# Ising networks for binary data

## What an Ising network is

An Ising network models the conditional dependence structure among binary variables. Each node represents a variable with two possible states, conventionally coded as 0 and 1, such as the absence or presence of a symptom, failure or success on an assessment, or non-endorsement and endorsement of a questionnaire item. The model estimates the conditional association between every pair of variables while accounting for all remaining variables in the network.

The edge parameters of an Ising model are expressed on the log-odds scale and quantify conditional dependence between binary variables. A positive edge indicates that endorsement of one variable is associated with an increased conditional probability of endorsing the other after adjustment for the remaining variables, whereas a negative edge indicates a reduced conditional probability. For example, in a network of study behaviours, a positive edge between planning and completing assignments indicates that students who report planning are more likely to complete assignments after accounting for their other recorded behaviours. The absence of an edge indicates that the conditional association between two variables is adequately explained by the remaining variables included in the model.

Ising models are particularly appropriate for psychometric and behavioural data that are intrinsically binary, including symptom checklists, diagnostic criteria, pass–fail outcomes, and dichotomously scored questionnaire items. Because the model estimates conditional rather than marginal associations, the resulting network distinguishes direct statistical dependencies from relationships that arise indirectly through other variables. As with Gaussian graphical models, the estimated edges represent conditional statistical dependence and should not be interpreted as evidence of causal relationships.

## The data

The worked example uses `SRL_GPT`, which contains 300 observations on cognitive strategy use (`CSU`), intrinsic value (`IV`), self-efficacy (`SE`), self-regulation (`SR`), and test anxiety (`TA`). The original variables are continuous composite scores. An Ising model requires binary data, so the example uses `dichotomize()` with `method = "rank"` to code the lower half of each score as 0 and the upper half as 1.

    head(binary_data)
    #>      CSU IV SE SR TA
    #> [1,]   0  0  0  0  1
    #> [2,]   1  1  1  1  1
    #> [3,]   1  1  1  1  0
    #> [4,]   0  1  1  0  1
    #> [5,]   0  0  0  0  1
    #> [6,]   0  0  0  0  0

The rank rule gives each variable an endorsement rate of 0.50 in these data. Here, a value of 1 means that the observation falls in the upper half of that construct. Dichotomization discards information from the original scale, so it should be used only when a binary research question is substantively justified. Naturally binary variables can be analyzed directly.

## Fitting the network with `psychnet()`

`psychnet()` estimates an Ising network when `method = "ising"`. The default AND rule retains an edge when both nodewise models select it. The function returns a fitted `psychnet` object containing the symmetric interaction weights, node thresholds, and numerical certificate.

    ising_net <- psychnet(data = binary_data, method = "ising", rule = "AND")
    ising_net
    #> <psychnet> ising network
    #>   nodes: 5   edges: 7   (undirected)
    #>   optimality (KKT residual): 3.74e-10

The fitted network contains 5 nodes and 7 edges. Three of the ten possible pairs are removed by the selected model.

## Inspecting interactions with `summary()`

`summary()` prints the fitted network and returns the edge table invisibly. The columns `from`, `to`, and `weight` identify the retained interactions. Positive weights indicate higher conditional odds of joint endorsement. Negative weights indicate lower conditional odds.

    summary(ising_net)
    #> <psychnet> ising network
    #>   nodes: 5   edges: 7   (undirected)
    #>   optimality (KKT residual): 3.74e-10
    #>   edge weight: range [-0.347, 2.242], mean 1.347

The six retained edges among `CSU`, `IV`, `SE`, and `SR` are positive. The largest is `IV`-`SR` at 2.242, followed by `CSU`-`SE` at 2.188. Students in the upper half of one construct have higher conditional odds of being in the upper half of the connected construct after the other binary variables are considered.

The `SR`-`TA` edge is negative (-0.347). Upper-half self-regulation is associated with lower conditional odds of upper-half test anxiety after cognitive strategy use, intrinsic value, and self-efficacy are taken into account. Test anxiety has no other retained edge in this model.

Ising weights are logistic interaction coefficients, not correlations. Their magnitudes should therefore be interpreted on the model’s log-odds scale and compared within the fitted network.

## Checking numerical optimality with `certificate()`

`certificate()` reports the largest stationarity residual across the fitted nodewise logistic models. It returns `method`, `certificate`, `kind`, and `certified`. A small residual indicates that the optimization satisfies its first-order conditions to the requested tolerance.

    certificate(ising_net)
    #>   method certificate kind certified
    #> 1  ising 3.74261e-10  kkt      TRUE

The residual is \(3.74 \times 10^{-10}\) and `certified` is `TRUE`. This value is close to machine precision and indicates that the penalized logistic fits satisfy their numerical conditions to numerical precision. The certificate checks optimization, not sampling stability, measurement validity, or causation.

## Describing node position with `net_centralities()`

`net_centralities()` returns `node`, `strength`, and `expected_influence` by default. Strength sums the absolute interaction weights. Expected influence sums the signed weights, so negative connections reduce the total.

    net_centralities(ising_net)
    #>   node  strength expected_influence
    #> 1  CSU 4.9459597          4.9459597
    #> 2   IV 5.3057980          5.3057980
    #> 3   SE 4.7737358          4.7737358
    #> 4   SR 4.8683790          4.1747307
    #> 5   TA 0.3468242         -0.3468242

Intrinsic value has the highest strength (5.306), followed by cognitive strategy use (4.946). Self-regulation has strength 4.868 and expected influence 4.175 because its negative connection with test anxiety reduces the signed sum. Test anxiety has one edge, giving strength 0.347 and expected influence -0.347.

