Linear Quantile Mixed Models Research Articles

A Bayesian quantile joint modeling of multivariate longitudinal and time-to-event data.

Linear mixed models are traditionally used for jointly modeling (multivariate) longitudinal outcomes and event-time(s). However, when the outcomes are non-Gaussian a quantile regression model is more appropriate. In addition, in the presence of some time-varying covariates, it might be of interest to see how the effects of different covariates vary from one quantile level (of outcomes) to the other, and consequently how the event-time changes across different quantiles. For such analyses linear quantile mixed models can be used, and an efficient computational algorithm can be developed. We analyze a dataset from the Acute Lymphocytic Leukemia (ALL) maintenance study conducted by Tata Medical Center, Kolkata. In this study, the patients suffering from ALL were treated with two standard drugs (6MP and MTx) for the first two years, and three biomarkers (e.g. lymphocyte count, neutrophil count and platelet count) were longitudinally measured. After treatment the patients were followed nearly for the next three years, and the relapse-time (if any) for each patient was recorded. For this dataset we develop a Bayesian quantile joint model for the three longitudinal biomarkers and time-to-relapse. We consider an Asymmetric Laplace Distribution (ALD) for each outcome, and exploit the mixture representation of the ALD for developing a Gibbs sampler algorithm to estimate the regression coefficients. Our proposed model allows different quantile levels for different biomarkers, but still simultaneously estimates the regression coefficients corresponding to a particular quantile combination. We infer that a higher lymphocyte count accelerates the chance of a relapse while a higher neutrophil count and a higher platelet count (jointly) reduce it. Also, we infer that across (almost) all quantiles 6MP reduces the lymphocyte count, while MTx increases the neutrophil count. Simulation studies are performed to assess the effectiveness of the proposed approach.

Effects of normal aging on the mouse retina assessed by full-field flash and flicker electroretinography

Changes in the full-field flash and flicker electroretinogram (ERG) that accompany normal aging were evaluated in mice. ERGs were recorded from a single cohort of C57BL/6J mice from 5 to 70 weeks of age using conventional techniques. Dark-adapted ERGs were recorded for flash luminances of − 3.0 to 1.5 log cd-s-m−2; a- and b-wave amplitude and implicit time (IT) were calculated from these responses. In addition, light-adapted flicker ERGs elicited by sinusoidally modulated light were measured for temporal frequencies of 2 to 31 Hz. Amplitudes and phases were extracted from the flicker responses using Fourier analysis. Linear quantile mixed models were used for statistical comparisons of the effects of age on amplitude and timing. There was a significant decrease in a-wave amplitude (p < 0.001) and b-wave amplitude (p < 0.001) over the 65 week study. From 5 to 70 weeks, the a- and b-wave amplitudes decreased by a factor of approximately 2. There was a small (2–14 ms), but significant (p < 0.001), delay in a- and b-wave IT over the 65 week study. There was also a significant decrease in fundamental amplitude (factor of 1.8, p < 0.001) and second harmonic amplitude (factor of 1.5, p < 0.001) over time. There were no significant age-related effects on the phase of these components (both p > 0.06). These results indicate that age scales the single flash and flicker ERG similarly, reducing response amplitude by a factor of approximately 2, from 5 to 70 weeks, with small or no effect on response timing. These data may be useful for guiding future longitudinal pre-clinical therapeutic studies.

Varying growth response of Central European tree species to the extraordinary drought period of 2018 – 2020

