# Understanding stroke with Bayesian networks

## Introduction

Stroke is a major source of morbidity worldwide, causing 5.78 million deaths per annum as per WHO global health estimates (1). An international effort is underway to improve outcomes in stroke by means of secondary and tertiary preventative measures. To maximise the efficacy of such interventions, we must fully understand the processes which lead to stroke-related morbidity and mortality. We propose to reframe stroke as a component of a network system, with multiple interacting causes and consequences. In real-world epidemiology, interactive systems are known to exist between social, behavioural and biological risk factors. The network paradigm accommodates such complexity well, and has demonstrated value in genetics (2), pathology (3-5) and therapeutics (6-8). We propose Bayesian network inference as a hypothesis-free method of characterising the causal processes of stroke outcomes.

### Causal networks in medicine

The sequence of events which follows an episode of stroke is a variable and complex process. Factors such as aetiology, time-to-treatment, comorbidities, and therapy vary between individual patients and presentations. The occult interplay of such phenomena greatly complicates the procedure of hypothesis testing on the true underlying pathological process. In each domain of clinical research problems may arise due to multidimensional confounding. Analyses such as multivariate regression and naïve Bayes assume independent sampling of all observed variables (9,10). In the medical field, where biological, clinical and social factors have multiple consequences, such broad assumptions of variable independence are often inappropriate. Unobserved interactions in the causal chain between predictor and outcome variables may subsequently influence the observed significance of their relationship (10). Several epidemiological studies have concluded that such confounding may be minimised with appropriate network modelling techniques (11-13). Ultimately, phenomena which are observed in the clinical setting may be better considered as a functional *system*, where disease states, treatments and clinical events each have causes and consequences. The “network” approach to causality modelling has proved insightful in genetics (2), pathology (3-5) and therapy (6-8). Graphical modelling methods make few assumptions about the dependence structure of the observed variables, allowing for the eventuality of dependent predictor variables.

### Bayesian networks overview

Bayesian network inference has multifaceted value in the interpretation of clinical data interpretation. The method was developed specifically to learn dependence structures from observational data (14,15). Generated graphs are intuitive and visually appealing, facilitating the comprehension of such outputs by non-statisticians. The structure of the inferred network may be used to infer causality, under some strong assumptions (16-18). Firstly, it is assumed that the observed data is representative of the true distribution. In a large random sample, this may be a reasonable assumption. Secondly, it is assumed that no unobserved factors influence the system. This is an inappropriate assumption in the medical field, as no hypothetical dataset of potential causative factors could ever be deemed exhaustive. It is assumed that each variable maintains independence of all other variables given, its direct causes and effects. This is analogous to the global Markov property (19). Finally, it is assumed that the true dependence structure of the studied variables may be represented by a directed acyclic graph. Some biological phenomena potentially violate this assumption. Feedback mechanisms, such as those found in metabolic and cardiovascular regulation are examples of graph cycles (20). Accordingly, we accept that the causal structure estimation we seek will provide a good approximation rather than a definitive result. Using cross-validation (21) and bootstrap approaches (22), a measure of confidence may be assigned to the reported network structure and parameters.

### Bayesian network inference

A Bayesian network is a directed acyclic graphical representation of the joint probability distribution of a set of *p* random variables *X* (22). Therefore, a Bayesian network *B* may be denoted by the tuple *G*(*V, Θ*), in which *G* is a graph of *V* vertices and *Θ* parameters. A vertex *V _{i}* exists for each variable in

*X*.

*Θ*encodes the network parameters, such that:

$$\begin{array}{c}{\Theta}_{{X}_{:,i}|pa\left({X}_{:,i}\right)}={P}_{B}\left({X}_{:,i}|pa\left({X}_{:,i}\right)\right)\forall i\in \left\{1,\dots ,p\right\}\end{array}$$
| [1] |

Where *pa*(*X*_{:,}* _{i}*) are the parent vertices of

*X*

_{:,}

*in*

_{i}*G*(22).