These indices describe connection patterns in the estimated binary network. They do not show which construct should be targeted by an intervention.

## Evaluating binary predictability with `net_predict()`

`net_predict()` evaluates how accurately each node is classified from the others. Binary nodes use normalized classification accuracy (`nCC`). An `nCC` of 0 means that prediction is no better than always choosing the more frequent category. An `nCC` of 1 means perfect classification. The `accuracy` column reports the unadjusted proportion classified correctly.

    net_predict(ising_net, data = binary_data)
    #>   node   type metric predictability  accuracy
    #> 1  CSU binary    nCC      0.7066667 0.8533333
    #> 2   IV binary    nCC      0.7266667 0.8633333
    #> 3   SE binary    nCC      0.7000000 0.8500000
    #> 4   SR binary    nCC      0.7133333 0.8566667
    #> 5   TA binary    nCC      0.0000000 0.5000000

Intrinsic value has the highest normalized accuracy (0.727) and a raw accuracy of 0.863. Cognitive strategy use, self-efficacy, and self-regulation have `nCC` values from 0.700 to 0.713 and raw accuracies near 0.85. Their neighbours provide substantial classification information beyond the 0.50 marginal baseline.

Test anxiety has `nCC = 0` and accuracy 0.50. Its neighbour does not improve classification beyond the marginal baseline in the penalized model. These are in-sample values; assessment on new observations requires validation data.

## Comparing an unregularized Ising estimator

`psychnet()` fits unregularized nodewise logistic models when `method = "ising_sampler"`. The argument `alpha = 0.05` removes edges whose Wald tests exceed the selected level. This comparison checks whether the main structure persists under a different estimation procedure.

    unregularized_net <- psychnet(data = binary_data, method = "ising_sampler", alpha = 0.05)
    unregularized_net
    #> <psychnet> ising_sampler network
    #>   nodes: 5   edges: 7   (undirected)
    #>   optimality (KKT residual): 1.38e-08

    summary(unregularized_net)
    #> <psychnet> ising_sampler network
    #>   nodes: 5   edges: 7   (undirected)
    #>   optimality (KKT residual): 1.38e-08
    #>   edge weight: range [-0.769, 2.309], mean 1.354

The unregularized model retains the same 7 edges. Its positive learning-cluster interactions are slightly larger, and the `SR`-`TA` edge is -0.769 compared with -0.347 in the penalized model. The shared structure provides evidence that the main qualitative pattern is not specific to the penalized estimator. The change in magnitude reflects the different estimation and pruning procedures.

    certificate(unregularized_net)
    #>          method  certificate kind certified
    #> 1 ising_sampler 1.384701e-08  kkt      TRUE

The residual is \(1.38 \times 10^{-8}\) and `certified` is `TRUE`, indicating that the unregularized nodewise fits satisfy their score conditions within the default tolerance.

## Visualizing the network with `cograph::splot()`

`cograph::splot()` draws the network when the optional `cograph` package is available. Psychological styling uses green for positive interactions and red for negative interactions. Predictability rings summarize node classification.

    cograph::splot(ising_net, psych_styling = TRUE, predictability = TRUE)

Node placement is determined by a layout algorithm and has no statistical scale. The edge table remains the primary source for numerical interpretation.

## Reporting the analysis

A report should define what 0 and 1 mean for every node, describe any dichotomization rule, state the sample size, estimator, EBIC setting, edge rule, and missing-data procedure, and report interaction weights numerically. Results should be described as conditional associations on a log-odds scale.

The present analysis used rank dichotomization, 300 complete observations, the EBIC-regularized Ising estimator, and the AND rule. The model retained 7 edges. The largest interaction was `IV`-`SR` (2.242), and `SR`-`TA` was negative (-0.347). The KKT residual was \(3.74 \times 10^{-10}\). An unregularized sensitivity model retained the same edge set.

## How the Ising network is estimated

Each binary node is modeled with logistic regression using the remaining nodes as predictors. The regularized estimator applies an \(\ell_1\) penalty and selects a penalty value by EBIC for every node. Small coefficients can be set to zero, which removes the corresponding neighbour from that node’s model.

Each pair receives two estimates because either node can be treated as the outcome. The AND rule retains a pair when both regressions select it and averages the two estimates into one undirected interaction. The OR rule retains a pair when either regression selects it.

## Mathematical foundations

For a binary vector \(\mathbf{x} \in \{0,1\}^p\), the Ising model is

\[ P(\mathbf{x}) = \frac{1}{Z}\exp\left( \sum_i \tau_i x_i + \sum_{i<j}\omega_{ij}x_ix_j \right), \]

where \(\tau_i\) is a node threshold, \(\omega_{ij}\) is an interaction, and \(Z\) is the normalizing constant. The conditional model for node \(i\) is

\[ \operatorname{logit}P(X_i=1\mid\mathbf{X}_{-i}) = \tau_i + \sum_{j\ne i}\omega_{ij}X_j. \]

The regularized estimator minimizes the negative nodewise log-likelihood plus \(\lambda\sum_{j\ne i}|\omega_{ij}|\). EBIC selects \(\lambda\). The certificate is the largest violation of the penalized logistic stationarity conditions across all nodes.

For binary predictability, normalized classification accuracy is

\[ \operatorname{nCC}_j = \frac{\operatorname{CC}_j-m_j}{1-m_j}, \]

where \(\operatorname{CC}_j\) is classification accuracy and \(m_j\) is the larger marginal class proportion.