Performance of Apriori Algorithm for Detecting Drug–Drug Interactions from Spontaneous Reporting Systems

Abstract

Drug–drug interactions (DDIs) can pose significant risks in clinical practice and pharmacovigilance. Although traditional association rule mining techniques, such as the Apriori algorithm, have been applied to drug safety signal detection, their performance in DDI detection has not been systematically evaluated, especially in the Spontaneous Reporting System (SRS), which contains a large number of drugs and AEs with a complex correlation structure and unobserved latent factors. This study fills that gap through comprehensive simulation studies designed to mimic key features of SRS data. We show that latent confounding can substantially distort detection accuracy: for example, when using the reporting ratio (RR) as a secondary indicator, the area under the curve (AUC) for detecting main effects dropped by approximately 30% and for DDIs by about 15%, compared to settings without confounding. A real-world application using 2024 VAERS data further illustrates the consequences of unmeasured bias, including a potentially spurious association between COVID-19 vaccination and infection. These findings highlight the limitations of existing methods and emphasize the need for future tools that account for latent factors to improve the reliability of safety signal detection in pharmacovigilance analyses.

1. Introduction

Pharmacovigilance [1], the post-marketing monitoring of drug safety, relies heavily on spontaneous reporting systems (SRSs) such as the FDA Adverse Event Reporting System (FAERS) [2] and the Vaccine Adverse Event Reporting System (VAERS) [3]. These systems collect voluntarily submitted reports of adverse events (AEs) and play a crucial role in identifying drug-associated adverse events (AEs, i.e., side effects), including those due to drug–drug interactions (DDIs), which are often not captured during clinical trials [4,5,6]. Detecting DDIs in SRS data is particularly challenging due to their high dimensionality, sparsity, and lack of structured covariate information.

Among the tools commonly used for drug safety signal detection in SRSs, the Apriori algorithm [7] stands out for its model-free nature, minimal reliance on assumptions, and reduced computation burden for large databases. Despite its popularity, the performance of the Apriori algorithm in detecting DDIs has not been thoroughly evaluated under realistic data settings. Prior studies [8,9,10,11,12] have generally applied Apriori to real datasets without examining how data characteristics, such as sparsity, outcome correlations, and hidden confounding, affect its accuracy.

Moreover, a critical gap in the literature is the absence of discussion regarding the impact of potential latent confounders. These latent confounders, also known as unobserved heterogeneity, are inherent in SRSs because these systems rely on voluntary, unstructured reports rather than systematically collected data. Such unobserved influences can affect both the likelihood that an adverse event is reported and its association with particular drugs or drug combinations, thereby introducing reporting bias and confounding conditions, and, consequently, significantly distorting detection results.

This methodological oversight represents a substantial gap between current analytical approaches and the complex reality of SRS data. To fill the above two critical gaps, the main contribution of this paper is therefore twofold: (1) to systematically examine the performance of the Apriori algorithm for DDI detection across a wide range of simulated conditions that mirror features of SRS data, and (2) to quantify, through rigorous simulation studies and real data analysis, how this oversight affects performance metrics in DDI detection across different confounding scenarios. To our knowledge, this is the first study to systematically evaluate how latent confounders affect the performance of association rule mining in the context of drug safety signal detection.

The remainder of this paper is structured as follows. In Section 2, we describe the Apriori method used for detecting both drug–AE association pairs and DDIs in SRSs. Section 3 presents a comprehensive simulation study that evaluates the performance of the Apriori approach in uncovering drug safety signals from SRSs under various confounding scenarios. In Section 4, we apply the Apriori method to the real-world VAERS database, highlighting how the presence of latent confounders affects results in practice. Conclusions and discussions regarding limitations, and future research directions to address the confounding challenge are provided in Section 5.

2. Apriori Method

The Apriori algorithm is selected as the framework to identify drug safety signals in this study based on several methodological and practical considerations that are relevant to the specific context of pharmacovigilance and spontaneous reporting systems (SRSs).

First, Apriori remains one of the most widely used and well-established algorithms for association rule mining, especially in applications involving large-scale, high-dimensional, and sparse data such as those in SRSs. Its popularity stems from its model-free nature, scalability, and transparent output structure, which makes it particularly suitable for exploratory pattern discovery. The algorithm’s ability to generate easily interpretable association rules, such as ‘Drug A and Drug B → AE X’, is especially valuable in the context of pharmacovigilance, where regulatory agencies and healthcare providers prioritize interpretability and accountability in decision making.

In addition, unlike many machine learning (ML) or deep learning (DL) methods that require labeled training data and are typically designed for single-outcome prediction tasks, the Apriori algorithm can flexibly accommodate multiple outcomes. This feature is essential in SRS datasets, where a single report may include multiple AEs, and the goal is to detect co-occurrence patterns without pre-specifying a target variable. While multi-label classification methods have been developed in the ML literature, they often suffer from scalability issues or lack the interpretability needed in drug safety applications [13]. Moreover, Apriori is computationally efficient and scalable to massive datasets like FAERIES or VAERS, while many ML or DL models are computationally expensive and prone to overfitting when applied to such types of datasets.