*B*encodes the probability distribution of

*X*such that:

$$\begin{array}{c}{P}_{B}\left({X}_{:,1},\dots ,{X}_{:,p}\right)={\displaystyle \prod}_{i=1}^{p}{P}_{B}\left({X}_{:,i}|pa\left({X}_{:,i}\right)\right)\end{array}$$
| [2] |

A graphical simplification of the Bayesian network is a representation as a graph *G*(*V, E*) of *V* vertices and *V* edges, such that an edge exists between *V _{i}* and

*V*if and only if they are dependent. The task of Bayesian network learning is to isolate the most likely probability distribution which may have generated a matrix of values

_{j}*X*. There are two primary subtasks associated with this problem, those of structure learning and parameter learning. Heuristic methods are employed to avoid the high computational complexity of exhaustive parameter search. Score-based models aim to maximise an objective based on a network score. We employ the Bayesian Information Criterion (BIC) and the Akaike Information Criterion (AIC) as network scoring methods.

In this paper, we apply Bayesian network inference to identify causative factors for several clinical phenomena associated with stroke. We examine data from the International Stroke Trial, detailing the clinical course of 19,000 episodes of stroke. We describe the feature-engineering, model assumptions and validation methods deployed to achieve and test our results.

## Methods

### Data preparation and cleaning

Data was mined from the repository of the International Stroke Trial 1 (IST-1) Database, hosted by the University of Edinburgh (23). IST-1 was a multicentre randomised trial of the efficacy of antithrombotic therapy in stroke. Patients were prescribed aspirin and/or unfractionated heparin according to a factorial design. The project was carried out at 437 centres in 36 countries. Detailed clinical documentation was performed during the trial, including specific features of the presentation, clinical examination, medication history and outcomes.

### Feature exclusion

Recorded information was screened for clinical relevance. Variables relating to the specific time and hospital of presentation were excluded. Non-specific variables such as “other side effect of anticoagulation” were excluded. Free-text comments and dates of secondary diagnoses were excluded.

### Feature modification

Categorical Features were extracted from the dataset. Two-class features such as “sex” were converted to binary variables. Features indicating clinical examination findings were assigned labels of class {“yes”, “no”, “cannot assess”}. These features were subsequently converted to binary features, such that positive labels were assigned in the case of “yes” or “cannot assess” and negative labels were assigned in the case of “no”. The etiological classification of the diagnosis was described by 4 mutually exclusive binary variables “ischaemic stroke”, “haemorrhagic stroke”, “unknown type”, “not stroke”. These features were encoded as a single categorical feature. 3-class categorical features were generated from numerical variables. Breakpoints between discrete factor levels were arbitrarily assigned to convenient intervals. Systolic blood pressure was converted to a 3-class categorical feature {“(0,140]”, “(140,220], “(220, 295]”}. A categorical feature was generated to describe mortality as follows:

$\begin{array}{c}f\left(x\right)=\{\begin{array}{l}period\text{\hspace{0.17em}}3,x=alive\text{\hspace{0.17em}}at\text{\hspace{0.17em}}6\text{\hspace{0.17em}}month\text{\hspace{0.17em}}follow\text{\hspace{0.17em}}up\\ period\text{\hspace{0.17em}}2,x=death\ge 14\text{\hspace{0.17em}}days\text{\hspace{0.17em}}AND\text{\hspace{0.17em}}death<6\text{\hspace{0.17em}}months\\ period\text{\hspace{0.17em}}1,x=death<14\text{\hspace{0.17em}}days\text{\hspace{0.17em}}of\text{\hspace{0.17em}}randomisation\end{array}\end{array}$
| [3] |

A categorical variable was available indicating the Bamford Classification (24) of the stroke {“PACS”, “TACS”, “POCS”, “LACS”, “OTH”}. This was encoded as a five-class categorical feature without modification. *Table 1* lists descriptions of all included variables.

### Restriction

Thrombolysis was performed in 13 cases of the cohort (0.066%). All patients who underwent thrombolysis were excluded from analysis and this feature was removed from the dataset. All observations from the pilot study were excluded due to a change in the set of observed variables in the full study.

### Missing values

One hundred and ninety-six observations (0.84%) contained at least one missing or unknown value. To determine whether the distribution of data absence was Missing at Random, 6-month mortality outcomes were modelled directly from data absence. This follows the approach of the dataset was converted according to the following function:

$\begin{array}{c}f\left(x\right)=\{\begin{array}{l}1,x=missing/unknown\\ 0,x=documented\end{array}\end{array}$
| [4] |

A generalised linear model was subsequently trained to predict death by 6-month follow up. The model failed to learn a significant association between missing data and death at 6 months (Accuracy =0.78, AccuracySD =0.00056, Kappa =0.0018, KappaSD =0.0028). Accordingly, missing values were assumed to be randomly distributed and all observations with missing values were excluded.