Droughts have intensified in Central Europe and will likely become a major driver of forest ecosystem change in the future. Regional studies are needed to understand drought impacts on forests and to provide guidance for adaptive management. In this study, we have investigated the effects of an extraordinary drought period in 2018 – 2020 on tree growth in the State of Bavaria (Germany). We (i) assessed overall changes in tree growth compared to pre-drought conditions, (ii) mapped the spatial configuration of drought and its impact on growth across ecoregions, (iii) quantified the standardized drought effect for the most abundant tree species, and (iv) analyzed the inter- and intraspecific variability in growth responses to drought.Covering a large environmental gradient, we compared 3,462 tree basal area growth records at 200 inventory plots with the pre-drought period of 2006 – 2017. We analyzed drought impacts on tree growth using nonparametric randomization tests and linear quantile mixed models (LQMMs).Median annual tree growth declined by 41.3% during drought. Decreases in growth were statistically significant for 13 out of 20 species. Growth reductions were most distinct in northwestern ecoregions of Bavaria. After controlling for other covariates of tree growth, the most abundant conifers, Picea abies [Karst.] and Pinus sylvestris [L.], were overall more susceptible than the common broadleaves, Fagus sylvatica [L.], Quercus robur [L.], and Quercus petraea [Matt.]. Yet, all species exhibited a high intraspecific variation in drought response.Altering tree species composition is a key strategy to adapt forest ecosystems. Our study reveals high potential to mitigate future drought impacts by admixing P. abies- and P. sylvestris-dominated forests with more drought-adapted broadleaved tree species. In addition, the high intraspecific variability in drought susceptibility indicates potential to promote provenances or individuals with low drought sensitivity to reduce the local loss of tree species.

A two-level copula joint model for joint analysis oflongitudinal and competing risks data.

In this article, we propose a two-level copula joint model to analyze clinical data with multiple disparate continuous longitudinal outcomes and multiple event-times in the presence of competing risks. At the first level, we use a copula to model the dependence between competing latent event-times, in the process constructing the submodel for the observed event-time, and employ the Gaussian copula to construct the submodel for the longitudinal outcomes that accounts for their conditional dependence; these submodels are glued together at the second level via the Gaussian copula to construct a joint model that incorporates conditional dependence between the observed event-time and the longitudinal outcomes. To have the flexibility to accommodate skewed data and examine possibly different covariate effects on quantiles of a non-Gaussian outcome, we propose linear quantile mixed models for the continuous longitudinal data. We adopt a Bayesian framework for model estimation and inference via Markov Chain Monte Carlo sampling. We examine the performance of the copula joint model through a simulation study and show that our proposed method outperforms the conventional approach assuming conditional independence with smaller biases and better coverage probabilities of the Bayesian credible intervals. Finally, we carry out an analysis of clinical data on renal transplantation for illustration.

Prediction of Travel Time Reliability on Interstates Using Linear Quantile Mixed Models

Under the Moving Ahead for Progress in the 21st Century Act (MAP-21), state Departments of Transportation (DOTs) are responsible for reporting travel time reliability and also setting targets and showing progress toward those targets. To know how to improve travel time reliability and what to expect from investments in transportation infrastructure, state DOTs need a better understanding of the factors that affect travel time reliability and methods to predict future travel time reliability. This paper proposes linear quantile mixed models (LQMMs) to quantify travel time reliability impact factors and predict selected reliability measures (level of travel time reliability [LOTTR] and the 90th percentile) to address these needs. The method was demonstrated using probe vehicle data from interstate segments in Virginia that had been partitioned into approximately homogeneous clusters based on the similarity of their cumulative distribution functions (CDFs) of travel times. Using clustered data meant that LQMMs were only necessary for a limited number of clusters rather than for hundreds of individual segments, thus making the process more efficient and manageable. The LQMMs showed that frequencies of non-recurrent events, such as incidents and weather, were correlated with higher travel time percentiles. The prediction performance of LQMMs was compared with trend line predictions, a common method used in practice. The results showed that LQMMs significantly improved prediction accuracy.

Parenting stress in caregiver-mediated interventions for toddlers with autism: An application of quantile regression mixed models.

