On the Use of Zero-Inflated and Hurdle Models for Modeling Vaccine Adverse Event Count Data

We compared several modeling strategies for vaccine adverse event count data in which the data are characterized by excess zeroes and heteroskedasticity. Count data are routinely modeled using Poisson and Negative Binomial (NB) regression but zero-inflated and hurdle models may be advantageous in this setting. Here we compared the fit of the Poisson, Negative Binomial (NB), zero-inflated Poisson (ZIP), zero-inflated Negative Binomial (ZINB), Poisson Hurdle (PH), and Negative Binomial Hurdle (NBH) models. In general, for public health studies, we may conceptualize zero-inflated models as allowing zeroes to arise from at-risk and not-at-risk populations. In contrast, hurdle models may be conceptualized as having zeroes only from an at-risk population. Our results illustrate, for our data, that the ZINB and NBH models are preferred but these models are indistinguishable with respect to fit. Choosing between the zero-inflated and hurdle modeling framework, assuming Poisson and NB models are inadequate because of excess zeroes, should generally be based on the study design and purpose. If the study's purpose is inference then modeling framework should be considered. For example, if the study design leads to count endpoints with both structural and sample zeroes then generally the zero-inflated modeling framework is more appropriate, while in contrast, if the endpoint of interest, by design, only exhibits sample zeroes (e.g., at-risk participants) then the hurdle model framework is generally preferred. Conversely, if the study's primary purpose it is to develop a prediction model then both the zero-inflated and hurdle modeling frameworks should be adequate.     

On the use of zero-inflated and hurdle models for modeling vaccine adverse event count data

We compared several modeling strategies for vaccine adverse event count data in which the data are characterized by excess zeroes and heteroskedasticity. Count data are routinely modeled using Poisson and Negative Binomial (NB) regression but zero-inflated and hurdle models may be advantageous in this setting. Here we compared the fit of the Poisson, Negative Binomial (NB), zero-inflated Poisson (ZIP), zero-inflated Negative Binomial (ZINB), Poisson Hurdle (PH), and Negative Binomial Hurdle (NBH) models. In general, for public health studies, we may conceptualize zero-inflated models as allowing zeroes to arise from at-risk and not-at-risk populations. In contrast, hurdle models may be conceptualized as having zeroes only from an at-risk population. Our results illustrate, for our data, that the ZINB and NBH models are preferred but these models are indistinguishable with respect to fit. Choosing between the zero-inflated and hurdle modeling framework, assuming Poisson and NB models are inadequate because of excess zeroes, should generally be based on the study design and purpose. If the study's purpose is inference then modeling framework should be considered. For example, if the study design leads to count endpoints with both structural and sample zeroes then generally the zero-inflated modeling framework is more appropriate, while in contrast, if the endpoint of interest, by design, only exhibits sample zeroes (e.g., at-risk participants) then the hurdle model framework is generally preferred. Conversely, if the study's primary purpose it is to develop a prediction model then both the zero-inflated and hurdle modeling frameworks should be adequate.     

A hierarchical Binomial-Poisson model for the analysis of a crossover design for correlated binary data when the number of trials is dose-dependent

The differential reinforcement of a low-rate 72-seconds schedule (DRL-72) is a standard behavioral test procedure for screening a potential antidepressant compound. The data analyzed in the article are binary outcomes from a crossover design for such an experiment. Recently, Shkedy et al. (2004) proposed to estimate the treatments effect using either generalized linear mixed models (GLMM) or generalized estimating equations (GEE) for clustered binary data. The models proposed by Shkedy et al. (2004) assumed the number of responses at each binomial observation is fixed. This might be an unrealistic assumption for a behavioral experiment such as the DRL-72 because the number of responses (the number of trials in each binomial observation) is expected to be influenced by the administered dose level. In this article, we extend the model proposed by Shkedy et al. (2004) and propose a hierarchical Bayesian binomial-Poisson model, which assumes the number of responses to be a Poisson random variable. The results obtained from the GLMM and the binomial-Poisson models are comparable. However, the latter model allows estimating the correlation between the number of successes and number of trials.     

Sample Size Calculation for Comparing Two Poisson or Negative Binomial Rates in Noninferiority or Equivalence Trials

Poisson and negative binomial models are frequently used to analyze count data in clinical trials. While several sample size calculation methods have recently been developed for superiority tests for these two models, similar methods for noninferiority and equivalence tests are not available. When a noninferiority or equivalence trial is designed to compare Poisson or negative binomial rates, an appropriate method is needed to estimate the sample size to ensure the trial is properly powered. In this article, several sample size calculation methods for noninferiority and equivalence tests have been derived based on Poisson and negative binomial models. All of these methods accounted for potential over-dispersion that commonly exists in count data obtained from clinical trials. The precision of these methods was evaluated using simulations. Supplementary materials for this article are available online.     

