Background and Objectives: Although Hispanic/Latino populations in the United States are remarkably diverse in terms of birthplace and age at migration, we poorly understand how these factors are associated with cognitive aging. Our research seeks to operationalize a life course perspective of migration and health and contribute new understanding of Alzheimer's disease/Alzheimer's disease-related dementias among U.S.-based Hispanic/Latino older adults. Research Design and Methods: Harnessing the Hispanic Community Health Study/Study of Latinos (n = 16,415) and the Study of Latinos-Investigation of Neurocognitive Aging (n = 6,377) data, we compare baseline cognition and 7-year cognitive change among U.S./mainland-born Hispanic/Latino adults relative to foreign/island-born immigrants by age of migration (4 groups: born in mainland United States, immigrated <16 years, 16-34 years, >34 years). Global cognition was calculated as a composite measure, and domain-specific measures were considered in secondary analyses. We employed linear regressions, ANOVA contrasts, and Blinder-Oaxaca decomposition techniques. Results: All Hispanic/Latino immigrant adults, regardless of age at migration, have a cognitive health disadvantage (at each visit and over time) relative to U.S./mainland-born Hispanic/Latino individuals. Differences did not endure the inclusion of covariates and were explained predominantly by first socioeconomic and then acculturative factors, and far less by health and health behaviors. Acculturative factors are particularly important for individuals who migrated after childhood. Discussion and Implications: Socioeconomic and acculturation factors have outsized roles in explaining gaps in cognitive aging among U.S.-born and migrant Hispanic/Latino adults. It is then vital to examine whether disrupting socioeconomic and acculturation inequalities closes such gaps in cognitive aging.

Objective: Spiritual well-being (SWB) has been shown to delay the onset of cognitive decline among older adults predisposed to Alzheimer's disease and related neurodegenerative dementias. It was, however, unknown if SWB is also associated with delay in disease manifestation ("phenoconversion") in rare, genetic neurodegenerative dementias, such as Huntington's disease (HD). Thus, we sought to evaluate the association between SWB and phenocovnersion in people at-risk for HD. Methods: The "Prospective Huntington At Risk Observation Study" (PHAROS), a large-scale national prospective research study, recruited a substantial cohort of 1001 participants. These participants, who were first-degree relatives of individuals diagnosed with HD and had a 50% chance of carrying the HD genetic mutation, were blinded to their genetic status and underwent repeated clinical assessments every 12 months. The study used Cox proportional models to examine the relationship between SWB and time to phenoconversion while also considering subcomponents of SWB and adjusting for age, sex, and CAG repeat length. Results: The study found no significant association between SWB and time to phenoconversion in individuals with the HD gene mutation. Conclusion: While existing data suggest that SWB may be an actionable target to improve health-related quality of life in HD and other serious illnesses, there is a lack of evidence supporting the role of SWB in attenuating phenoconversion in people with the HD genetic mutation. Unlike Alzheimer's, HD may be less responsive to analogous factors in delaying disease onset.

BACKGROUND:Prior studies report associations between periconceptional exposure to natural and synthetic oestrogen and progesterone and autism spectrum disorder (ASD). Hormonal contraception contains synthetic forms of one or both hormones. Although hormonal contraception is highly effective when consistently used, unintended pregnancy can occur with irregular use. Given the popularity of hormonal contraception, foetal exposure in utero is possible, yet the potential consequences are unknown. OBJECTIVES:We investigated the association between periconceptional hormonal contraception use and the development of ASD in offspring. METHODS:We analysed data from the Study to Explore Early Development (SEED), a population-based case-control study conducted in select US states, from 2007 to 2020. Children with and without ASD were identified from clinical/education sources and vital records, respectively, and enrolled at ages 2.5-5 years. We confirmed the ASD case status by in-person developmental assessment. We assessed hormonal contraception via a structured interview. We assessed the associations between ASD and hormonal contraception exposure separately for contraception discontinued in the 3 months prior to pregnancy and contraception continued during pregnancy using logistic models to estimate odds ratios (OR) adjusted for biological mother age, education, parity, pre-pregnancy body mass index (BMI), and presence of gynaecologic conditions and 95% confidence intervals (CI). RESULTS:Of 5210 participants, 9.9% reported discontinuing hormonal contraception use before pregnancy and 2.3% reported continuing use during pregnancy. A suggestive association was found between ASD and hormonal contraception use during pregnancy (aOR 1.38,95% CI 0.93, 2.05). There was no association with use prior to pregnancy (aOR 1.02, 95% CI 0.84, 1.25). CONCLUSIONS:Discontinuation of hormonal contraception prior to conception was not associated with ASD. The prevalence of hormonal contraception use during pregnancy was low. Results were imprecise and may be impacted by recall bias and unmeasured confounding by indication and health behaviours related to planning pregnancy.