### Temporal information

Temporal reasoning is a fundamental component of medical diagnosis and causal deduction (25), such that it is a criterion of the Bradford Hill criteria (26). Natural temporal relationships exist between many of the variables measured in the dataset. Each variable was assigned a clinical timepoint as appropriate. Using blacklisting (27), the model was prevented from learning arcs directed from latter timepoints to an earlier timepoints. *Table 1* lists the timepoints of all included variables.

$\begin{array}{c}timepoint\left(var\right)\in \{\begin{array}{c}sex\\ age\\ aetiology\\ symptom\\ prehospital\\ presentation\\ examination\\ randomisation\\ syndromediagnosis\\ clinical\text{\hspace{0.17em}}course\\ day\text{\hspace{0.17em}}14\\ 6\text{\hspace{0.17em}}month\text{\hspace{0.17em}}followup\end{array}\end{array}$
| [5] |

### Graph inference

A discrete Bayesian network structure was chosen to represent the data. The model was generated using the R package “bnlearn” (27). Both the Hill-Climbing and Tabu algorithms were employed to perform score-based network inference. Each search algorithm was implemented using each of four network score functions – Bayesian Information Criterion, Akaike Information Criterion, Bayesian Dirichlet equivalent and K2. Each model was evaluated using 10 runs of 10-fold cross-validation. The optimal was selected as that achieving the highest cross validation likelihood.

Arc confidence was assessed using the non-parametric bootstrap technique described by Friedman *et al.* (22), in which a set of models is learned on bootstrap samples of the data. Equivalent networks are identified as graphs which contain the same set of conditional independence statements (22,28,29).

The non-parametric bootstrap procedure is carried out on a dataset *D* as follows (22,30):

Let *G*_{0}(*V, E*_{0}) be a Bayesian network representing the true dependence structure of *X*, composed of vertices *V* and edges *E* such that *E* ={*e*_{1},..*e _{k}*}.

FOR *b* in {1,..*m*}:

(I) *D _{b}* sampled using non-parametric bootstrap from

*X*.

(II) A graphical model ${\widehat{G}}_{b}\left(V,{E}_{b}\right)$ is estimated from *D _{b}*.

(III) END FOR

FOR EACH edge *e _{i,j}*:

(I) Estimate the probability that each *e _{i}* is present in the network structure by the frequency of its occurrence in the bootstrap sample graphs. Let 1

_{{a}}be the indicator function that condition

*α*is true.

$\begin{array}{c}\widehat{P}\left({e}_{i}\right)={m}^{-1}{\displaystyle \sum}_{b=1}^{m}{1}_{\left\{{e}_{i}\in {E}_{b}\right\}}\end{array}$
| [6] |

(III) END FOR EACH

Empirical probabilities of each edge, $\widehat{P}\left({e}_{i}\right)$ are known as arc strengths (30). They quantify confidence that *e _{i,j}* is present in

*G*

_{0}. Scutari presents a method to estimate the threshold of arc strength which optimally separates true edges from spurious ones (30). Let $\widehat{P}\left({e}_{i}\right)={1}_{\left({e}_{i}\in {E}^{0}\right)}$. Therefore $\tilde{P}\left(E\right)$ is an indicator of each edge’s presence in

*G*

_{0}.

Let:

$\begin{array}{c}{F}_{\widehat{P}\left(E\right)}\left(x\right)={m}^{-1}{\displaystyle \sum}_{b=1}^{m}{1}_{\left\{\widehat{P}\left({e}_{i}\right)x\right\}}\end{array}$
| [7] |

and

$\begin{array}{c}{F}_{\tilde{P}\left(E\right)}\left(x\right)=\{\begin{array}{l}0,x\in \left(-\infty ,0\right)\\ t,x\in \left[0,1\right)\\ 1,x\in \left[1,\infty \right)\end{array}\end{array}$
| [8] |

Therefore, *t* corresponds to the frequency of non-significant edges. Furthermore, *t* is the threshold above which an edge is present in *G _{true}*, such that:

$\begin{array}{c}{e}_{i}\in {E}_{0}\iff \widehat{P}\left({e}_{i}\right){F}_{\tilde{P}\left(E\right)}^{-1}\left(t\right)\end{array}$
| [9] |

Estimation of *t* is performed by minimisation of the following *L*_{1} normalised expression.