Traditional longitudinal modeling approaches require normally distributed data and do not account for sample heterogeneity. Parenting stress, in particular, can be difficult to model across time without transforming the data as it is usually high for caregivers of children with ASD. This study used novel linear quantile mixed models (LQMMs) to model non-normal parent stress scores across two caregiver-mediated interventions involving toddlers with ASD. The sample included 86 caregiver-child dyads who were randomized to either a parent-only psychoeducational intervention or hands-on parent training in a naturalistic developmental intervention. Child and parent-related domains of the Parenting Stress Index (PSI) were the primary outcomes in this study. The PSI was collected at entry, 10-week exit, 3-month follow-up, and 6-month follow-up periods. Separate LQMMs were used to model five specific quantiles ( of the two PSI domains across the complete intervention timeline. These five quantiles effectively modeled the entire conditional distribution of parenting stress scores. The LQMMs indicated that child-related parenting stress decreased across all quantiles within both interventions, with no difference in the rate of parenting stress change between the intervention groups. For parent-related parenting stress, the effect of intervention depended on the group's stress level; some parents increased their perceived stress within the hands-on intervention at the 3-month follow-up. Overall, this study demonstrated that the use of LQMMs yielded additional information, beyond traditional longitudinal models, regarding the relationship between parenting stress within two caregiver-mediated intervention protocols. This study also discussed the methodological contributions and potential future applications of LQMMs. LAY SUMMARY: This study used a newer longitudinal modeling technique to examine how parenting stress changed across two caregiver-mediated interventions for toddlers with ASD. Results showed that certain parents in the JASPER condition might require additional support as they exit the study and enter into their first follow-up period. It was also determined that this new modeling technique could be a valuable tool to analyze highly variable data often present in ASD intervention studies.

Constrained Bayesian doubly elastic net Lasso for linear quantile mixed models

In this article, we propose a novel constrained Bayesian elastic net approach for linear quantile mixed model shrinkage. A partially collapsed Gibbs sampling algorithm is developed for efficient posterior computation based on a modified Cholesky decomposition for the covariance matrix of random effects and an asymmetric Laplace distribution for the error distribution. We demonstrate the proposed method based on simulated data and an experimental dataset from a longitudinal study of age-related macular degeneration trial. Both simulation studies and real data analysis indicate that the proposed constrained Bayesian elastic net approach is competitive with the existing methods under a variety of scenarios, such as presence of a large number of covariates and collinearity.

Shrinkage estimation of fixed and random effects in linear quantile mixed models

ABSTRACT This paper presents a Bayesian analysis of linear mixed models for quantile regression using a modified Cholesky decomposition for the covariance matrix of random effects and an asymmetric Laplace distribution for the error distribution. We consider several novel Bayesian shrinkage approaches for both fixed and random effects in a linear mixed quantile model using extended penalties. To improve mixing of the Markov chains, a simple and efficient partially collapsed Gibbs sampling algorithm is developed for posterior inference. We also extend the framework to a Bayesian mixed expectile model and develop a Metropolis–Hastings acceptance–rejection (MHAR) algorithm using proposal densities based on iteratively weighted least squares estimation. The proposed approach is then illustrated via both simulated and real data examples. Results indicate that the proposed approach performs very well in comparison to the other approaches.

A species-specific, site-sensitive maximum stand density index model for Pacific Northwest conifer forests

Maximum stand density index (SDIMAX) models were developed for important Pacific Northwest conifers of western Oregon and Washington, USA, based on site and species influences and interactions. Inventory and monitoring data from numerous federal, state, and private forest management groups were obtained throughout the region to ensure a wide coverage of site characteristics. These observations include information on tree size, number, and species composition. The effects and influence on the self-thinning frontier of plot-specific factors such as climate, topography, soils, and geology, as well as species composition, were evaluated based on geographic location using a multistep approach to analysis involving linear quantile mixed models, random forest, and stochastic frontier functions. The self-thinning slope of forest stands dominated by Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco) was found to be –1.517 and that of stands dominated by western hemlock (Tsuga heterophylla (Raf.) Sarg.) was found to be –1.461, leading to regionwide modelled SDIMAX values at the 95th percentile of 1728 and 1952 trees per hectare, respectively. The regional model of site-specific SDIMAX will support forest managers in decision-making regarding density management and species selection to more efficiently utilize site resources toward healthy, productive forests.

