Efficient Estimation for Patient-Specific Rates of Disease Progression Using Nonnormal Linear Mixed Models

Rates of Glaucoma Progression Derived from Linear Mixed Models Using Varied Random Effect Distributions

Purpose

To compare the ability of linear mixed models with different random effect distributions to estimate rates of visual field loss in glaucoma patients.

Methods

Eyes with five or more reliable standard automated perimetry (SAP) tests were identified from the Duke Glaucoma Registry. Mean deviation (MD) values from each visual field and associated timepoints were collected. These data were modeled using ordinary least square (OLS) regression and linear mixed models using the Gaussian, Student's t, or log-gamma (LG) distributions as the prior distribution for random effects. Model fit was compared using the Watanabe–Akaike information criterion (WAIC). Simulated eyes of varying initial disease severity and rates of progression were created to assess the accuracy of each model in predicting the rate of change and likelihood of declaring progression.

Results

A total of 52,900 visual fields from 6558 eyes of 3981 subjects were included. Mean follow-up period was 8.7 ± 4.0 years, with an average of 8.1 ± 3.7 visual fields per eye. The LG model produced the lowest WAIC, demonstrating optimal model fit. In simulations, the LG model declared progression earlier than OLS (P < 0.001) and had the greatest accuracy in predicted slopes (P < 0.001). The Gaussian model significantly underestimated rates of progression among fast and catastrophic progressors.

Conclusions

Linear mixed models using the LG distribution outperformed conventional approaches for estimating rates of SAP MD loss in a population with glaucoma.

Translational Relevance

Use of the LG distribution in models estimating rates of change among glaucoma patients may improve their accuracy in rapidly identifying progressors at high risk for vision loss.

Biomarkers of Kidney Tubule Health, CKD Progression, and Acute Kidney Injury in SPRINT (Systolic Blood Pressure Intervention Trial) Participants

RATIONALE & OBJECTIVE

The Systolic Blood Pressure Intervention Trial (SPRINT) compared the effect of intensive versus standard systolic blood pressure targets on cardiovascular morbidity and mortality. In this ancillary study, we evaluated the use of exploratory factor analysis (EFA) to combine biomarkers of kidney tubule health in urine and plasma and then study their role in longitudinal estimated glomerular filtration rate (eGFR) change and risk of acute kidney injury (AKI).

STUDY DESIGN

Observational cohort nested in a clinical trial.

SETTING & PARTICIPANTS

2,351 SPRINT participants with eGFR < 60 mL/min/1.73 m² at baseline.

EXPOSURE

Levels of neutrophil gelatinase-associated lipocalin (NGAL), interleukin 18 (IL-18), chitinase-3-like protein (YKL-40), kidney injury molecule 1 (KIM-1), monocyte chemoattractant protein 1 (MCP-1), α₁-microglobulin (A1M) and β₂-microglobulin (B2M), uromodulin (UMOD), fibroblast growth factor 23 (FGF-23), and intact parathyroid hormone (PTH).

OUTCOME

Longitudinal changes in eGFR and risk of AKI.

ANALYTICAL APPROACH

We performed EFA to capture different tubule pathophysiologic processes. We used linear mixed effects models to evaluate the association of each factor with longitudinal changes in eGFR. We evaluated the association of the tubular factors scores with AKI using Cox proportional hazards regression.

RESULTS

From 10 biomarkers, EFA generated 4 factors reflecting tubule injury/repair (NGAL, IL-18, and YKL-40), tubule injury/fibrosis (KIM-1 and MCP-1), tubule reabsorption (A1M and B2M), and tubule reserve/mineral metabolism (UMOD, FGF-23, and PTH). Each 1-SD higher tubule reserve/mineral metabolism factor score was associated with a 0.58% (95% CI, 0.39%-0.67%) faster eGFR decline independent of baseline eGFR and albuminuria. Both the tubule injury/repair and tubule injury/fibrosis factors were independently associated with future risk of AKI (per 1 SD higher, HRs of 1.18 [95% CI, 1.10-1.37] and 1.23 [95% CI, 1.02-1.48], respectively).