$\begin{array}{c}{L}_{1}\left(t;\widehat{P}\left(E\right)\right)=\int \left|{F}_{\widehat{P}\left(E\right)}\left(x\right)-{F}_{\tilde{P}\left(E\right)}\left(x;t\right)\right|dx\end{array}$
| [10] |

A set of 1,001 bootstrap network models were learned. A network average model was generated from this set, including arcs which were present in more than *t* of the generated models. This process was implemented with the “bnlearn” package (27).

### Conditional independence testing

Mutual information, *I*, is a quantification of the co-dependence of two variables (31), such that:

$\begin{array}{c}I\left({X}_{i},{X}_{j}\right)={\displaystyle \sum}_{{X}_{i},{X}_{j}}p\left({X}_{i},{X}_{j}\right)\mathrm{log}\frac{p\left({X}_{i},{X}_{j}\right)}{p\left({X}_{i}\right)p\left({X}_{j}\right)}\end{array}$
| [11] |

Mutual Information may be considered as the reduction of the uncertainty in *X _{i}* achieved by observation of

*X*. It is symmetrical, such that

_{j}*I*(

*X*) =

_{i}, X_{j}*I*(

*X*). Given a vertex

_{j}, X_{i}*X*with a single parent

_{i}*X*in a Bayesian network

_{j}*B*, The mutual information between

*X*and

_{i}*X*describes

_{j}*X*’s influence of on

_{i}*X*and vice versa.

_{j}Let Ω(*X _{i}*) be the state space of

*X*and

_{i}*p*(

_{pr}*X*=

_{i}*a*) be the prior probability of

*X*to be in state

_{i}*a*. The arc weight of ${\Theta}_{{X}_{i},{X}_{j}}$ is calculated as follows (31):

$$\begin{array}{c}{\Theta}_{{X}_{i},{X}_{j}}={\displaystyle \sum}_{a\in \text{\Omega}\left({X}_{i}\right)}{p}_{pr}\left({X}_{i}=a\right){\displaystyle \sum}_{b\in \text{\Omega}\left({X}_{j}\right)}p\left({X}_{j}=b\text{|}{X}_{i}=a\right)\mathrm{log}\frac{p\left({X}_{j}=b\text{|}{X}_{i}=a\right)}{{p}_{pr}\left({X}_{j}=b\right)}\end{array}$$
| [12] |

In the case that *X _{i}* has multiple parents {

*X*,

_{j}*X*

_{k}_{:}

*} (31)*

_{n}
$$\begin{array}{c}{\Theta}_{{X}_{i},{X}_{j}}={\displaystyle \sum}_{c\in \text{\Omega}\left({X}_{i}\right)}{p}_{pr}\left({X}_{k:n}=c\right){\displaystyle \sum}_{a\in \text{\Omega}\left({X}_{i}\right)}{p}_{pr}\left({X}_{i}=a\right)\end{array}$$
| [13] |

$\begin{array}{c}{\displaystyle \sum}_{b\in \text{\Omega}\left({X}_{j}\right)}p\left({X}_{j}=b\text{|}{X}_{i}=a,{X}_{k:n}=c\right)\mathrm{log}\frac{p\left({X}_{j}=b\text{|}{X}_{i}=a,{X}_{k:n}=c\right)}{{p}_{pr}\left({X}_{j}=b\text{|}{X}_{k:n}=c\right)}\end{array}$
| [14] |

Inference testing was performed on the average-network graph using the asymptotic chi-squared test of mutual information.

## Results

### Model selection

The tabu algorithm, using the k2 score demonstrated the optimal cross validation loglikelihood (tabu_k2: Mean = 1.52e+01, SD = 7.23e-04, CI.lower = 1.52e+01, CI.upper = 1.52e+01). Hill Climbing and Tabu search algorithms exhibited similar performance (*Figure 1*, *Table 2*). Scutari’s L1 normal approximation method (30) selected an arc strength significance cut-off of 0.5. Accordingly, 50.3% of all potential arcs were included in one or more network bootstraps (*Figure 2*). The final bayesian network estimate is illustrated in *Figure 3*.

**Figure 1**Model cross validation performance. Repeated cross validation was performed with 10 repeats of 10-fold cross validation. Evaluated model scores were Bayesian Information Criterion (bic), Akaike Information Criterion (aic), Bayesian Dirichlet equivalent (bde) and K2. Each model score function was evaluated in with Hill-climbing (hc) and TABU search algorithms. The optimal method was selected as that producing the highest validation loglikelihood.