Finally, the objective of this paper is not to propose a fully optimized prediction framework, but rather to systematically study the effect of latent confounding on drug safety signal detection from SRSs. The Apriori algorithm, with its transparent rule structure and modular form, offers a clear platform for this foundational investigation and is more acceptable compared with black-box models like DL.

2.1. Apriori Algorithm for Drug–AE Association Detection

Association rule mining (ARM) is one of the well-established unsupervised learning techniques, frequently used to discover interesting relationships between variables hidden in large databases. The Apriori method, first proposed by Agrawal and Srikant (1994) [7], is a type of ARM and has wide applications, especially in the field of marketing analysis. Consider a dataset containing n transactions or reports, where each transaction/report T consists of a set of items. Let X and Y be two disjointed sets of items. An association rule is expressed as an implication of the form $X\to Y$. For example, in the scenario of PhV, the antecedent X can be a set of drugs, and the consequent Y is a set of AEs.

There are several indicators that are frequently used in ARM. The first one is Support, which is defined as the percentage of all reports that include both X and Y, that is,

$$Support(X\to Y)={\displaystyle \frac{n_{xy}}{n}},$$

(1)

where $n_{xy}$ is the number of reports in the dataset that have both sets of elements X and Y.

Second, the Confidence is denoted as the conditional probability of Y given X, i.e., the proportion of occurrences of Y among the reports containing occurrences of X in the dataset:

$$Confidence(X\to Y)={\displaystyle \frac{Support(X\to Y)}{Support\left(X\right)}}.$$

(2)

The higher the confidence, the stronger the association between X and Y.

Third, the Lift represents the ratio between the Confidence of the rule and the Support of the item set in the consequent of the rule. That is,

$$Lift(X\to Y)={\displaystyle \frac{Confidence(X\to Y)}{Support\left(Y\right)}}={\displaystyle \frac{Support(X\to Y)}{Support\left(X\right)\times Support\left(Y\right)}}.$$

(3)

The Lift score measures the relative magnitude of the probability of observing Y under the condition of X, compared to the overall probability of observing Y. If Lift = 1, X and Y are independent of each other. When Lift is greater than 1, X and Y are not independent, and the higher the Lift value, the greater the relevance of X and Y [14].

Finally, Conviction is interpreted as the expected frequency that X occurs without Y if they are independent, divided by the observed frequency of incorrect predictions, which is mathematically defined as

$$Conviction(X\to Y)={\displaystyle \frac{1-Support\left(Y\right)}{1-Confidence(X\to Y)}}.$$

(4)

Conviction is an indicator that illustrates whether a wrong prediction is made. If the value of Conviction is large, it is unlikely that Y is not true for the premise of X [15].

The classic Apriori algorithm operates in two steps. The first step searches for item sets that exceed a minimum Support threshold, while in the second step, association rules are mined from these frequent item sets based on thresholds from a secondary indicator, usually a Confidence score. The Apriori algorithm leverages the downward closure property of frequency to efficiently prune the search space. This property states that if an item set is infrequent, then any larger item set containing it will also be infrequent and can be removed from the search space. This “a priori” knowledge of frequency patterns significantly reduces computational complexity, making the algorithm particularly suitable for spontaneous reporting systems, which typically contain thousands or even millions of reports with hundreds to thousands of variables (drugs/vaccines).

However, using Confidence as the secondary indicator has proven to be inappropriate in drug safety surveillance [9,10,11]. This limitation arises because frequently reported AEs, such as headaches or nausea, often yield high Confidence scores irrespective of their true association with a given drug. In contrast, rare AEs tend to produce lower Confidence values, even when strong associations with specific drugs exist. In Sarkar and Sun (2024) [16], the authors demonstrate that employing disproportionality measures, such as the proportional reporting ratio (PRR) [17], the relative reporting ratio (RR) [9] and the reporting odds ratio (ROR) [18], as secondary indicators yields better performance than Confidence in terms of detection sensitivity, specificity, and overall accuracy.

2.2. Apriori Algorithm for DDI Detection

Typically, the variables in one report from the spontaneous reporting system can be classified into three categories. One is the clinical (or covariate) variables, including the patient’s age, gender, and so on. Another group contains the variables recording the drug usage information, such as which drug(s) is(are) used and the corresponding dosage(s). The third category usually includes the AE information, such as what adverse events are reported here.

If the classic Apriori was applied directly to the spontaneous reporting system data with three categories of variables as a whole, then some of the selected rules might contain variables from only one category, or from only clinical and AE categories, consequently missing the interesting DDIs. Therefore, to identify DDIs, it is natural to modify the classic Apriori by adding constraints to the variable categories. That is, one should only consider the variables belonging to drug and AE categories, ignoring clinical variables, and restrict the search space to include rules with at least one drug and one AE. This modification only reduces the search space, and consequently reduces the computation burden for the Apriori algorithm, but also ensures the generated rules from this modified Apriori contains both drug and AE information, avoiding the situations of selecting only {drug, drug} or {AE, AE} item sets. In addition, this modification inherits the hierarchical nature, i.e., the downward closure property of frequency, of the classic Apriori, so as to further prune the search spaces and decrease the computation cost. Consequently, the steps for detecting DDIs by using Apriori with this modification are as follows:

• Choose all individual drugs and AEs that have Support larger than a pre-defined threshold, minsup. Denote the set for all selected drugs as $D_1$ and the set for all selected AEs as $A_1$.
• For all possible combinations of 2-item pairs $(d,ae)$, where $d\in D_1$ and $ae\in A_1$, calculate the corresponding Support and remove the pairs with Support less than minsup. Denote the set for the remaining pairs as $DA$. Use $D_2$ and $A_2$ to represent the set that contains all unique drugs and unique AEs in $DA$, respectively.
• For each 2-item pair $(d_1,d_2)$ in the set $D_2$, where $d_1\ne d_2$, compute the Support scores for all 3-item triplets $(d_1,d_2,ae)$, with $ae\in A_2$. The set for the triplets that have Support larger than minsup is denoted as $DDA$.
• For each triplet in the set $DDA$, compute the score for a secondary Apriori indicator. The triplets with this secondary score larger than the pre-defined threshold are then identified as the desired DDIs associated with AEs.

Previous studies that used a similar approach as above to identify DDIs employed various secondary indicators for the purpose of identification in the final step of the analysis. These indicators include Lift [12], as well as disproportionality measures PRR [8,10,11] and RR [9]. For any drug–drug–AE combination, disproportionality measures quantify the difference between the observed and expected number of reports for that AE associated with a given drug–drug pair, relative to the overall reporting frequency in the database. These measures can be calculated using the $2\times 2$ contingency table in

Table 1. Contingency table for disproportionality measures.

	Suspected AE	Other AEs	Total
Suspected drug–drug pair	a	b	$a+b$
Other drug combinations	c	d	$c+d$
Total	$a+c$	$b+d$	$a+c+b+d=n$

.

In particular, the Proportional Reporting Ratio (PRR) is the ratio between the proportion of a suspected AE among individuals who consume a particular drug–drug combination and the proportion of this AE among subjects that consume other drug combinations. That is,

$$PRR={\displaystyle \frac{a/(a+b)}{c/(c+d)}}={\displaystyle \frac{a(c+d)}{c(a+b)}}$$

(5)

A PRR value exceeding one suggests an elevated prevalence of reports featuring the suspected AE in association with the specified drug–drug pair relative to reports of this AE occurring with alternative medications [17].

The Relative Reporting Ratio (RR) represents the ratio between a rule’s observed frequency and a baseline expected frequency for all reports in the database. It can be calculated as

$$RR={\displaystyle \frac{Pr\left(\left\{AE\right\}\cap \{Drug-Drug\}\right)}{Pr\left(AE\right)·Pr\left(Drug-Drug\right)}}={\displaystyle \frac{na}{(a+c)(a+b)}}.$$

(6)

An RR value larger than one indicates that the co-occurrence of the suspected AE and the suspected drug–drug pair is more frequently reported than would be expected if they were independent of each other [9].

The Reporting Odds Ratio (ROR) is the ratio between the reporting odds of a suspected AE among individuals exposed to a specific drug and the reporting odds of this AE among subjects that are not exposed to the specific drug. That is

$$ROR={\displaystyle \frac{a/b}{c/d}}={\displaystyle \frac{ad}{bc}}$$

(7)

The ROR measures the likelihood of an AE being reported in the presence of a certain drug pair compared to the likelihood of that AE being reported in the absence of that drug pair. An ROR value exceeding one provides strong evidence of the association between the suspected AE and suspected drug pair [18].

It is important to note that all these metrics—Support, Confidence, Lift, ROR, and PRR—are fundamentally calculated from $2\times 2$ contingency tables. Each table represents a simplified model examining the relationship between a single drug (or drug combination) and a single adverse event. This approach can be viewed as equivalent to fitting a series of simple logistic regression models, each with one outcome (AE) and one covariate (either a single drug or a drug–drug interaction term).

While computationally efficient, this framework inherently makes several strong assumptions that rarely hold in real-world spontaneous reporting data. First, it treats each drug–AE or DDI-AE association as independent, ignoring potential correlations among adverse events. Second, and perhaps more critically, this approach disregards the presence of latent confounders—unmeasured variables that may influence both drug administration patterns and adverse event reporting. Such confounders could include patient demographics, underlying health conditions, reporting behaviors of different healthcare providers, or temporal trends in reporting—none of which are typically accounted for in the standard Apriori implementation. As our simulation study will demonstrate, these methodological limitations can significantly impact the performance of the Apriori algorithm in detecting true drug safety signals, particularly in the complex, heterogeneous data environment of spontaneous reporting systems.

