Set Of Random Effects Research Articles

Improved inference for areal unit count data using graph-based optimisation

Spatio-temporal count data relating to a set of non-overlapping areal units are prevalent in many fields, including epidemiology and social science. The spatial autocorrelation inherent in these data is typically modelled by a set of random effects that are assigned a conditional autoregressive prior distribution, which is a special case of a Gaussian Markov random field. The autocorrelation structure implied by this model depends on a binary neighbourhood matrix, where two random effects are assumed to be partially autocorrelated if their areal units share a common border, and are conditionally independent otherwise. This paper proposes a novel graph-based optimisation algorithm for estimating either a static or a temporally varying neighbourhood matrix for the data that better represents its spatial correlation structure, by viewing the areal units as the vertices of a graph and the neighbour relations as the set of edges. The improved estimation performance of our methodology compared to the commonly used border sharing rule is evidenced by simulation, before the method is applied to a new respiratory disease surveillance study in Scotland between 2011 and 2017.

Hospital Emergency Room Savings via Health Line S24 in Portugal

Hospital emergency departments are often overused by patients that do not really need urgent care. These admissions are one of the major factors contributing to hospital costs, which should not be allowed to compromise the response and effectiveness of the National Health Services (SNS). The aim of this study is to perform a detailed spatial health econometrics analysis of the non-urgent emergency situations (classified by Manchester triage) by area, linking them with the efficient use of the national health line, the Saude24 line (S24 line). This is evaluated through the S24 savings calls, using a savings index and its spatial effectiveness in solving the non-urgent emergency situations. A savings call is a call by a user whose initial intention was to go to an urgency department, but who. after calling the S24 line. changed his/her mind. Given the spatial nature of the data, and resorting to INLA in a Bayesian paradigm, the number of non-urgent cases in the Portuguese urgency hospital departments is modeled in an autoregressive way. The spatial structure is accounted for by a set of random effects. The model additionally includes regular covariates and a spatially lagged covariate savings index, related with the S24 savings calls. Therefore, the response in a given area depends not only on the (weighted) values of the response in its neighborhood and of the considered covariates, but also on the (weighted) values of the covariate savings index measured in each neighbor, by means of a Bayesian Poisson spatial Durbin model.

An alternative characterization of MAR in shared parameter models for incomplete longitudinal data and its utilization for sensitivity analysis

Dropout is a common complication in longitudinal studies, especially since the distinction between missing not at random (MNAR) and missing at random (MAR) dropout is intractable. Consequently, one starts with an analysis that is valid under MAR and then performs a sensitivity analysis by considering MNAR departures from it. To this end, specific classes of joint models, such as pattern-mixture models (PMMs) and selection models (SeMs), have been proposed. On the contrary, shared-parameter models (SPMs) have received less attention, possibly because they do not embody a characterization of MAR. A few approaches to achieve MAR in SPMs exist, but are difficult to implement in existing software. In this article, we focus on SPMs for incomplete longitudinal and time-to-dropout data and propose an alternative characterization of MAR by exploiting the conditional independence assumption, under which outcome and missingness are independent given a set of random effects. By doing so, the censoring distribution can be utilized to cover a wide range of assumptions for the missing data mechanism on the subject-specific level. This approach offers substantial advantages over its counterparts and can be easily implemented in existing software. More specifically, it offers flexibility over the assumption for the missing data generating mechanism that governs dropout by allowing subject-specific perturbations of the censoring distribution, whereas in PMMs and SeMs dropout is considered MNAR strictly.

Social Deprivation, Waiting Time for Diagnosis and Therapeutic Management of Patients With Prostatic Cancer in the Somme Area, France