LIMITATIONS

The factors require validation in other settings.

CONCLUSIONS

EFA allows parsimonious subgrouping of biomarkers into factors that are differentially associated with progressive eGFR decline and AKI. These subgroups may provide insights into the pathological processes driving adverse kidney outcomes.

Two-part models for repeatedly measured ordinal data with "don't know" category

Ordinal data (eg, "low," "medium," "high"; graded response on a Likert scale) with an additional "don't know" category are frequently encountered in the medical, social, and behavioral science literature. The handling of a "don't know" option presents unique challenges as it often "destroys" the ordinal nature of the data. Commonly, nominal models are employed which ignore the partial ordering and have a complicated interpretation, especially in situations with repeatedly measured outcomes. We propose two-part models that easily accommodate longitudinal partially ordered (semiordinal) data. The most easily interpretable formulation consists of a random effect logistic submodel for "don't know" vs all the other categories combined, and a random effect ordinal submodel for the ordered categories. Correlated random effects account for statistical dependence within individual. An extension allowing for nonproportionality of odds for the predictor effects in the ordinal submodel is also considered. Maximum likelihood estimation is performed using adaptive Gaussian quadrature in SAS PROC NLMIXED. A simulation study is performed to evaluate the performance of the estimation algorithm in terms of bias and efficiency, and to compare the results of joint and separate models of the two parts, and of proportional and nonproportional model formulations. The methods are motivated and illustrated on a dataset from a study of adolescents' perceptions of nicotine strength of JUUL e-cigarettes. Using the proposed approach we show that adolescents perceive 5% nicotine content as relatively low, a misconception more pronounced among past month nonusers than among past month users of JUUL e-cigarettes.

Perils and pitfalls of mixed-effects regression models in biology

Biological systems, at all scales of organisation from nucleic acids to ecosystems, are inherently complex and variable. Biologists therefore use statistical analyses to detect signal among this systemic noise. Statistical models infer trends, find functional relationships and detect differences that exist among groups or are caused by experimental manipulations. They also use statistical relationships to help predict uncertain futures. All branches of the biological sciences now embrace the possibilities of mixed-effects modelling and its flexible toolkit for partitioning noise and signal. The mixed-effects model is not, however, a panacea for poor experimental design, and should be used with caution when inferring or deducing the importance of both fixed and random effects. Here we describe a selection of the perils and pitfalls that are widespread in the biological literature, but can be avoided by careful reflection, modelling and model-checking. We focus on situations where incautious modelling risks exposure to these pitfalls and the drawing of incorrect conclusions. Our stance is that statements of significance, information content or credibility all have their place in biological research, as long as these statements are cautious and well-informed by checks on the validity of assumptions. Our intention is to reveal potential perils and pitfalls in mixed model estimation so that researchers can use these powerful approaches with greater awareness and confidence. Our examples are ecological, but translate easily to all branches of biology.

A consistent estimator for logistic mixed effect models

We propose a consistent and locally efficient estimator to estimate the model parameters for a logistic mixed effect model with random slopes. Our approach relaxes two typical assumptions: the random effects being normally distributed, and the covariates and random effects being independent of each other. Adhering to these assumptions is particularly difficult in health studies where in many cases we have limited resources to design experiments and gather data in long-term studies, while new findings from other fields might emerge, suggesting the violation of such assumptions. So it is crucial if we could have an estimator robust to such violations and then we could make better use of current data harvested using various valuable resources. Our method generalizes the framework presented in Garcia & Ma (2016) which also deals with a logistic mixed effect model but only considers a random intercept. A simulation study reveals that our proposed estimator remains consistent even when the independence and normality assumptions are violated. This contrasts from the traditional maximum likelihood estimator which is likely to be inconsistent when there is dependence between the covariates and random effects. Application of this work to a Huntington disease study reveals that disease diagnosis can be further improved using assessments of cognitive performance.