**Figure 2**Cumulative distribution of arc strength in network bootstraps. Arc strength is the frequency in which a conditional dependency is inferred between a given pair of variables in network bootstraps. The L1 norm approximation is a binary approximation of arc strength, such that all arcs with arc strength above the L1 norm threshold are considered significant and included in the network structure.

**Figure 3**Estimated conditional dependency structure of the causation network in stroke. Abbreviated variables are described in

*Table 1*. Arrows represent causal relationships, such that arrows originate from the cause and point to the effect.

### Demography

Sex was found to influence age at presentation (Mutual Info: P value <2e-16), prescription of aspirin in the 72 hrs prior to presentation (Mutual Info: P value <2e-16), atrial fibrillation on presentation (Mutual Info: P value <2e-16) and systolic blood pressure at randomisation (Mutual Info: P value <2e-16). Age at presentation influenced prescription of aspirin in the 72hrs prior to presentation (Mutual Info: P value <2e-16), atrial fibrillation on presentation (Mutual Info: P value <2e-16), stroke aetiology (Mutual Info: P value <2e-16), conscious level at presentation (Mutual Info: P value <2e-16), 14-day mortality (Mutual Info: P value <2e-16) and outcome at 6-month follow up (Mutual Info: P value <2e-16). Age at presentation was found to influence systolic blood pressure at randomisation (Mutual Info: P value <2e-16), visuospatial disorder risk (Mutual Info: P value <2e-16), prescription of other anticoagulants (Mutual Info: P value <2e-16) and discharge within 14 days (Mutual Info: P value <2e-16).

### Recent history

Prescription of aspirin in the 72 hours prior to presentation influenced delay prior to randomisation (Mutual Info: P value <2e-16), prescription of glycerol or mannitol (Mutual Info: P value <2e-16) and haemodilution (Mutual Info: P value <2e-16). Prescription of heparin in the 24 hrs prior to presentation was found to influence atrial fibrillation on presentation (Mutual Info: P value =0.0022), delay prior to randomisation (Mutual Info: P value <2e-16) and prescription of glycerol or mannitol (Mutual Info: P value <2e-16). Atrial fibrillation on presentation influenced delay prior to randomisation (Mutual Info: P value <2e-16), conscious level at presentation (Mutual Info: P value <2e-16), dysphasia risk (Mutual Info: P value <2e-16) and hemianopia risk (Mutual Info: P value <2e-16). Atrial fibrillation on presentation affected visuospatial disorder risk (Mutual Info: P value <2e-16) and prescription of other anticoagulants (Mutual Info: P value <2e-16).

### Event

Stroke aetiology was found to influence onset of symptoms on waking (Mutual Info: P value <2e-16), delay prior to randomisation (Mutual Info: P value <2e-16), infarct visibility (Mutual Info: P value <2e-16), upper limb deficit risk (Mutual Info: P value <2e-16), completion of aspirin prescription (Mutual Info: P value <2e-16) and discharge within 14 days (Mutual Info: P value <2e-16). Stroke aetiology affected prescription of corticosteroids (Mutual Info: P value <2e-16), haemorrhagic recurrence/transformation of stroke within 14 days (Mutual Info: P value <2e-16), recurrence of stroke of unknown aetiology within 14 days (Mutual Info: P value <2e-16) and cheparin prescription completion (Mutual Info: P value <2e-16).

### Presentation

Delay prior to randomisation influenced infarct visibility (Mutual Info: P value <2e-16) and conscious level at presentation (Mutual Info: P value <2e-16). Infarct visibility was found to influence hemianopia risk (Mutual Info: P value <2e-16) and visuospatial disorder risk (Mutual Info: P value <2e-16). Conscious level at presentation was found to influence infarct visibility (Mutual Info: P value <2e-16), facial deficit risk (Mutual Info: P value <2e-16), upper limb deficit risk (Mutual Info: P value <2e-16) and lower limb deficit risk (Mutual Info: P value <2e-16). Conscious level at presentation influenced dysphasia risk (Mutual Info: P value <2e-16), hemianopia risk (Mutual Info: P value <2e-16), visuospatial disorder risk (Mutual Info: P value <2e-16) and brainstem and cerebellar deficit risk (Mutual Info: P value <2e-16). Conscious level at presentation affected Bamford classification (Mutual Info: P value <2e-16), prescription of corticosteroids (Mutual Info: P value <2e-16), discharge within 14 days (Mutual Info: P value <2e-16) and outcome at 6-month follow up (Mutual Info: P value <2e-16). Systolic blood pressure at randomisation influenced prescription of calcium channel blockers (Mutual Info: P value <2e-16). Presence of a facial deficit on examination exerted influence on Bamford classification (Mutual Info: P value <2e-16), prescription of glycerol or mannitol (Mutual Info: P value <2e-16) and discharge within 14 days (Mutual Info: P value <2e-16).