Background: In France, prostate cancer is at the top of the list of the most common cancers in men. The morbidity and mortality of this cancer were found to be related to the geographic level of socioeconomic deprivation with a higher rate of mortality and more frequent aggressive cases among men with low socioeconomic level, this was associated with health disparities in the management of this cancer. Our study region is considered as an economically deprived area with a poverty rate significantly higher than the national average. Aim: The aim of our study was to assess the impact of the socioeconomic level on the incidence, mortality, aggressiveness and management of prostate cancer, using data from a population-based cancer registry. Methods: For this research, prostate cancer data, between 2006 and 2010, were obtained from the Somme area cancer registry. Social economic status was assessed using the European Deprivation Index (EDI). This index has been used to classify each geographical unit (IRIS) according to social deprivation. IRIS is the smallest submunicipal geographical entity for which census data are available. Each prostate cancer case was allocated to the corresponding IRIS by geolocalizing the addresses using geographic information system (GIS). For spatial analysis, hierarchical generalized linear modeling was fitted. To assess for spatial autocorrelation, Moran's I test was conducted and then spatial autocorrelation was modeled by a set of random effects that are assigned a conditional autoregressive (CAR) prior distribution. Results: A total of 2405 incident cases of prostate cancer were registered in the Somme area. The age-standardized rate was 98.2 cases per 100,000 person-years (PY). The standardized mortality rate was 28.1 deaths per 105 PY. The coefficient associated with the EDI obtained from the spatial analysis of prostate cancer incidence was negative (-0.348; 95% CI: −0.0831) which indicates that prostate cancer incidence was more important in the less deprived areas. The relative risk of prostate cancer mortality associated with the quintile 5 of the EDI relatively to quintile 1 was 3.09; 95% CI: [1.70-5.59]. For the aggressiveness, the coefficient associated with the EDI was 0.0493 with a 95% CI: [0.0162-0.0810], and the Q5/Q1 RR was equal to 1.36 95% CI: [1.09-1.73]. EDI estimated coefficient for proportion of cases who received curative treatment versus patients who received palliative treatment was −0.1089, 95 CI%: [−0.1505 to −0.0693]. EDI coefficient for waiting time was not significant. Conclusion: Our study showed a significant association between socioeconomic deprivation and prostate cancer with worse outcomes among men with the lowest socioeconomic status. Geographical differences in screening rate could explain this pattern. More in-depth research with a source data review is required to know precisely the determinism of this association and therefore adjust the eventual disparities.

Jointly Modelling Single Nucleotide Polymorphisms With Longitudinal and Time-to-Event Trait: An Application to Type 2 Diabetes and Fasting Plasma Glucose.

In observational cohorts, longitudinal data are collected with repeated measurements at predetermined time points for many biomarkers, along with other variables measured at baseline. In these cohorts, time until a certain event of interest occurs is reported and very often, a relationship will be observed between some biomarker repeatedly measured over time and that event. Joint models were designed to efficiently estimate statistical parameters describing this relationship by combining a mixed model for the longitudinal biomarker trajectory and a survival model for the time until occurrence of the event, using a set of random effects to account for the relationship between the two types of data. In this paper, we discuss the implementation of joint models in genetic association studies. First, we check model consistency based on different simulation scenarios, by varying sample sizes, minor allele frequencies and number of repeated measurements. Second, using genotypes assayed with the Metabochip DNA arrays (Illumina) from about 4,500 individuals recruited in the French cohort D.E.S.I.R. (Data from an Epidemiological Study on the Insulin Resistance syndrome), we assess the feasibility of implementing the joint modelling approach in a real high-throughput genomic dataset. An alternative model approximating the joint model, called the Two-Step approach (TS), is also presented. Although the joint model shows more precise and less biased estimators than its alternative counterpart, the TS approach results in much reduced computational times, and could thus be used for testing millions of SNPs at the genome-wide scale.

A More Robust Random Effects Model for Disease Mapping

Disease mapping studies have found wide applications within geographical epidemiology and public health and are typically analysed within a Bayesian hierarchical model formulation. The most popular disease mapping model is the Besag-York-Molli´e model. A distinguishing feature of this model is the use of two sets of random effects: one spatially structured to model spatial autocorrelation and the other spatially unstructured to describe residual unstructured heterogeneity. Very often the spatially unstructured random effect is assumed to be normally distributed. Under practical situations, this normality assumption is found to be over restrictive. In this study, we investigate a more robust spatially unstructured random effect distribution by considering the Inverse Gaussian (IG) distribution in the disease mapping problem. The distribution has the normal distribution as special case. The inferences under this model are carried out within a bayesian hierarchical model formulation implemented in WinBUGS. The IG distribution is introduced in WinBUGS using zero tricks. The usefulness of the proposed model is investigated with a simulation study and applied in real data; mapping HIV in Kenya. In this work we showed that the IG distribution can produce better results when the normality assumption is violated due to the skewness of the data. For the case of data in which the random effects are truly normal, the IG distribution adjusts to a normal distribution as dictated by the data itself. On the other hand, the spatially structured random effect is normally modelled using the intrinsic conditional autoregressive (iCAR) prior. This prior is improper and has the undesirable large scale property of leading to a negative pairwise correlation for regions located further apart. In addition, the BYM model presents some identifiability problems of the spatial and non-spatial effects. In this work, we consider Leroux CAR (named lCAR hereafter) prior, a less widely used prior in disease mapping, as the prior distribution for the spatially structured random effects.

