A Bayesian Hierarchical Model for Characterizing the Diffusion of New Antipsychotic Drugs

Summary:

New prescription medications are a primary driver of spending growth in the United States. For patients with severe mental illnesses, second generation antipsychotic (SGA) medications feature prominently. However, many SGAs are costly, particularly before generic entry, and some may increase the risk of diabetes. Because physicians play a prominent role in new prescription adoption, understanding their prescribing behaviors is policy-relevant. Several features of prescription data, such as different antipsychotic choice sets over time, variable physician prescription volumes, and correlation among drug choices within physicians, complicate inferences. We propose a multivariate Bayesian hierarchical model with piecewise random effects to characterize the diffusion of new antipsychotic drugs. This model captures the complex prescriber-specific relationships among the different diffusion processes and takes advantage of the Bayesian paradigm to quantify uncertainty for all parameters straightforwardly. To evaluate the prescribing patterns for each physician, we propose various indices to identify early new SGA adopters. A sample of nearly 17,000 U.S. physicians whose antipsychotic drug prescribing information was collected between January 1, 1997 and December 31, 2007 illustrates the methods. Determinants of high prescription rates and adoption speeds of new SGAs included physician sex, age, hospital affiliation, physician specialty, and office location. Large within- and between-provider variations in prescribing patterns of new SGAs were identified. Early adopters for one drug were not early adopters for another drug.

1. Introduction

New medical technologies, including new prescription medications, are a primary driver of spending growth in the United States, with physicians playing a pivotal role in their adoption. Studies of physician prescribing behavior indicate that the placement of a particular physician on a diffusion curve for one drug does not necessarily predict where that physician falls for other drugs (Anderson et al., 2018). Innovations in prescription drugs to treat chronic diseases, such as diabetes, heart disease, asthma, and mental illness, have been particularly welcomed. For patients with severe mental illnesses, second generation antipsychotic (SGA) medications feature prominently in their treatment. Despite their clinical importance, many SGAs are costly, particularly before generics entered the U.S. market in 2008, and some have controversial value related to increasing cardiovascular morbidity. Second generation antipsychotics differ from one another in route of administration, dosing, side effect profile, among other features. In the last two decades dramatic changes in the use of SGAs, most of which are financed by public programs in the U.S., such as Medicaid and Medicare, have been observed. Between 1999 and 2005, antipsychotic prescriptions per Medicaid beneficiary increased 30% (Slade and Simoni-Wastila, 2015). Few studies have examined how physician characteristics impact the adoption speed across several new, therapeutically-similar drugs over time.

1.1. Motivating example

Our work is motivated by the desire to understand new antipsychotic drug prescribing behaviors using dispensing information for nearly 17,000 U.S. physicians between January 1, 1997 and December 31, 2007. Characterizing adoption patterns of new SGAs requires addressing several complex features of prescription data. During this period three new brand-name SGAs were introduced into the U.S. market: quetiapine in October 1997, ziprasidone in March 2001, and aripiprazole in November 2002. At the same time, several other SGA drugs were already available including clozapine, risperidone, olanzapine, and several first generation antipsychotic drugs. Thus, the antipsychotic drug choice sets vary over our study period. Physicians may substitute one antipsychotic drug with another or may supplement the new antipsychotic to their choice set. Figure 1 illustrates the monthly market shares of each second generation antipsychotic among all filled antipsychotics prescribed by a randomly selected physician. We have grouped first generation antipsychotic drugs and included them in the figure. Several features in Figure 1 are noteworthy. First, very few prescriptions for clozapine were filled regardless of month. Second, by the end of our observation period, the monthly market share of first generation antipsychotic drugs filled (as a share of all antipsychotics) declined from 92% to 34%. Third, olanzapine (an SGA) prescription fills increased and then decreased over the 10-year period.

Figure 1.

Antipsychotic prescription shares, by drug, for a randomly sampled prescriber. The y-axis depicts the monthly share of specific antipsychotic prescription fills relative to all antipsychotic prescription fills during the month; the x-axis reflects the number of months since January 1997. Total antipsychotic prescription fills for the prescriber is 10,863. Data from IQVIA Xponent™.

For each physician, we know whether a particular drug was filled, and if so, the month and the number of prescriptions filled. Because the new SGAs were introduced at different points during the observation period, the number of months observed for each new SGA prescription once launched varied from 62 (aripiprazole) to 123 (quetiapine). Most drugs reach a saturation level and some may leave the market entirely. Thus, the diffusion of new SGAs through the market may not have reached saturation levels for all new SGAs we study. The average share of a physician’s total antipsychotic prescription fills for the new SGAs over the 10-year period was 19% for quetiapine, while the maximum share observed was 84% (Web Appendix A). The total volume of filled antipsychotic prescriptions written by physicians also varied greatly, ranging from 143 to 105,300, with 50% of physicians with more than 4,622 filled antipsychotic prescriptions. The amount of information for each physician thus also varies. Virtually all physicians prescribed quetiapine while 5% and 3% did not prescribe ziprasidone and aripiprazole respectively. Finally, given the dependence structure of the dispensing information within physicians, it is computationally challenging to model prescription data for all antipsychotic drugs from all physicians simultaneously.

1.2. Existing approaches for modeling diffusion of new drugs

Several models have been proposed to characterize the speed and extent of adoption of new medications among physicians and to identify physicians’ characteristics associated with early versus late adoption. Huskamp et al. (2013) estimated drug-specific Cox proportional hazards model to study the association between time to first adoption of nine SGAs and physicians’ characteristics but ignored between-drug within-physician dependence. Donohue et al. (2016) proposed a method to construct a composite measure of physician adoption of three SGAs through a shared frailty model of the time to the first prescription for each antipsychotic. The frailty model accommodates the correlation among several antipsychotics through the introduction of a random physician effect. While use of overall time to first adoption provides information about early innovators, it masks the substantial heterogeneity in physicians’ longitudinal prescription patterns and hence provides only a partial understanding of the diffusion processes of new medications. Lo-Ciganic et al. (2016) developed a group-based trajectory model of log-transformed dabigatron (an anticoagulant) prescription counts to identify differential adoption trajectories of individual physician monthly shares, and to characterize physician subgroups more likely to follow certain trajectories. While a valuable tool to understand heterogeneity in how drugs and other new medical technologies diffuse among physicians, the model is restricted to one drug and the log-transformation complicates parameter interpretation. A different approach is required to characterize the relationship among the diffusion processes of several drugs.