Dropouts in longitudinal studies: definitions and models

The widely used distinction of Little and Rubin (1) about types of randomness for missing data presents difficulties in its application to dropouts in longitudinal repeated measurement studies. In its place, a new typology of randomness for dropouts is proposed that relies on using a survival model for the dropout process. In terms of a stochastic process, dropping out is a change of state. Then, the longitudinal measures and dropout processes can be modeled simultaneously, each conditional on the complete previous history of both repeated measures and states. In this context, Poisson regression is used to fit various proportional hazards models, some of which are new, to the dropout process using the longitudinal measurements responses as time-varying covariates. As examples of longitudinal measurement studies displaying nonrandom dropout processes, a dental study of testosterone production in rats and clinical trials for treatment of gallstones and of depression are analyzed.     

Dose as Instrumental Variable in Exposure–Safety Analysis Using Count Models

Confounding bias often occurs in the analysis of the exposure–safety relationship due to confounding factors that have impacts on both drug exposure and safety outcomes. Instrumental variable (IV) methods have been widely used to eliminate or to reduce the bias in observational studies in, for example, epidemiology. Recently applications of IV methods can also be found in clinical trials to deal with problems such as treatment non-compliance. IV methods have rarely been used in pharmacokinetic/pharmacodynamic analyses in clinical trials, although in a randomized trial with multiple dose levels dose may be a powerful IV. We consider modeling the relationship between pharmacokinetics as a measure of drug exposure and risk of adverse events with Poisson regression models and dose as an IV. We show that although IV methods for nonlinear models are in general complex, simple approaches are available for the combination of Poisson regression models and routinely used dose-exposure models. We propose two simple methods that are intuitive and easy to implement. Both methods consist of two stages with the first stage fitting the dose-exposure model; then the fitted model is used in fitting the Poisson regression model in two different ways. The properties of the two methods are compared under several practical scenarios with simulation. A numerical example is used to illustrate an application of the methods.     

Dose as instrumental variable in exposure-safety analysis using count models

Confounding bias often occurs in the analysis of the exposure-safety relationship due to confounding factors that have impacts on both drug exposure and safety outcomes. Instrumental variable (IV) methods have been widely used to eliminate or to reduce the bias in observational studies in, for example, epidemiology. Recently applications of IV methods can also be found in clinical trials to deal with problems such as treatment non-compliance. IV methods have rarely been used in pharmacokinetic/pharmacodynamic analyses in clinical trials, although in a randomized trial with multiple dose levels dose may be a powerful IV. We consider modeling the relationship between pharmacokinetics as a measure of drug exposure and risk of adverse events with Poisson regression models and dose as an IV. We show that although IV methods for nonlinear models are in general complex, simple approaches are available for the combination of Poisson regression models and routinely used dose-exposure models. We propose two simple methods that are intuitive and easy to implement. Both methods consist of two stages with the first stage fitting the dose-exposure model; then the fitted model is used in fitting the Poisson regression model in two different ways. The properties of the two methods are compared under several practical scenarios with simulation. A numerical example is used to illustrate an application of the methods.     

Missing data handling in chronic pain trials

In chronic pain trials, proper handling of missing data due to dropout is an important issue because the dropout rate is high and the study conclusion may depend on the method chosen. The intent-to-treat (ITT) principle usually requires imputations for missing data to include the dropouts as well as completers in the statistical analysis. However, a statistical analysis with imputation might lead to a misinterpretation of clinical data. In chronic pain trials, treatment-related dropouts are clinical outcomes themselves. For example, an early dropout due to toxicity usually indicates a treatment failure, as does a dropout due to lack of efficacy. Problems with traditional methods such as last observation carried forward (LOCF) or baseline observation carried forward (BOCF) are identified especially in the chronic pain setting. Alternative methods, such as continuous responder analysis and two-part model analysis, treating dropouts as clinical events, are introduced with an example of osteoarthritis clinical trial data.     

Flexible parametrization of variance functions for quantal response data derived from counts

Although the Poisson model has been widely used to fit count data, a well-known drawback is that the Poisson mean equals its variance. Many alternative models for counts that are overdispersed relative to Poisson have been developed to solve this issue, including the negative binomial model. In this article, the negative binomial model with a four-parameter logistic mean is proposed to handle these types of counts, with variance that flexibly depends on the mean. Various parameterizations for the variance are considered, including extra-Poisson variability modeled as an exponentiated B-spline. Thus, the proposed model ably captures the leveling off of the mean, i.e., the “lazy-S” shape often encountered for overdispersed dose–response counts, simultaneously taking into account both overdispersion and natural mortality. Two real datasets illustrate the merits of the proposed approach: media colony counts after tuberculosis decontamination, and the number of monkeys killed by Ache hunters over several hunting trips in the Paraguayan tropical forest.     

