# 3. Ordinary VAR

    library(idiographic)
    data(srl)
    vars <- c("efficacy", "value", "planning", "monitoring", "effort")
    has_cograph <- requireNamespace("cograph", quietly = TRUE)

The first-order vector autoregression, VAR(1), is the reference model for person-specific network estimation from intensive longitudinal data. It models one person’s multivariate series as a first-order Gaussian process, in which each occasion’s measurements depend jointly on the values of all variables one occasion earlier and on the associations that remain among the variables within the same occasion. It is an idiographic model, estimated for a single individual, on the premise that within-person dynamics need not match the between-person structure of the group. It presumes weak stationarity, meaning that the mean, variance, and autocovariance of every series are constant across the observation window; linear, lag-one dynamics; approximately Gaussian fluctuations; and equally spaced measurement occasions.

The model yields two networks over the same variables (Epskamp et al. 2018). The temporal network is directed within-person lag-one prediction: an edge `from -> to` states that the value of `from` at occasion \(t-1\) predicts the value of `to` at occasion \(t\) within the same person, holding the other lagged variables constant, which is predictability in the Granger sense (Bringmann et al. 2013). The self-loop of each variable is its autoregression — the inertia, or carry-over, of that variable from one occasion to the next. The contemporaneous network is undirected within-occasion partial correlation: after the previous occasion has explained what it can, the part of each variable left unexplained — its innovation — may still be associated with the innovations of the others, and the contemporaneous edges are the partial correlations among these remainders, the within-occasion associations that lagged prediction does not account for.

`fit_var()` estimates both networks by ordinary least squares, one regression per variable, without regularization. Each equation carries one coefficient per lagged predictor plus an intercept, so the size of the temporal network grows quadratically in the number of variables and the sampling variance of the estimates grows with it. Under a correctly specified linear VAR with exogenous innovations, the coefficient estimates are unbiased; every cell of both networks receives a coefficient, and no edge is removed on the estimator’s behalf, which leaves the analyst to judge which small values are noise. This makes ordinary VAR the transparent unregularized baseline: the appropriate estimator when the number of occasions is large relative to the number of parameters and every estimate is wanted for inspection, and the reference against which the regularized `fit_graphical_var()` trades variance for sparsity by shrinking weak edges to exactly zero under an extended-BIC penalty. For several people, `fit_var_each()` fits a separate ordinary VAR per person and `fit_mlvar()` pools information across people through random effects.

# Data and preprocessing

The estimator expects long format: 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 a single student, Grace, on five indicators: `efficacy`, `value`, `planning`, `monitoring`, and `effort`. `fit_var()` standardizes each variable to unit variance (`scale = TRUE`), so that coefficients are comparable across indicators, and centres within person (`center_within = TRUE`), so that the intercept absorbs the person mean; it does not remove trends. Because a violated assumption is absorbed silently — a trend, for instance, inflates the autoregressive diagonal rather than producing an error — the stationarity screen precedes the fit.

    preprocess(srl, vars = vars, id = "name")
    #> Idiographic Preprocessing
    #>   Variables:      5 (efficacy, value, planning, monitoring, effort)
    #>   Ordered rows:   5616
    #>   Retained pairs: 5548
    #>   Trend flags:    10
    #>   High AR flags:  0
    #>   Drift flags:    1
    #>   Unit-root risk: 0
    #>   Zero variance:  0
    #>   Tables:         x$pairs | x$counts | x$diagnostics
    #>
    #> 10 of 180 subject-series show a trend or unit-root that can bias the temporal network. preprocess() only diagnosed this; to clean just the series that need it, re-run with:
    #>   preprocess(data = srl, vars = vars, id = "name", detrend = "auto")

Across all 36 students the five indicators give 5616 ordered rows, of which 5548 survive as complete current/lagged pairs. Ten subject-series trip the linear-trend flag and one a drift flag, while the high-autoregression, unit-root, and zero-variance screens are clear. Grace’s own five series are among the clean ones: her 156 ordered occasions yield 155 complete pairs and trip no flag, so the model is fitted on her series as they stand. Where a flagged series is to be modelled, `preprocess(..., detrend = )` constructs and rechecks the transformed lag design. The same transformation must then be applied to the long data supplied to `fit_var()`; `preprocess()` does not mutate the source data. A persistent trend left in place can masquerade as lag-one structure.