In this paper, we propose a Bayesian hierarchical multivariate Poisson regression model to characterize the diffusion processes for three new SGAs that describes the diffusion curve for each physician and antipsychotic, accommodates the complex correlation structure among the three diffusion processes, and provides effect estimates of prescriber characters on diffusion. For each antipsychotic diffusion curve, the proposed model uses the monthly number of prescriptions for the antipsychotic as the outcome and the monthly total number of prescription fills of all antipsychotics prescribed by the physician as an offset. Each physician-specific prescription rate for each antipsychotic is modeled as a piecewise linear function added to a smoothing antipsychotic-specific baseline function (e.g. penalized spline) of time. The physician-specific piecewise random effects of all three diffusion processes are assumed to be correlated so that we capture the dependence among the antipsychotic diffusion processes within physician, and these effects are assumed to follow a multivariate normal distribution. Physicians’ characteristics are included in the model of random effects to determine which set of characteristics is related to adoption rates. We use Markov Chain Monte Carlo (MCMC) methods to fit the model and to obtain posterior inferences. Posterior predictive checks are conducted for assessing model fit.

We also propose three classes of indices to identify early versus late new SGA adopters. These indices are based on posterior tail probabilities of relevant model parameters that indicate the degree of early adoption by a prescriber. We estimate our indices to profile prescribers on the basis of time to a fixed number of prescriptions for a new antipsychotic, such as time to the 5% cumulative share of filled prescriptions, and share of prescriptions of a particular new SGA in a given time interval. such as the share of prescription fills in the first six months after new drug introduction. We further use these indices to (1) evaluate the diffusion process of one existing antipsychotic before and after an introduction of new antipsychotic; (2) compare the diffusion processes between existing antipsychotics with respect to a new antipsychotic; and (3) identify which subgroup of physicians are early adopters for all three new antipsychotics.

The proposed method overcomes several limitations in the existing approaches: (1) it can simultaneously characterize the speed and extent of adoption of new antipsychotics among prescribers; (2) because it is a joint model for all new antipsychotics, the correlation among many diffusion processes can be estimated; (3) the monthly number of prescription fills is modeled using Poisson regression model, a generalized linear model, that provides a straightforward interpretation of the coefficients for policy makers; (4) the proposed hierarchical model applies a smooth function to capture the complicated mean structure of time; and (5) the model facilitates estimation of indices to identify early adopters for each antipsychotic and an overall antipsychotic early adopter.

The remainder of the article is organized as follows. Section 2 presents the Bayesian hierarchical multivariate Poisson regression model. Section 3 describes the measures of early adoption. Section 4 applies the proposed model and indices to identify early adopters for the three new SGAs in our cohort. Section 5 summarizes our methods and discusses the implications of our proposed method.

2. Modeling and Inference

2.1. Model Specification

Suppose that i denotes prescriber and j denotes monthly observation. We assume that there are N prescribers in the study and a total study duration of T months. We assume that the ith prescriber has n_ik monthly observations for the kth new antipsychotic (k = 1 for quetiapine, k = 2 for ziprasidone, and k = 3 aripiprazole). Let y_ijk denote the number of prescriptions for the kth new antipsychotic associated with the ith prescriber at study month t_ij for each monthly observation j = 1, …, n_ik for each prescriber i = 1, …, N. We assume that y_ijk follows a Poisson distribution with parameter λ_ijk = E(y_ijk) arising from a joint Poisson regression model of the form

$${\lambda }_{ij1}=O_{ij}\exp \left(f_1\left(t_{ij}\right)+b_{i10}+b_{i11}{\delta }_{11}\left(t_{ij}\right)+b_{i12}{\delta }_{12}\left(t_{ij}\right)+b_{i13}{\delta }_{13}\left(t_{ij}\right)\right)$$ (1)

$${\lambda }_{ij2}=O_{ij}\exp \left(f_2\left(t_{ij}\right)+b_{i20}+b_{i21}{\delta }_{21}\left(t_{ij}\right)+b_{i22}{\delta }_{22}\left(t_{ij}\right)\right)$$ (2)

$${\lambda }_{ij3}=O_{ij}\exp \left(f_3\left(t_{ij}\right)+b_{i30}+b_{i31}{\delta }_{31}\left(t_{ij}\right)\right),$$ (3)

where the offset term O_ij is the total number of prescription fills of all antipsychotic drugs for the ith prescriber in month t_ij and f_k(t_ij) is a smooth antipsychotic-specific baseline function of time.

As depicted in Figure 2, we divide the observation period into three segments based on the introduction dates of ziprasidone, τ, and aripiprazole, ψ. The terms b_i10, b_i11, b_i12 and b_i13 represent the prescriber-level random intercept and three period-specific slopes for the diffusion of quetiapine. The functions δ₁₁(t), δ₁₂(t) and δ₁₃(t) represent indicator functions such that

$${\delta }_{11}(t)=\{\begin{array}{ll}t\hfill & 0<t⩽\tau \hfill \\ \tau \hfill & \tau <t\hfill \end{array},{\delta }_{12}(t)=\{\begin{array}{ll}t-\tau \hfill & \tau <t⩽\psi \hfill \\ \psi -\tau \hfill & \psi <t\hfill \\ 0\hfill & t⩽\tau \hfill \end{array},{\delta }_{13}(t)=\{\begin{array}{ll}t-\psi \hfill & \psi <t\hfill \\ 0\hfill & t⩽\psi \hfill \end{array}.$$

Similarly, b_i20, b_i21 and b_i22 represent the prescriber-level random intercept and two period-specific slopes for the diffusion of ziprasidone, and the indicator functions δ₂₁(t) and δ₂₂(t) represent

$${\delta }_{21}(t)=\{\begin{array}{ll}t-\tau \hfill & \tau ⩽t⩽\psi \hfill \\ \psi -\tau \hfill & \psi <t\hfill \end{array},{\delta }_{22}(t)=\{\begin{array}{ll}t-\tau \hfill & \psi <t\hfill \\ 0\hfill & \tau ⩽t⩽\psi \hfill \end{array}.$$

Lastly, b_i30 and b_i31 represent the prescriber-level random intercept and one period-specific slope for the diffusion curve of aripiprazole, and the indicator function δ₃₁(t) represents

$${\delta }_{31}(t)=\{\begin{array}{ll}t-\psi \hfill & \psi <t\hfill \\ 0\hfill & t=\psi \hfill \end{array}.$$

The piecewise random effects capture the complex physician-level relationship among the three diffusion processes, e.g., examination of whether the introduction of aripiprazole slows the growth of ziprasidone but accelerates the growth of quetiapine.

