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7. Unified SEM
Unified structural equation modelling (uSEM) specifies a
person-specific structural model for a single individual’s multivariate
time series, in which each variable at the current occasion is regressed
simultaneously on the lagged values of all variables — its own and the
others’ — and on the other variables measured at the same occasion. It
is an idiographic model, built on the premise that within-person
dynamics need not match the between-person structure of the group: every
coefficient describes how one person’s process unfolds around that
person’s own means, not how people differ from one another. Like the
other lag-one estimators in this package, it presumes weak stationarity
— constant mean, variance, and autocovariance across the observation
window — linear lag-one dynamics, and equally spaced occasions; to these
it adds the identification requirements of a structural model, since the
within-occasion paths must be estimable for each person, which
constrains how many paths can be entertained relative to the length of
the series.
The model yields three networks over the same variables. The temporal
network is directed and within-person: an edge
from -> to states that the person’s value of
from at occasion \(t-1\)
predicts their value of to at occasion \(t\), holding the other lagged variables
constant. The contemporaneous network collects the within-occasion
relations, and this layer is what separates uSEM from VAR and graphical
VAR: where those models summarize same-occasion association as
undirected partial correlations among residuals, uSEM resolves each
within-occasion relation into a directed structural path, so an
edge from -> to records a directed same-occasion
coefficient from from to to for that person.
What the directed paths leave unexplained is carried by the third layer,
an undirected residual-covariance network among the innovations. The
directed contemporaneous reading is warranted when theory or design
implies a within-occasion ordering among the indicators; where no
ordering is defensible, the undirected graphical-VAR contemporaneous
network is the more conservative summary. GIMME, treated in the next
vignette, extends the uSEM equation with a group-level search that
recovers paths shared across people.
fit_usem() estimates one SEM per selected person and
returns temporal, directed contemporaneous, and residual-covariance
networks. With one selected person, as below, every coefficient is
idiographic. With multiple people the function also averages across
converged fits; that average summarizes the sample and is not any one
person’s model. Failed fits are reported rather than silently
included.
Data and preprocessing
The estimator takes the same long-format panel as the other
estimators: one row per person-occasion, an id column, and numeric
time-varying indicators ordered within person. The bundled
srl data hold self-regulated-learning indicators for 36
students measured over 156 occasions each; this vignette fits Grace on
five indicators: efficacy, value,
planning, monitoring, and effort.
Grace is chosen because her five series pass the input audit, not
because of the network returned later. Because uSEM is a dynamic lag-one
model that absorbs assumption violations silently — a trending series
inflates its lagged coefficients rather than producing an error — the
stationarity screen precedes the fit.
preprocess(srl, vars = vars, id = "name", subject = "Grace")
#> Idiographic Preprocessing
#> Variables: 5 (efficacy, value, planning, monitoring, effort)
#> Ordered rows: 156
#> Retained pairs: 155
#> Trend flags: 0
#> High AR flags: 0
#> Drift flags: 0
#> Unit-root risk: 0
#> Zero variance: 0
#> Tables: x$pairs | x$counts | x$diagnostics
Grace’s 156 rows yield 155 complete lagged pairs. None of the five
series trips the trend, high-autoregression, mean-shift, variance-shift,
unit-root, or zero-variance screen, so the series is fitted as
supplied.
Fitting the model
The substantive arguments are time (orders occasions
within id), temporal ("ar" for
autoregressions only, "all" for candidate cross-lags),
contemporaneous ("none" or "all"
candidate directed same-occasion paths), and trim. A
reciprocal all-path same-occasion model is not identified as a final
SEM. Therefore trim = TRUE starts from an identified base
model and uses the documented fit and significance criteria to add and
prune candidate paths; the vignette does not present the underidentified
untrimmed model as a result. The fit and all accessors below are
evaluated when the suggested lavaan package is installed
and the build environment permits core detection. This second guard
handles restricted builders where lavaan cannot initialize its options;
no static output is substituted when execution is unavailable.
usem_fit <- fit_usem(srl, vars = vars, id = "name", time = "day",
subject = "Grace", temporal = "all",
contemporaneous = "all", trim = TRUE)
usem_fit
#> uSEM Result
#> Subjects: 1 (1 converged)
#> Variables: 5 (efficacy, value, planning, monitoring, effort)
#> Observations: median 155 (range 155-155)
#>
#> Temporal [directed]
#> no non-zero edges
#> efficacy value planning monitoring effort
#> efficacy 0 0 0 0 0
#> value 0 0 0 0 0
#> planning 0 0 0 0 0
#> monitoring 0 0 0 0 0
#> effort 0 0 0 0 0
#>
#> Contemporaneous [directed]
#> weights [0.348, 0.456] | +3 / -0 edges
#> efficacy value planning monitoring effort
#> efficacy 0 0 0.00 0.42 0.00
#> value 0 0 0.00 0.00 0.00
#> planning 0 0 0.00 0.00 0.00
#> monitoring 0 0 0.00 0.00 0.46
#> effort 0 0 0.35 0.00 0.00
#>
#> Residual_cov [undirected]
#> no non-zero edges
#> efficacy value planning monitoring effort
#> efficacy 0 0 0 0 0
#> value 0 0 0 0 0
#> planning 0 0 0 0 0
#> monitoring 0 0 0 0 0
#> effort 0 0 0 0 0
#>
#> plot(x) | plot(x, layer = "temporal") | plot(x, layer = "contemporaneous")
#> edges(x) | nodes(x) | summary(x) | coefs(x) | matrices(x)
Grace’s model converges on 155 usable lagged pairs. The search
retains three directed contemporaneous paths and no temporal or
residual-covariance edges. That zero is an executed selection result: it
says no candidate temporal path survived this uSEM search for Grace, not
that the temporal layer was disabled.
