Mixture Drug-Count Response Model for the High Dimensional Drug Combinatory Effect on Myopathy

Abstract

Drug-drug interactions (DDIs) are a common cause of adverse drug events (ADEs). The electronic medical record (EMR) database and the FDA's Adverse Event Reporting System (FAERS) database are the major data sources for mining and testing the ADE associated DDI signals. Most DDI data mining methods focus on pair-wise drug interactions, and methods to detect high-dimensional DDIs in medical databases are lacking. In this paper, we propose two novel mixture drug-count response models for detecting high dimensional drug combinations that induce myopathy. The “count” indicates the number of drugs in a combination. One model is called fixed probability mixture drug-count response model with a maximum risk threshold (FMDRM-MRT). The other model is called count-dependent probability mixture drug-count response model with a maximum risk threshold (CMDRM-MRT), in which the mixture probability is count-dependent. Compared to the previous mixture drug-count response model (MDRM) developed by our group, these two new models show a better likelihood in detecting high dimensional drug combinatory effects on myopathy. CMDRM-MRT identified and validated (54; 374; 637; 442; 131) 2-way to 6-way drug interactions, respectively, which induce myopathy in both EMR and FAERS databases. We further demonstrate FAERS data capture much higher maximum myopathy risk than EMR data do. The consistency of two mixture models’ parameters and local false discovery rate estimates are evaluated through statistical simulation studies.

1. Introduction

Adverse drug events (ADEs) are a significant cause of morbidity and mortality. ADEs lead to 125,000 hospital admissions each year; prolong hospital stays by nearly 1.7 to 4.6 days [1]; and result in as many as 4.6% of deaths in the United States [2]. It has been reported that 26% to 59.1% of ADEs are related to DDIs [3–5]. DDIs occur due to pharmacokinetic or pharmacodynamic interactions between co-administrated drugs. The risk of DDI-induced ADEs increases exponentially with the number of drugs taken by a patient [6]. A study from National Center for Health statistics (NCHS) showed that the number of patients taking more than 3 drugs and 5 drugs has increased 1.8- and 2.5-fold in the past decade, respectively [7]. Therefore, the evaluation of DDIs’ clinical impact, especially for high dimensional drug interactions, is an important issue. Some pre-marketing clinical trials focus on two-way drug interactions and often are limited to specific populations, in which adverse drug events are usually not primary hypotheses. In addition, the clinical trial data collected during the premarketing phase 3 trials are typically not large enough to capture less common combinations of drugs. Routinely, researchers rely on pharmaco-epidemiology studies on large-scale health record databases to investigate drug interactions [8]. The spontaneous reporting system (SRS) and the electronic medical record (EMR) are two major types of health record data sources for post-marketing pharmacovigilance [9–12]. Recently, as these big health record data sets become increasingly available to the general research community, novel data mining algorithms have shown promise in detecting potential drug- or DDI-induced ADEs [13–15].

Most data mining methods were developed to identify single drug induced ADEs. The salient examples include the information component (IC), a Bayesian confidence propagation neural network [16] used by World Health Organization (WHO), and the empirical Bayes geometric mean (EBGM) [17], which has been adopted by the United States Food and Drug Administration (FDA). There have been some recent developments in studying DDI induced ADEs. Noren et al. [18] developed a Ω shrinkage measure approach to screen potential pair-wise DDIs in the entire WHO database. It calculates a shrinked observed to expected ratio of disproportionality for a DDI induced ADE relative report rates. Huang et al. [19] proposed a likelihood ratio test method (LRT) for detecting ADE signals from the FAERS database. Later, in order to handle the extensive zeros in the FAERS database, a zero-inflated Poisson model based LRT (ZIP_LRT) was proposed [20]. LRT and its extension can be used to detect signals for single drug (or ADE) or signals including a class of drugs (or ADEs) [21, 22]. Though these methods were originally developed to analyze the FAERS database, their extensions were derived to analyze longitudinal drug safety data (longitudinal LRT) as well [23]. LRT and its extensions can control the type-I error and false discovery rate (FDR) while retaining good power and sensitive for identifying signals. Thakrar et al. [24] proposed multiplicative and additive models to detect DDIs in the FDA’s Adverse Event Reporting System (FAERS) database. These two model assumptions characterize the relationship between the relative risk of the two-drug combination and the relative risk from two single drugs. In the DDI detecting algorithm outlined by Tatonetti et al. [25], they considered and adjusted the confounding variables by using propensity score derived from logistic regression analysis. Harpaz et al. [26] applied an association rule mining (ARM) to detect the multi-item ADE associations in the FAERS. In order to overcome the computational challenge of the ARM, Xiang et al. [27] proposed a Frequent Closed Item set Mining and filtering (FCI-filter) based on UMLS mapping for mining multiple drug interactions; and FCI-filter has been applied to FAERS data as well.

