Nonlinear Random Effects Models Research Articles

머신러닝과 통계 모형을 이용한 미술품 가격 예측

The art market is comprised of complex cultural, economic, and social characteristics and has recently received a lot of attention. Art transactions are carried out in a variety of ways, of which art auctions account for a large portion. We collected auction data on Post-War and Contemporary Art artists born after 1920 from Artnet and implemented an art price prediction model. In addition to the hedonic price model commonly used in existing studies, price prediction performance was compared using various machine learning models such as regression model, random effects model, regularization model, tree-based model, and SVM. To compare various machine learning models, RMSE and MAE were compared using two cross-validation methods. As a result, the nonlinear random effects model using author information showed the good performance. The impact of explanatory variables on price was analyzed using the optimal model. In addition, we were able to obtain price curves of various patterns according to the passage of time for each author. Among machine learning models that do not use author information, Bagging showed the best performance. As a result of measuring the importance of variables through Bagging, it was found that the size of the work had the most influence in predicting the price, followed by the artist's year of birth, the method of painting, the material with which the painting was painted, and the year of production.

Short-term effect of particulate air pollutant on the risk of tuberculosis outpatient visits: A multicity ecological study in Anhui, China

Tuberculosis (TB) causes about 10 million new cases and 1.5 million deaths every year, as the World Health Organization's Global Tuberculosis Report shows. We aimed to investigate the association between short-term exposure to particulate pollutants and the risk of tuberculosis outpatient visits in 16 cities of Anhui Province, China, during the period 2015–2020, and to provide more evidence for the relationship between tuberculosis and air pollution. In this study, we analyzed the data using a variety of models, including Poisson's generalized linear regression model, distributed lag non-linear model (DLNM) and random effects model. We assessed the relationship between particulate matter exposure and tuberculosis using a 28-day cumulative lag effect for every 10 unit increases in pollutant concentration and 95% CI. We performed stratified analyses for gender and age. Age stratification included <18 years old group, 19–64 years old group and >64 years old group and gender stratification included male group and female group, and then we performed a sensitivity analysis to test the stability of the results. The overall effect values of PM2.5 and PM10 were RR = 1.031 (95%CI: 1.019–1.043) and RR = 1.023 (95%CI: 1.016–1.030), respectively. Estimation for individual cities showed that the strongest association between PM2.5 and the risk of TB outpatient visits was observed in Tongling, with an RR of 1.136 (95%CI: 1.0261–1.2583), also the PM10 in Suzhou, with an RR of 1.0629 (95%CI: 1.0421–1.0842). Stratified analysis showed that particulate matter exposure had a more significant impact on males and the elderly, and that there was little association between particulate matter exposure and TB risk in children and adolescents. Exposure to PM2.5 and PM10 increases the risk of TB outpatient visits. Males and the elderly are vulnerable groups, and different prevention and control strategies should be developed for different groups to reduce the burden of TB.

Variation in Radiologists' Follow-Up Imaging Recommendations for Small Cystic Pancreatic Lesions.

This study aimed to determine the incidence, identify imaging and patient factors, and measure individual radiologist variation associated with follow-up recommendations for small focal cystic pancreatic lesions (FCPLs), a common incidental imaging finding. This institutional review board-approved retrospective study analyzed 146,709 reports from abdominal CTs and MRIs performed in a large academic hospital from July 1, 2016, to June 30, 2018. A trained natural language processing tool identified 4,345 reports with FCPLs, which were manually reviewed to identify those containing one or more <1.5-cm pancreatic cysts. For these patients, patient, lesion, and radiologist features and follow-up recommendations for FCPL were extracted. A nonlinear random-effects model estimated degree of variation in follow-up recommendations across radiologists at department and division levels. Of 2,872 reports with FCPLs < 1.5 cm, 708 (24.7%) had FCPL-related follow-up recommendations. Average patient age was 67 years (SD ± 11). In all, 1,721 (60.0%) reports were for female patients; 59.3% of patients had only one cyst. In multivariable analysis, older patients had slightly lower follow-up recommendation rates (odds ratio [OR]: 0.98 [0.98-1.00] per additional year), and lesions associated with main duct dilatation and septation were more likely to have a follow-up recommendation (ORs: 1.93 [1.11-3.36] and 2.88 [1.89-4.38], respectively). Radiologist years in practice (P= .51), trainee presence (P= .21), and radiologist gender (P= .52) were not associated with increased follow-up recommendations. There was significant interradiologist variation in the Abdominal Imaging Division (P= .04), but not in Emergency Radiology (P= .31) or Cancer Imaging Divisions (P= .29). Interradiologist variation significantly contributes to variability in follow-up imaging recommendations for FCPLs.