Figure 2.

Representation of model. The y-axis represents the percentage of fills for a specific drug relative to all antipsychotic fills in the month; the x-axis represents months from October 1997, the date of market introduction of quetiapine, and is divided into 3 segments: October 1997 to March 2001 when zipradidone was introduced; post-March 2001 to November 2002 when aripiprazole was introduced; and post-November 2002 to December 2007, the end of our observational period. Prescriber-specific random intercepts include b₁₀, b₂₀, and b₃₀ and period-specific slopes, b_kp; k = 1, 2, 3; p = 1, 2.

We further assume that the vector of prescriber-specific random effects

$${\mathit{b}}_i={\left(b_{i10},b_{i11},b_{i12},b_{i13},b_{i20},b_{i21},b_{i22},b_{i30},b_{i31}\right)}^{\top }~\mathcal{N}\left(\mathbf{\Gamma }{\mathit{w}}_i^{\top },{\mathbf{\Sigma }}_b\right),$$ (4)

where $\mathcal{N}$ denotes a multivariate Normal distribution, w_i represents the q-dimensional vector of prescriber-level characteristics, Γ the corresponding 9×q matrix of regression coefficients, and Σ_b is the variance-covariance matrix. The diagonal elements of the 9 × 9 matrix Σ_b characterize variation across physicians in prescription patterns at the date of introduction and in each period defined by the introduction dates τ and ψ, that is unexplained by physician covariates included in w. The off-diagonal elements of Σ_b permit covariation among the nine piecewise random effects across the prescribers. These parameters quantify prescribing behaviors such as substitution or addition of drugs.

To accommodate nonlinear structure in the mean diffusion curve, the antipsychotic-specific smooth function f_k(t_ij) is modeled by Bayesian penalized spline with low-rank thin-plate bases (Ruppert et al., 2003). The low-rank thin-plate representation of a smooth function is

$$f_k\left(t_{ij}\right)={\alpha }_{k0}+{\alpha }_{k1}{t}_{ij}+\sum _{l=1}^{L_k}{\xi }_{kl}{\left|t_{ij}-{\nu }_{kl}\right|}^3,$$ (5)

where α_k = (α_k0, α_k1)^⊤ is a vector of fixed regression coefficients, ${\mathit{\xi }}_k={\left({\xi }_{k1},\dots ,{\xi }_{k{L}_k}\right)}^{\top }$ is a vector of coefficients associated with the thin-plate bases, and ${\nu }_{k1}<\cdots <{\nu }_{k{L}_k}$ are fixed knots. To avoid potential non-identifiability issues, we set α_k0 = 0, for k = 1, 2, 3. We set ν_kl at the l_k/(L_k +1) sample of quantile of t_ij’s, with L_k = min(max_i(n_ik), 35) (Ruppert, 2002). While other options are available for specifying the spline functions (Ruppert et al., 2003), this structure balances flexibility, computational stability, and computational convenience.

Let $\mathcal{D}=\left\{{\mathcal{D}}_{ijk}\right\}$ where ${\mathcal{D}}_{ijk}$ denote the observed data at observation j, prescriber i for the kth new antipsychotic. The likelihood function for the model implied by Equation (1) - Equation (3) involves the parameters (α_k, Γ, ξ_k, Σ_b). Integrating over the prescriber-specific random effects b in Equation (4), the observed likelihood can be written as

$$L\left(\alpha ,\mathbf{\Gamma },\xi ,{\mathbf{\Sigma }}_b|\mathcal{D}\right)=\int \prod _{i=1}^N\prod _{k=1}^3\prod _{j=1}^{n_{ik}}{\lambda }_{ijk}^{y_{ijk}}\exp \left(-{\lambda }_{ijk}\right){\left(y_{ijk}!\right)}^{-1}f\left({\mathit{b}}_i|\mathbf{\Gamma },{\mathbf{\Sigma }}_b\right)d{\mathit{b}}_i.$$ (6)

2.2. Prior Distributions

We assume that α_k, Γ, ξ_k and Σ_b are independent a priori. For each regression coefficient contained in α_k and Γ, we adopt an independent diffuse Normal prior $\mathcal{N}\left(0,{10}^6\right)$. We assume that the coefficients associated with thin-plate spline bases ξ_k are independent Normal-Gamma variables $\mathcal{N}(0,{\sigma }_{\xi }^2)$ with ${\sigma }_{\xi }^{-2}~\mathcal{G}\left({10}^{-6},{10}^{-6}\right)$.

While a common choice for the prior distribution of Σ_b is the inverse-Wishart prior distribution, a single parameter controls the precision of all its elements. However, we would like to estimate the nine variance components of the prescriber-specific random effects. Changing the degrees of freedom of the inverse-Wishart prior allows the variance components to be estimated more freely, but at the cost of constraining the correlation parameters. We circumvent this problem by assigning a scaled-inverse-Wishart prior (O’Malley and Zaslavsky, 2008) for the variance-covariance matrix Σ_b. We expand the inverse-Wishart model with a new vector of scale parameters ζ, that is, Σ_b = Diag(ζ)QDiag(ζ), with the unscaled covariance matrix Q being given the inverse-Wishart model: Q ∼ Inv-Wishart₁₀(I), where I is an identity matrix. For each of the scale parameters ζ, we assume independent uniform distributions $\mathcal{U}(0,100)$.