Data mining methods to detect the single drug and two-drug DDI induced ADEs cannot be easily expanded to evaluate high-dimensional drug interactions. In our FAERS and EMR database, the report frequency for most of 5-way to 6- way drug combinations is no more than 20. Although ARM and FCI-filter methods have the advantage of handling high-dimensional drug and ADE combinations freely, they are indeed constructed and limited to ADE cases only drugs/ADEs combinations. These structural limited methods cannot easily be expanded to handle drug combinations whose ADE frequencies are moderate or low.

To address these challenges in detecting high-dimensional drug interactions, our group proposed a drug-count response model [28], where “count” indicates the number of drugs in a combination, and in which the same dimensionality of drug combinations shared the same ADE risk model. In this risk model, the same dimensionality of drug combinations either share the baseline risk that doesn’t depend on the dimensionality of combination drugs, or follows a drug-count response model that depends on the dimensionality of combination drugs. This model allows high-dimensional drug combinations share their ADE risks, such that they can borrow data strength from each other and make up the small sample size deficiency. Using the empirical Bayes mixture model framework, this model will give each drug combination a probability of belonging to a constant risk model, and a probability of following the drug-count response model. This drug combination specific probability allows us to evaluate, interpret and rank the high dimensional drug interaction evidence from the data. This probability also has a local discovery rate interpretation. Using the EMR data, we successfully identified 2- to 6-way drug combinations that increased myopathy risk at a low local false discovery rate [28]. However, while this model is highly powerful in detecting high dimensional drug interactions, it possesses intrinsic deficiencies and needs further improvement. Statistically, the baseline model and drug-count response model do not meet continuity when the number of drug equals to one. Also, the mixture probability (i.e. the proportion of drug combinations belonging to the drug-count response model) is fixed and assumed to be the same regardless of the number of drug combinations. From the pharmaco-vigilance point of view, the drug-count response model was tested in only one EMR database, and it was not externally validated. Thus, the top ranked myopathy associated high-dimensional drug interactions identified by the method were not yet validated. In this paper, our novel mixture drug-count response models address these statistical and pharmacology challenges. In order to identify which drug combination follows the drug-count response model or constant risk model, we use both EMR and FAERS datasets to derive their drug-count response models, and evaluate and validate top myopathy associated high-dimensional drug interactions.

2. Methods

2.1 Data Sources

The data sources used in this analysis are the FDA’s Adverse Event Reporting System (FAERS) and the Indiana Network of Patient Care data, which is an Electronic Medical Record (EMR) database.

2.1.1 FAERS Data Set

FAERS contains spontaneous adverse drug event reports from healthcare professionals, consumers, and pharmaceutical manufactures. The data used in this paper were from the FAERS 2004Q1 to 2012Q3. Duplicated reports that had the same primary record ID were removed. ADEs in the FAERS were annotated using MedDRA’s PT code [29]. The drug names in the FAERS may contain abbreviations, brand names, synonyms, and sometimes contain spell mistakes. Therefore, they were normalized through a drug name mapping scheme implemented in the DrugBank. Un-mapable names due to spelling errors (i.e. drug names that are only one letter different from a generic name, a brand name, or a synonym) and with the reporting frequency greater than 1000 were manually checked and mapped. After data clean-up, the FAERS dataset contained 4,280,322 reports with 1,753 generic drug names and 15,445 MedDRA PT ADE names.

2.1.2 Indiana Network of Patient Care Data Set

Indiana Network for Patient Care (INPC) is a local health information infrastructure that has been approved as exempt research by institutional review board (IRB) [30]. A subset of INPC called Common Data Model (CDM) are de-identified and extracted. This data set contains coded prescription medications, diagnoses, and lab tests for 2.2 million patients between 2004 and 2009. The CDM data have been processed with the Observational Medical Outcomes Partnership Common Data Model [31].

2.2 Case and Control Definitions

2.2.1 Myopathy Case and Control Definitions in FAERS

From 4,280,322 reports in the FAERS dataset, we defined myopathy “cases” as those reports listing myositis, myoglobinuria, muscle fatigue, muscle spasms, myalgia, muscle injury, muscular weakness, polymyositis and rhabdomyolysis (Table S1). All other reports that do not contain these ADEs are defined as controls. Based on this definition, we identified 140,071 cases and 4,140,251 controls in the FAERS database.

2.2.2 Myopathy Case and Control Definitions in INPC

The myopathy cases (Table S1) in the INPC are similar to the myopathy cases defined in the FAERS.

For the EMR database, we defined two types of myopathy events: 1) the first event of myopathy that occurs more than 6 months after the start of the database (01/01/2004), and 2) any additional myopathy event(s) that occur(s) more than 6 months after the previous myopathy event. In another word, for patients with multiple myopathy events, a 6-month myopathy free window was used for selecting any additional myopathy event(s).

Patients who experienced a myopathy event are considered cases. For each case, a drug exposure window is set as 1 month prior to the index event, and the drug(s) prescribed during this time period are classified as being associated with myopathy. For the control group, we randomly selected 50 patients who did not experience a myopathy event during the same time interval as the case. Drugs prescribed to these patients during the one month period before the index date are classified as not being associated with myopathy [32, 33].