Construction of Windows for Pharmacokinetic Sampling

This paper describes a method for the construction of pharmacokinetic sampling windows so that they are around the $D$-optimum time points. Here we consider the situation where a pharmacokinetic (PK) study is accompanied by a dose-finding study in phase I clinical trial. The D-optimal criterion is often used to determine the optimal time for collecting blood samples so that they provide maximum information regarding the population PK parameters. However, collecting blood samples at the D-optimal time points is often difficult. Instead, the sampling time point chosen from a suitable time interval or window can ease the process. The proposed method is conceptually simple and considers the average value and standard deviation of D-optimal time points up to create sampling windows. Random time points can be chosen from these windows then to collect blood samples from the next cohort. The nonlinear random-effects model has been used to model the PK data. Also, we employ the continual reassessment method for dose allocation to the patients. Comparisons of the accuracy and precision for the PK parameter estimates obtained at the D-optimal and random time points are also presented. The results are convincing enough to suggest the proposed method as a useful tool for blood sample collection.

Longitudinal Growth in Single-Word Intelligibility Among Children With Cerebral Palsy From 24 to 96 Months of Age: Effects of Speech-Language Profile Group Membership on Outcomes.

Purpose We examined whether there were differences among speech-language profile groups of children with cerebral palsy (CP) in age of crossing 25%, 50%, and 75% intelligibility thresholds; age of greatest intelligibility growth; rate of intelligibility growth; maximum attained intelligibility at 8 years; and how well intelligibility at 36 months predicts intelligibility at 96 months when group membership is accounted for. Profile groups were children with no speech motor impairment (NSMI), those with speech motor impairment and language comprehension that is typically developing (SMI-LCT), and those with speech motor impairment and language comprehension impairment (SMI-LCI). Method Sixty-eight children with CP were followed longitudinally between 24 and 96 months of age. A total of 564 time points were examined across children (M = 8.3 time points per child, SD = 2.6). We fitted a nonlinear random effects model for longitudinal observations, allowing for differences between profile groups. We used the fitted model trajectories to generate descriptive analyses of intelligibility growth by group and to generate simulations to analyze how well 36-month intelligibility data predicted 96-month data accounting for profile groups. Results Children with CP who have NSMI have different growth and better intelligibility outcomes than those with speech motor impairment. Children with SMI-LCT tend to have better outcomes but similar intelligibility growth as children with SMI-LCI. There may be a subset of children that cut across SMI-LCI and SMI-LCT groups who have severe speech motor involvement and show limited growth in intelligibility. Conclusions Intelligibility outcomes for children with CP are affected by profile group membership. Intelligibility growth tends to be delayed in children with speech motor impairment. Intelligibility at 3 years is highly predictive of later outcomes regardless of profile group. Intervention decision making should include consideration of early intelligibility, and treatment directions should include consideration of augmentative and alternative communication.

Estimation of Dynamic Nonlinear Random Effects Models with Unbalanced Panels

AbstractThis paper presents estimation methods for dynamic nonlinear models with correlated random effects (CRE) when having unbalanced panels. Unbalancedness is often encountered in applied work and ignoring it in dynamic nonlinear models produces inconsistent estimates even if the unbalancedness process is completely at random. We show that selecting a balanced panel from the sample can produce efficiency losses or even inconsistent estimates of the average marginal effects. We allow the process that determines the unbalancedness structure of the data to be correlated with the permanent unobserved heterogeneity. We discuss how to address the estimation by maximizing the likelihood function for the whole sample and also propose a Minimum Distance approach, which is computationally simpler and asymptotically equivalent to the Maximum Likelihood estimation. Our Monte Carlo experiments and empirical illustration show that the issue is relevant. Our proposed solutions perform better both in terms of bias and RMSE than the approaches that ignore the unbalancedness or that balance the sample.

Longitudinal Growth in Single Word Intelligibility Among Children With Cerebral Palsy From 24 to 96 Months of Age: Predicting Later Outcomes From Early Speech Production.