2.3. Posterior Inference

The joint posterior distribution of α, Γ, ξ, and Σ_b conditional on the observed data $\mathcal{D}$ is

$$\pi \left(\alpha ,\mathbf{\Gamma },\xi ,{\mathbf{\Sigma }}_b|\mathcal{D}\right)\propto L\left(\alpha ,\mathbf{\Gamma },\xi ,{\mathbf{\Sigma }}_b|\mathcal{D}\right)\pi (\alpha )\pi (\mathbf{\Gamma })\pi (\xi )\pi \left({\mathbf{\Sigma }}_b\right),$$

where $L\left(\alpha ,\mathbf{\Gamma },\xi ,{\mathbf{\Sigma }}_b|\mathcal{D}\right)$ is defined in Equation (6). We use a Gibbs sampling algorithm, implemented using the JAGS software, to generate samples from the joint posterior distribution. Starting values for the regression coefficients of the multivariate Poisson regression model were obtained by fitting separate Poisson regression models for each new antipsychotic drug. Choices for the starting values of Σ_b had little effect on model estimation and thus we adopted an identity matrix. For each model fitted, multiple parallel chains of length 10,000 were simulated with a burn-in period of length 5,000. Standard MCMC convergence diagnostics such as Gelman-Rubin statistic (Gelman et al., 1992), trace plots, and autocorrelation plots were examined for a small sample of the draws. These diagnostics indicated convergence.

2.4. Model Evaluation

Model fit was assessed using posterior predictive checking (PPC) (Gelman et al., 1996). PPC aims to detect systematic differences between the model and observed data by comparing functions of the observed data with functions of replicated datasets generated from the posterior predictive distribution of the fitted model. The functions of data and model parameters used to compare to the model are called discrepancy variables (Gelman et al., 1996). We examined the cumulative number of prescription fills of each new antipsychotic drug in the first year after a new antipsychotic drug was introduced. This discrepancy variable permits examination of the overall model fit to the entire observation period separated by the two introduction dates of ziprasidone and aripiprazole.

We assessed the sensitivity of inference to the prior distribution of the coefficients associated with thin-plate spline bases ξ_k by comparing inferences when assuming the coefficients are independent Normal-Gamma variables $\mathcal{N}(0,{\sigma }_{\xi }^2)$ with ${\sigma }_{\xi }^{-2}~\mathcal{G}\left({10}^{-6},{10}^{-6}\right)$ to assuming ${\sigma }_{\xi }^{-2}~\mathcal{G}\left({10}^{-2},{10}^{-2}\right)$. We also conducted a small simulation study to characterize bias and coverage of our multivariate model compared to a univariate model.

3. Profiling Prescribers

We propose two types of new antipsychotic adoption measures to characterize filling patterns associated with each prescriber. The first measures characterize time of adoption and amount of prescriptions. For each new SGA, we determine the number of months, $T_{ik}^{*}$, needed to reach a specific number of prescription fills, $y_{ik}^{*}$. Months to first fill corresponds to $y_{ik}^{*}=1$. We also determine the cumulative number of new antipsychotic prescription fills, C_ik, over a fixed period of time, $\mathcal{H}$. For each measure, we compare each physician to a ”standardized” set of physicians defined as physicians having the same characteristics as the physician of interest. The second type of measures compare the adoption rates, b_ka; a ≠ 0, of one time segment relative to another time segment (see Figure 2).

3.1. Time and Amount of Adoption

Following Normand et al. (1997), we determine the expected adoption time (or cumulative amount) for new antipsychotic k for prescriber i as $T_{ik}^A\left(C_{ik}^A\right)$ and the standardized adoption time (cumulative amount) of the antipsychotic k for the prescriber i as $T_{ik}^S\left(C_{ik}^S\right)$. The expected adoption time of antipsychotic k for the prescriber i, $T_{ik}^A$, is defined as the conditional expectation of the prescriber’s average time to a fixed number of prescriptions given the prescriber effect, and determined by identifying the value of $T_{ik}^A$ such that

$$y_{ik}^{*}⩽\sum _{t=1}^{T_{ik}^A}E\left(y_{ijk}|t_i,w_i,\mathbf{\Lambda }\right)=\sum _{t=1}^{T_{ik}^A}E\left(y_{ijk}|t_i,b_i,\alpha ,\xi \right),$$ (7)

where we have emphasized that y is a function of t = (t_ij;j = 1, · · · n_ik) and where Λ = {Γ, α, ξ, Σ_b, b_i; i = 1, …, N} denotes the full set of parameters. The last equality holds because the sampling distribution, f(y_ijk|w_i, Λ), depends only on b_i, α and ξ. In Equation (7), $y_{ik}^{*}=1$ could correspond to the first prescription or $y_{ik}^{*}$ could correspond to the 5% cumulative share of prescriptions. The standardized adoption time of antipsychotic k for prescriber i, $T_{ik}^S$, is the expected average adoption time among the set of physicians comparable to physician i. This standardized time is obtained by averaging the expected outcome over the physician-specific parameters and identifying the value of $T_{ik}^S$ such that

$$y_{ik}^{*}⩽\sum _{t=1}^{T_{ik}^S}\int E\left(y_{ijk}|t_i,w_i,\mathbf{\Gamma },\alpha ,\xi ,{\mathbf{\Sigma }}_b,b_i\right)g\left(b_i|w_i,\mathbf{\Gamma },{\mathbf{\Sigma }}_b\right)d{b}_i=\sum _{t=1}^{T_{ik}^S}E\left(y_{ijk}|t_i,w_i,\mathbf{\Gamma },{\mathbf{\Sigma }}_b,\alpha ,\xi \right).$$ (8)

To determine the number of months, T_ik, needed to reach a specific number of prescription fills, $y_{ik}^{*}$, we adopted the following procedure. First, we estimated the expected and standardized monthly number of prescription fills of one antipsychotic for one prescriber over the study period. Next, we conducted a grid-search beginning with the first month of the study period. This was repeated for each posterior draw of the model parameters to obtain the posterior distribution of the expected number of months.