Tissue transcriptome-driven identification of epidermal growth factor as a chronic kidney disease biomarker

Chronic kidney disease (CKD) affects 8 to 16% people worldwide, with an increasing incidence and prevalence of end-stage kidney disease (ESKD). The effective management of CKD is confounded by the inability to identify patients at high risk of progression while in early stages of CKD. To address this challenge, a renal biopsy transcriptome-driven approach was applied to develop noninvasive prognostic biomarkers for CKD progression. Expression of intrarenal transcripts was correlated with the baseline estimated glomerular filtration rate (eGFR) in 261 patients. Proteins encoded by eGFR-associated transcripts were tested in urine for association with renal tissue injury and baseline eGFR. The ability to predict CKD progression, defined as the composite of ESKD or 40% reduction of baseline eGFR, was then determined in three independent CKD cohorts. A panel of intrarenal transcripts, including epidermal growth factor (EGF), a tubule-specific protein critical for cell differentiation and regeneration, predicted eGFR. The amount of EGF protein in urine (uEGF) showed significant correlation (P < 0.001) with intrarenal EGF mRNA, interstitial fibrosis/tubular atrophy, eGFR, and rate of eGFR loss. Prediction of the composite renal end point by age, gender, eGFR, and albuminuria was significantly (P < 0.001) improved by addition of uEGF, with an increase of the C-statistic from 0.75 to 0.87. Outcome predictions were replicated in two independent CKD cohorts. Our approach identified uEGF as an independent risk predictor of CKD progression. Addition of uEGF to standard clinical parameters improved the prediction of disease events in diverse CKD populations with a wide spectrum of causes and stages.

Log-gamma linear-mixed effects models for multiple outcomes with application to a longitudinal glaucoma study

Glaucoma is a progressive disease due to damage in the optic nerve with associated functional losses. Although the relationship between structural and functional progression in glaucoma is well established, there is disagreement on how this association evolves over time. In addressing this issue, we propose a new class of non-Gaussian linear-mixed models to estimate the correlations among subject-specific effects in multivariate longitudinal studies with a skewed distribution of random effects, to be used in a study of glaucoma. This class provides an efficient estimation of subject-specific effects by modeling the skewed random effects through the log-gamma distribution. It also provides more reliable estimates of the correlations between the random effects. To validate the log-gamma assumption against the usual normality assumption of the random effects, we propose a lack-of-fit test using the profile likelihood function of the shape parameter. We apply this method to data from a prospective observation study, the Diagnostic Innovations in Glaucoma Study, to present a statistically significant association between structural and functional change rates that leads to a better understanding of the progression of glaucoma over time.

Statistical methods for modeling repeated measures of maternal environmental exposure biomarkers during pregnancy in association with preterm birth

BACKGROUND

It is of critical importance to evaluate the role of environmental chemical exposures in premature birth. While a number of studies investigate this relationship, most utilize single exposure measurements during pregnancy in association with the outcome. The studies with repeated measures of exposure during pregnancy employ primarily cross-sectional analyses that may not be fully leveraging the power and additional information that the data provide.

METHODS

We examine 9 statistical methods that may be utilized to estimate the relationship between a longitudinal exposure and a binary, non-time-varying outcome. To exemplify these methods we utilized data from a nested case-control study examining repeated measures of urinary phthalate metabolites during pregnancy in association with preterm birth.

RESULTS

The methods summarized may be useful for: 1) Examining sensitive windows of exposure in association with an outcome; 2) Summarizing repeated measures to estimate the relationship between average exposure and an outcome; 3) Identifying acute exposures that may be relevant to the outcome; and 4) Understanding the contribution of temporal patterns in exposure levels to the outcome of interest. In the study of phthalates, changes in urinary metabolites over pregnancy did not appear to contribute significantly to preterm birth, making summary of average exposure across gestation optimal given the current design.

CONCLUSIONS

The methods exemplified may be of great use in future epidemiologic research projects intended to: 1) Elucidate the complex relationships between environmental chemical exposures and preterm birth; 2) Investigate biological mechanisms in prematurity using repeated measures of maternal factors throughout pregnancy; and 3) More generally, address the relationship between a longitudinal predictor and a binary, non-time-varying outcome.