Purpose Children with cerebral palsy (CP) are at risk for significant communication problems. Reduced speech intelligibility is common, even for those who do not have speech motor deficits. Development of intelligibility has not been comprehensively quantified in children with CP; as a result, we are currently unable to predict later speech outcomes. Such information would advance treatment decision making. We sought to examine growth in speech intelligibility among children with CP using a prospective longitudinal design, with a focus on age of crossing target intelligibility thresholds, age of greatest intelligibility growth, and how well intelligibility at 36 months predicted intelligibility at 96 months. Method Sixty-nine children with CP were followed longitudinally between 24 and 96 months of age. A total of 566 time points were examined across children ( M = 8.2 time points per child, SD = 2.6). We fitted a nonlinear random effects model for longitudinal observations and then used the fitted model trajectories to generate descriptive analyses of growth. We used results of the model to generate a set of simulations, which we analyzed to determine how well 36-month intelligibility data predicted 96-month data. Results Half of children crossed 25% and 50% intelligibility thresholds at 36 and 49 months of age, respectively. Slightly more than half of children did not reach 75% intelligibility by 96 months of age. Age of crossing 25%, 50%, and 75% intelligibility thresholds was highly negatively correlated with intelligibly at 96 months. Children had the steepest intelligibility growth at 36 months, followed by 48 and 60 months. Intelligibility at 36 months was highly predictive of intelligibility at 96 months. Conclusions The developmental window from 3 to 5 years constitutes a time of rapid growth in speech intelligibility in children with CP. Children who cross intelligibility thresholds of 25%, 50%, and 75% at earlier ages have better outcomes when they are older; early performance is highly predictive of later speech intelligibility outcomes. Children with CP as a group have delayed speech intelligibility development but are still growing through 96 months of age.

A Bayesian nonlinear random effects model for identification of defective batteries from lot samples

Abstract Numerous materials and processes go into the manufacture of lithium-ion batteries, resulting in variations across batteries' capacity fade measurements. Accounting for this variability is essential when determining whether batteries are performing satisfactorily. Motivated by a real manufacturing problem, this article presents an approach to assess whether lithium-ion batteries from a production lot are not representative of a healthy population of batteries from earlier production lots, and to determine, based on capacity fade data, the earliest stage (in terms of cycles) that battery anomalies can be identified. The approach involves the use of a double exponential function to describe nonlinear capacity fade data. To capture the variability of repeated measurements on a number of individual batteries, the double exponential function is then embedded as the individual batteries' trajectories in a Bayesian random effects model. The model allows for probabilistic predictions of capacity fading not only at the underlying mean process level but also at the individual battery level. The results show good predictive coverage for individual batteries and demonstrate that, for our data, non-healthy lithium-ion batteries can be identified in as few as 50 cycles.

Semiparametric models of longitudinal and time-to-event data with applications to HIV viral dynamics and CD4 counts

We propose a semiparametric approach based on proportional hazards and copula method to jointly model longitudinal outcomes and the time-to-event. The dependence between the longitudinal outcomes on the covariates is modeled by a copula-based times series, which allows non-Gaussian random effects and overcomes the limitation of the parametric assumptions in existing linear and nonlinear random effects models. A modified partial likelihood method using estimated covariates at failure times is employed to draw statistical inference. The proposed model and method are applied to analyze a set of progression to AIDS data in a study of the association between the human immunodeficiency virus viral dynamics and the time trend in the CD4/CD8 ratio with measurement errors. Simulations are also reported to evaluate the proposed model and method.

Alcohol drinking and cutaneous melanoma risk: a systematic review and dose-risk meta-analysis

It has been suggested that alcohol intake increases sunburn severity, a major risk factor for cutaneous melanoma (CM). Several epidemiological studies have investigated the relationship between alcohol consumption and CM, but the evidence is inconsistent. Therefore, we aimed to quantify this relationship better, using a meta-analytical approach. The dose-risk relationship was also modelled through a class of flexible nonlinear meta-regression random effects models. The present meta-analysis included 16 studies (14 case-control and two cohort investigations) with a total of 6251 cases of CM. The pooled relative risk (RR) for any alcohol drinking compared with no/occasional drinking was 1·20 [95% confidence interval (CI) 1·06-1·37]. The risk estimate was similar in case-control (RR 1·20, 95% CI 1·01-1·44) and cohort studies (RR 1·26, 95% CI 1·19-1·35). The pooled RR was 1·10 (95% CI 0·96-1·26) for light alcohol drinking (≤1 drink per day) and 1·18 (95% CI 1·01-1·40) for moderate-to-heavy drinking. The pooled RR from 10 studies adjusting for sun exposure was 1·15 (95% CI 0·94-1·41), while the RR from six unadjusted studies was 1·27 (95% CI 1·20-1·35). No evidence of publication bias was detected. This meta-analysis of published data reveals that alcohol consumption is positively associated with the risk of CM. However, caution in interpreting these results is required, as residual confounding by sun exposure cannot be ruled out.