2.3 Drug and Drug Combination Selections

For this analysis, we limited the number of drugs studied to the 20 most frequent drugs associated with myopathy in the EMR dataset (Table S2) [28]. Among these 20 drugs, 17 are associated with myopathy (the myopathy definition are shown in the Table S1) side effect in the Side Effect Resource database [34].

For the 20 drugs, we selected all their possible 2-way to 6-way drug combinations in EMR and FAERS resulting in 60,460 possible drug combinations. To avoid false positive signals, both the FAERS and EMR datasets were filtered so that only those drug combinations with a total report number (case number plus control number) greater than 4 (n_ij > 4) were evaluated. This filtering step reduced the number of drug combinations in the EMR to 20,161 and FAERS to 31,476 combinations (Figure S1).

2.4 Mixture Drug-Count Response Models

2.4.1. Previously Defined Drug-Count Response Model

Our group has previously described a mixture drug-count response model (MDRM) [28] for identifying myopathy induced by high-dimensional drug interactions. In this model, “count” indicates the number of drug combinations. The primary novelty of this model was a mixture of two model components: one component represents a constant myopathy risk regardless of the dimensionality of drug combinations while the other component characterizes an increasing drug-count response relationship between the dimensionality of drug combinations and the myopathy risk.

In the mixture drug-count response model, i indicates the number of drugs for i-way drug combinations; j is the jth i-way drug combinations; N_ij is the total number of patients taking jth i-way drug combination; and Y_ij is the number of cases among those N_ij patients. Additionally, let Z_ij be the underlying binary random variable. Specifically, if Y_ij follows the drug-count response model, Z_ij equals to 1; otherwise Z_ij equals to 0 when Y_ij follows the constant model. The joint distribution of (Y_ij, Z_ij) is

$$\mathrm{P}(y_{\mathit{\text{ij}}},z_{\mathit{\text{ij}}})={[(1-\pi )\times \text{Bin}(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},q_0)]}^{1-z_{\mathit{\text{ij}}}}\times {[\mathrm{\pi }\times \text{Bin}(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},q_1)]}^{z_{\mathit{\text{ij}}}}.$$ (1)

Where π is the proportion of drug combinations that follow the drug-count response component; $q_0=\frac{\text{exp}({\beta }_0)}{1+\text{exp}({\beta }_0)}$ represents the constant ADE risk; and $q_1=\frac{\text{exp}({\beta }_0+{\beta }_{1}i)}{1+\text{exp}({\beta }_0+{\beta }_{1}i)}$ represents the drug-count response ADE risk. Then, the marginal distribution of Y_ij can be written as a two-component mixture distribution (2):

$$P(y_{\mathit{\text{ij}}})=(1-\pi )\text{Bin}(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},q_0)+\pi \text{ Bin}(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},q_1).$$ (2)

2.4.2 Novel Mixture Drug-Count Response Models

Here, we propose two novel drug-count response models to identify the myopathy risk induced by high-dimensional drug combinations: a fixed probability mixture drug-count response model with a maximum risk threshold (FMDRM-MRT) model, and a count-dependent probability mixture drug-count response model with a maximum risk threshold (CMDRM-MRT) model. As in our previous model, “count” indicates the number of drug combinations.

Fixed Probability Mixture Drug-Count Response Model with a Maximum Risk Threshold (FMDRM-MRT)

In the FMDRM-MRT, the definitions of i (1<i<6) and j; the random variables, N_ij, Y_ij and Z_ij; and the parameter π are the same as in the MDRM. We also assume that the marginal distribution function of Y_ij follows a two-component mixture distribution (same as equation 1). However, in the FMDRM-MRT model q₀ and q₁ are defined as: $q_0=\frac{\text{exp}({\beta }_0)}{1+\text{exp}({\beta }_0)}\times c$ and $q_1=\frac{\text{exp}({\beta }_0+{\beta }_1(i-1))}{1+\text{exp}({\beta }_0+{\beta }_1(i-1))}\times c,c\in (0,1)$.

The FMDRM-MRT has two noticeable differences from the MDRM. First, the β₁(i − 1) is revised on the β₁i in the MDRM, when i = 1, q₀ and q₁ are equal. This follows a continuity assumption. Second the maximum ADE risk of FMDRM-MRT is bounded by c, while the maximum risk of MDRM is 1.

Under FMDRM-MRT, the joint distribution function of (Y_ij, Z_ij) can be written as

$$P(y_{\mathit{\text{ij}}},z_{\mathit{\text{ij}}})={[(1-\pi )\mathit{\text{Bin }}(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},\frac{\mathit{\text{exp}}({\beta }_0)\times c}{1+\mathit{\text{exp}}({\beta }_0)}\left)\right]}^{1-z_{\mathit{\text{ij}}}}\times {\left[\pi \mathit{\text{ Bin }}\right(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},\frac{\mathit{\text{exp}}[{\beta }_0+{\beta }_1(i-1)]\times c}{1+\mathit{\text{exp}}({\beta }_0+{\beta }_1(i-1))}\left)\right]}^{z_{\mathit{\text{ij}}}}.$$ (3)