In longitudinal studies, the devices used to measure exposures can change from visit to visit. Calibration studies, wherein a subset of participants is measured using both devices at follow-up, may be used to assess between-device differences (ie, errors). Then, statistical methods are needed to adjust for between-device differences and the missing measurement data that often appear in calibration studies. Regression calibration and multiple imputation are two possible methods. We compared both methods in linear regression with a simulation study, considering various real-world scenarios for a longitudinal study of pulse wave velocity. Regression calibration and multiple imputation were both essentially unbiased, but correctly estimating the standard errors posed challenges. Multiple imputation with predicted mean matching produced close agreement with the empirical standard error. Fully stochastic multiple imputation underestimated the standard error by up to 50%, and regression calibration with bootstrapped standard errors performed slightly better than fully stochastic multiple imputation. Regression calibration was slightly more efficient than either multiple imputation method. The results suggest use of multiple imputation with predictive mean matching over fully stochastic imputation or regression calibration in longitudinal studies where a new device at follow-up might be error-prone compared to the device used at baseline.

Modeling symptom progression to identify ideal subjects for a Huntington's disease clinical trial is problematic since time to diagnosis, a key covariate, can be heavily censored. Imputation is an appealing strategy that replaces the censored covariate with its conditional mean, but existing methods saw over 200% bias under heavy censoring. Calculating conditional means well requires estimating and then integrating over the survival function of the censored covariate from the censored value to infinity. To estimate the survival function flexibly, existing methods use the semiparametric Cox model with Breslow's estimator, leaving the integrand for the conditional means (the survival function) undefined beyond the observed data. The integral is then estimated up to the largest observed covariate value, and this approximation can cut off the tail of the survival function and lead to severe bias. We combine the semiparametric survival estimator with a parametric extension to approximate the integral up to infinity. In simulations, our proposed extrapolation-before-imputation approach substantially reduces the bias seen with existing imputation methods, sometimes even when the parametric extension was misspecified. We further demonstrate how imputing with corrected conditional means can prioritize subjects for clinical trials. The R code to reproduce results is available in the Supplementary Material.

Despite its drawbacks, the complete case analysis is commonly used in regression models with incomplete covariates. Understanding when the complete case analysis will lead to consistent parameter estimation is vital before use. Our aim here is to demonstrate when a complete case analysis is consistent for randomly right-censored covariates and to discuss the implications of its use even when consistent. Across the censored covariate literature, different assumptions are made to ensure a complete case analysis produces a consistent estimator, which leads to confusion in practice. We make several contributions to dispel this confusion. First, we summarize the language surrounding the assumptions that lead to a consistent complete case estimator. Then, we show a unidirectional hierarchical relationship between these assumptions, which leads us to one sufficient assumption to consider before using a complete case analysis. Lastly, we conduct a simulation study to illustrate the performance of a complete case analysis with a right-censored covariate under different censoring mechanism assumptions, and we demonstrate its use with a Huntington disease data example.

BACKGROUND:Wastewater monitoring data can be used to estimate disease trends to inform public health responses. One commonly estimated metric is the rate of change in pathogen quantity, which typically correlates with clinical surveillance in retrospective analyses. However, the accuracy of rate of change estimation approaches has not previously been evaluated. OBJECTIVES:We assessed the performance of approaches for estimating rates of change in wastewater pathogen loads by generating synthetic wastewater time series data for which rates of change were known. Each approach was also evaluated on real-world data. METHODS:Smooth trends and their first derivatives were jointly sampled from Gaussian processes (GP) and independent errors were added to generate synthetic viral load measurements; the range hyperparameter and error variance were varied to produce nine simulation scenarios representing different potential disease patterns. The directions and magnitudes of the rate of change estimates from four estimation approaches (two established and two developed in this work) were compared to the GP first derivative to evaluate classification and quantitative accuracy. Each approach was also implemented for public SARS-CoV-2 wastewater monitoring data collected January 2021-May 2023 at 25 sites in North Carolina, USA. RESULTS:All four approaches inconsistently identified the correct direction of the trend given by the sign of the GP first derivative. Across all nine simulated disease patterns, between a quarter and a half of all estimates indicated the wrong trend direction, regardless of estimation approach. The proportion of trends classified as plateaus (statistically indistinguishable from zero) for the North Carolina SARS-CoV-2 data varied considerably by estimation method but not by site. DISCUSSION:Our results suggest that wastewater measurements alone might not provide sufficient data to reliably track disease trends in real-time. Instead, wastewater viral loads could be combined with additional public health surveillance data to improve predictions of other outcomes.