Early life growth trajectories and future risk for overweight

Objective:Standard approaches have found that rapid growth during the first 2 years of life is a risk factor for overweight in later childhood. Our objective was to test whether growth velocity, independent of concurrent size, was associated with overweight using a nonlinear random-effects model that allows for enhanced specifications and estimations.Methods:Longitudinal data from a birth cohort in Mexico (n=586) were used to estimate growth trajectories over 0–24 months for body mass index (BMI), length and weight using the SuperImposition by Translation and Rotation (SITAR) models. The SITAR models use a nonlinear random-effects model to estimate an average growth curve for BMI, length and weight and each participant's deviation from this curve on three dimensions—size, velocity and timing of peak velocity. We used logistic regression to estimate the association between overweight status at 7–9 years and size, velocity and timing of BMI, length and weight trajectories during 0–24 months. We tested whether any association between velocity and overweight varied by relative size during 0–24 months or birth weight.Results:SITAR models explained the majority of the variance in BMI (73%), height (86%) and weight (85%) between 0–24 months. When analyzed individually, relative BMI/length/weight (size) and BMI/length/weight velocity during 0–24 months were each associated with increased odds of overweight in late childhood. Associations for timing of peak velocity varied by anthropometric measure. However, in the mutually adjusted models, only relative BMI/length/weight (size) remained statistically significant. We found no evidence that any association between velocity and overweight varied by size during 0–24 months or birth weight.Conclusions:After mutual adjustment, size during 0–24 months of life (as opposed to birth size), but not velocity or timing of peak velocity, was most consistently associated with overweight in later childhood.

Statistical methods in randomised controlled trials for delirium

ObjectiveThe analysis of clinical trials in delirium is typically complicated by treatment dropouts and by the fact that even untreated individuals may have a good prognosis. These features of delirium trials warrant special statistical attention; implications for each stage of a trial planning process are described. MethodsChoice of outcome, patient sample, and data collection in delirium trials are discussed. Descriptions are given, together with examples, of time-to-event, imputation-based, linear and nonlinear models for the analysis of randomised controlled trials for delirium. ResultsImputation allows evaluation of the plausibility of individual recovery trajectories, but some simple imputations are found to be unsuitable for delirium research. Time-to-event and nonlinear models encourage a global perspective on analysis, which is often preferable to cross-sectional end-of-trial assessments. It is suggested that nonlinear random effects models for longitudinal trajectories are particularly suitable in a delirium context. ConclusionIt is hoped that the methods described, and nonlinear models in particular, will play a part in convincing analyses of future delirium research.

A Geometric Characterization of Optimal Designs for Regression Models with Correlated Observations

We consider the problem of optimal design of experiments for random effects models, especially population models, where a small number of correlated observations can be taken on each individual, while the observations corresponding to different individuals are assumed to be uncorrelated. We focus on c-optimal design problems and show that the classical equivalence theorem and the famous geometric characterization of Elfving (1952) from the case of uncorrelated data can be adapted to the problem of selecting optimal sets of observations for the n individual patients. The theory is demonstrated by finding optimal designs for a linear model with correlated observations and a nonlinear random effects population model, which is commonly used in pharmacokinetics.

Sickness Absence, Employment History, and High-Risk Employees: A 10-Year Longitudinal Study

To investigate the impact of employment characteristics on the occurrence of long-term sickness absence taking socioeconomic covariates into account. A longitudinal analysis was conducted for 62,745 employees aged 18 to 64 during 1990-1999. A nonlinear random-effects model was used. In particular, the missing data patterns and their relationship with sickness absence were studied. Strong relationship of socioeconomic background with occurrence of long-term sickness absence (≥ 3 days) was found. The missing data patterns significantly predicted sickness absence for female employees. In subsequent analysis, the discontinuous employment history with high occurrence of short-term sickness absence (<3 days) appeared to be strongly linked to long-term sickness absence in young women. An investigation of missing values and short-term sickness absence as predictors for long-term sickness absence uncovered that young women with irregular employment history constituted high-risk employees.