Similarly, we define the cumulative expected share of antipsychotic k for prescriber i, $C_{ik}^A$, as

$$C_{ik}^A=\sum _{t\in \mathcal{H}}E\left(y_{ijk}|t_i,w_i,\mathbf{\Lambda }\right)/\sum _{t\in \mathcal{H}}{O}_{ij}=\sum _{t\in \mathcal{H}}E\left(y_{ijk}|t_i,b_i,\alpha ,\xi \right)/\sum _{t\in \mathcal{H}}{O}_{ij},$$ (9)

where $\mathcal{H}$ represents a fixed time interval such as the first six months after the entry of a new antipsychotic. The standardized cumulative share of antipsychotic k for prescriber i, $C_{ik}^S$, is defined as

$$C_{ik}^S=\sum _{t\in \mathcal{H}}E\left(y_{ijk}|t_i,w_i,\mathbf{\Gamma },{\mathbf{\Sigma }}_b,\alpha ,\xi \right)/\sum _{t\in \mathcal{H}}{O}_{ij}.$$ (10)

The first type of indices is defined as $T_{ik}^A-T_{ik}^S$ or $C_{ik}^A-C_{ik}^S$, which is the comparison between the average expected adoption time or amount of antipsychotic k for prescriber i and the average expected adoption time or amount of the antipsychotic for a comparable set of prescribers. Small values of $T_{ik}^A-T_{ik}^S$ correspond to faster adoption while large values of $C_{ik}^A-C_{ik}^S$ correspond to more fills. We may calculate the posterior probability that $T_{ik}^A-T_{ik}^S$ is small; if this probability is large, then the prescriber is an early adoptor. Similarly we may determine the posterior probability of $C_{ik}^A-C_{ik}^S$ being large. This motivates the following index. Let $T_k^{A-S}=\left\{T_{1k}^A-T_{1k}^S,\dots ,T_{Nk}^A-T_{Nk}^S\right\}$ denote the vector of deviations between expected and standardized prescriber outcomes. Let B(·) denote a benchmark function of $T_k^{A-S}$. Then

$$P_{ik}^{A-S}=\mathrm{Pr}\left(T_{ik}^A-T_{ik}^S<B\left(T_k^{A-S}\right)|\mathcal{D}\right)$$ (11)

is the posterior probability that the difference between expected and standardized adoption times for prescriber i to have at least $y_{ik}^{*}$ is smaller than some relevant function of the deviations across the sample of prescribers. For example, B(·) could be the 5th percentile. The function B(·) could also be a constant function so that the adoption time of the physician, $P_{ik}^{A-S}$, is determined without reference to the distribution of other prescribers.

We can also sort the differences $T_{ik}^A-T_{ik}^S$ from lowest (negative) to highest (positive), which thus ranks the prescribers by their relative adoption speed. Using the MCMC samples, we can approximate the marginal posterior distribution of the rank for each prescriber, and further derive any statistic of interest. For instance, if we are interested in the probability that a prescriber belongs to a certain quantile for antipsychotic k, we can simply approximate the probability that the ith prescriber is in the corresponding rank interval R_k as

$$\frac{1}{M}\sum _{m=1}^{M}I(\mathrm{rank}{[i]}_k^{(m)}\in R_k)\to \mathrm{Pr}\left(\mathrm{rank}{[i]}_k\in R_k|\mathcal{D}\right)\text{as}M\to \infty ,$$ (12)

where $\mathrm{rank}{[i]}_k^{(m)}$ is the rank of the ith prescriber for antipsychotic k at the mth draw of the associated sampling chain and M is the total number of iterations.

3.2. Adoption Rates

We can further evaluate the diffusion curve of one existing antipsychotic drug or compare the diffusion curves between existing antipsychotic drugs before and after the introduction of a new antipsychotic drug for a prescriber. For example, we can examine (1) whether the diffusion of quetiapine accelerates or decelerates before and after the introduction of ziprasidone through a comparison of the cumulative share of quetiapine prescriptions before and after ziprasidone introduction; and (2) whether the introduction of aripiprazole slows the growth of ziprasidone but increases the growth of quetiapine. Suppose we are interested in investigating the diffusion path of quetiapine before and after the introduction of ziprasidone. The expected difference in adoption speed of quetiapine between pre- and post-introduction of ziprasidone is

$$D_{i,kl}^A=\frac{{\sum }_{t\in {\mathcal{H}}_l^{post}}E\left(y_{ijk}|{\mathit{t}}_i,\mathit{\alpha },\mathit{\xi }\right)}{{\sum }_{t\in {\mathcal{H}}_l^{post}}{O}_{ij}}-\frac{{\sum }_{t\in {\mathcal{H}}_l^{pre}}E\left(y_{ijk}|{\mathit{t}}_i,\mathit{\alpha },\mathit{\xi }\right)}{{\sum }_{t\in {\mathcal{H}}_l^{pre}}{O}_{ij}},$$ (13)

where ${\mathcal{H}}_l^{pre}$ and ${\mathcal{H}}_l^{post}$ represents a fixed time interval before and after the introduction of the lth antipsychotic. A negative value of $D_{i,kl}^A$ represents a decline in the prescriber’s quetiapine share whereas a positive value of $D_{i,kl}^A$ implies an increase in share. The standardized difference in cumulative shares of quetiapine between pre- and post-introduction of ziprasidone is

$$\begin{gathered} D_{i,kl}^S=\frac{{\sum }_{t\in {\mathcal{H}}_l^{post}}\int E\left(y_{ijk}|{\mathit{t}}_i,{\mathit{w}}_i,\mathbf{\Gamma },\mathit{\alpha },\mathit{\xi },\mathbf{\Sigma },{\mathit{b}}_i\right)g\left({\mathit{b}}_i|{\mathit{w}}_i,\mathbf{\Gamma },\mathbf{\Sigma }\right)d{\mathit{b}}_i}{{\sum }_{t\in {\mathcal{H}}_l^{post}}{O}_{ij}} \\ -\frac{{\sum }_{t\in {\mathcal{H}}_l^{pre}}\int E\left(y_{ijk}|{\mathit{t}}_i,{\mathit{w}}_i,\mathbf{\Gamma },\alpha ,\xi ,\mathbf{\Sigma },{\mathit{b}}_i\right)g\left({\mathit{b}}_i|{\mathit{w}}_i,\mathbf{\Gamma },\mathbf{\Sigma }\right)d{\mathit{b}}_i}{{\sum }_{t\in {\mathcal{H}}_l^{pre}}{O}_{ij}}. \end{gathered}$$ (14)

The second class of indices is defined as $D_{i,kl}^A-D_{i,kl}^S$. A negative difference of $D_{i,kl}^A-D_{i,kl}^S$ reflects that the prescriber’s adoption of antipsychotic k is accelerating less than expected or decelerating more than expected after the introduction of antipsychotic l.