3. Simulation Studies

Although several studies have applied the Apriori algorithm to detect DDIs from spontaneous reporting systems (SRSs) [9,11,12], they have primarily focused on implementation aspects, with limited evaluation of the algorithm’s performance under realistic conditions. Most critically, no previous work has systematically assessed how the presence of latent confounders—a common feature in SRS data—affect detection accuracy. To address this gap, we conducted a series of comprehensive simulation studies to evaluate the performance of the modified Apriori approach described in Section 2.2 for identifying drug–drug interactions under various scenarios that mimic the challenges inherent in spontaneous reporting systems. These challenges include high dimensionality, data sparsity, weak signal strength, complex correlation structures, and—central to our investigation—the presence of latent confounding factors. In this section, we first present our simulation design, followed by detailed results that demonstrate the substantial impact of confounding on DDI detection performance

3.1. Data Generation Model

Assume that we have collected data $\{({\overrightarrow{x}}_i,{\overrightarrow{y}}_i);i=1,\dots ,n\}$ from n subjects, where ${\overrightarrow{y}}_i={(y_{i1},\dots ,y_{id})}^T$ denotes d responses (AEs) and ${\overrightarrow{x}}_i={(x_{i1},\dots ,x_{ip})}^T$ is a $p\times 1$ vector of covariates (drugs and drug–drug interactions) for subject i. Here, p is the number of drugs plus the number of all possible drug–drug interactions. That is, if there are m drugs, then $p=m+\left({\displaystyle \genfrac{}{}{0pt}{}{m}{2}}\right)$. Given that $y_{ij}\in \{0,1\}$, we assume a generalized linear mixed model for ${\overrightarrow{y}}_i$ given ${\overrightarrow{x}}_i$ and conditional on random effects $b_{ij}$,

$$h\left({\mu }_{ij}\right)={\overrightarrow{x}}_i^T{\overrightarrow{\beta }}_j+b_{ij},i=1,\dots ,n;j=1,\dots ,d.$$

(8)

where ${\mu }_{ij}=E\left(y_{ij}\right|{\overrightarrow{x}}_i,{\overrightarrow{\beta }}_j,b_{ij})$ is the conditional mean of $y_{ij}$ given ${\overrightarrow{x}}_i$ and $b_{ij}$, with $h\left({\mu }_{ij}\right)$ being the logit link function; ${\overrightarrow{\beta }}_j\in {\mathcal{R}}^p$ are the regression coefficients; and $b_{ij}$ represents a subject-specific random effect for $y_{ij}$, the occurrence of the jth AE for subject i.

To model the dependence among AEs, we assume a latent factor model for ${\overrightarrow{b}}_i={(b_{i1},\dots ,b_{id})}^T$. That is

$${\overrightarrow{b}}_i=\mathrm{\Lambda }{\overrightarrow{\eta }}_i,$$

(9)

where $\mathrm{\Lambda }=\left({\lambda }_{jk}\right)$ is a $d\times K$ factor loading matrix with K being the number of latent factors, and ${\overrightarrow{\eta }}_i\in {\mathcal{R}}^K$ is the potential latent factors for subject i following a multivariate normal distribution with variance covariance matrix ${\mathrm{I}}_K$. The purpose of including this subject-specific random effect, ${\overrightarrow{b}}_i$, is to induce dependence among the binary outcomes conditional on ${\mathbf{X}}_{n\times p}={({\overrightarrow{x}}_1,\dots ,{\overrightarrow{x}}_n)}^T$. We are not interested in estimating $\mathrm{\Lambda }$, but we still require that the upper triangular part of $\mathrm{\Lambda }$ is 0, i.e., ${\lambda }_{jk}=0$ if $j<k$, to ensure the identifiability of $\mathrm{\Lambda }$ as discussed in [19].

Combining Equations (8) and (9), the resulting multivariate logistic regression model can be viewed as a generalized linear mixed effects model (GLMM) for repeatedly measured binary outcomes, but with a specific covariance structure described by a factor analysis model.

To systematically evaluate the impact of latent confounders, we consider three distinct scenarios for generating the subject-specific random effects ${\overrightarrow{b}}_i$ via Equation (9):

• Scenario 1 (No Random Effects): We assume no random effects, setting $\mathrm{\Lambda }=\mathbf{0}$. This represents the idealized case assumed by standard Apriori implementations.
• Scenario 2 (Random Effects Without Confounding): We introduce subject-specific random effects ${\overrightarrow{b}}_i=\mathrm{\Lambda }{\overrightarrow{\eta }}_i$, where $\mathrm{\Lambda }=\left({\lambda }_{jk}\right)$ with ${\lambda }_{jk}=0$ if $j<k$ and ${\lambda }_{jk}\sim N(0,1)$ if $\ge k$, representing dependence among AEs, and ${\overrightarrow{\eta }}_i\sim N(\mathbf{0},{\mathbf{I}}_K)$. Under this setup, the random effects ${\overrightarrow{b}}_i$ are independent of $\mathbf{X}$, and thus, ${\overrightarrow{\eta }}_i$ do not function as latent confounders.
• Scenario 3 (Random Effects With Confounding): We maintain the same structure for $\mathrm{\Lambda }$ as in Scenario 2, but crucially, we introduce correlation between ${\overrightarrow{\eta }}_i$ and $\mathbf{X}$ by defining ${\overrightarrow{\eta }}_i=\mathbf{A}{\mathbf{x}}_i^T+\mathbf{W}$, where both ${\mathbf{A}}_{K\times p}$ and ${\mathbf{W}}_{K\times 1}$ are generated from $N(\mathbf{0},{\mathbf{I}}_K)$. In this scenario, ${\overrightarrow{\eta }}_i$ represent true latent confounders, creating spurious associations between drugs and adverse events.