BACKGROUND AND OBJECTIVES:Prospective measures of plasma and cerebral MRI biomarkers of Alzheimer disease (AD) and vascular neuropathology provide an opportunity to investigate possible mechanisms linking liver disease and dementia. We aimed to quantify the association of midlife nonalcoholic fatty liver disease (NAFLD) with change in plasma and brain MRI biomarkers of AD and vascular neuropathology. METHODS:We included participants from the Atherosclerosis Risk in Communities Study with brain MRI measurements of white matter hyperintensity (WMH) volume and temporal-parietal lobe cortical thickness meta region of interest (ROI) at up to 2 different visits, in 2011-13 and 2016-19, and plasma biomarkers of β-amyloid (Aβ)42:40, phosphorylated tau at threonine 181, and neurofilament light (NfL) were measured up to 3 times in 1993-95, 2011-13, and 2016-19. NAFLD was categorized using the fatty liver index in 1990-92. Multivariate linear regression was performed for associations between midlife NAFLD and change in plasma and brain MRI biomarkers of AD and vascular neuropathology. The primary models adjusted for demographics, Apolipoprotein E, alcohol use, and kidney function. RESULTS:Among 1,706 participants (mean age 56 years, 62% female, 28% Black), midlife NAFLD vs no NAFLD was associated with greater late-life WMH volume (difference per SD 0.19, 95% CI 0.06-0.31) and faster late-life WMH increase over 6 years (difference in annual change, SD 0.28, 95% CI 0.05-0.51), suggesting accumulating vascular pathology. Midlife NAFLD vs no NAFLD was also associated with AD biomarkers in midlife (lower Aβ42:40 [SD -0.21, 95% CI -0.39 to -0.04] measured in 1993-95) and late life (lower Aβ42:40 [SD -0.13, 95% CI -0.23 to -0.03] and lower temporal-parietal lobe cortical thickness meta ROI [SD -0.16, 95% CI -0.28 to -0.05] measured in 2011-13). Although midlife NfL was lower in individuals with vs without midlife NAFLD, those with NAFLD exhibited a faster rate of NfL increase that accelerated over time. DISCUSSION:Midlife NAFLD shows associations with AD and accumulating vascular pathology, revealing potential pathways linking liver function to dementia. Plasma biomarkers of neuropathology and neuronal injury may serve as easily measurable and dynamic indicators for monitoring the impacts of impaired liver function on brain health.