The marginal distribution function of Y_ij is

$$P(y_{\mathit{\text{ij}}})=(1-\pi )\text{Bin }(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},\frac{\text{exp}({\beta }_0)\times \mathrm{c}}{1+\text{exp}({\beta }_0)})+\pi \text{ Bin }(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},\frac{\text{exp}[{\beta }_0+{\beta }_1(i-1)]\times \mathrm{c}}{1+\text{exp}({\beta }_0+{\beta }_1(i-1))}).$$ (4)

The log-likelihood function based on (4) is

$$l^f({\mathit{n}}_{\mathit{\text{ij}}},{\mathit{y}}_{\mathit{\text{ij}}};\mathit{\theta })={\sum }_i{\sum }_j\text{log P}(Y_{\mathit{\text{ij}}}),\mathit{\theta }=(\pi ,{\beta }_0,{\beta }_1,c).$$ (5)

Count-dependent Probability Mixture Drug-Count Response Model with a Maximum Risk Threshold (CMDRM-MRT)

In this model, we assume that the proportion of drug combinations following the drug-count response model will depend on the dimensionality of drug combinations. Therefore, the joint distribution function of (Y_ij, Z_ij) changes to (6).

$$P(y_{\mathit{\text{ij}}},z_{\mathit{\text{ij}}})={[(1-{\pi }_i)\mathit{\text{Bin }}(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},\frac{\mathit{\text{exp}}({\beta }_0)\times c}{1+\mathit{\text{exp}}({\beta }_0)}\left)\right]}^{1-z_{\mathit{\text{ij}}}}\times {\left[{\pi }_i\mathit{\text{ Bin }}\right(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},\frac{\mathit{\text{exp}}[{\beta }_0+{\beta }_1(i-1)]\times c}{1+\mathit{\text{exp}}({\beta }_0+{\beta }_1(i-1))}\left)\right]}^{z_{\mathit{\text{ij}}}}.$$ (6)

where π_i (i = 1,2, …, 6 and 0 < π_i < 1) is the proportion of i-way drug combinations that follow the drug-count response component. The marginal distribution function for (6) is

$$P(y_{\mathit{\text{ij}}})=(1-{\pi }_i)\text{Bin }(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},\frac{\text{exp}({\beta }_0)\times c}{1+\text{exp}({\beta }_0)})+{\pi }_i\text{ Bin }(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},\frac{\text{exp}[{\beta }_0+{\beta }_1(i-1)\times c}{1+\text{exp}({\beta }_0+{\beta }_1(i-1))}).$$ (7)

The log-likelihood function for (7) can be written as

$$l^c({\mathit{n}}_{\mathit{\text{ij}}},{\mathit{y}}_{\mathit{\text{ij}}};\mathit{\theta })={\sum }_i{\sum }_j\text{log P}(y_{\mathit{\text{ij}}}),\mathit{\theta }=({\pi }_1,\dots ,{\pi }_6,{\beta }_0,{\beta }_1,c).$$ (8)

2.5 Expectation-Maximization Algorithm

As Z_ij is not observed, the maximum likelihood estimators (MLEs) of parameters in equations (3) and (6) can be obtained through an expectation-maximization (EM) algorithm. Hence, we define l(n_ij, y_ij, z_ij; θ) = Σ_iΣ_j log P(y_ij, z_ij) as the log-likelihood for equations (3) and (6). The EM algorithm is an iterative method and after the tth iteration, θ^t is the estimator of θ. First, in the E-step, Q(n_ij, y_ij, w_ij; θ) = E_Z|Y[l(n_ij, y_ij, z_ij; θ)|y_ij, θ^t] is computed, and w_ij is the estimator of Z_ij.

For CMDRM-MRT, the w_ij can be written as

$$w_{\mathit{\text{ij}}}=E(z_{\mathit{\text{ij}}}|y_{\mathit{\text{ij}}},\mathit{\theta })=\frac{\mathrm{\pi }\text{ Bin}(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},\frac{\text{exp}[{\beta }_0+{\beta }_1(i-1)]\times \mathrm{c}}{1+\text{exp}({\beta }_0+{\beta }_1(i-1))})}{(1-\mathrm{\pi })\text{Bin}(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},\frac{\text{exp}({\beta }_0)\times \mathrm{c}}{1+\text{exp}({\beta }_0)})+\mathrm{\pi }\text{ Bin}(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},\frac{\text{exp}[{\beta }_0+{\beta }_1(i-1)]\times \mathrm{c}}{1+\text{exp}({\beta }_0+{\beta }_1(i-1))})}.$$ (9)

Correspondingly, for DMDRM-MRT w_ij is

$$w_{\mathit{\text{ij}}}=E(z_{\mathit{\text{ij}}}|y_{\mathit{\text{ij}}},\mathit{\theta })=\frac{{\pi }_i\text{ Bin}(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},\frac{\text{exp}[{\beta }_0+{\beta }_1(i-1)]\times \mathrm{c}}{1+\text{exp}({\beta }_0+{\beta }_1(i-1))})}{(1-{\pi }_i)\text{Bin}(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},\frac{\text{exp}({\beta }_0)\times \mathrm{c}}{1+\text{exp}({\beta }_0)})+{\pi }_i\text{ Bin}(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},\frac{\text{exp}[{\beta }_0+{\beta }_1(i-1)]\times \mathrm{c}}{1+\text{exp}({\beta }_0+{\beta }_1(i-1))})}.$$ (10)

Second, we find θ^t+1 in the M-step, where ${\mathit{\theta }}^{\mathit{t}+\mathbf{1}}=\text{arg}\underset{\theta }{\text{max }}Q({\mathit{n}}_{\mathit{\text{ij}}},{\mathit{y}}_{\mathit{\text{ij}}},{\mathit{w}}_{\mathit{\text{ij}}};\mathit{\theta })$.

