluation functions, and choices over outgoing ties.

TERGM

A probability model for networks conditional on past networks and covariates.

The choice should follow the research question, temporal resolution, actor control, and desired interpretation, not a claim that one family is universally more realistic.

The Referee Checklist

Ask of any SAOM application:

Why is actor-oriented change defensible for this relation?
How were composition, missingness, and structural zeros handled?
Did the model converge across effects and repeated runs?
Do simulations reproduce relevant network and behavior features?
Are selection, influence, and causal language matched to the assumptions?
Would a TERGM or relational event model answer a different and possibly better question?

Open the full checklist

After the Model Runs, Describe What Changed

Separate change opportunities in the rate function from preferences in the evaluation function.
Interpret an evaluation effect as a comparison among candidate microsteps, not an unconditional tie probability.
Keep selection in the network equation and influence in the behavior equation.
Use convergence, repeated runs, and GOF before stating the substantive conclusion.

“Across three observations of 50 Scottish adolescents, the fitted friendship process gives greater weight to reciprocal and transitive friendship states. The drinking-similarity estimate is 1.44 with a standard error of 0.61, which is consistent with selection into friendships with similar peers. The peer-drinking influence estimate is 1.36 with a standard error of 0.95, so this panel does not distinguish the influence pattern from zero.”

Read the full interpretation guide

What to Carry Into Day 13

A SAOM fills the intervals between panel waves with actor-oriented micro-steps.
Rate and evaluation parameters describe opportunities and preferences, not the same quantity.
Co-evolution represents selection and influence jointly but does not identify them causally without strong assumptions.
Convergence and goodness-of-fit are separate requirements.
SAOM and TERGM encode different temporal questions.

Tomorrow we separate two problems that are often blurred together: interference changes the estimand, while dyadic dependence changes uncertainty.