Meta‐analysis of diagnostic accuracy studies accounting for disease prevalence: Alternative parameterizations and model selection

In a meta-analysis of diagnostic accuracy studies, the sensitivities and specificities of a diagnostic test may depend on the disease prevalence since the severity and definition of disease may differ from study to study due to the design and the population considered. In this paper, we extend the bivariate nonlinear random effects model on sensitivities and specificities to jointly model the disease prevalence, sensitivities and specificities using trivariate nonlinear random-effects models. Furthermore, as an alternative parameterization, we also propose jointly modeling the test prevalence and the predictive values, which reflect the clinical utility of a diagnostic test. These models allow investigators to study the complex relationship among the disease prevalence, sensitivities and specificities; or among test prevalence and the predictive values, which can reveal hidden information about test performance. We illustrate the proposed two approaches by reanalyzing the data from a meta-analysis of radiological evaluation of lymph node metastases in patients with cervical cancer and a simulation study. The latter illustrates the importance of carefully choosing an appropriate normality assumption for the disease prevalence, sensitivities and specificities, or the test prevalence and the predictive values. In practice, it is recommended to use model selection techniques to identify a best-fitting model for making statistical inference. In summary, the proposed trivariate random effects models are novel and can be very useful in practice for meta-analysis of diagnostic accuracy studies.

Abstract 4396: Clinical Response to Recombinant Acid α-Glucosidase is Predicted by Cardiac Outcome Measures in Children with Pompe Disease

Background : Pompe disease is an autosomal-recessive glycogen storage disease due to deficiency of acid α-glucosidase (GAA). Clinical findings in infantile onset disease are due to glycogen accumulation leading to hypertrophic cardiomyopathy, skeletal myopathy, respiratory failure and death. Enzyme replacement therapy (ERT) with recombinant human GAA (rhGAA; Myozyme®) has been shown to prolong survival in infants. We assessed the hypothesis that myocardial response to rhGAA predicts clinical outcome in Pompe. Methods: Echos during one year of ERT were reviewed from patients enrolled in open label trials of rhGAA (n=38). A standardized echo protocol was used at specific time points (wks 0, 4, 8, 12, 26, 38, 52). A central reader made measurements off-line, blinded to study information. Variables including LV mass, volumes, and ejection fraction (EF) were determined. Clinical outcomes at study-end were analyzed in 2 groups: Group A, alive and free of assisted ventilation (AV) or Group B, died or requiring AV. Z-scores were calculated for all echo variables and median z-scores between outcome groups at wk 0 and wk 52 were compared using median scores tests. Non-linear random effects models were used to examine changes over time. Results: Mean age at first infusion = 10.9±9.8 months (range 1.2 – 43.1). At study onset, 7/38 received AV. At study end, mean time to last infusion = 115±44 weeks with 16/38 in Group A and 22/38 in Group B. For Group A vs. Group B, median z-scores at week 0 for LV mass = 6.11 vs. 7.41, mass to volume ratio (MVR) = 4.5 vs. 8.8, and EF = 0.8 for both groups; p≥ 0.12 for each measure. Median z-scores for the same outcome groups, respectively, at week 52 for LV mass = 0.57 vs. 3.41 (p=0.06), for MVR = 0.48 vs. 3.54 (p=0.001) and EF = −0.6 in both groups. In the subgroup not requiring AV at study onset (n=31), MVR is higher at all times in Group B (p≤ 0.03). Conclusion: rhGAA therapy can prolong life in young patients with Pompe, improving outcome compared to an untreated natural history cohort. Even with ERT, there is significant morbidity and mortality. Diminished myocardial response with persistence of significant LV hypertrophy predicts progression to poor clinical outcomes. These and other predictors are required to fully understand response to ERT in Pompe disease.