In this study, the maximization is carried out by using the R function nlminb, which is an unconstrained and box-constrained optimization using PORT routines which is a Newton-like method.[35]

2.6 Local False Discovery Rate

The local false discovery rate (lfdr) was introduced by Efron et al. [36] for analyzing data from microarray experiments, and was defined as the posterior probability of a gene’s expression belonging to “null distribution” [37]. In both FMDRM-MRT and CMDRM-MRT, drug combinations have either a constant myopathy risk (“null distribution”) or a drug-count response risk. Thus, both models follow the same model framework of Efron et al. [37]. The lfdrs for FMDRM-MRT and CMDRM-MRT are defined in (11) and (12), respectively:

$$\mathit{\text{lfdr}}(y_{\mathit{\text{ij}}})=\frac{(1-\pi )\text{Bin}(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},q_0)}{(1-\pi )\mathit{\text{Bin}}(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},q_0)+\pi \mathit{\text{Bin}}(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},q_1)}.$$ (11)

$$\mathit{\text{lfdr}}(y_{\mathit{\text{ij}}})=\frac{(1-{\pi }_i)\text{Bin}(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},q_0)}{(1-{\pi }_i)\text{Bin}(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},q_0)+{\pi }_i\text{Bin}(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}},q_1)}.$$ (12)

lfdrs in (11) and (12) represent the posterior probabilities that a drug combination has a constant myopathy risk; i.e. lfdr represents the odds that myopathy risk will be constant as the dimensionality of drug combination increases.

2.7 Likelihood Ratio Test

Because FMDRM-MRT and CMDRM-MRT are nested models, the likelihood-ratio test is used to test and compare the goodness of fit between two models. Let the FMDRM-MRT be considered the null model, and the CMDRM-MRT be the alternative model. The likelihood ratio can then be defined as

$$\mathrm{\Lambda }=\frac{l^f(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}};\theta )}{l^c(n_{\mathit{\text{ij}}},y_{\mathit{\text{ij}}};\theta )}.$$ (13)

According to Wilks’ theorem [38], the test statistic −2log(Λ) can be assumed to follow a chi-squared distribution. Our model has 4 degrees of freedom.

$$-2\text{ log}(\mathrm{\Lambda })~{\chi }^2(4)$$ (14)

2.8 Simulation Study

To evaluate the performance of our models, a simulation study was conducted to assess local false discovery rate estimates. In the simulation, λ_i is the mean of N_ij in i-way drug combinations, and k_i is the number of i-way drug combinations in the simulation. N_ij is the number of patients taking the jth i-way drug combinations and it follows a Poisson distribution with the mean equals to λ_i. Y_ij is the number of drug combinations with myopathy cases in N_ij. Let Z_ij be the random binary variable. Specifically, Z_ij = 1 if Y_ij follows a drug-count response myopathy risk, otherwise Z_ij = 0 when Y_ij has a constant myopathy risk. Z_ij is generated using a Bernoulli distribution with a probability π_i. Given N_ij, we assume Y_ij follows a binomial distribution with size N_ij and probability equal to q₀ or q₁, it depends on the value of Z_ij, either follows the drug-count response myopathy risk or the constant myopathy risk.

In order to assess the consistency of the lfdr estimate, we calculate the model based ${\widehat{\mathit{\text{lfdr}}}}_{\mathit{\text{ij}}}$ estimate and the empirical lfdr_ij estimate in the simulation study. The simulation data {n_ij, y_ij, z_ij, ${\widehat{\mathit{\text{lfdr}}}}_{\mathit{\text{ij}}}$}, are divided into 100 intervals according to the value of n_ij and y_ij/n_ij. In each interval, we calculate the model based lfdr which is defined as the mean of ${\widehat{\mathit{\text{lfdr}}}}_{\mathit{\text{ij}}}s$ and the empirical lfdr which is the proportion of z_ij = 0.