OBJECTIVE: To evaluate whether hypertensive disorders of pregnancy, including gestational hypertension, preeclampsia, and eclampsia, are associated with cognitive decline later in life among U.S. Hispanic/Latina individuals. METHODS: The HCHS/SOL (Hispanic Community Health Study/Study of Latinos) is a prospective population-based study of Hispanic/Latino individuals aged 18–74 years from four U.S. communities. This analysis included parous individuals aged 45 years or older who participated in the HCHS/SOL clinic study visit 1 (2008–2011) neurocognitive assessment and subsequently completed a repeat neurocognitive assessment as part of the Study of Latinos–Investigation of Neurocognitive Aging ancillary study visit 2 (2015–2018). Hypertensive disorders of pregnancy were assessed retrospectively by self-report of any gestational hypertension, preeclampsia, or eclampsia. Cognitive functioning was measured at both study visits with the Brief Spanish-English Verbal Learning Test, Digit Symbol Substitution, and Word Fluency. A regression-based approach was used to define cognitive decline at visit 2 as a function of cognition at visit 1 after adjustment for age, education, and follow-up time. Linear regression models were used to determine whether hypertensive disorders of pregnancy or their component diagnoses were associated with standardized cognitive decline after adjustment for sociodemographic characteristics, clinical and behavioral risk factors, and follow-up time. RESULTS: Among 3,554 individuals included in analysis, the mean age was 56.2 years, and 467 of individuals (13.4%) reported at least one hypertensive disorder of pregnancy. Individuals with hypertensive disorders of pregnancy compared with those without were more likely to have higher mean systolic blood pressure, fasting glucose, and body mass index. After an average of 7 years of follow-up, in fully adjusted models, gestational hypertension was associated with a 0.17-SD relative decline in Digit Symbol Substitution scores (95% CI, −0.31 to −0.04) but not other cognitive domains (Brief Spanish-English Verbal Learning Test or Word Fluency). Neither preeclampsia nor eclampsia was associated with neurocognitive differences. CONCLUSION: The presence of preeclampsia or eclampsia was not associated with interval neurocognitive decline. In this cohort of U.S. Hispanic/Latina individuals, gestational hypertension alone was associated with decreased processing speed and executive functioning later in life.

The landscape of survival analysis is constantly being revolutionized to answer biomedical challenges, most recently the statistical challenge of censored covariates rather than outcomes. There are many promising strategies to tackle censored covariates, including weighting, imputation, maximum likelihood, and Bayesian methods. Still, this is a relatively fresh area of research, different from the areas of censored outcomes (i.e., survival analysis) or missing covariates. In this review, we discuss the unique statistical challenges encountered when handling censored covariates and provide an in-depth review of existing methods designed to address those challenges. We emphasize each method's relative strengths and weaknesses, providing recommendations to help investigators pinpoint the best approach to handling censored covariates in their data.

While right-censored time-to-event outcomes have been studied for decades, handling time-to-event covariates, also known as right-censored covariates, is now of growing interest. So far, the literature has treated right-censored covariates as distinct from missing covariates, overlooking the potential applicability of estimators to both scenarios. We bridge this gap by establishing connections between right-censored and missing covariates under various assumptions about censoring and missingness, allowing us to identify parallels and differences to determine when estimators can be used in both contexts. These connections reveal adaptations to five estimators for right-censored covariates in the unexplored area of informative covariate right-censoring and to formulate a new estimator for this setting, where the event time depends on the censoring time. We establish the asymptotic properties of the six estimators, evaluate their robustness under incorrect distributional assumptions, and establish their comparative efficiency. We conducted a simulation study to confirm our theoretical results, and then applied all estimators to a Huntington disease observational study to analyze cognitive impairments as a function of time to clinical diagnosis.

In Huntington's disease research, a current goal is to understand how symptoms change prior to a clinical diagnosis. Statistically, this entails modeling symptom severity as a function of the covariate 'time until diagnosis', which is often heavily right-censored in observational studies. Existing estimators that handle right-censored covariates have varying statistical efficiency and robustness to misspecified models for nuisance distributions (those of the censored covariate and censoring variable). On one extreme, complete case estimation, which utilizes uncensored data only, is free of nuisance distribution models but discards informative censored observations. On the other extreme, maximum likelihood estimation is maximally efficient but inconsistent when the covariate's distribution is misspecified. We propose a semiparametric estimator that is robust and efficient. When the nuisance distributions are modeled parametrically, the estimator is doubly robust, i.e., consistent if at least one distribution is correctly specified, and semiparametric efficient if both models are correctly specified. When the nuisance distributions are estimated via nonparametric or machine learning methods, the estimator is consistent and semiparametric efficient. We show empirically that the proposed estimator, implemented in the R package sparcc, has its claimed properties, and we apply it to study Huntington's disease symptom trajectories using data from the Enroll-HD study.