HIV Viral Dynamic Models With Censoring and Informative Dropouts

HIV viral dynamic models have received much interest in the literature in recent years. These models are useful for modeling the viral load trajectories during an anti-HIV treatment and for evaluating the efficacy of the treatment. In AIDS studies, patients may drop out of the study early due possibly to drug side-effects, and viral load measurements often have a lower limit of detection. Statistical analyses are therefore complicated by the censoring and dropouts in the data. We propose a joint likelihood method which addresses censoring and dropouts in a mixed effects model simultaneously. A real AIDS dataset is analyzed, and a simulation is conducted to evaluate the proposed method.     

Modeling change trajectories with count and zero-inflated outcomes: Challenges and recommendations

The goal of this article is to describe models to examine change over time with an outcome that represents a count, such as the number of alcoholic drinks per day. Common challenges encountered with this type of data are: (1) the outcome is discrete, may have a large number of zeroes, and may be overdispersed, (2) the data are clustered (multiple observations within each individual), (3) the researchers needs to carefully consider and choose an appropriate time metric, and (4) the researcher needs to identify both a proper individual (potentially nonlinear) change model and an appropriate distributional form that captures the properties of the data. In this article, we provide an overview of generalized linear models, generalized estimating equation models, and generalized latent variable (mixed-effects) models for longitudinal count outcomes focusing on the Poisson, negative binomial, zero-inflated, and hurdle distributions. We review common challenges and provide recommendations for identifying an appropriate change trajectory while determining an appropriate distributional form for the outcome (e.g., determining zero-inflation and overdispersion). We demonstrate the process of fitting and choosing a model with empirical longitudinal data on alcohol intake across adolescence collected as part of the National Longitudinal Survey of Youth 1997.     

Conditional modeling of antibody titers using a zero-inflated poisson random effects model: application to Fabrazyme<Superscript>®</Superscript>

Patients that are exposed to biotechnology-derived therapeutics often develop antibodies to the therapeutic, the magnitude of which is assessed by measuring antibody titers. A statistical approach for analyzing antibody titer data conditional on seroconversion is presented. The proposed method is to first transform the antibody titer data based on a geometric series using a common ratio of 2 and a scale factor of 50 and then analyze the exponent using a zero-inflated or hurdle model assuming a Poisson or negative binomial distribution with random effects to account for patient heterogeneity. Patient specific covariates can be used to model the probability of developing an antibody response, i.e., seroconversion, as well as the magnitude of the antibody titer itself. The method was illustrated using antibody titer data from 87 male seroconverted Fabry patients receiving Fabrazyme^®. Titers from five clinical trials were collected over 276 weeks of therapy with anti-Fabrazyme IgG titers ranging from 100 to 409,600 after exclusion of seronegative patients. The best model to explain seroconversion was a zero-inflated Poisson (ZIP) model where cumulative dose (under a constant dose regimen of dosing every 2 weeks) influenced the probability of seroconversion. There was an 80% chance of seroconversion when the cumulative dose reached 210 mg (90% confidence interval: 194–226 mg). No difference in antibody titers was noted between Japanese or Western patients. Once seroconverted, antibody titers did not remain constant but decreased in an exponential manner from an initial magnitude to a new lower steady-state value. The expected titer after the new steady-state titer had been achieved was 870 (90% CI: 630–1109). The half-life to the new steady-state value after seroconversion was 44 weeks (90% CI: 17–70 weeks). Time to seroconversion did not appear to be correlated with titer at the time of seroconversion. The method can be adequately used to model antibody titer data.     

Expression of P-glycoprotein in killifish (Fundulus heteroclitus) exposed to environmental xenobiotics