3. Results

3.1 Model Performance Comparisons among CMDRM-MRT, FMDRM-MRT and MDRM

All three models are fitted to the EMR dataset and FAERS dataset. Their parameter estimates are shown in Table I. In fitting the EMR data, CMDRM-MRT shows an increasing trend of probability, i.e. from π₁ to π₆, (0.50, 0.67, 0.73, 0.82, 0.90, 0.94) respectively, that drug combinations follow drug-count response model (Figure 1). The likelihood ratio test between CMDRM-MRT to FMDRM-MRT has a p-value of 8.3 × 10⁻³⁵ suggesting that CMDRM-MRT fit the data better than FMDRM-MRT. This is a piece of strong evidence that the mixture probability of drug-count response model is indeed drug-count dependent. Both CMDRM-MRT and FMDRM-MRT models show comparable maximum myopathy risk, 0.448 vs 0.460 (Table I, Figure 1 and 2), respectively. On the other hand, MDRM shows a relatively higher mixture probability of drug-count response compared to the other models. This is likely because the MDRM allows a discontinuous drug-count response model and the constant risk model for the single drug, and drug-count response model has a higher myopathy risk than the constant risk model (Figure 3).

Table I.

Parameters estimated for CMDRM-MRT, FMDRM-MRT and MDRM using EMR and FAERS dataset. {π₁, π₂, π₃, π₄, π₅, π₆} are the proportions of 2 way to 6 way drug combinations that follow drug-count response component. c is the maximum Myopathy risk.

a.
Dataset	EMR
Parameter	CMDRM-MRT [95%CI]	FMDRM-MRT [95% CI]	MDRM [95% CI]
π1	-	0.869 [0.858, 0.880]	0.929 [0.920, 0.939]
π2	0.667 [0.564, 0.770]	0.869 [0.858, 0.880]	0.929 [0.920, 0.939]
π3	0.726 [0.683, 0.768]	0.869 [0.858, 0.880]	0.929 [0.920, 0.939]
π4	0.823 [0.801, 0.845]	0.869 [0.858, 0.880]	0.929 [0.920, 0.939]
π5	0.904 [0.890, 0.919]	0.869 [0.858, 0.880]	0.929 [0.920, 0.939]
π6	0.937 [0.920, 0.952]	0.869 [0.858, 0.880]	0.929 [0.920, 0.939]
c	0.448 [0.441, 0.456]	0.460 [0.443, 0.478]	-
β0	−1.084 [−1.107, −1.060]	−1.120 [−1.168, −1.072]	−2.269 [−2.284, −2.254]
β1	0.843 [0.820, 0.866]	0.810 [0.767, 0.853]	−1.908 [−1.926, −1.889]
β2	-	-	0.304 [0.298, 0.309]

b.
Dataset	FAERS
Parameter	CMDRM-MRT [95% CI]	FMDRM-MRT [95% CI]	MDRM [95% CI]
π1	-	0.798 [0.791, 0.805]	0.675 [0.667, 0.684]
π2	0.532 [0.423, 0.641]	0.798 [0.791, 0.805]	0.675 [0.667, 0.684]
π3	0.781 [0.742, 0.821]	0.798 [0.791, 0.805]	0.675 [0.667, 0.684]
π4	0.808 [0.788, 0.827]	0.798 [0.791, 0.805]	0.675 [0.667, 0.684]
π5	0.790 [0.778, 0.803]	0.798 [0.791, 0.805]	0.675 [0.667, 0.684]
π6	0.803 [0.793, 0.813]	0.798 [0.791, 0.805]	0.675 [0.667, 0.684]
c	0.999 [0.989, 1.000]	0.999 [0.991, 1.000]	-
β0	−2.554 [−2.567, −2.541]	−2.562 [−2.579, −2.546]	−1.594 [−1.605, −1.583]
β1	0.695 [0.691, 0.699]	0.697 [0.693, 0.700]	−3.874 [−3.886, −3.862]
β2	-	-	0.881 [0.877, 0.884]

Figure 1.

The CMDRM-MRT fit myopathy risks, distribution of the proportion of drug combinations that follow the drug-count response component (π_i) and the number of drug combinations in the EMR and FAERS datasets.

Figure 2.

The FMDRM-MRT fit myopathy risks in the EMR and FAERS datasets.

Figure 3.

The MDRD fit myopathy risks in the EMR and FAERS datasets.

In fitting the FAERS data, CMDRM-MRT also shows an increasing trend of probability of drug-count response model when the drug combination goes from 1 to 3, i.e. (0.50, 0.53, 0.78) respectively, and this probability stabilizes around 0.80 when the drug combination goes from 4 to 6 (Figure 1). The likelihood ratio test between CMDRM-MRT to FMDRM-MRT has a p-value of 1.9 × 10⁻⁶, suggesting that CMDRM-MRT fits the data better than FMDRM-MRT does. The mixture probability of drug-count response model thus appears drug-count dependent in FAERS. Comparing CMDRM-MRT to FMDRM-MRT, both models show the same maximum myopathy risk, 0.999 (Table I, Figure 1 and 2). However, MDRM shows lower mixture probability of drug-count response model than the other models. Because the MDRM allows a discontinuous drug-count response model and the constant risk model for the single drug, the drug-count response model has a lower myopathy risk than the constant risk model. (Figure 3).

Comparing CMDRM-MRT’s fitting and performance between FAERS and EMR data sets (Figure 1), FAERS’s drug-count response model shows a much steeper increase of myopathy risk than EMR’s drug-count response model does. FAERS has a much higher maximum myopathy risk, 0.999 than the EMR has (i.e. 0.448) when drug count goes high. Similarly, both EMR and FAERS have the similar increasing trend of mixture probability of drug count response model.