Lastly, we can identify subgroups of prescribers who are early adopters for all three new antipsychotics (overall early adopters) using

$$P_{i,\forall k}^{A-S}=\mathrm{Pr}\left(T_{ik}^A-T_{ik}^S<B_k\left({\mathit{T}}_k^{A-S}\right)\forall k|\mathcal{D}\right),$$ (15)

where B_k(·), k = 1, 2, 3, are the “benchmark” functions of the vector of deviations between expected and standardized prescriber outcomes for quetiapine, ziprasidone and aripiprazole, respectively. The posterior probability that ith prescriber is in the pre-specified rank intervals of all antipsychotics is,

$$\mathrm{Pr}\left(\mathrm{rank}{[i]}_k\in R_k\forall k\right)=\frac{1}{M}\sum _{m=1}^{M}I\left(\mathrm{rank}{[i]}_k^{(m)}\in R_k\forall k|\mathcal{D}\right),$$ (16)

where $\mathrm{rank}{[i]}_k^{(m)}$ k = 1, 2, 3, are the ranks of the ith prescriber for quetiapine, ziprasidone, and aripiprazole at the mth draw of the associated sampling chain.

4. Application

4.1. The Data

We use physician-level information from the Xponent™ database, a prescriber-level database maintained by IQVIA Health. IQVIA Xponent™ directly captures over 70% of all U.S. prescriptions filled in retail pharmacies, and uses a patented projection methodology to represent 100% of prescriptions filled in these outlets. The data include monthly counts of prescriptions filled for each antipsychotic. The monthly prescribing information was linked to physician information in the American Medical Association Masterfile in order to select key sub-groups of physicians with more homogeneous patient case-mix. Variables include physician age, sex, specialty, sub-specialty training, office-based vs. hospital-based practice, major non-patient care activity (e.g., administration, research, teaching), percentage of practice time spent in hospital settings, medical school attended and year of graduation, residency program, admitting hospital, and data on retirement and death. The data also identify a practice mailing and office address that we linked to census data. The sample was selected to include physicians having at least 10 prescriptions in each year between 1997 and 2007 inclusive. Thus, we do not capture practice patterns of new physicians, those who retired before 2007, and those who died before 2007. Unfortunately, we have no patient characteristics. The prescribers’ characteristics are summarized in Table 1. All variables listed in Table 1 were included in w_i in the analysis.

Table 1.

Prescriber characteristics for 16,932 prescribers practicing within the U.S. between January 1, 1997 and December 31, 2007. Data from IQVIA Xponent™.

Characteristic	Percent	Characteristic	Percent
Male	74.1	Top 25 Medical School	13.9
Age		Urban Location	91.3
< 50	10.0	Census Area
50–59 yrs	37.8	New England	9.3
⩾ 60 yrs	52.2	East North Central	14.1
Foreign Trained	27.4	West North Central	5.5
Practice Type		Mountain	5.1
Solo or 2-Person	42.4	Pacific	14.9
Group	27.2	Middle Atlantic	21.5
Other	21.7	South Atlantic	17.5
unknown	8.7	East South Central	4.6
No Hospital Affiliation	50	West South Central	7.4
Specialty
Psychiatrist	80.7	Prescription Volume	Mean [SD]
Child Psychiatrist	15.0	No. New Antipsychotics	3547 [3997]
Family/Internal	4.3	No. All Antipsychotics	7243 [8099]

4.2. Model Estimation

Each chain of 10,000 iterations required approximately 50 hours of computing time on a Dell workstation with a 3.6-GHz Intel Core i7 processor for a random sample of 1,500 prescribers in our data. The overall computing time for the sample of more than 15,000 prescribers was too long to obtain valid estimates given limited computing resources. Thus, we divided the overall sample into nine more manageable subsamples based on prescribers’ census area information. The number of prescribers in the region-specific samples ranges from 739 to 3,310. Region-specific estimates were obtained by fitting the proposed model in each of nine region-specific subsamples.

Figure 1 in Web Appendix B displays posterior predictive checks for 1,442 prescribers in the New England area using discrepancy variables that estimate the average cumulative share of filled quetiapine prescriptions over 3 different periods: 1) during the first six months following the introduction of quetiapine; 2) during the first six months after the introduction of ziprasidone; and during the first six months following aripiprazole entrance. These distributions are based on 1,000 draws. Compared to their observed values, the posterior predictive p-values are 0.12, 0.35, and 0.07 for discrepancy variables 1 to 3 respectively. Figure 2 in Web Appendix B displays similar graphs for additional discrepancy variables. Predictive checks are also performed for all new antipsychotic drugs for all prescribers from the remaining census areas (not shown). Our findings suggested no systematic difference existed between the model and observed data.

4.3. Prescriber Characteristics Associated with Diffusion

Table 2 displays posterior means and 95% credible intervals for the regression coefficients relating prescriber characteristics to the random intercepts and slopes for ziprasidone using 1,442 New England prescribers. Several features are noteworthy. First, male prescribers had a higher monthly fill rate relative to female prescribers at market entry: males prescribed ziprasidone nearly twice as often as females. Second, relative to prescribers less than 50 years of age, those older than 60 years were associated with lower ziprasidone fill rates in the month of its entry. Third, those without hospital affiliations prescribed ziprasidone 1.7 times that of those with an affiliation at its market entry but slower after that. Fourth, psychiatrists had twice the monthly fill rate of child psychiatrists at entry of ziprasidone whereas family or internal medicine physicians were 5.7% faster than child psychiatrists after aripiprazole entered the market. Fifth, prescribers in urban areas prescribed ziprasidone half as often as those in urban areas when ziprasidone entered the market. Results for ziprasidone in other census areas are reported in Web Appendix C (see Tables 2–9 and Figures 3–11).

Table 2.

Summary of regression coefficients for ziprasidone (exponential scale) for the 1,442 prescribers in the New England area.