P-glycoproteins (P-gp) are transmembrane efflux flippases that prevent the cellular accumulation of moderately hydrophobic compounds and are responsible for certain multidrug resistance phenotypes in tumor cell lines and human patients. We investigated whether P-gps could be involved in a contaminant resistant phenotype observed in a population of fish exposed over generations to high levels of planar halogenated aromatic hydrocarbons (PHAHs). Hepatic and intestinal epithelial P-gp expression was examined by immunoblot and immunohistochemistry in killifish (Fundulus heteroclitus) from New Bedford Harbor, MA (NBH), a Superfund site highly contaminated with PHAHs, and from Scorton Creek on Cape Cod, MA (SC), a relatively unpolluted site. The NBH population has developed resistance to the toxicity of PHAHs. Hepatic P-gp levels were more than 40% greater in fish freshly collected from SC than in fish freshly collected from NBH. When killifish from either site were maintained in clean water for up to 78 days to permit depuration of bioaccumulated contaminants, hepatic P-gp levels decreased approximately 50% by day 8. P-glycoprotein expression was detected in the intestinal epithelium in 55% of freshly collected NBH fish. However, depurated NBH fish and freshly caught and depurated SC fish rarely expressed P-gp in the intestine. In an effort to determine whether environmental chemicals at the two sites might contribute to altered P-gp expression, depurated fish were exposed either to sediment collected from SC or 2,3,7,8-tetrachlorodibenzofuran, a contaminant found at the NBH site and a model aryl hydrocarbon receptor agonist. Neither exposure affected hepatic P-gp levels in killifish. Elevated intestinal P-gp in NBH fish might counter the absorption of P-gp substrates/inducers and thus limit the amount of these compounds reaching the liver, which might account for the lower hepatic P-gp levels in NBH fish compared to SC fish. The differences in hepatic P-gp levels (SC>NBH) and intestinal P-gp (NBH>SC) in freshly collected fish also might reflect environmental exposure to different anthropogenic contaminants or microbial, algal, plant or other natural products via the water column, sediment, or diet at each site.     

PROBLEMATIC FORMULATIONS OF SAS PROC.MIXED MODELS FOR REPEATED MEASUREMENTS

The work reported in this article was undertaken to evaluate the utility of SAS PROC.MIXED for testing hypotheses concerning GROUP and TIME × GROUP effects in repeated measurements designs with dropouts. If dropouts are not completely at random, covariate control over informative individual differences on which dropout data patterns depend is widely recognized to be important. However, the inclusion of baseline scores and time-in-study as between-subject covariates in an otherwise well formulated SAS PROC.MIXED model resulted in inadequate control over type I error in simulated data with or without dropouts present. The inadequate model formulations and resulting deviant test sizes are presented here as a warning for others who might be guided by the same information sources to employ similar model specifications when analyzing data from actual clinical trials. It is important that the complete model specification be provided in detail when reporting applications of the general linear mixed-model procedure. A single random-coefficients model produced appropriate test sizes, but it provided inferior power when informative covariates were added in the attempt to adjust for dropouts. As an alternative, the incorporation of covariate controls in simpler two-stage endpoint or random regression analyses is documented to be effective in dealing with dropouts under specifiable conditions.     

Isolation and characterization of two cytochrome P450 aromatase forms in killifish (Fundulus heteroclitus): differential expression in fish from polluted and unpolluted environments

Populations of killifish (Fundulus heteroclitus) persist in many different highly polluted environment indicative of adaptation or tolerance. In this study, we sought to determine whether long term, multigenerational exposures to environmental contaminants has affected reproductively relevant genes and biological processes. A homology cloning strategy was used to isolate the killifish cytochrome P450 aromatase (P450arom, estrogen synthetase) cDNAs. Consistent with previous fish studies, killifish were found to have two P450arom forms, which segregated into A- and B-gene clades and were differentially expressed in brain (B > A) and ovary (A > B). Comparison of killifish from highly polluted (New Bedford Harbor, NBH) and unpolluted (Scorton Creek, SC) environments revealed no site-related differences in P450arom coding sequences or in overall tissue distribution patterns. As measured by real-time quantitative PCR (QPCR) analysis, however, P450arormB (a known marker of estrogen effect) was approximately two-fold higher in the brain of NBH than of SC fish, a difference seen in reproductively active and inactive males and females. Providing further evidence of exposure to estrogen-like pollutants or metabolites in NBH, vitellogenin (vtg) mRNA and protein were elevated in seasonally active and inactive males, and in reproductively inactive females, when compared to SC fish. By contrast, during the period of reproductive activity, NBH females had a lower gonadosomatic index, lower plasma estrogen, a decreased hepatosomatic index, and reduced vtg expression as compared to SC females, indicating that the female hypothalamic-pituitary-gonadal (HPG)-liver axis is impaired in the polluted environment. As measured by a decrease in plasma androgen (but not GSI), the male HPG axis was impaired in reproductively active NBH versus SC fish. In agreement with reports that NBH killifish are resistant to dioxin-like chemicals (DLC) that activate arylhydrocarbon receptor (AhR) signaling, ovarian P450aromA (a marker of dioxin-like effect in zebrafish embryos) did not differ in SC and NBH fish. In conclusion, the killifish population at the NBH Superfund site maintains a level of reproductive competence in the face of evidence of exposure to estrogen-like pollutants and endocrine disruption.     