Analysis of risk factors associated with renal function trajectory over time: a comparison of different statistical approaches

BACKGROUND

The most commonly used methods to investigate risk factors associated with renal function trajectory over time include linear regression on individual glomerular filtration rate (GFR) slopes, linear mixed models and generalized estimating equations (GEEs). The objective of this study was to explain the principles of these three methods and to discuss their advantages and limitations in particular when renal function trajectories are not completely observable due to dropout.

METHODS

We generated data from a hypothetical cohort of 200 patients with chronic kidney disease at inclusion and seven subsequent annual measurements of GFR. The data were generated such that both baseline level and slope of GFR over time were associated with baseline albuminuria status. In a second version of the dataset, we assumed that patients systematically dropped out after a GFR measurement of <15 mL/min/1.73 m(2). Each dataset was analysed with the three methods.

RESULTS

The estimated effects of baseline albuminuria status on GFR slope were similar among the three methods when no patient dropped out. When 32.7% dropped out, standard GEE provided biased estimates of the mean GFR slope in normo-, micro- and macroalbuminuric patients. Linear regression on individual slopes and linear mixed models provided slope estimates of the same magnitude, likely because most patients had at least three GFR measurements. However, the linear mixed model was the only method to provide effect estimates on both slope and baseline level of GFR unaffected by dropout.

CONCLUSION

This study illustrates that the linear mixed model is the preferred method to investigate risk factors associated with renal function trajectories in studies, where patients may dropout during the study period because of initiation of renal replacement therapy.

Estimation of covariate effects in generalized linear mixed models with a misspecified distribution of random intercepts and slopes

Generalized linear mixed models with random intercepts and slopes provide useful analyses of clustered and longitudinal data and typically require the specification of the distribution of the random effects. Previous work for models with only random intercepts has shown that misspecifying the shape of this distribution may bias estimates of the intercept, but typically leads to little bias in estimates of covariate effects. Very few papers have examined the effects of misspecifying the joint distribution of random intercepts and slopes. However, simulation results in a recent paper suggest that misspecifying the shape of the random slope distribution can yield severely biased estimates of all model parameters. Using analytic results, simulation studies and fits to example data, this paper examines the bias in parameter estimates due to misspecification of the shape of the joint distribution of random intercepts and slopes. Consistent with results for models with only random intercepts, and contrary to the claims of severe bias in a recent paper, we show that misspecification of the joint distribution typically yields little bias in estimates of covariate effects and is restricted to covariates associated with the misspecified random effects distributions. We also show that misspecification of the distribution of random effects has little effect on confidence interval performance. Coverage rates based on the model-based standard errors from fitted likelihoods were generally quite close to nominal.

Semiparametric Stochastic Modeling of the Rate Function in Longitudinal Studies

In longitudinal biomedical studies, there is often interest in the rate functions, which describe the functional rates of change of biomarker profiles. This paper proposes a semiparametric approach to model these functions as the realizations of stochastic processes defined by stochastic differential equations. These processes are dependent on the covariates of interest and vary around a specified parametric function. An efficient Markov chain Monte Carlo algorithm is developed for inference. The proposed method is compared with several existing methods in terms of goodness-of-fit and more importantly the ability to forecast future functional data in a simulation study. The proposed methodology is applied to prostate-specific antigen profiles for illustration. Supplementary materials for this paper are available online.

Prediction of random effects in linear and generalized linear models under model misspecification

Statistical models that include random effects are commonly used to analyze longitudinal and correlated data, often with the assumption that the random effects follow a Gaussian distribution. Via theoretical and numerical calculations and simulation, we investigate the impact of misspecification of this distribution on both how well the predicted values recover the true underlying distribution and the accuracy of prediction of the realized values of the random effects. We show that, although the predicted values can vary with the assumed distribution, the prediction accuracy, as measured by mean square error, is little affected for mild-to-moderate violations of the assumptions. Thus, standard approaches, readily available in statistical software, will often suffice. The results are illustrated using data from the Heart and Estrogen/Progestin Replacement Study using models to predict future blood pressure values.