3.2. Simulation Setups

To mimic the real-world characteristics of spontaneous reporting systems, we simulated $\mathbf{X}$ by sampling from the VAERS vaccine data from 2018, by viewing vaccines as a type of drug. For each replication, we sampled n records (reports), retained the drug (i.e., vaccine) usage information, and expanded it as $\mathbf{X}$ by including both single drugs and all possible two-way interactions. Given $\mathbf{X}$, we generated the adverse event data ${\mathbf{Y}}_{n\times d}=\left(y_{ij}\right)$ following the multivariate logistic regression model described previously.

We set sample sizes at $n=5000$ and 10,000. Due to computational constraints, we focused on low to moderate dimensions rather than high-dimensional scenarios. Specifically, we set the number of drugs (m) and adverse events (d) to be $m=d=10$ and $m=d=20$. It is important to note that with $m=10$ drugs, there are 45 potential two-way interaction terms, resulting in $p=55$ total covariates; with $m=20$ drugs, there are 190 possible two-way interaction terms, yielding $p=210$ total covariates.

When generating adverse events ($\mathbf{Y}$) using Equation (8), for a fixed number of drugs m, there are $m\times d$ coefficients ${\beta }_{jl}$ representing main effects, and $(p-m)·d$ coefficients representing all two-way interaction effects, for $j=1,\dots ,d$ and $l=1,\dots ,p$. Among the main effect coefficients, we designated five to be non-zero (set to 1.0), indicating significant drug–AE associations. Similarly, among all interaction coefficients, we designated three to be non-zero (also set to 1.0), representing significant DDIs. We included an intercept term ${\beta }_{j0}$ calibrated to maintain an overall AE prevalence of approximately $10\%$.

To comprehensively evaluate the algorithm’s performance under different effect structures, we considered four distinct scenarios for generating the relationships between main effects and interactions:

• Main effects only: no two-way interactions exist.
• Interaction effects only: no main effects exist.
• Independent effects: both main effects and two-way interactions exist but are generated independently.
• Hierarchical effects: both main effects and two-way interactions exist, with a hierarchical structure—if drugs i and j have a significant interaction effect, then both also have significant main effects.

For each simulation setting, we performed $N=200$ replications. In each replication, we implemented the Apriori algorithm described in Section 2.2 and computed scores for various secondary indicators: Confidence, Lift, PRR, ROR, and RR for both main effects and DDIs. The value of minsup was set to 3 to exclude extreme cases where a drug–AE pair or DDI appears in only one or two reports, which are unlikely to provide reliable evidence. In practice, the choice of minsup can be tailored to the dataset. For instance, one can set minsup as a small proportion of the total sample size, such as 1% of n.

While Sarkar and Sun (2024) [16] evaluated performance based on sensitivity, specificity, and overall accuracy using pre-selected thresholds, we recognized that threshold selection is inherently subjective and that comparing different indicators at the same threshold value can be misleading since each captures distinct aspects of association. Therefore, we employed the area under the receiver operating characteristic curve (AUC) as our primary performance metric, which provides a threshold-independent measure of each indicator’s ability to discriminate between true and false associations for both main effects and drug–drug interactions.

3.3. Performance Evaluation

3.3.1. No Random Effect

In this scenario, we assume no subject-specific random effects exist, setting $\mathrm{\Lambda }=\mathbf{0}$ in Equation (9). This represents the idealized case most commonly assumed in standard Apriori implementations.

Table 2. AUC for detecting main effects when $\mathrm{\Lambda }=\mathbf{0}$.

n	$\mathit{m}=\mathit{d}$	Confidence	Lift	PRR	ROR	RR
$\beta$ generation setting 1
5000	10	1.000	0.997	0.997	0.998	0.997
5000	20	0.999	0.915	0.931	0.943	0.915
10,000	10	1.000	0.993	0.995	0.997	0.993
10,000	20	1.000	1.000	1.000	1.000	1.000
$\beta$ generation setting 3
5000	10	1.000	0.998	0.998	0.998	0.998
5000	20	0.999	0.919	0.935	0.947	0.919
10,000	10	1.000	0.993	0.995	0.996	0.993
10,000	20	1.000	1.000	1.000	1.000	1.000
$\beta$ generation setting 4
5000	10	1.000	0.997	0.997	0.998	0.997
5000	20	1.000	0.922	0.937	0.949	0.922
10,000	10	1.000	0.994	0.997	0.997	0.994
10,000	20	1.000	0.999	0.999	1.000	0.999

summarizes the AUC values for detecting main effects across different sample sizes (n), numbers of drugs (m), numbers of adverse events (d), coefficient generation settings, and secondary Apriori indicators. All indicators perform well in this scenario, consistently achieving high AUC values. When the sample size is moderate ($n=5000$), we observe a slight decrease in AUC values as the dimensionality (m and d) increases. However, with a larger sample size (n = 10,000), this dimensionality effect becomes negligible, and performance remains robust. Note that for the second coefficient generation setting (interaction effects only), AUC values are not reported since this setting assumes no true main effects exist.