### Clinical examination

Presence of an upper limb deficit on examination exerted influence on facial deficit risk (Mutual Info: P value <2e-16), lower limb deficit risk (Mutual Info: P value <2e-16), dysphasia risk (Mutual Info: P value <2e-16) and brainstem and cerebellar deficit risk (Mutual Info: P value <2e-16). Presence of an upper limb deficit on examination influenced Bamford classification (Mutual Info: P value <2e-16) and discharge within 14 days (Mutual Info: P value =0.0496). Presence of a lower limb deficit on examination influenced facial deficit risk (Mutual Info: P value <2e-16), dysphasia risk (Mutual Info: P value <2e-16), brainstem and cerebellar deficit risk (Mutual Info: P value <2e-16) and Bamford classification (Mutual Info: P value <2e-16). Presence of a lower limb deficit on examination influenced likelihood of carotid endarterectomy within 14 days (Mutual Info: P value <2e-16), discharge within 14 days (Mutual Info: P value <2e-16) and outcome at 6-month follow up (Mutual Info: P value <2e-16). Dysphasia affected facial deficit risk (Mutual Info: P value <2e-16), brainstem and cerebellar deficit risk (Mutual Info: P value <2e-16) and Bamford classification (Mutual Info: P value <2e-16). Hemianopia exerted influence on facial deficit risk (Mutual Info: P value <2e-16), lower limb deficit risk (Mutual Info: P value <2e-16), dysphasia risk (Mutual Info: P value <2e-16) and brainstem and cerebellar deficit risk (Mutual Info: P value <2e-16). Hemianopia influenced Bamford classification (Mutual Info: P value <2e-16), pulmonary embolism within 14 days (Mutual Info: P value <2e-16) and outcome at 6-month follow up (Mutual Info: P value <2e-16). Presence of visuospatial disorders on examination exerted influence on upper limb deficit risk (Mutual Info: P value <2e-16), lower limb deficit risk (Mutual Info: P value <2e-16), dysphasia risk (Mutual Info: P value <2e-16) and hemianopia risk (Mutual Info: P value <2e-16). Presence of visuospatial disorders on examination was found to influence brainstem and cerebellar deficit risk (Mutual Info: P value <2e-16), Bamford classification (Mutual Info: P value <2e-16) and discharge within 14 days (Mutual Info: P value <2e-16). Presence of brainstem/cerebellar deficits on examination influenced facial deficit risk (Mutual Info: P value <2e-16) and Bamford classification (Mutual Info: P value <2e-16). Allocation to the aspirin trial arm influenced completion of aspirin prescription (Mutual Info: P value <2e-16).

### Management

Allocation to the heparin trial arm affected risk of major non-cerebral haemorrhage within 14 days (Mutual Info: P value <2e-16), completion of heparin prescription (Mutual Info: P value <2e-16) and completion of aspirin prescription (Mutual Info: P value <2e-16). Prescription of other anticoagulants affected recurrence risk for ischaemic stroke within 14 days (Mutual Info: P value <2e-16), completion of heparin prescription (Mutual Info: P value <2e-16) and completion of aspirin prescription (Mutual Info: P value <2e-16). Prescription of glycerol or mannitol was found to influence prescription of corticosteroids (Mutual Info: P value <2e-16) and prescription of calcium channel blockers (Mutual Info: P value <2e-16). Prescription of calcium channel blockers was found to influence prescription of corticosteroids (Mutual Info: P value <2e-16). Haemodilution influenced prescription of calcium channel blockers (Mutual Info: P value <2e-16). Carotid endarterectomy within 14 days influenced prescription of other anticoagulants (Mutual Info: P value =0.0024), prescription of corticosteroids (Mutual Info: P value =0.0002) and recurrence of ischaemic stroke within 14 days (Mutual Info: P value <2e-16).