Reading the output
The summary() method reports one row per network layer,
with the edge count, density, and mean absolute weight.
summary(usem_fit)
#> network n_nodes n_edges density mean_abs_weight n_positive n_negative
#> 1 temporal 5 0 0.00 0.0000000 0 0
#> 2 contemporaneous 5 3 0.15 0.4091158 3 0
#> 3 residual_cov 5 0 0.00 0.0000000 0 0
The contemporaneous network has three of the 20 possible directed
edges (density 0.15) and mean absolute weight 0.409. The temporal and
residual layers have zero selected edges.
edges(usem_fit, network = "temporal", n = 5)
#> [1] network from to weight
#> <0 rows> (or 0-length row.names)
The empty table makes the selection outcome explicit. It should not
be read as proof of no lagged process; a different candidate set,
trimming rule, or person can give a different selected model.
edges(usem_fit, network = "contemporaneous", n = 5)
#> network from to weight
#> 1 contemporaneous monitoring effort 0.4560845
#> 2 contemporaneous efficacy monitoring 0.4237397
#> 3 contemporaneous effort planning 0.3475233
The directed contemporaneous network retains monitoring to effort
(0.456), efficacy to monitoring (0.424), and effort to planning (0.348).
These arrows are directed SEM coefficients conditional on the selected
specification; causal interpretation still requires a defensible
within-occasion ordering.
nodes(usem_fit)
#> network node strength out_strength in_strength self
#> 1 temporal efficacy 0.0000000 0.0000000 0.0000000 0
#> 2 temporal value 0.0000000 0.0000000 0.0000000 0
#> 3 temporal planning 0.0000000 0.0000000 0.0000000 0
#> 4 temporal monitoring 0.0000000 0.0000000 0.0000000 0
#> 5 temporal effort 0.0000000 0.0000000 0.0000000 0
#> 6 contemporaneous efficacy 0.4237397 0.4237397 0.0000000 0
#> 7 contemporaneous value 0.0000000 0.0000000 0.0000000 0
#> 8 contemporaneous planning 0.3475233 0.0000000 0.3475233 0
#> 9 contemporaneous monitoring 0.8798242 0.4560845 0.4237397 0
#> 10 contemporaneous effort 0.8036079 0.3475233 0.4560845 0
#> 11 residual_cov efficacy 0.0000000 NA NA 0
#> 12 residual_cov value 0.0000000 NA NA 0
#> 13 residual_cov planning 0.0000000 NA NA 0
#> 14 residual_cov monitoring 0.0000000 NA NA 0
#> 15 residual_cov effort 0.0000000 NA NA 0
Because the contemporaneous layer is directed, nodes()
separates outgoing from incoming weight. Monitoring has the largest
total contemporaneous strength (0.880); it receives the efficacy path
and sends the effort path. The full path and residual matrices are
available from coefs() and matrices().
matrices(usem_fit)
#>
#> $temporal
#> efficacy value planning monitoring effort
#> efficacy 0 0 0 0 0
#> value 0 0 0 0 0
#> planning 0 0 0 0 0
#> monitoring 0 0 0 0 0
#> effort 0 0 0 0 0
#>
#> $contemporaneous
#> efficacy value planning monitoring effort
#> efficacy 0.000 0 0 0.000 0.000
#> value 0.000 0 0 0.000 0.000
#> planning 0.000 0 0 0.000 0.348
#> monitoring 0.424 0 0 0.000 0.000
#> effort 0.000 0 0 0.456 0.000
#>
#> $residual_cov
#> efficacy value planning monitoring effort
#> efficacy 0 0 0 0 0
#> value 0 0 0 0 0
#> planning 0 0 0 0 0
#> monitoring 0 0 0 0 0
#> effort 0 0 0 0 0
Visualizing the network
The temporal layer is already documented as empty, so the vignette
does not draw an empty showcase panel. The contemporaneous panel draws
the three selected directed paths, with edge width scaled to absolute
weight and colour encoding sign.
plot(usem_fit, layer = "contemporaneous")
The within-occasion structure can also be drawn as a mixed network.
Directed contemporaneous paths appear as curved arrows and any residual
covariances as straight edges.
plot(usem_fit, mixed = TRUE)
The monitoring-to-effort arrow is the largest retained coefficient.
No residual covariance survives this search. The direction of each arrow
is only as credible as the within-occasion ordering assumption behind
it, which is the consideration that should govern the choice between
uSEM and the undirected graphical-VAR contemporaneous summary.
References
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.