3.2 Common Myopathy Associated Drug Combinations Identified from EMR and FAERS Data Sets

Using an lfdr threshold of 0.00001, significant drug combinations are selected from both EMR and FAERS data sets. Figure 4 displays the overlapped drug combinations with lfdr<0.00001 (red dots) and lfdr>0.00001 (black dots) between two data sets. CMDRM-MRT and FMDRM-MRT have very similar pattern, while MDRM shows different trend. As shown in Figure 5, MDRM identifies more two-way drug combinations than the other two models, but fewer high-dimensional drug combinations (3-way to 5-way). This is mainly because of the mis-specified discontinuity assumption between drug-count response model and constant risk model in the MDRM.

Figure 4.

The distribution of −log₁₀ (lfdr) for the 2 way to 6 way common drug combinations in the EMR and FAERS datasets. The purple line is a threshold with −log₁₀ (lfdr) =5 (lfdr =0.00001), red plot means the common drug combinations with the condition of −log₁₀ (lfdr) > 5 (lfdr <0.00001) in two datasets.

Figure 5.

Statistic the number of common drug combinations for 2 way to 6 way drug combinations in the EMR and FAERS dataset with the condition of lfdr<0.00001 for the three models, respectively.

3.3 Common Myopathy Associated 6-Way Drug Combinations

Using the CMDRM-MRT model, 131 six-way drug combinations were identified in both the FAERS and EMR databases with lfdr < 0.00001. The FMDRM-MRT model yielded 97 six-way drug combinations with lfdr < 0.00001. Table II presents several examples of the 6-way drug combinations detected by CMDRM-MRT model that were associated with increased myopathy risk in both the FAERS and EMR databases. Myopathy risk associated with these combinations ranged from 0.38 to 0.73 in the EMR and 0.43 to 0.76 in the FAERS analyses. Only one of the 14 drugs that are presented in these 6-way drug combinations have not been shown to have myopathy risk in the SIDER database [34]. Of note, three of these 6-way combinations include simvastatin or atorvastatin and drugs known to inhibit their metabolism. These combinations can lead to increased exposure of the statin drugs, which are commonly known to cause myopathy. Considering the baseline risk estimated from the constant risk model in CMDRM-MRT, these 6-way drug combinations have 3.45- to 10.85-fold increase in myopathy risk.

Table II.

Common 6-way drug combinations detected by CMDRM-MRT with the condition of lfdr<0.00001. RR is the relative risk, it is calculated as risk/constant risk.

a.
Database	EMR
Drug1	Drug2	Drug3	Drug4	Drug5	Drug6	Risk [95% CI]	RR
atorvastatin	omeprazole	zolpidem	acetaminophen	hydrocodone	duloxetine	0.73 [0.47, 0.99]	6.64
simvastatin	venlafaxine	zolpidem	acetaminophen	hydrocodone	tramadol	0.59 [0.36, 0.82]	5.36
simvastatin	venlafaxine	omeprazole	acetaminophen	hydrocodone	duloxetine	0.56 [0.32, 0.80]	5.09
ondansetron	omeprazole	promethazine	acetaminophen	hydrocodone	alprazolam	0.38 [0.23, 0.53]	3.45
ondansetron	omeprazole	promethazine	acetaminophen	alprazolam	oxycodone	0.43 [0.25, 0.61]	3.91
escitalopram	omeprazole	zolpidem	acetaminophen	oxycodone	duloxetine	0.69 [0.44, 0.94]	6.27

b.
Database	FAERS
Drug1	Drug2	Drug3	Drug4	Drug5	Drug6	Risk [95%CI]	RR
atorvastatin	omeprazole	zolpidem	acetaminophen	hydrocodone	duloxetine	0.63 [0.49, 0.76]	9.00
simvastatin	venlafaxine	zolpidem	acetaminophen	hydrocodone	tramadol	0.70 [0.53, 0.87]	10.00
simvastatin	venlafaxine	omeprazole	acetaminophen	hydrocodone	duloxetine	0.76 [0.58, 0.94]	10.85
ondansetron	omeprazole	promethazine	acetaminophen	hydrocodone	alprazolam	0.48 [0.36, 0.60]	6.85
ondansetron	omeprazole	promethazine	acetaminophen	alprazolam	oxycodone	0.43 [0.30, 0.56]	6.14
escitalopram	omeprazole	zolpidem	acetaminophen	oxycodone	duloxetine	0.71 [0.61, 0.81]	10.14

3.4 Assess lfdr Estimate through A Simulation Study

The CMDRM-MRT was further evaluated in a simulation study. Parameter values were selected using estimates from the FAERS data analysis. In the simulation, 15 drugs were used to generate the drug combinations, resulting in 105 2-way combinations; 455 3-way combinations; 1,365 4-way combinations; 3,003 5-way combinations, and 5,005 6-way combinations. 500 simulated data sets were generated. Each time, the EM algorithm was used to estimate the parameters in the CMDRM-MRT. Table III presents the model-based estimates, their SDs, 95% CI, SD/estimates and relative biases. The relative biases of these estimates ranged from 0 to 1%. The SDs estimated from the simulations are also very small compared to the estimates, suggesting a high confidence on these parameter estimates. Figure 6 further demonstrates the consistency of the model based lfdr estimate (y-axis) and empirical lfdr estimate (x-axis) estimated from the simulation data.