The results for detecting interaction effects are presented in

Table 3. AUC for detecting interaction effects when $\mathrm{\Lambda }=\mathbf{0}$.

n	$\mathit{m}=\mathit{d}$	Confidence	Lift	PRR	ROR	RR
$\beta$ generation setting 2
5000	10	0.629	0.628	0.628	0.629	0.628
5000	20	0.720	0.720	0.720	0.720	0.720
10,000	10	0.936	0.935	0.936	0.936	0.935
10,000	20	0.990	0.990	0.990	0.990	0.990
$\beta$ generation setting 3
5000	10	0.774	0.774	0.774	0.774	0.774
5000	20	0.998	0.993	0.993	0.995	0.993
10,000	10	0.812	0.811	0.811	0.812	0.811
10,000	20	0.812	0.811	0.811	0.811	0.811
$\beta$ generation setting 4
5000	10	0.801	0.801	0.801	0.801	0.801
5000	20	0.869	0.867	0.867	0.868	0.867
10,000	10	0.998	0.996	0.996	0.998	0.996
10,000	20	0.999	0.999	0.999	0.999	0.999

. For the first coefficient generation setting (main effects only), no AUC values are reported since this setting assumes no true interaction effects exist. Compared to main effect detection, the AUC values for identifying interaction effects are generally lower, likely due to the low frequency of reports with relevant drug pairs as well as relative rarity of true DDIs in our simulation design. As expected, larger sample sizes substantially improve detection performance. Interestingly, we observe that performance actually improves as the dimensionality (m and d) increases, contrary to what might be intuitively expected. One possible explanation is that the increased correlations among drugs or AEs in higher-dimensional settings can enhance the Apriori algorithm’s ability to detect meaningful interactions. Furthermore, comparing across coefficient generation settings, we find that setting 3 (independent main and interaction effects) outperforms setting 2 (interaction effects only), and setting 4 (hierarchical effects) yields the best performance overall. This suggests that the coexistence of main and interaction effects improves DDI detection, with particularly strong performance when these effects follow a hierarchical structure—a pattern that may reflect real-world drug interaction mechanisms.

3.3.2. Random Effects Without Confounding

In this scenario, we introduce subject-specific random effects ${\overrightarrow{b}}_i$ that are uncorrelated with the covariates $\mathbf{X}$. While these random effects create dependence among adverse events, they do not function as true confounders since they are independent of drug exposures.

Table 4. AUC for detecting main effects when ${\overrightarrow{b}}_i$ exists but is not correlated with ${\mathbf{x}}_i$.

n	$\mathit{m}=\mathit{d}$	Confidence	Lift	PRR	ROR	RR
$\beta$ generation setting 1
5000	10	0.958	0.991	0.991	0.991	0.991
5000	20	0.940	0.890	0.906	0.914	0.890
10,000	10	0.963	0.985	0.989	0.990	0.985
10,000	20	0.976	0.999	0.999	1.000	0.999
$\beta$ generation setting 3
5000	10	0.965	0.993	0.993	0.993	0.993
5000	20	0.952	0.899	0.915	0.922	0.899
10,000	10	0.979	0.988	0.992	0.993	0.988
10,000	20	0.980	0.999	0.999	0.999	0.999
$\beta$ generation setting 4
5000	10	0.969	0.994	0.994	0.994	0.994
5000	20	0.950	0.896	0.912	0.919	0.896
10,000	10	0.972	0.985	0.989	0.991	0.985
10,000	20	0.985	0.999	0.999	0.999	0.999

and

Table 5. AUC for detecting interaction effects when ${\overrightarrow{b}}_i$ exists but is not correlated with ${\mathbf{x}}_i$.

n	$\mathit{m}=\mathit{d}$	Confidence	Lift	PRR	ROR	RR
$\beta$ generation setting 2
5000	10	0.584	0.586	0.586	0.586	0.586
5000	20	0.684	0.687	0.687	0.687	0.687
10,000	10	0.873	0.885	0.885	0.884	0.885
10,000	20	0.940	0.949	0.949	0.948	0.949
$\beta$ generation setting 3
5000	10	0.675	0.676	0.676	0.676	0.676
5000	20	0.994	0.992	0.992	0.993	0.992
10,000	10	0.812	0.811	0.811	0.811	0.811
10,000	20	0.811	0.811	0.811	0.811	0.811
$\beta$ generation setting 4
5000	10	0.722	0.722	0.722	0.722	0.722
5000	20	0.826	0.826	0.826	0.826	0.826
10,000	10	0.995	0.994	0.995	0.995	0.994
10,000	20	0.999	0.999	0.999	0.999	0.999

summarize the AUC values for detecting main effects and interaction effects, respectively.