Bayesian analysis of longitudinal dyadic data with informative missing data using a dyadic shared-parameter model.

Analyzing longitudinal dyadic data is a challenging task due to the complicated correlations from repeated measurements and within-dyad interdependence, as well as potentially informative (or non-ignorable) missing data. We propose a dyadic shared-parameter model to analyze longitudinal dyadic data with ordinal outcomes and informative intermittent missing data and dropouts. We model the longitudinal measurement process using a proportional odds model, which accommodates the within-dyad interdependence using the concept of the actor-partner interdependence effects, as well as dyad-specific random effects. We model informative dropouts and intermittent missing data using a transition model, which shares the same set of random effects as the longitudinal measurement model. We evaluate the performance of the proposed method through extensive simulation studies. As our approach relies on some untestable assumptions on the missing data mechanism, we perform sensitivity analyses to evaluate how the analysis results change when the missing data mechanism is misspecified. We demonstrate our method using a longitudinal dyadic study of metastatic breast cancer.

A shared parameter model of longitudinal measurements and survival time with heterogeneous random-effects distribution

ABSTRACTTypical joint modeling of longitudinal measurements and time to event data assumes that two models share a common set of random effects with a normal distribution assumption. But, sometimes the underlying population that the sample is extracted from is a heterogeneous population and detecting homogeneous subsamples of it is an important scientific question. In this paper, a finite mixture of normal distributions for the shared random effects is proposed for considering the heterogeneity in the population. For detecting whether the unobserved heterogeneity exits or not, we use a simple graphical exploratory diagnostic tool proposed by Verbeke and Molenberghs [34] to assess whether the traditional normality assumption for the random effects in the mixed model is adequate. In the joint modeling setting, in the case of evidence against normality (homogeneity), a finite mixture of normals is used for the shared random-effects distribution. A Bayesian MCMC procedure is developed for parameter estimation and inference. The methodology is illustrated using some simulation studies. Also, the proposed approach is used for analyzing a real HIV data set, using the heterogeneous joint model for this data set, the individuals are classified into two groups: a group with high risk and a group with moderate risk.

On the specification of prior distributions for variance components in disease mapping models

In this paper, we consider the problem of specifying priors for the variance components in the Bayesian analysis of the Besag-York-Mollie model, a model that is popular among epidemiologists for disease mapping. The model encompasses two sets of random effects: one spatially structured to model spatial autocorrelation and the other spatially unstructured to describe residual heterogeneity. In this model, prior specification for variance components is an important problem because these priors maintain their influence on the posterior distributions of relative risks when mapping rare diseases. We propose using generalised inverse Gaussian priors, a broad class of distributions that includes many distributions commonly used as priors in this context, such as inverse gammas. We discuss the prior parameter choice with the aim of balancing the prior weight of the two sets of random effects on total variation and controlling the amount of shrinkage. The suggested prior specification strategy is compared to popular alternatives using a simulation exercise and applications to real data sets.

Rapid model exploration for complex hierarchical data: application to pharmacokinetics of insulin aspart.

We consider situations, which are common in medical statistics, where we have a number of sets of response data, from different individuals, say, potentially under different conditions. A parametric model is defined for each set of data, giving rise to a set of random effects. Our goal here is to efficiently explore a range of possible ‘population’ models for the random effects, to select the most appropriate model. The range of possible models is potentially vast, because the random effects may depend on observed covariates, and there may be multiple credible ways of partitioning their variability. Here, we consider pharmacokinetic (PK) data on insulin aspart, a fast acting insulin analogue used in the treatment of diabetes. PK models are typically nonlinear (in their parameters), often complex and sometimes only available as a set of differential equations, with no closed‐form solution. Fitting such a model for just a single individual can be a challenging task. Fitting a joint model for all individuals can be even harder, even without the complication of an overarching model selection objective. We describe a two‐stage approach that decouples the population model for the random effects from the PK model applied to the response data but nevertheless fits the full, joint, hierarchical model, accounting fully for uncertainty. This allows us to repeatedly reuse results from a single analysis of the response data to explore various population models for the random effects. This greatly expedites not only model exploration but also cross‐validation for the purposes of model criticism. © 2015 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.