	Introduction, b20			Change in Rate,† b21			Change in Rate,‡ b22
	Est	SE	95% CI	Est	SE	95% CI	Est	SE	95% CI
Male	1.953	0.401	(1.292, 2.862)	0.979	0.010	(0.959, 0.999)	0.999	0.002	(0.994, 1.003)
Age vs < 50
50–59 yrs	0.841	0.240	(0.419, 1.344)	1.009	0.016	(0.980, 1.047)	1.005	0.003	(.998, 1.012)
⩾60 yrs	0.458	0.127	(0.226, 0.713)	1.039	0.016	(1.010, 1.075)	1.002	0.003	(0.996, 1.008)
Foreign Trained	0.770	0.172	(0.479, 1.150)	1.013	0.012	(0.992, 1.038)	1.001	0.002	(0.997, 1.006)
Practice vs < 3-Person
Group	1.276	0.432	(0.620, 2.311)	0.977	0.017	(0.944, 1.011)	1.005	0.004	(0.999, 1.012)
Unknown	0.854	0.189	(0.537, 1.283)	1.006	0.011	(0.985, 1.028)	1.001	0.002	(0.996, 1.006)
Other	0.752	0.176	(0.459, 1.129)	1.011	0.012	(0.987, 1.034)	1.004	0.002	(0.999, 1.009)
No Hospital Affiliation	1.707	0.304	(1.188, 2.384)	0.976	0.009	(0.960, 0.993)	1.000	0.002	(0.996, 1.004)
Specialty vs Child Psychiatrist
Psychiatrist	2.060	0.483	(1.277, 3.162)	0.971	0.011	(0.950, 0.993)	1.003	0.003	(0.998, 1.008)
Family/Internal	0.214	0.663	(0.001, 1.103)	1.003	0.115	(0.828, 1.263)	1.057	0.015	(1.025, 1.087)
Top 25 Medical School	1.418	0.313	(0.908, 2.156)	0.979	0.011	(0.958, 0.999)	1.002	0.002	(0.997, 1.007)
Urban	0.554	0.160	(0.303, 0.919)	1.014	0.015	(0.987, 1.044)	1.003	0.003	(0.998, 1.010)

^†Prior to entry of aripiprazole

^‡after introduction of aripiprazole.

Data from IQVIA Xponent™.

4.4. Between-Prescriber Variation

Table 3 provides the posterior estimates of the standard deviations and correlations of the prescriber-specific piecewise random effects (Σ_b) for the New England area. The estimated covariance matrix of piecewise random effects in other regions are shown in Web Appendix D, Tables 10–17. The estimated correlation matrix reveals both substitution and complementary effects among antipsychotic fills. After ziprasidone entry in March 2001 but prior to aripiprazole entry, prescribers who were more likely to prescribe ziprasidone had an increased rate of monthly fills of quetiapine in most of the regions: the posterior mean correlation coefficient between b₁₂ and b₂₁, $\widehat{\rho }\left(b_{12},b_{21}\right)$, is 0.11 with 95% credible interval (CrI) of (0.03, 0.18) in the West North Central area (Web Appendix D, Table 16). After aripiprazole entry in December 2007, prescribers in the South Atlantic area, who were associated with more quetiapine fills, had a lower monthly rate of ziprasidone prescription fills ($\widehat{\rho }\left(b_{13},b_{22}\right)=-0.12$; 95% CrI = (−0.16, −0.08)) (Web Appendix D, Table 15). Prescribers in the New England area (Table 3), who were associated with more quetiapine fills had a lower aripiprazole fill rate: $\widehat{\rho }\left(b_{13},b_{31}\right)=-0.06$; 95% CrI = (−0.12, −0.00). However, prescribers in the Middle Atlantic area who were more likely to have aripiprazole fills were associated with an increased monthly ziprasidone fill rate during the period: $\widehat{\rho }\left(b_{31},b_{22}\right)=0.05$; 95% CrI = (0.01, 0.09) (Web Appendix D, Table 12).

Table 3.

Standard deviations and correlations of piecewise random effects for the 1,442 prescribers in the New England area: Estimated Posterior Means and 95% Credible Intervals (CIs). Data from IQVIA Xponent™.

		Standard deviation: mean (CI)	Correlation: mean (CI)
			Aripiprazole		Ziprasidone			Quetiapine
			b30	b31	b20	b21	b22	b10	b11	b12	b13
Aripiprazole	b30	1.387 (1.325, 1.449)	1	(−0.82, −0.77)	(0.21, 0.33)	(−0.23, −0.10)	(−0.13, −0.02)	(0.11, 0.22)	(−0.17, −0.05)	(−0.17, −0.05)	(−0.13, −0.02)
	b31	0.026 (0.025, 0.027)	−0.80	1	(−0.26, −0.14)	(0.08, 0.20)	(−0.02, 0.10)	(−0.22, −0.12)	(0.09, 0.20)	(0.04, 0.16)	(−0.12, 0.00)
Ziprasidone	b20	2.935 (2.804, 3.067)	0.27	−0.20	1	(−0.88, −0.85)	(−0.19, −0.07)	(0.14, 0.25)	(−0.19, −0.07)	(−0.15, −0.03)	(−0.16, −0.04)
	b21	0.149 (0.143, 0.156)	−0.17	0.14	−0.86	1	(−0.26, −0.14)	(−0.20, −0.08)	(0.04, 0.16)	(−0.02, 0.10)	(0.02, 0.15)
	b22	0.033 (0.031, 0.034)	−0.07	0.04	−0.13	−0.20	1	(−0.06, 0.05)	(−0.05, 0.07)	(−0.03, 0.08)	(−0.11, 0.01)
Quetiapine	b10	2.495 (2.384, 2.605)	0.17	−0.17	0.19	−0.14	−0.01	1	(−0.91, −0.88)	(−0.16, −0.04)	(−0.14, −0.03)
	b11	0.062 (0.059, 0.065)	−0.11	0.15	−0.13	0.10	0.01	−0.89	1	(−0.29, −0.17)	(−0.13, −0.01)
	b12	0.052 (0.050, 0.054)	−0.11	0.10	−0.09	0.04	0.02	−0.10	−0.23	1	(−0.16, −0.05)
	b13	0.015 (0.014, 0.016)	−0.07	−0.06	−0.10	0.09	−0.05	−0.09	−0.07	−0.10	1

After accounting for prescriber characteristics, substantial between-prescriber variation remained. For example, an aripiprzole prescriber 1 standard deviation above the mean among all prescribers is $16=\exp \left(2\times \sqrt{\mathrm{Var}\left(b_{30}\right)}\right)$ times more likely to prescribe aripiprazole as an aripiprazole prescriber 1 standard deviation below the mean in the New England area. Sensitivity analyses examining the impact of changing the prior specification of the thin-plate spline bases coefficient to ${\sigma }_{\xi }^{-2}~\mathcal{G}\left({10}^{-2},{10}^{-2}\right)$ were consistent with the results in the primary analysis (Tables 18 and 19 in Web Appendix E). Our simulation study suggested that the multivariate hierarchical Poisson model compared to a univariate hierarchical model had smaller absolute bias for regression coefficients and slightly larger coverage (Web Appendix F, Table 20).