Similar trends to Scenario 1 are observed, but with an important distinction: the presence of subject-specific random effects reduces the AUC values for both main effect and interaction effect detection. Notably, this reduction is substantially more pronounced for DDI identification than for main effect identification. This differential impact likely stems from the fact that interaction effects represent higher-order relationships that are inherently more sensitive to noise. When random effects introduce additional variability in the data, the already subtle signals of drug–drug interactions become more difficult to distinguish from background noise, whereas the stronger signals of main effects remain relatively more detectable. Additionally, since each interaction effect involves multiple drugs, the propagation of noise through these complex relationships may have a multiplicative effect on detection performance.

3.3.3. Random Effects with Confounding

In this scenario, we introduce true latent confounders by generating subject-specific random effects ${\overrightarrow{b}}_i$ that are correlated with the covariates ${\mathbf{X}}_i$.

Table 6. AUC for detecting main effects when ${\overrightarrow{b}}_i$ exists and is correlated with ${\mathbf{x}}_i$.

n	$\mathit{m}=\mathit{d}$	Confidence	Lift	PRR	ROR	RR
$\beta$ generation setting 1
5000	10	0.811	0.810	0.804	0.804	0.810
5000	20	0.778	0.644	0.655	0.658	0.644
10,000	10	0.784	0.711	0.718	0.720	0.711
10,000	20	0.839	0.837	0.837	0.835	0.837
$\beta$ generation setting 3
5000	10	0.788	0.793	0.787	0.786	0.793
5000	20	0.785	0.648	0.660	0.663	0.648
10,000	10	0.816	0.749	0.758	0.760	0.749
10,000	20	0.839	0.839	0.839	0.837	0.839
$\beta$ generation setting 4
5000	10	0.815	0.819	0.815	0.814	0.819
5000	20	0.774	0.644	0.656	0.659	0.644
10,000	10	0.821	0.748	0.754	0.758	0.748
10,000	20	0.849	0.851	0.851	0.849	0.851

and

Table 7. AUC for detecting interaction effects when ${\overrightarrow{b}}_i$ exists and is correlated with ${\mathbf{x}}_i$.

n	m = d	Confidence	Lift	PRR	ROR	RR
$\beta$ generation setting 2
5000	10	0.570	0.572	0.572	0.571	0.572
5000	20	0.658	0.661	0.661	0.660	0.661
10,000	10	0.795	0.804	0.804	0.801	0.804
10,000	20	0.871	0.885	0.885	0.880	0.885
$\beta$ generation setting 3
5000	10	0.642	0.644	0.644	0.642	0.644
5000	20	0.935	0.928	0.929	0.931	0.928
10,000	10	0.774	0.771	0.771	0.772	0.771
10,000	20	0.768	0.771	0.771	0.769	0.771
$\beta$ generation setting 4
5000	10	0.712	0.714	0.714	0.712	0.714
5000	20	0.820	0.816	0.816	0.818	0.816
10,000	10	0.940	0.933	0.934	0.936	0.933
10,000	20	0.976	0.980	0.981	0.978	0.980

summarize the detection performance for main effects and interaction effects, respectively.

Compared with the previous two scenarios, the presence of latent confounders substantially degrades the detection performance for both types of effects, but with a notable pattern: the decline in AUC values is considerably more pronounced for main effect detection than for DDI detection. This asymmetric impact likely stems from the fundamental difference in how confounding affects single-drug versus multi-drug associations. When a latent variable confounds the relationship between drugs and adverse events, it directly creates spurious associations at the main effect level. However, for interaction effects to be confounded, the latent variable would need to be associated not just with individual drugs but with specific drug combinations. This higher-order confounding structure is less common and typically weaker than first-order confounding. Additionally, interaction effects benefit from an implicit form of adjustment—the detection algorithms already account for the constituent main effects, which may partially absorb confounding bias, leaving interaction effects as relatively more robust against confounding than main effects.

4. Real Data Example

Since vaccines are a type of drug, except for the simulation studies, we also implement the proposed modified Apriori approach to the VAERS dataset to select both drug–AE (i.e., vaccine–AE) association pairs and DDIs (i.e., vaccine–vaccine interactions). In particular, we choose the reports from the year of 2024, which include a total of 51,984 reports with 5894 different AEs and 123 unique vaccine names. The data files, and VAERS Symptoms and VAERS Vaccine files for the year of 2024 can be downloaded at https://vaers.hhs.gov/data/datasets.html, accessed on 5 March 2025.

The top 10 (by ROR) identified vaccine–AE associations and vaccine interactions are reported in

Table 8. Identified AE-associated vaccines from VAERS reports in the year of 2024.

Vaccine	AE
COVID19 (PFIZER-BIONTECH)	COVID-19
MENINGOCOCCAL CONJUGATE (MENQUADFI)	Syncope
HPV (GARDASIL 9)	Syncope
MENINGOCOCCAL CONJUGATE (MENVEO)	Syncope
TDAP (BOOSTRIX)	Syncope
ZOSTER (NO BRAND NAME)	Pain
MENINGOCOCCAL CONJUGATE (MENQUADFI)	Dizziness
PNEUMO (PREVNAR20)	Injection site erythema
PNEUMO (PREVNAR20)	Erythema
INFLUENZA (SEASONAL) (FLUZONE HIGH-DOSE QUADRIVALENT)	Injection site pain

and

Table 9. Identified AE-associated vaccine interactions from VAERS reports in the year of 2024.