A Case Study of Modeling and Exposure-Response Prediction for Count Data

Even with two doses of an experimental drug in Phase III studies, with the commonly used approach for assessing treatment effects of individual doses, it may still be difficult to determine the final commercial dose. In such a scenario, with plasma concentration data collected in the studies, a modeling approach can be applied to predict treatment effects at different plasma concentration levels. Through an established relationship between plasma concentration and dose, the treatment effects of doses not studied in the Phase III studies can then be predicted. The results can further be applied to justify the final dose confirmation or selection. In this article, a Phase III program example with count data as the primary endpoint in the multiple sclerosis area is used to illustrate the application of such a technique for dose confirmation. Several models, such as the overdispersion Poisson model, negative binomial model, and recurrent event models, are considered. The negative binomial model is preferable due to better data fitting and the capability of within-treatment assessment and between-treatment comparison.     

A Generalized Continual Reassessment Method for Two-Agent Phase I Trials

Numerous models have been suggested for Phase I adaptive designs for identifying the maximum tolerated combination (MTC) of two agents. However, these designs have yet to be adopted as the standard approach to use in actual clinical trials, which we posit is mostly due to the complexity of the models that are used. Given that the continual reassessment method (CRM) is gradually being adopted as a standard for single-agent Phase I trials, we propose a generalized version of the CRM, which we denote by gCRM, in hopes of providing such a standard for two-agent trials. For each dose of one agent, we apply the traditional CRM to study doses of the other agent; each of these CRM designs assumes the same dose-toxicity model, as well as the value of the parameter used in the model. However, each model includes a second parameter that varies among the models in an effort to allow flexibility when modeling the probability of dose-limiting toxicity (DLT) of all combinations, yet borrow strength among neighboring combinations as well. We incorporate an adaptive Bayesian algorithm to sequentially assign each patient to the most appropriate dose combination, as well as focus patient assignments to a dose combination that has a DLT probability closest to a prespecified target rate. We test the performance of our method via extensive simulations in various scenarios that are likely to arise in two-agent Phase I trials. We also directly compare the operating characteristics of our model to the alternate published models.     

Responder cell frequency estimation and binomial three-level nonlinear mixed effects model in limiting dilution assays

Responder cell frequencies (RCF), which describe vaccine-boosted immune responses in herpes zoster (HZ) prevention studies, have been estimated by using limiting dilution assays (LDA). The theoretical linearity assumption between the logarithm of the proportion of nonresponding wells (s) and the cell concentration (N) (or dilution level) in LDA, based on the single-hit Poisson model, is often violated with observed data resulting in biased estimates of RCF. In this article, the Poisson assumption is modified by applying a mixture of Poisson and gamma distributions, resulting in a negative binomial assumption, which presents a better fit between s and N. In LDA for HZ prevention studies, binary responses (responder or non-responder wells) are measured repeatedly at different cell concentrations and over time. To account for the correlation between responses to varying dilution levels from individuals, and the correlation between repeated assays of individuals over time simultaneously, a binomial three-level nonlinear mixed-effects model is proposed. For parameter estimation, a maximum likelihood method is applied via adaptive Gaussian quadrature. There is a lack of non-Gaussian multilevel nonlinear mixed-effects software, which can execute such a complicated fit. In this article, an algorithm for the three-level nonlinear mixed-effects model, which can be inserted into the code in the SAS procedure NLMIXED, is suggested.     

Problematic formulations of SAS PROC.MIXED models for repeated measurements

The work reported in this article was undertaken to evaluate the utility of SAS PROC.MIXED for testing hypotheses concerning GROUP and TIME x GROUP effects in repeated measurements designs with drop-outs. If dropouts are not completely at random, covariate control over informative individual differences on which dropout data patterns depend is widely recognized to be important. However, the inclusion of baseline scores and time-in-study as between-subject covariates in an otherwise well formulated SAS PROC.MIXED model resulted in inadequate control over type I error in simulated data with or without drop-outs present. The inadequate model formulations and resulting deviant test sizes are presented here as a warning for others who might be guided by the same information sources to employ similar model specifications when analyzing data from actual clinical trials. It is important that the complete model specification be provided in detail when reporting applications of the general linear mixed-model procedure. A single random-coefficients model produced appropriate test sizes, but it provided inferior power when informative covariates were added in the attempt to adjust for dropouts. As an alternative, the incorporation of covariate controls in simpler two-stage endpoint or random regression analyses is documented to be effective in dealing with dropouts under specifiable conditions.