Evaluating the rainwater killifish (Lucania parva) as an indicator of Everglades restoration

Comprehensive Everglades Restoration Plan (CERP) objectives include the return of more natural salinity regimes to the bays and estuaries of southern Florida (U.S.A.). We examined for spatiotemporal patterns in rainwater killifish (Lucania parva) density and size with emphasis on relations with salinity, salinity variation, submerged aquatic vegetation (SAV), temperature, and depth. Using linear quantile mixed models (LQMM) and cluster analyses, we analyzed 18 consecutive wet‐dry seasons (9 years) of enclosure trap data from 44 fixed stations along mainland shoreline of southern Biscayne Bay, Florida. The LQMM results suggested that SAV was the strongest influencing, and (potentially) most limiting, habitat variable on killifish density, and that decreases in mean salinity and salinity variation may result in higher killifish densities in our study domain. Killifish size‐habitat relationships were few. Cluster analysis of stations, based on killifish densities across time, failed to reveal clear spatial fish density gradients, consistent density hot‐spots, or grouping according to median habitat conditions. However, cluster analysis of seasons, based on killifish densities across space, indicated that high densities were associated with the wet season and low densities with the dry, although this depended on year. Our analyses suggest that realization of lower salinities and lower salinity variation in our study domain, if they are not accompanied by SAV decreases, may enhance killifish populations and/or their role as prey. Owing to its high stress ‐tolerance and extreme variability in density, L. parva is not an ideal indicator species of CERP‐modified salinity conditions, but warrants tracking as restoration activities ensue.

Short-term Exposure to Ambient Particulate Elements and Epigenome-wide DNA Methylation in Older Men: the Normative Aging Study

Background: Current research suggests that short- and intermediate- term exposure to ambient particulate matter is associated with global and gene-specific DNA methylation (DNAm), but little is known regarding the changes in epigenome-wide caused by particulate elements (PEs). Additionally, the potential pathways remain unknown.Objective: We evaluated short- to intermediate-term associations between ambient PEs and DNAm in epigenome-wide and identified potential biological pathways.Method: We collected whole blood samples from 691 elderly men in the Normative Aging Study (1999-2013) and measured epigenome-wide DNAm with the Illumina Infinium HumanMethylation450 BeadChip. Ambient PEs were measured daily at a fixed monitoring station and 5 different moving averages from short- to intermediate-term (current day to 28-day) were considered. Linear quantile mixed models were used to estimate the associations between 15 elements and DNAm (N=1, 262 visits), adjusting for fine particle mass, cell type, batch effects, age, body mass index, smoking, etc. Pathway analysis was also performed to explore the potential biologic mechanisms.Results: Short- to intermediate-term exposure to PEs was significantly associated with epigenome-wide DNAm, especially for 28-d moving average. We observed different Bonferroni significant CpGs for different PE, most of them related with aluminum, iron, silicon, lead, and, calcium. The pathway analyses suggest that the detrimental effects might be relevant to cell cycle, biosynthetic process, and signal transduction. We also found that there were several common genes significantly associated with calcium, silicon and lead, such as, neurodegenerative disease related gene tubulin folding cofactor D.Conclusions: Ours is the first study that find short- to intermediate-term effects on DNAm in epigenome-wide by ambient PEs, especially elements associated with road dust, oil combustion, and wood burning. Our findings provide insight into potential mechanistic pathways.

Bayesian variable selection in linear quantile mixed models for longitudinal data with application to macular degeneration.

This paper presents a Bayesian analysis of linear mixed models for quantile regression based on a Cholesky decomposition for the covariance matrix of random effects. We develop a Bayesian shrinkage approach to quantile mixed regression models using a Bayesian adaptive lasso and an extended Bayesian adaptive group lasso. We also consider variable selection procedures for both fixed and random effects in a linear quantile mixed model via the Bayesian adaptive lasso and extended Bayesian adaptive group lasso with spike and slab priors. To improve mixing of the Markov chains, a simple and efficient partially collapsed Gibbs sampling algorithm is developed for posterior inference. Simulation experiments and an application to the Age-Related Macular Degeneration Trial data to demonstrate the proposed methods.

Toxic iron species in lower-risk myelodysplastic syndrome patients: course of disease and effects on outcome

Toxic iron species in lower-risk myelodysplastic syndrome patients: course of disease and effects on outcome