Vaccine	Vaccine	AE
HPV (GARDASIL 9)	MENINGOCOCCAL B (BEXSERO)	Syncope
COVID19 (COVID19 (PFIZER-BIONTECH))	COVID19 (COVID19 (MODERNA))	COVID-19
HPV (GARDASIL 9)	MENINGOCOCCAL CONJUGATE (MENVEO)	Syncope
VARICELLA (VARIVAX)	MENINGOCOCCAL CONJUGATE (MENQUADFI)	Syncope
TDAP (BOOSTRIX)	MEASLES + MUMPS + RUBELLA (MMR II)	Syncope
MENINGOCOCCAL B (BEXSERO)	MENINGOCOCCAL CONJUGATE (MENVEO)	Syncope
TDAP (BOOSTRIX)	MENINGOCOCCAL CONJUGATE (MENQUADFI)	Syncope
VARICELLA (VARIVAX)	TDAP (BOOSTRIX)	Syncope
HPV (GARDASIL 9)	TDAP (BOOSTRIX)	Syncope
TDAP (BOOSTRIX)	HEP A (HAVRIX)	Syncope

, respectively. To save space, we list the corresponding secondary indicators values in the Supplementary Materials.

Syncope (fainting), dizziness, and pain are common side effects observed with the administration of injectable vaccines. It is, therefore, not surprising to see these AEs in the identified associations listed in the tables above. However, the identified associations between the COVID-19 vaccine and COVID-19 infection were unexpected and warrant careful interpretation. This finding likely demonstrates the real-world impact of latent confounding factors that our simulation studies predicted. Several plausible latent confounding mechanisms could explain this observation: (1) temporal confounding, where vaccination coincided with periods of high community transmission; (2) behavioral confounding, where recently vaccinated individuals might have changed their risk behaviors; (3) detection bias, where vaccinated individuals were more likely to be tested for COVID-19; or (4) reporting bias, where individuals who experienced COVID-19 after vaccination were more likely to submit reports than those with other post-vaccination experiences. This real-world example directly illustrates how the Apriori algorithm can generate potentially misleading associations when latent confounders exist—precisely the phenomenon our simulation studies demonstrated. These results underscore the importance of developing modified approaches that can account for or adjust for such confounding factors when mining spontaneous reporting data.

5. Conclusions

In this study, we conducted a systematic evaluation of the Apriori algorithm for detecting drug–drug interactions using spontaneous reporting systems data, with a focus on assessing its performance under realistic data conditions, particularly to assess and quantify the impact of latent confounders, which is a critical but previously overlooked issue.

Our comprehensive simulation studies revealed several important findings. First, various secondary indicators (Confidence, Lift, PRR, ROR, and RR) showed comparable performance in detecting both drug–AE associations and drug–drug interactions in idealized settings. Second, and most critically, the presence of latent factors significantly degraded algorithm performance, with differential impacts on main effects versus interaction effects. For example, when using RR as the secondary indicator, the AUC for detecting main effects dropped by approximately 30% in the presence of latent confounders (Scenario 3) compared to scenarios without confounding (Scenario 1), while the AUC for identifying DDIs declined by about 15%. This observation provides the first systematic evidence that latent confounding—an inherent feature of real-world spontaneous reporting data—substantially compromises the reliability of current Apriori implementations for drug safety signal detection.

The real-data application using VAERS data not only confirmed the practical feasibility of our approach but also provided a concrete example of potentially spurious associations (COVID-19 vaccine and COVID-19 infection) that likely result from unaccounted latent confounders. This finding validates our simulation results and underscores the real-world implications of our methodological investigation.

While this paper does not focus on directly estimating or modeling latent factors, it highlights the critical consequences of ignoring them in drug safety signal detection. Our simulation studies and real-world analysis show that latent confounding can substantially distort the detection of drug–AE associations and DDIs, often resulting in false-positive signals. Given that post-marketing drug safety monitoring is of great interest to clinicians, pharmaceutical companies, and regulatory professionals, especially as rare or delayed adverse effects may only appear after widespread use, our findings highlight the need for caution when interpreting signals from existing tools that overlook unmeasured confounders.

Moreover, our findings underscore three important directions for future research. First, methodological development should focus on integrating approaches to account for latent confounders without compromising the computational efficiency that makes Apriori attractive for large-scale pharmacovigilance. Potential strategies might include incorporating propensity score adjustments, using instrumental variables, or developing hybrid approaches that combine association rule mining with more sophisticated causal inference methods. Second, extending the current framework to capture higher-order interactions would address another limitation of current approaches. Third, incorporating additional observed covariates, such as age, sex, and comorbidities, could partially mitigate confounding effects.

The R code used to implement the modified Apriori algorithm and the supporting code and data files necessary to conduct the simulation studies are available at https://github.com/TXMUNC/DDI_MDPI_Mathematics_2025, accessed on 22 May 2025. These files are intended to facilitate reproducibility and provide a clear reference for researchers and practitioners seeking to apply or extend the proposed method.