### Clinical course

Major non-cerebral haemorrhage within 14 days affected likelihood of haemorrhagic recurrence/transformation of stroke within 14 days (Mutual Info: P value =0.0002), completion of heparin prescription (Mutual Info: P value <2e-16), completion of aspirin prescription (Mutual Info: P value <2e-16) and 14-day mortality (Mutual Info: P value <2e-16). Recurrence of ischaemic stroke within 14 days influenced prescription of corticosteroids (Mutual Info: P value <2e-16), discharge within 14 days (Mutual Info: P value <2e-16) and outcome at 6-month follow up (Mutual Info: P value <2e-16). Haemorrhagic recurrence/transformation of stroke within 14 days affected prescription of glycerol or mannitol (Mutual Info: P value =0.0066), prescription of corticosteroids (Mutual Info: P value =0.0002), completion of heparin prescription (Mutual Info: P value <2e-16) and outcome at 6-month follow up (Mutual Info: P value <2e-16). Recurrence of stroke of unknown aetiology within 14 days was found to influence discharge within 14 days (Mutual Info: P value <2e-16) and outcome at 6-month follow up (Mutual Info: P value <2e-16). Pulmonary Embolism within 14 days was found to influence prescription of other anticoagulants (Mutual Info: P value <2e-16). Completion of heparin prescription affected completion of aspirin prescription (Mutual Info: P value <2e-16). Discharge within 14 days affected outcome at 6-month follow up (Mutual Info: P value <2e-16). Fourteen-day mortality affected outcome at 6-month follow up (Mutual Info: P value <2e-16).

### Specific outcomes: morbidity and mortality

Fourteen-day mortality risk was highest in the (75,99] age bracket (RR =2.8806, ARI =0.0296, P value ≤2.22e-16), and lowest in under 55s (RR =0.272, ARI =-2.20e-02, P value =7.78e-143). Major non-cerebral haemorrhage also increased mortality risk (RR =6.44e+00, ARI =1.47e-01, P value <2.22e-16). Higher age at presentation adversely affected 6-month outcomes (Mutual information P value: <2.22e-16) independently of its effect on short term mortality. This effect included an increased risk of death prior to 6-month follow up in the 55–75 years group (RR =7.39e-01, ARI =-1.53e-01, P value <2.22e-16) and the 75–99 years group (RR =1.999, ARI =0.256, P value <2.22e-16). Conscious level at presentation also adversely affected 6-month mortality independent of 14-day mortality (Mutual information P value: <2.22e-16). Patients who were unconscious at presentation accepted an increased risk of dependence (RR =23.576, ARI =0.0201, P value <2.22e-16) and death (RR =11.809, ARI =0.0434, P value <2.22e-16) at follow up. Stroke recurrence was associated with poorer outcomes regardless of whether the aetiology was ischaemic (Mutual information P value <2.22e-16), haemorrhagic (Mutual information P value <2.22e-16) or unknown (Mutual information P value: <2.22e-16). Patients who were fit to be discharged within 14 days of presentation experienced favourable 6-month outcomes (Mutual information P value <2.22e-16), including a decreased risk of dependency (RR =0.387, ARI =-0.348, P value <2.22e-16) and death (RR =2.72e-01, ARI =−3.02e-01, P value <2.22e-16).

### Specific outcomes: recurrence

Stroke recurrence risk varied with aetiology. Presentation with haemorrhagic stroke greatly increased risk of recurrence haemorrhage (RR =8.05e+00, ARI =2.11e-01, P value <2.22e-16). Although presentation with ischaemic stroke increased risk of recurrence of stroke of unknown aetiology (RR =4.02e+00, ARI =1.19e-01, P value <2.22e-16) it was not found to independently influence ischaemic stroke recurrence. Patients who suffered major non cerebral haemorrhage undertook an increased risk of haemorrhagic stroke recurrence (RR =9.05, ARI =0.06, P value =1.54e-05).

## Discussion

### Consistent arcs

The structure inference algorithm successfully learned a network which is largely consistent with clinical knowledge. Vertices representing clinical examination findings were densely interconnected, and clusters within this region reflected clinical syndromes.