Table III.

The parameters and relative statistical variables estimated in the simulation study.

Parameters	True value	Estimate (SD)	95% CI	SD/ Estimate	Relative bias
π2	0.532	0.527 (0.049)	[0.513, 0.541]	0.093	−0.010
π3	0.781	0.782 (0.018)	[0.776, 0.788]	0.023	0.001
π4	0.808	0.807 (0.011)	[0.804, 0.810]	0.014	−0.001
π5	0.790	0.790 (0.007)	[0.788, 0.792]	0.009	0
π6	0.803	0.804 (0.006)	[0.802, 0.806]	0.007	0
β0	−2.554	−2.550 (0.006)	[−2.553, −2.547]	−0.002	−0.001
β1	0.695	0.696 (0.002)	[0.695, 0.697]	0.003	0.002
c	0.999	0.995 (0.006)	[0.992, 0.998]	0.006	−0.004

Figure 6.

Comparison of the model based lfdr and the empirical lfdr.

4. Discussion

In this article, we propose two novel mixture drug-count response models, FMDRM-MRT and CMDRM-MRT to characterize relationship between the counts of drug combinations and the myopathy risks. Unlike MDRM [28], these two models speculate the maximum myopathy risk as one model parameter, and allow drug-count response model and constant risk model share the same myopathy risk when drug count is 1. In addition, CMDRM-MRT further allows the mixture probability to be drug count dependent. Using EMR and FAERS datasets, we demonstrate that CMDRM-MRT fits data better than FMDRM-MRT, p =8.3 × 10⁻³⁵ and 1.9 × 10⁻⁶, respectively. Interestedly, both CMDRM-MRT and FMDRM-MRT suggest that maximum myopathy risk reaches to 0.999 in FAERS and 0.45 in EMR when the drug counts in drug combinations goes high. This difference of maximum myopathy risk between two databases make sense, because FAERS is designed to capture the adverse drug events, while EMR keeps tracks all the medical info for patients. Nevertheless, these maximum myopathy risk estimates are strikingly high, comparing to the background myopathy risk estimated from the constant risk model (q₀), which are 0.11 and 0.07 in EMR and FAERS, respectively.

Due to the accuracy of the high dimensional drug interactions that detected by our models can be improved by combining the FAERS database and EMR database [39], all the myopathy associated 2-way to 6-way drug interactions are validated between two databases. Based on an lfdr threshold of 0.00001, we further select six 6-way drug combinations among atorvastatin, simvastatin, ondansetron, escitalopram, omeprazole, venlafaxine, zolpidem, promethazine, acetaminophen, hydrocodone, alprazolam, oxycodone, duloxetine, tramadol. Among these drugs, only ondansetron does not have myopathy side effect in SIDER database [34]. It should also be noted that a number of the drugs identified in our six-way drug combinations may also be used to treat pain associated with myopathy (e.g. acetaminophen, hydrocodone, oxycodone, tramadol). Since the FAERS database does not distinguish between drugs taken prior to the diagnosis of myopathy and those taken after the diagnosis of myopathy, we do not know whether these drugs lead to the myopathy event or if they were given as treatment for the event. However, the EMR database is capable of separating drugs prescribed before the myopathy event from those administered as treatment for myopathy. As this data set also supports the correlation between pain relievers and myopathy, the co-administration of these drugs is likely to be associated with increased risk of myopathy. Considering their 6-way drug interaction induced myopathy risk ranging from 0.38 to 0.76 in two databases, it is essential to recognize that these risks are 3.45 to 10.85 fold higher than the background risk. Therefore, for the first time, in population level (i.e. considering all the drug combinations) and individual level (i.e. drug combination specific), our newly proposed drug-count response models characterize and select the high dimensional drug interactions and estimate their myopathy risks. Our follow-up simulation studies further show the consistency of parameters and lfdr estimates.

FMDRM-MRT and CMDRM-MRT, however, have not been able to incorporate the other confounding variables in the current mixture model framework. Therefore, the interpretation of the data needs to be cautious before it can be done. As the number of drug combinations will increase exponentially as the number of drug rise, computation is another limiting factor that we right now can not apply our models to all 1000 plus drugs in EMR and FAERS databases. Another interesting issue is that some patients take extreme high number of co-medications. For examples, we have observed patients who took >90 drugs in the FAERS database. Each of these patients will contribution a great number of drug combinations, and the usage of this patient data will be tremendously out-weight some drug combinations that are only taken by a few patients. This issue needs to be further addressed more carefully and systemically in the future.

Supplementary Material

Supp FigS1

Figure S1. Flowchart for FAERS and EMR data selection.