Slower CD4 cell decline following cessation of a 3 month course of HAART in primary HIV infection: findings from an observational cohort

To investigate the effect of a short course of HAART during primary HIV infection (PHI) on rate of CD4 cell and viral load change. Data following HAART cessation from 89 individuals (seroconverting 1999-2003) who chose to take a 3 month course of HAART at PHI were compared with 179 untreated controls in CASCADE, using linear and nonlinear random effects models. Participants were non-randomized but frequency matched for age, sex, risk factor, year of seroconversion and presentation within the first 6 months of seroconversion. Time to CD4 cell count < 350 cells/microl and initiation of clinically indicated antiretroviral therapy (ART) were also compared as competing risks. The rate of CD4 cell decline following therapy cessation appeared significantly slower among treated participants than untreated controls: losses of 51 cells/microl [95% confidence interval (CI), 32-69] and 77 cells/microl (95% CI, 65-89), respectively, 3 years after seroconversion (P = 0.011). Based on extrapolated data, viral loads also differed significantly at this point (4.09 and 4.53 copies/ml, respectively). At 2 years, there was no significant difference in mean viral load levels: 4.31 copies/ml (95% CI, 4.14-4.48) and 4.47 copies/ml (95% CI, 4.28-4.66), respectively. CASCADE seroconverters were more likely to reach CD4 cell count < 350 cells/microl or initiate clinically indicated ART (hazard ratio, 1.45; 95% CI, 1.02-2.05; P = 0.039). A short course of ART at PHI may delay CD4 cell decline. Confirmation of this requires a randomized clinical trial powered to address definitively the role of ART intervention in PHI (currently underway through SPARTAC).

Nonlinear random effects mixture models: Maximum likelihood estimation via the EM algorithm

Nonlinear random effects models with finite mixture structures are used to identify polymorphism in pharmacokinetic/pharmacodynamic (PK/PD) phenotypes. An EM algorithm for maximum likelihood estimation approach is developed and uses sampling-based methods to implement the expectation step, that results in an analytically tractable maximization step. A benefit of the approach is that no model linearization is performed and the estimation precision can be arbitrarily controlled by the sampling process. A detailed simulation study illustrates the feasibility of the estimation approach and evaluates its performance. Applications of the proposed nonlinear random effects mixture model approach to other population PK/PD problems will be of interest for future investigation.

Systematic review on the recurrence of postoperative nausea and vomiting after a first episode in the recovery room – implications for the treatment of PONV and related clinical trials

BackgroundDespite the presence of a plethora of publications on the prevention of postoperative nausea and vomiting (PONV) only little is known how to treat established symptoms. Besides the high effort of performing these efficacy trials (much more patients must give their consent than are actually included in a study) and ethical concerns, little is known about the rate of re-occurring PONV/vomiting after placebo. As a consequence investigators will have difficulties defining a clinically relevant effect for the new treatment which is crucial for any planning. A quantitative systematic review was performed in order to provide more reliable estimates of the incidence of re-occurring PONV/vomiting after placebo and to help investigators defining a clinically relevant treatment effect.MethodsA systematic search of the literature was performed using an extended search strategy of a previous review. Data on the recurrence of PONV (any nausea or emetic symptom) and vomiting (retching or vomiting) was extracted from published reports treating PONV with placebo and unpublished results from two observational trials where no treatment was given. A nonlinear random effects model was used to calculate estimates of the recurrence of symptoms and their 95%-confidence intervals (95%-CI).ResultsA total of 29 trials (including the unpublished data) were eligible for the calculations. Depending on the length of observation after administering placebo or no treatment the recurrence rate of PONV was between 65% (95%-CI: 53%...75%) and 84% (95%-CI: 73%...91%) and that of vomiting was between 65% (95%-CI: 44%...81%) and 78% (95%-CI: 59%...90%).ConclusionAlmost all trials showed a considerable and consistently high rate of recurrence of emetic symptoms after placebo highlighting the need for a consequent antiemetic treatment. Future (placebo) controlled efficacy trials may use the presented empirical estimates for defining clinically relevant effects and for statistical power considerations.

Non-linear random effects model for multivariate responses with missing data

The use of random-effects models for the analysis of longitudinal data with missing responses has been discussed by several authors. In this paper, we extend the non-linear random-effects model for a single response to the case of multiple responses, allowing for arbitrary patterns of observed and missing data. Parameters for this model are estimated via the EM algorithm and by the first-order approximation available in SAS Proc NLMIXED. The set of equations for this estimation procedure is derived and these are appropriately modified to deal with missing data. The methodology is illustrated with an example using data coming from a study involving 161 pregnant women presenting to a private obstetrics clinic in Santiago, Chile.