For instance, the presence of brainstem or cerebellar deficits on clinical examination was found to depend on the presence of visuospatial disorders. Visuospatial disorders indicate occipital lobe pathology and therefore posterior circulation deficiency. The cerebellum is also supplied by the posterior circulatory system. Applying causal reasoning to this interaction, visuospatial disorders and cerebellar deficits share a common causative factor—posterior circulatory disruption. As this factor is unobserved, confounding occurs between the two child nodes. Consequently, an apparent causal relationship is observed from visuospatial disorders to cerebellar disorders. Nonetheless, detection of the association between these clinical features appropriately identifies their special relationship.

Each neurological deficit exerted some influence on the Bamford Classification. Bamford Classification is an objective clinical sign which is assigned by a clinician based on the cluster of neurological deficits found on exam. Thus, these dependencies are consistent with the true clinical process. Visibility of the infarct depended on delay prior to randomisation aetiology of the stroke, this is consistent with the increasing visibility of infarcts over time. Stroke aetiology also influenced the visibility of the stroke, this is likely due to the exclusion of all patients with visible intracranial haemorrhage at the time of recruitment.

Prescription of calcium antagonists was influenced by systolic blood pressure and concomitant prescription of glycerol or mannitol. As these features represent the indication and alternatives to calcium antagonist medications respectively, this association is clinically consistent. Prescription of other anticoagulants depended on occurrence of pulmonary embolism.

Age at presentation depended on sex of the patient. This is a likely consequence of the known difference between the age profile of strokes in males and females, with males suffering earlier events (32). Delay prior to randomisation was lower in patients who received heparin within the previous 24 hours. As this relationship is independent of stroke aetiology, it may indicate earlier detection of stroke in patients previously interacting with health services.

### Inconsistent arcs and latent features

Infarct visibility depended directly on conscious level at presentation. Although it is conceivable that these events are associated, it is probable that some other latent feature such as infarct size influenced both of these variables. Carotid endarterectomy was directly linked to the presence of a lower limb deficit. Carotid endarterectomy is indicated due to carotid stenosis, a cause of ischaemic stroke—therefore, it is likely that aetiology d-separates carotid endarterectomy and lower limb deficits in the true causality network.

### Model selection and regularisation

Bayesian network inference detected 119 conditional dependencies in the International Stroke Trial dataset. In the majority cases, these associations were consistent with known clinical phenomena. Few associations were detected which were not clinically explicable. It is notable that the selection of the arc strength significance cut-off critically affected the resultant network structure, as several arcs had strengths near the threshold. Approximately half of all potential arcs were deemed significant in more than one bootstrap network estimate, indicating moderate variability of the network estimate.

## Conclusions

We organise the pathogenesis, management and sequelae as a single functional system, in which clinical events are assumed to influence one another naturally. We have demonstrated the utility of the method to form and test multiple hypotheses in a highly objective fashion.

This feature efficiently addresses the dual problems of hypothesis search and multiple hypothesis testing. Though true evidence of causality may only be attained through interventional research (17,18), complete observation of the causative process allow its improved approximation. In this analysis, we demonstrate the interdependent nature of many factors which are known to be associated with stroke risk. This methodology is general and may theoretically be applied to various observational datasets across the health sciences.

## Acknowledgments

The author would like to thank Professor Peter Sandercock and the IST1 collaboration for provision of the International Stroke Trial data. The author would like to thank Reecha Sofat and Dionisio Acosta Mena for their guidance.

This research was funded by National Institute for Health Research Biomedical Research Centre and a University College London Hospital Levi Fellowship. The project was supported by the National Institute for Health Research University College London Hospitals Biomedical Research Centre. The funders of this analysis had no role in the study design; gathering, analysis and interpretation of these data; writing of the report; and decision to submit the report for publication. The corresponding author had full access to all data; takes responsibility for the integrity of these data and the accuracy of the analysis; and takes final responsibility for the decision to submit for publication.

## Footnote

*Conflicts of Interest:* The author has no conflicts of interest to declare.

*Ethical Statement:* All patient data used in this analysis is fully anonymised and available via the University of Edinburgh Data Share repository. This analysis is performed on data which was originally collected and analysed during the International Stroke Trial by the IST1 collaboration. This data was accessed and reanalysed with the permission and review of Peter Sandercock, principal investigator of the International Stroke Trial. The author is accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.

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**Cite this article as:**O’Shea R. Understanding stroke with Bayesian networks. J Med Artif Intell 2020;3:2.