### I. Introduction

### II. Methods

### 1. Novel Sleep Apnea Parameters

### 2. Patients

*Unisalkku*[2141516], and they were reanalyzed using standard respiratory rules developed by the American Academy of Sleep Medicine (AASM) [13], as in our previous studies [141516171819].

### 3. Bayesian Network Analysis

_{KL}). It provides a natural method for this study to compare distributions of two connected variables [26]. That is to say, we estimate the strength of a specific arc as D

_{KL}in the context of the entire DAG. What if this arc were removed but all the others remained? Furthermore, direct effect (DE) is calculated between each variable and the outcome variable to compare the causal strength of the variables on the outcome variable. DE is based on Jouffe's proprietary likelihood matching algorithm [12], and it estimates the causal dependency between two variables by measuring the impact of a conditional mean of each state of variable A on the mean of variable B (outcome variable) with Kullback's minimum cross-entropy method MinxEnt [27] and by keeping the values of all other variables fixed. DE is especially suitable for situations where the dependency between two variables is linear.

*Outcome total*were included in the analysis. Sleep apnea parameters and CPAP treatment were selected by using a local SC value of 0.4 for them.

*Outcome total*as the target, or (2) clinically commonly used thresholds (when using a decision tree algorithm was not possible). The discretized values as well as the total data set are presented in Table 2.

*age*had a TI = 1 (oldest known measured value), and

*Outcome total*had a TI = 8 (last measurement at the end of the follow-up period), for instance.

*DesSev*→

*AHI*do not indicate causality because both variables were measured at the same time and they have the same TI value. Expert opinion was used to determine causality in a case with an obvious wrong direction of the arc. As an example, an arc direction of

*CHD*→

*Diabetes*was manually forbidden, but the opposite direction and no arc were allowed.

*CPAP*was considered to have a DE on

*Outcome total*, even though this hypothesis was not supported by the DDBN. The model was simplified by limiting the number of variables to include only the most prominent ones (nine variables). The variable

*Diuretic*was dropped because it was considered to be a marker, not a causal factor for

*Outcome total*.

*CPAP*to

*Outcome total*. In the inference phase (i.e., when the constructed model was used), the variable

*CPAP*was set to be an intervention. In this way, real causal dependencies between CPAP and the outcome variable could be identified when this model was purged of unwanted associational backdoor paths between them [12].

### III. Results

*Outcome total*is presented in Figure 2. In the DDBN model, variables

*CHD*,

*Diuretic*, and

*CHF*were causally associated with the outcome. No causal association between sleep apnea parameters or CPAP and the outcome variable was seen. Instead, there was a path between

*CPAP*and

*Outcome total*consisting of associational dependencies. There was a weak association between

*AHI*and

*Outcome total*due to common causes

*BMI*and

*Gender*.

*Recruitment time*was included in the DDBN model (Figure 2) because the patient recruitment period was long (11 years), and

*Recruitment time*was considered a potential source of bias. There was an association between

*Recruitment time*and the variable

*CHF*, indicating that congestive heart failure was a more common finding in patients before the year 2000 than after. However, there was no DE between

*Recruitment time*and

*Outcome total*.

*AHI*→

*ODI*had the strongest association).

*Outcome total*is presented in Table 4. Variables

*CHF*,

*CHD*, and

*Diuretic*have strong direct effects on

*Outcome total*. Sleep apnea parameters and

*CPAP*have only a minimal DE on the target. In other words, based on the DDBN approach, there is no causal relationship between

*CPAP*and the outcome variable.

*CPAP*to the target were found; only one path is causal, i.e., the direct link from

*CPAP*to

*Outcome total*. All other paths from

*CPAP*to

*Outcome total*are associated with

*BMI*or

*Gender*as a common cause.

*CPAP*set as an intervention is presented in Figure 4. When

*CPAP*is an intervention, this intervention variable is separated from all non-causal associations. This model was fixed independently for each value of the variables, and the results are presented in Table 7. In general, CPAP treatment showed a 5.3 percentage points improvement in

*Outcome total*in comparison with no treatment. The most improvement was seen in patients aged 55 years or less (8.4% improvement with CPAP in comparison with no treatment). In patients with CHF, CPAP treatment showed a 10.2% increase in risk of mortality, AMI, or CVI (HDBN models number 16–17 in Table 7).

### IV. Discussion

*docalculus*-equation? The implemented causal analysis follows the guidelines in [30]. Therefore, we can claim that, within the observed variables, the dependences are causal. However, due to weak dependences between multiple variables, causal dependences similarly are weak. Therefore, this has led to some differences between data-driven and hypothesis-driven networks when MDL scoring has been used.

*AHI*and

*Outcome total*can explain the results reported by Rich et al. [3]. BMI was clearly a common cause for both CPAP treatment (and for all the variables in the path from BMI to CPAP) and

*Outcome total*.

*CPAP*and

*Outcome total*consists of a causal dependency and spurious associational dependencies enabled by

*BMI*and

*Gender*as common causes. This result can be compared with the study by Jennum et al. [34] who found that CPAP therapy is associated with reduced all-cause mortality in males, but not significantly in females.