INTRODUCTION:We examined midlife (1990-1992, mean age 57) and late-life (2011-2013, mean age 75) nonalcoholic fatty liver disease (NAFLD) and aminotransferase with incident dementia risk through 2019 in the Atherosclerosis Risk in Communities (ARIC) Study. METHODS:We characterized NAFLD using the fatty liver index and fibrosis-4, and we categorized aminotransferase using the optimal equal-hazard ratio (HR) approach. We estimated HRs for incident dementia ascertained from multiple data sources. RESULTS:Adjusted for demographics, alcohol consumption, and kidney function, individuals with low, intermediate, and high liver fibrosis in midlife (HRs: 1.45, 1.40, and 2.25, respectively), but not at older age, had higher dementia risks than individuals without fatty liver. A U-shaped association was observed for alanine aminotransferase with dementia risk, which was more pronounced in late-life assessment. DISCUSSION:Our findings highlight dementia burden in high-prevalent NAFLD and the important feature of late-life aminotransaminase as a surrogate biomarker linking liver hypometabolism to dementia. Highlights Although evidence of liver involvement in dementia development has been documented in animal studies, the evidence in humans is limited. Midlife NAFLD raised dementia risk proportionate to severity. Late-life NAFLD was not associated with a high risk of dementia. Low alanine aminotransferase was associated with an elevated dementia risk, especially when measured in late life.

Objective: In longitudinal studies, devices used to measure exposures, like pulse wave velocity (PWV), can change from visit to visit. Calibration studies, where a subset of participants receive measurements from both devices at follow-up, are often used to assess differences in the device measurements. Regression calibration and multiple imputation are common statistical methods to correct for those differences, but no study yet exists to compare the two when the quantity of interest is change in the exposure over time. We compared both methods in a hypothetical study of change in PWV and its association with total brain volume. Methods: We simulated true values of PWV at baseline and follow up, as well as imperfect measurements of PWV using an “old” device and “new” device. Two statistical methods were compared: regression calibration , which calibrates the new device measurements at follow up to the old device using linear regression in a calibration study; and multiple imputation , which imputes the (mostly) missing old device measurements at follow up.We varied the bias and measurement error of each device and for each scenario simulated 1,000 datasets of size n=2,500. Two percent of participants in each iteration were chosen to participate in the calibration study, and thus had measurements on the old and new devices at follow up. We used 200 bootstrap replicates to calculate the standard errors for the regression calibration method and 50 imputed datasets for the multiple imputation method. To compare the methods we used bias of the estimated association and how well the standard errors approximated the empirical standard errors. Results: Regression calibration was virtually unbiased for the association between change in PWV and total brain volume when the old device had larger measurement error than the new device. The maximum bias for regression calibration across all scenarios was still small (6%). When the old device had more measurement error or the two devices had equal measurement error, multiple imputation underestimated the association by more than 10%. This underestimation was reduced to approximately 2% when the new device had a larger measurement error than the old device. In all scenarios, regression calibration underestimated the empirical standard error by approximately 35%, while multiple imputation underestimated it by only 2-5%. Conclusions: In analyses of change in PWV and total brain volume, when unbiased estimation is the main objective, regression calibration is favorable to multiple imputation. When null hypothesis significance testing is the main objective, multiple imputation may be favorable in order to not underestimate the standard errors. We expect these conclusions to apply to other change in exposure and outcome relationships with similar ratios between the association’s magnitude and the amount of measurement error.

A positive definite estimator of a covariance matrix with zero entries provides a valid covariance matrix that can be used an input in almost any area of multivariate statistical analysis. However, most current approaches do not yet guarantee positive definiteness or deal with the asymptotic efficiency of the covariance estimator. Focusing on the classical setting when the number of Gaussian variables is fixed and the sample size increases, we construct a positive definite and asymptotically efficient estimator by the iterative conditional fitting algorithm (Chaudhuri et al., 2007) when the location of the zero entries is known. If the location of the zero entries is unknown, we further construct a positive definite thresholding estimator by combining the iterative conditional fitting algorithm with thresholding. We prove our thresholding estimator is asymptotically efficient with probability tending to one. In simulation studies, we show our estimator more closely matches the true covariance and more correctly identifies the non-zero entries than competing estimators. We apply our estimator to a neuroimaging study of Huntington disease to detect non-zero correlations among brain regional volumes. Such correlations are timely for ongoing treatment studies to inform how different brain regions are likely to be affected by these treatments.