Structural and Functional Abnormalities in Early-stage Diabetic Retinopathy

ABSTRACT Purpose To evaluate the relationship between microperimetric (MP) sensitivity and retinal thickness measured at co-registered retinal locations in individuals who have mild or no diabetic retinopathy. Methods Fifty non-diabetic control subjects and 50 type-2 diabetic subjects participated (25 had no clinically apparent DR [NDR] and 25 had mild nonproliferative DR [MDR]). MP sensitivity was measured at 36 retinal locations that were arranged in three concentric rings centered on the fovea (radii of 3°, 6°, and 12°). Optical coherence tomography (OCT) images were obtained, and total retinal thickness (TRT), inner retinal thickness (IRT), and outer retinal thickness (ORT) were quantified from the OCT images at locations that matched the MP measures. Linear quantile mixed models (LQMMs) and linear quantile models (LQMs) were used to compare MP and thickness values for the three subject groups and to quantify structure–function relationships. Results The statistical models indicated significant TRT and IRT reductions in the NDR and MDR groups, relative to the controls, that were most apparent in the 3° ring. By contrast, ORT was not reduced significantly for either diabetic group. MP sensitivity was reduced significantly within each ring and for both diabetic groups. Despite reductions in both thickness and sensitivity, the structure–function associations were generally weak with borderline statistical significance. For example, a TRT or IRT reduction of approximately 27 µm was predicted to result in approximately 1 dB of MP sensitivity loss for the MDR group (p = .03 and 0.05, respectively) Conclusions The results support previous findings of early retinal neurodegeneration in diabetics who have NDR or MDR. Interestingly, the structural and functional deficits appear to be only weakly associated, suggesting that mechanisms in addition to retinal thinning underlie the functional defects in early-stage DR.

Nonlinearities in the flicker electroretinogram: A tool for studying retinal dysfunction applied to early-stage diabetic retinopathy

The flicker electroretinogram (ERG) is typically analyzed in terms of peak-to-trough amplitude and implicit time. However, additional important information may be captured by spectral-domain analysis of the ERG harmonics (responses that occur at multiples of the stimulus frequency). This study describes an approach to analyze the harmonic components of the flicker ERG and its application to patients who have early-stage non-proliferative diabetic retinopathy (NPDR). Of particular interest were the sub-harmonic components occurring at 1.5x and 2.5x the stimulus frequency that produce cycle-to-cycle variation in amplitude termed “period doubling.” Twenty visually-normal subjects, 20 diabetic subjects who have no clinically-apparent retinopathy (NDR), and 20 diabetic subjects who have mild NPDR participated. ERGs were recorded in response to sinusoidal flicker (27–63 Hz) and Fourier analysis was performed to extract fundamental and harmonic response amplitudes. Linear quantile mixed models (LQMMs) were used to compare the amplitude of the response components among the three subject groups. Results indicated that the maximum sub-harmonic amplitude occurred in the stimulus frequency range of 33–38 Hz for all subjects. The LQMMs showed a significant sub-harmonic amplitude reduction for the mild NPDR subjects compared to the controls; sub-harmonic amplitude for the NDR subjects did not differ significantly from the controls. In contrast, the fundamental response did not differ among the groups for stimulus frequencies between 33 and 38 Hz. Modeling these results indicated that subharmonic amplitude loss in mild NPDR subjects may be attributed to attenuated feedback occurring early in the retina, possibly at the synapse of cone photoreceptors and OFF bipolar cells.

Letter to the Editor.

Galarza, Lachos and Bandyopadhyay (2017) have recently proposed a method of estimating linear quantile mixed models (Geraci and Bottai, 2014) based on a Monte Carlo EM algorithm. They assert that their procedure represents an improvement over the numerical quadrature and non-smooth optimization approach implemented by Geraci (2014). The objective of this note is to demonstrate that this claim is incorrect. We also point out several inaccuracies and shortcomings in their paper which affect other results and conclusions that can be drawn.

Depressive symptomatology in school-aged children with obstructive sleep apnea syndrome: incidence, demographic factors, and changes following a randomized controlled trial of adenotonsillectomy