Modeling Spatial Heterogeneity with Excess Zeroes from School Absenteeism dSata

Absenteeism has great advantages in promoting the early detection of epidemics. The spatial patterns of the data generally are polytropy and heterogeneity. The public health experts pay more attention to whether an outbreak will happen or/and how large the epidemic will be of school absenteeism data in spatial patterns. We construct simultaneously two set of random effects (u1, u2) in RE-ZIP to quantify this two kind spatial heterogeneity for 62 schools.

A Marginalized Combined Gamma Frailty and Normal Random-effects Model for Repeated, Overdispersed, Time-to-event Outcomes

This article proposes a marginalized model for repeated or otherwise hierarchical, overdispersed time-to-event outcomes, adapting the so-called combined model for time-to-event outcomes of Molenberghs et al. (in press), who combined gamma and normal random effects. The two sets of random effects are used to accommodate simultaneously correlation between repeated measures and overdispersion. The proposed version allows for a direct marginal interpretation of all model parameters. The outcomes are allowed to be censored. Two estimation methods are proposed: full likelihood and pairwise likelihood. The proposed model is applied to data from a so-called comet assay and to data from recurrent asthma attacks in children. Both estimation methods perform very well. From simulation results, it follows that the marginalized combined model behaves similarly to the ordinary combined model in terms of point estimation and precision. It is also observed that the pairwise likelihood required more computation time on the one hand but is less sensitive to starting values and stabler in terms of bias with increasing sample size and censoring percentage than full likelihood, on the other, leaving room for both in practice.

Empirical Bayes estimates for correlated hierarchical data with overdispersion.

An extension of the generalized linear mixed model was constructed to simultaneously accommodate overdispersion and hierarchies present in longitudinal or clustered data. This so-called combined model includes conjugate random effects at observation level for overdispersion and normal random effects at subject level to handle correlation, respectively. A variety of data types can be handled in this way, using different members of the exponential family. Both maximum likelihood and Bayesian estimation for covariate effects and variance components were proposed. The focus of this paper is the development of an estimation procedure for the two sets of random effects. These are necessary when making predictions for future responses or their associated probabilities. Such (empirical) Bayes estimates will also be helpful in model diagnosis, both when checking the fit of the model as well as when investigating outlying observations. The proposed procedure is applied to three datasets of different outcome types.

The Combined Model: A Tool for Simulating Correlated Counts with Overdispersion

The combined model as introduced by Molenberghs et al. (2007, 2010) has been shown to be an appealing tool for modeling not only correlated or overdispersed data but also for data that exhibit both these features. Unlike techniques available in the literature prior to the combined model, which use a single random-effects vector to capture correlation and/or overdispersion, the combined model allows for the correlation and overdispersion features to be modeled by two sets of random effects. In the context of count data, for example, the combined model naturally reduces to the Poisson-normal model, an instance of the generalized linear mixed model in the absence of overdispersion and it also reduces to the negative-binomial model in the absence of correlation. Here, a Poisson model is specified as the parent distribution of the data conditional on a normally distributed random effect at the subject or cluster level and/or a gamma distribution at observation level. Importantly, the development of the combined model and surrounding derivations have relevance well beyond mere data analysis. It so happens that the combined model can also be used to simulate correlated data. If a researcher is interested in comparing marginal models via Monte Carlo simulations, a necessity to generate suitable correlated count data arises. One option is to induce correlation via random effects but calculation of such quantities as the bias is then not straightforward. Since overdispersion and correlation are simultaneous features of longitudinal count data, the combined model presents an appealing framework for generating data to evaluate statistical properties, through a pre-specification of the desired marginal mean (possibly in terms of the covariates and marginal parameters) and a marginal variance-covariance structure. By comparing the marginal mean and variance of the combined model to the desired or pre-specified marginal mean and variance, respectively, the implied hierarchical parameters and the variance-covariance matrices of the normal and Gamma random effects are then derived from which correlated Poisson data are generated. We explore data generation when a random intercept or random intercept and slope model is specified to induce correlation. The data generator, however, allows for any dimension of the random effects although an increase in the random-effects dimension increases the sensitivity of the derived random effects variance-covariance matrix to deviations from positive-definiteness. A simulation study is carried out for the random-intercept model and for the random intercept and slope model, with or without the normal and Gamma random effects. We also pay specific attention to the case of serial correlation.