Background and Aims: To investigate associations between avocado intake and glycemia in adults with Hispanic/Latino ancestry.Methods and Results: The associations of avocado intake with measures of insulin and glucose homeostasis were evaluated in a cross-sectional analysis of up to 14,591 Hispanic/Latino adults, using measures of: average glucose levels (hemoglobin A1c; HbA1c), fasting glucose and insulin, glucose and insulin levels after an oral glucose tolerance test (OGTT), and calculated measures of insulin resistance (HOMA-IR, and HOMA-%0), and insulinogenic index. Associations were assessed using multivariable linear regression models, which controlled for sociodemographic factors and health behaviors, and which were stratified by dysglycemia status. In those with normoglycemia, avocado intake was associated with a higher insulinogenic index (0 = 0.17 +/- 0.07, P = 0.02).In those with T2D (treated and untreated), avocado intake was associated with lower hemoglobin A1c (HbA1c; 0 =-0.36 +/- 0.21, P = 0.02), and lower fasting glucose (0 =-0.27 +/- 0.12, P = 0.02). In the those with untreated T2D, avocado intake was additionally associated with HOMA-%0 (0 = 0.39 +/- 0.19, P = 0.04), higher insulin values 2-h after an oral glucose load (0 = 0.62 +/- 0.23, P = 0.01), and a higher insulinogenic index (0 = 0.42 +/- 0.18, P = 0.02). No associations were observed in participants with prediabetes.Conclusions: We observed an association of avocado intake with better glucose/insulin homeostasis, especially in those with T2D.(c) 2023 The Author(s). Published by Elsevier B.V. on behalf of The Italian Diabetes Society, the Italian Society for the Study of Atherosclerosis, the Italian Society of Human Nutrition and the Department of Clinical Medicine and Surgery, Federico II University. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

To select outcomes for clinical trials testing experimental therapies for Huntington disease, a fatal neurodegenerative disorder, analysts model how potential outcomes change over time. Yet, subjects with Huntington disease are often observed at different levels of disease progression. To account for these differences, analysts include time to clinical diagnosis as a covariate when modeling potential outcomes, but this covariate is often censored. One popular solution is imputation, whereby we impute censored values using predictions from a model of the censored covariate given other data, then analyze the imputed dataset. However, when this imputation model is misspecified, our outcome model estimates can be biased. To address this problem, we developed a novel method, dubbed "ACE imputation." First, we model imputed values as error-prone versions of the true covariate values. Then, we correct for these errors using semiparametric theory. Specifically, we derive an outcome model estimator that is consistent, even when the censored covariate is imputed using a misspecified imputation model. Simulation results show that ACE imputation remains empirically unbiased even if the imputation model is misspecified, unlike multiple imputation which yields >100% bias. Applying our method to a Huntington disease study pinpoints outcomes for clinical trials aimed at slowing disease progression.

Introduction: Associations of nonalcoholic fatty liver disease (NAFLD) with dementia are controversial. The full spectrum of NAFLD from simple steatosis to fibrosis has been less investigated. Hypothesis: The severity of NAFLD, especially with the stage of liver fibrosis, is associated with dementia. Methods: The associations between midlife (1990-92, N=9283, mean age 57, female 55%) and late-life (2011-13, N=5138, mean age 75, female 58%) NAFLD and all cause dementia were quantified by Cox regression. NAFLD was characterized using the fatty liver index and liver fibrosis assessment model (FIB-4). Dementia was adjudicated and ascertained through Dec. 31, 2019 from neuropsychological assessments, annual participant or informant contact, and medical record surveillance. We used Cox models with adjustment for demographics, APOE ε4, alcohol, and kidney function. Additional adjustments were made for metabolic factors to explore the independent contribution of NAFLD to dementia. Results: During a median follow-up of 24.5 years after midlife NAFLD assessment, 1854 participants developed dementia. During a median of 6.3 years in late-life, there were 893 dementia cases. A graded increase in dementia risk across the spectrum of NAFLD was observed at midlife (p-trend <0.001), but not at older age ( Figure ). In fact, NAFLD in late-life tended to be protective against dementia. After adjusting for the metabolic factors, the associations remained statistically significant in some categories. Conclusions: Remodeling of hepatic architecture and dysregulation in hepatocellular function in NAFLD at midlife plausibly induce peripheral promoters of neurodegeneration and may expose a large segment of the population to increased risk of dementia. The inverse association of NAFLD with dementia at late-life requires further investigation, possibly reflecting the depletion of susceptibles, the reversibility of NAFLD due to weight loss, and the lack of accuracy in identifying NAFLD using clinical prediction models at older age.