Depressive symptoms following adenotonsillectomy (AT) relative to controls were examined in children with obstructive sleep apnea syndrome (OSAS). The Childhood Adenotonsillectomy Trial (CHAT) multisite study examined the impact of AT in 453 children aged 5 to 9.9 years with polysomnographic evidence of OSAS without prolonged desaturation, randomized to early adenotonsillectomy (eAT) or watchful waiting with supportive care (WWSC). One hundred seventy-six children (eAT n = 83; WWSC n = 93) with complete evaluations for depressive symptomatology between baseline and after a 7-month intervention period were included in this secondary analysis. Exact binomial test assessed proportion of depressive symptomatology relative to norms, while effects of AT and OSAS resolution were assessed through linear quantile mixed-models. Treatment group assignment did not significantly impact depression symptoms, although self-reported depression symptoms improved over time (p < 0.001). Resolution of OSAS symptoms demonstrated a small interaction effect in an unexpected direction, with more improvement in parent ratings of anxious/depressed symptoms for children without resolution (p = 0.030). Black children reported more severe depressive symptoms (p = 0.026) and parents of overweight/obese children reported more withdrawn/depressed symptoms (p = 0.004). Desaturation nadir during sleep was associated with self-report depressed (r = -0.17, p = 0.028), parent-reported anxious/depressed (r = -0.15, p = 0.049), and withdrawn/depressed (r = -0.24, p = 0.002) symptoms. Increased risk for depressed and withdrawn/depressed symptoms was detected among children with OSAS, and different demographic variables contributed to risk in self-reported and parent-reported depression symptoms. Arterial oxygen desaturation nadir during sleep was strongly associated with depressed symptoms. However, despite improvements in child-reported depressed symptoms over time, changes were unrelated to either treatment group or OSAS resolution status. Childhood Adenotonsillectomy Study for Children with OSAS (CHAT), https://clinicaltrials.gov/show/NCT00560859, NCT00560859.

Bayesian quantile regression joint models: Inference and dynamic predictions.

In the traditional joint models of a longitudinal and time-to-event outcome, a linear mixed model assuming normal random errors is used to model the longitudinal process. However, in many circumstances, the normality assumption is violated and the linear mixed model is not an appropriate sub-model in the joint models. In addition, as the linear mixed model models the conditional mean of the longitudinal outcome, it is not appropriate if clinical interest lies in making inference or prediction on median, lower, or upper ends of the longitudinal process. To this end, quantile regression provides a flexible, distribution-free way to study covariate effects at different quantiles of the longitudinal outcome and it is robust not only to deviation from normality, but also to outlying observations. In this article, we present and advocate the linear quantile mixed model for the longitudinal process in the joint models framework. Our development is motivated by a large prospective study of Huntington's disease where primary clinical interest is in utilizing longitudinal motor scores and other early covariates to predict the risk of developing Huntington's disease. We develop a Bayesian method based on the location-scale representation of the asymmetric Laplace distribution, assess its performance through an extensive simulation study, and demonstrate how this linear quantile mixed model-based joint models approach can be used for making subject-specific dynamic predictions of survival probability.

Analyzing Variation in Prior Record Penalties Across Conviction Offenses

The focal concerns perspective suggests that criminal history and the nature of the offense interact to influence judicial assessments of community threat, yet this question has not been subject to systematic empirical examination. Drawing on 4 years of data (2007-2010) from the Pennsylvania Commission on Sentencing ( N = 75,676), we utilize linear quantile mixed models (LQMM) to examine the impact of prior record on the conditional distribution of sentence lengths across violent, property, drug, and sex offenders, controlling for the effects of important individual and judicial district-level covariates. The results indicate that prior record penalties differ both between and within conviction offense types across the conditional sentence length distribution. Substantive, theoretical, and methodological implications are discussed.

Modeling normative kinetic perimetry isopters using mixed-effects quantile regression.

Kinetic perimetry is used to quantify visual field size/sensitivity. Clinically, perimetry can be used to diagnose and monitor ophthalmic and neuro-ophthalmic disease. Normative data are integral to the interpretation of these findings. However, there are few computational developments that allow clinicians to collect and analyze normative data from kinetic perimeters. In this article we describe an approach to fitting kinetic responses using linear quantile mixed models. Analogously to traditional linear mixed-effects models for the mean, linear quantile mixed models account for repeated measurements taken from the same individual, but differently from linear mixed-effects models, they are more flexible as they require weaker distributional assumptions and allow for quantile-specific inference. Our approach improves on parametric alternatives based on normal assumptions. We introduce the R package kineticF, a freely available and open-access resource for the analysis of perimetry data. Our proposed approach can be used to analyze normative data from further studies.