4.5. Profiling Prescribers

The posterior mean number of months (posterior standard deviation) from market entry to 5% cumulative share of the new SGAs are: 10.95 (9.27) for quetiapine; 16.56 (17.10) for ziprasidone; and 17.50 (13.59) for aripiprazole. Figure 3 displays the posterior probabilities that the prescriber deviation between expected and standardized months to 5% cumulative share of prescriptions of each antipsychotic is smaller (e.g., faster) than the 5^th percentile of the deviations across all New England prescribers. Prescribers who are early adopters for one antipsychotic drug do not appear be early adopters for another antipsychotic drug. This observation is consistent with the results from the previous studies of physician prescribing behavior indicating that the placement of a particular physician on an adoption curve for one drug does not necessarily predict where the physician falls for other drugs (Anderson et al., 2018). The red triangles in the figure denote prescribers who are among the fastest 1% in adopting all three new antipsychotic drugs. The small proportion of the overall early adopters indicates large variations in adoption patterns of new antipsychotic drugs.

Figure 3.

Posterior probability that the deviation between a prescriber’s expected and standardized number of months to 5% cumulative share of her prescription fills is less than the 5^th percentile of the deviations across all prescribers in the New England Area. The red triangles identifies prescribers who are among the top 1% in posterior probability that the deviation for all three new antipsychotic drugs is small. Data from IQVIA Xponent™.

Figure 12 in Web Appendix G displays the posterior probabilities that the prescriber’s cumulative monthly share of prescription fills for each new antipsychotic drug during the first year after their introduction ranks in the top 5% among New England prescribers. The red triangles denote prescribers who are among the top 1% for all three new antipsychotic drugs. Similar with the findings in Figure 3, we identify the early adopters for one antipsychotic drug who may not be early adopters for another new antipsychotic drug.

Figure 13 Web Appendix G displays the posterior mean deviations between expected and standardized differences in the monthly quetiapine fill rates six months pre (b₁₁) and six months post (b₁₂) ziprasidone market entry, $\left(D_{i,kl}^A-D_{i,kl}^S\right)$, among New England prescribers. Examination of the 95% credible intervals of $D_{i,kl}^A-D_{i,kl}^S$ indicates that the monthly increase in filled quetiapine prescriptions was faster than the standardized rate for 44 prescribers (3.1% among 1,442 prescribers) and slower for 1 prescriber. Post-ziprasidone market entry, 112 prescribers (7.8% among 1,442 prescribers) differed from the standardized adoption rate. Sixty prescribers were slower to adopt and 52 prescribers had monthly decreases in quetiapine larger than the standardized rate.

5. Discussion

We proposed a Bayesian hierarchical multivariate model for characterizing the diffusion of three new antipsychotic drugs. The model allows researchers to capture the complex prescriber-specific longitudinal patterns of relationships among the different diffusion processes and takes advantage of the Bayesian paradigm to quantify uncertainty for all parameters straightforwardly. To evaluate the prescribing patterns for each physician, we used random effects and proposed functions of the random effects to identify early new SGA adopters.

Our work is motivated by a collaboration investigating the speed with which prescribers adopt several new, therapeutically-similar drugs. Toward this, we applied the proposed model to a sample of nearly 17,000 U.S. physicians whose antipsychotic drug fills were collected between January 1, 1997 and December 31, 2007. The results from our analysis indicate policy makers should tailor informational strategies by prescriber characteristics, including their urbanicity of their office locations. The analyses also revealed large variations in prescribing patterns of the new drugs and prescribers who are early adopters for one antipsychotic drug may not be early adopters for another antipsychotic drug.

Our findings have important policy implications given state (Medicaid) and federal (Medicare) programs are the largest purchasers of SGAs. Off-label SGA prescribing in the U.S. has contributed to the growth of SGA use. This is concerning because the risks associated with SGA use compared to first generation antipsychotic use often outweigh the benefits. Our findings suggest utilization management strategies, such as drug formularies that include prescription drugs approved by health insurers or prior authorization requirements, should be targeted to specific states, rural practices, and practice types. Physician educational programs about risks/benefits of SGAs should be directed to specific specialties and instituted early in physician training. Finally, providing prescribers with information regarding how they compare to peer physicians in terms of their prescription trajectories, such as speed and amount prescribed, can promote discussions about best practices.

Our approach has some limitations. Our method involves a growing number of parameters in order to accommodate new drug entrants. In our setting, 3 new entrants resulted in 3 random intercepts and 6 random slopes. Because of the extensive computing time required, we divided the sample into nine manageable subsamples based on census area information. Geographically stratified analyses are sensible in our setting because many studies have documented geographic variations in the utilization of medical services. With that said, we note that direct comparisons of model results between census areas may not be sensible. More computationally efficient sampling techniques will be required to perform estimation and inference for large datasets and this type of complicated Bayesian hierarchical model. We assumed that random effects accounted for within-prescriber and between-drugs-within-prescriber correlations. Serial correlation could be included in addition to the random effects. While we did not include this feature, its addition is conceptually straightforward.

A limitation specific to our application involves the absence of patient characteristics. It is likely that the types of patients filling antipsychotic prescriptions would differ among prescribers, and among drugs within prescribers. For instance, a clear example involves child psychiatrists who generally treat children rather than adults. Including patient characteristics is straightforward methodologically but will be computationally expensive. We also note that many contextual factors may impact new technology adoption to which we do not have access. These factors include, among others, pharmaceutical promotional activities conducted by companies and a prescriber’s network of professional contacts. Methods to combine social network analysis techniques with approaches to modeling multivariate longitudinal count data will be needed in order to provide a more complete understanding of new technology uptake. Finally, our measure of diffusion uses prescriptions that were filled rather than written. This limitation implies we have underestimated written prescriptions and these unfilled prescriptions may be related to patient characteristics.

In conclusion, the proposed model significantly expands the set of statistical tools available to researchers to study diffusion patterns, particularly the relationship among multiple diffusion processes. While our focus has been on new SGA drugs, the proposed model is broadly applicable to diffusion patterns of other medical technologies.

Data Availability Statement

The data that support the findings in this paper are available from IQVIA. Restrictions apply to the availability of these data, which were used under license for this study. Data are available from the authors only with the permission of IQVIA.