# Fitting the model

The estimator takes the data, the variable set, the id column, and the subject. The `subject =` argument selects the person inside the call, so the occasion ordering that the lag-one design requires is preserved rather than reconstructed by the caller.

    var_fit <- fit_var(srl, vars = vars, id = "name", subject = "Grace",
                       scale = TRUE)
    var_fit
    #> OLS VAR Result
    #>   Variables:      5 (efficacy, value, planning, monitoring, effort)
    #>   Observations:   155
    #>   Temporal edges: 25 / 25
    #>   Contemp edges:  10 / 10
    #>
    #>   Temporal [directed]
    #>     weights [-0.160, 0.159]  |  +11 / -14 edges
    #>                efficacy value planning monitoring effort
    #>     efficacy      -0.13  0.04    -0.01      -0.04  -0.09
    #>     value         -0.10  0.10     0.09      -0.12   0.13
    #>     planning      -0.04 -0.11    -0.01      -0.06   0.16
    #>     monitoring     0.05 -0.16    -0.03       0.00   0.00
    #>     effort         0.06  0.07     0.11       0.01  -0.02
    #>
    #>   Contemporaneous [undirected]
    #>     weights [-0.060, 0.467]  |  +6 / -4 edges
    #>                efficacy value planning monitoring effort
    #>     efficacy       0.00  0.06    -0.06       0.38  -0.03
    #>     value          0.06  0.00     0.11       0.10  -0.01
    #>     planning      -0.06  0.11     0.00      -0.06   0.33
    #>     monitoring     0.38  0.10    -0.06       0.00   0.47
    #>     effort        -0.03 -0.01     0.33       0.47   0.00
    #>
    #>   plot(x) | plot(x, layer = "temporal")
    #>   edges(x) | nodes(x) | summary(x) | coefs(x) | matrices(x)

The call returns a `var_result` object whose print method reports both estimated networks with their weight ranges and signed-edge counts. Grace contributes 155 usable occasions, the single lost row being the first occasion, which has no predecessor. All 25 temporal coefficients and all 10 contemporaneous partial correlations are estimated — least squares retains every edge — with the temporal weights confined between −0.160 and 0.159 while the strongest contemporaneous weight reaches 0.467, a first indication that Grace’s within-occasion co-regulation is stronger than her lag-one carry-over.

# Reading the output

The `summary()` method reports one row per network layer, with the edge count, density, mean absolute weight, and the split of positive and negative edges.

    summary(var_fit)
    #>           network n_nodes n_edges density mean_abs_weight n_positive n_negative
    #> 1        temporal       5      20       1      0.07457677          9         11
    #> 2 contemporaneous       5      10       1      0.16039547          6          4

Both layers are fully dense, as they must be under least squares, so the comparison is carried by the weights: the contemporaneous mean absolute weight of 0.160 is roughly twice the temporal 0.075. Within-occasion association is larger than lagged prediction in this fitted series; no claim of typicality is made from one participant.

The `edges()` accessor returns one row per edge in decreasing magnitude; `network =` selects a layer and `n =` keeps the strongest edges. For the temporal layer the `from` and `to` columns read as `from` at occasion \(t-1\) predicting `to` at occasion \(t\).

    edges(var_fit, network = "temporal", n = 5)
    #>    network       from         to     weight
    #> 1 temporal monitoring      value -0.1604470
    #> 2 temporal   planning     effort  0.1591502
    #> 3 temporal      value     effort  0.1346800
    #> 4 temporal      value monitoring -0.1204242
    #> 5 temporal   planning      value -0.1098008

Grace’s strongest lag-one coefficients are small. Monitoring on one occasion predicts lower task value on the next (−0.160), and planning predicts higher subsequent effort regulation (0.159), conditional on the other lagged variables. These are predictive associations, not evidence that planning causes effort or monitoring causes a later change in value. At this magnitude, and with 155 occasions, the individual temporal coefficients can be high-variance, so the temporal layer is better read as a pattern than as a set of point claims. Where a sparse, multiplicity-controlled temporal network is required, `fit_graphical_var()` is the estimator of choice.