A skew-normal random effects model for longitudinal ordinal categorical responses with missing data

Missing values are common in longitudinal data studies. The missing data mechanism is termed non-ignorable (NI) if the probability of missingness depends on the non-response (missing) observations. This paper presents a model for the ordinal categorical longitudinal data with NI non-monotone missing values. We assumed two separate models for the response and missing procedure. The response is modeled as ordinal logistic, whereas the logistic binary model is considered for the missing process. We employ these models in the context of so-called shared-parameter models, where the outcome and missing data models are connected by a common set of random effects. It is commonly assumed that the random effect follows the normal distribution in longitudinal data with or without missing data. This can be extremely restrictive in practice, and it may result in misleading statistical inferences. In this paper, we instead adopt a more flexible alternative distribution which is called the skew-normal distribution. The methodology is illustrated through an application to Schizophrenia Collaborative Study data [19] and a simulation.

Modelling multivariate, overdispersed binomial data with additive and multiplicative random effects

When modelling multivariate binomial data, it often occurs that it is necessary to take into consideration both clustering and overdispersion, the former arising from the dependence between data, and the latter due to the additional variability in the data not prescribed by the distribution. If interest lies in accommodating both phenomena at the same time, we can use separate sets of random effects that capture the within-cluster association and the extra variability. In particular, the random effects for overdispersion can be included in the model either additively or multiplicatively. For this purpose, we propose a series of Bayesian hierarchical models that deal simultaneously with both phenomena. The proposed models are applied to bivariate repeated prevalence data for hepatitis C virus (HCV) and human immunodeficiency virus (HIV) infection in injecting drug users in Italy from 1998 to 2007.

Regional Variation in Relative Survival—Quantifying the Effects of the Competing Risks of Death by Using a Cure Fraction Model with Random Effects

SummaryWe extend a mixture cure fraction model with random effects to allow estimation of relative survival of cancer patients by region in a country with a parsimonious number of parameters. The heterogeneity in the expected survival was taken into account such that the expected mortality rate was considered as a random quantity varying across regions. Two sets of random effects were used to describe regional variation, both in the cure fraction and in the relative survival of the non-cured patients. This hierarchical model was fitted in a Bayesian framework to the follow-up data of Finnish colon cancer patients diagnosed in 1975–2004. The public health importance of the regional differences was described by the extra survival time and avoidable deaths that could be gained, if the regional variation were eliminated and all the hospital districts in Finland achieved the most favourable level of relative survival. The differences in relative survival between the districts were quite small during the three decades. If all the districts had achieved the most favourable level of relative survival, the mean survival times of the female patients diagnosed in 1975–1984, 1985–1994 and 1995–2004 would have increased by 5%, 4% and 2% respectively.

Boundary detection in disease mapping studies

In disease mapping, the aim is to estimate the spatial pattern in disease risk over an extended geographical region, so that areas with elevated risks can be identified. A Bayesian hierarchical approach is typically used to produce such maps, which represents the risk surface with a set of random effects that exhibit a single global level of spatial smoothness. However, in complex urban settings, the risk surface is likely to exhibit localized rather than global spatial structure, including areas where the risk varies smoothly over space, as well as boundaries separating populations that are geographically adjacent but have very different risk profiles. Therefore, this paper proposes an approach for capturing localized spatial structure, including the identification of such risk boundaries. The effectiveness of the approach is tested by simulation, before being applied to lung cancer incidence data in Greater Glasgow, UK, between 2001 and 2005.

Cure fraction model with random effects for regional variation in cancer survival

Assessing regional differences in the survival of cancer patients is important but difficult when separate regions are small or sparsely populated. In this paper, we apply a mixture cure fraction model with random effects to cause-specific survival data of female breast cancer patients collected by the population-based Finnish Cancer Registry. Two sets of random effects were used to capture the regional variation in the cure fraction and in the survival of the non-cured patients, respectively. This hierarchical model was implemented in a Bayesian framework using a Metropolis-within-Gibbs algorithm. To avoid poor mixing of the Markov chain, when the variance of either set of random effects was close to zero, posterior simulations were based on a parameter-expanded model with tailor-made proposal distributions in Metropolis steps. The random effects allowed the fitting of the cure fraction model to the sparse regional data and the estimation of the regional variation in 10-year cause-specific breast cancer survival with a parsimonious number of parameters. Before 1986, the capital of Finland clearly stood out from the rest, but since then all the 21 hospital districts have achieved approximately the same level of survival.

Shared parameter models under random effects misspecification

SUMMARY A common objective in longitudinal studies is the investigation of the association structure between a longitudinal response process and the time to an event of interest. An attractive paradigm for the joint modelling of longitudinal and survival processes is the shared parameter framework, where a set of random effects is assumed to induce their interdependence. In this work, we propose an alternative parameterization for shared parameter models and investigate the effect of misspecifying the random effects distribution in the parameter estimates and their standard errors.