    edges(var_fit, network = "contemporaneous", n = 5)
    #>           network       from         to     weight
    #> 1 contemporaneous monitoring     effort 0.46674337
    #> 2 contemporaneous   efficacy monitoring 0.37844327
    #> 3 contemporaneous   planning     effort 0.33179869
    #> 4 contemporaneous      value   planning 0.10632926
    #> 5 contemporaneous      value monitoring 0.09825036

The contemporaneous layer is where Grace’s process is organized. Monitoring and effort share a partial correlation of 0.467, and monitoring also couples with efficacy (0.378) and planning with effort (0.332). Within a single occasion, monitoring, effort, and efficacy move together after the other indicators are partialled out; these are the same three edges retained when the clean-room vignette applies graphical VAR to Grace, there with shrinkage-biased weights.

Node-level structure follows from `nodes()`, whose strength column sums the absolute weights incident to each node within a layer, split into out- and in-strength for the directed temporal network, with the self-loop reported separately.

    nodes(var_fit)
    #>            network       node  strength out_strength in_strength         self
    #> 1         temporal   efficacy 0.4379985    0.1821870   0.2558116 -0.130248472
    #> 2         temporal      value 0.8318268    0.4468467   0.3849801  0.103987599
    #> 3         temporal   planning 0.6027558    0.3639382   0.2388176 -0.006545312
    #> 4         temporal monitoring 0.4756466    0.2470050   0.2286416  0.004077164
    #> 5         temporal     effort 0.6348432    0.2515586   0.3832846 -0.024338368
    #> 6  contemporaneous   efficacy 0.5314378           NA          NA  0.000000000
    #> 7  contemporaneous      value 0.2755102           NA          NA  0.000000000
    #> 8  contemporaneous   planning 0.5548748           NA          NA  0.000000000
    #> 9  contemporaneous monitoring 1.0030222           NA          NA  0.000000000
    #> 10 contemporaneous     effort 0.8430642           NA          NA  0.000000000

In the temporal layer `value` has the highest strength (0.832), driven by both outgoing and incoming effects; in the contemporaneous layer `monitoring` is the hub (1.003), consistent with its position at the junction of the two strongest within-occasion edges. The autoregressions are weak throughout — the largest, for efficacy, is −0.13 — so Grace’s states carry little inertia from one occasion to the next. The full coefficient tables, including intercepts and self-loops, are available from `coefs(var_fit)`, and the underlying matrices — the lagged coefficient matrix, the residual covariance, and the partial-correlation matrix derived from it — from `matrices(var_fit)`.

# Visualizing the network

Plotting the fit draws both layers side by side: arrows in the temporal panel denote lag-one prediction, edge width scales with absolute weight, and colour encodes sign.

    plot(var_fit)

    plot(var_fit, layer = "temporal")

The temporal graph is diffuse, matching the narrow weight range: no single lagged driver dominates Grace’s next-occasion state.

    plot(var_fit, layer = "contemporaneous")

The contemporaneous graph is concentrated: the thick monitoring–effort edge and its links to efficacy and planning form the within-occasion co-regulation core that the summary statistics quantified. Because the estimates are person-specific, both graphs describe Grace’s process and carry no claim about other students.

The two layers together make a mixed network — directed lag-one effects and undirected contemporaneous partial correlations — and `plot(var_fit, mixed = TRUE)` draws them in one graph, the temporal edges as curved arrows and the contemporaneous edges as straight lines.

    plot(var_fit, mixed = TRUE)

# References

Bringmann, Laura F., Nathalie Vissers, Marieke Wichers, et al. 2013. “A Network Approach to Psychopathology: New Insights into Clinical Longitudinal Data.” PLoS ONE 8 (4): e60188.
Epskamp, Sacha, Lourens J. Waldorp, René Mõttus, and Denny Borsboom. 2018. “The Gaussian Graphical Model in Cross-Sectional and Time-Series Data.” Multivariate Behavioral Research 53 (4): 453–80.