Regression with autocorrelated data: A study of time trend of global infant mortality rate

The infant mortality rate (IMR) is considered to be one of the most important indices of a country's well-being. Countries around the world and other health organizations like the World Health Organization are dedicating their resources, knowledge and energy to reduce the infant mortality rates. The well-known Millennium Development Goal 4 (MDG 4), whose aim is to archive a two thirds reduction of the under-five mortality rate between 1990 and 2015, is an example of the commitment. ^ In this study our goal is to model the trends of IMR between the 1950s to 2010s for selected countries. We would like to know how the IMR is changing overtime and how it differs across countries. ^ IMR data collected over time forms a time series. The repeated observations of IMR time series are not statistically independent. So in modeling the trend of IMR, it is necessary to account for these correlations. We proposed to use the generalized least squares method in general linear models setting to deal with the variance-covariance structure in our model. In order to estimate the variance-covariance matrix, we referred to the time-series models, especially the autoregressive and moving average models. Furthermore, we will compared results from general linear model with correlation structure to that from ordinary least squares method without taking into account the correlation structure to check how significantly the estimates change.^

Correlates of tobacco cessation counseling among Hispanic physicians in the US: a cross-sectional survey study.

BACKGROUND: Physician advice is an important motivator for attempting to stop smoking. However, physicians' lack of intervention with smokers has only modestly improved in the last decade. Although the literature includes extensive research in the area of the smoking intervention practices of clinicians, few studies have focused on Hispanic physicians. The purpose of this study was to explore the correlates of tobacco cessation counseling practices among Hispanic physicians in the US. METHODS: Data were collected through a validated survey instrument among a cross-sectional sample of self-reported Hispanic physicians practicing in New Mexico, and who were members of the New Mexico Hispanic Medical Society in the year 2001. Domains of interest included counseling practices, self-efficacy, attitudes/responsibility, and knowledge/skills. Returned surveys were analyzed to obtain frequencies and descriptive statistics for each survey item. Other analyses included: bivariate Pearson's correlation, factorial ANOVAs, and multiple linear regressions. RESULTS: Respondents (n = 45) reported a low level of compliance with tobacco control guidelines and recommendations. Results indicate that physicians' familiarity with standard cessation protocols has a significant effect on their tobacco-related practices (r = .35, variance shared = 12%). Self-efficacy and gender were both significantly correlated to tobacco related practices (r = .42, variance shared = 17%). A significant correlation was also found between self-efficacy and knowledge/skills (r = .60, variance shared = 36%). Attitudes/responsibility was not significantly correlated with any of the other measures. CONCLUSION: More resources should be dedicated to training Hispanic physicians in tobacco intervention. Training may facilitate practice by increasing knowledge, developing skills and, ultimately, enhancing feelings of self-efficacy.

Hispanic physicians' tobacco intervention practices: a cross-sectional survey study

Background U.S. Hispanic physicians constitute a considerable professional collective, and they may be most suited to attend to the health education needs of the growing U.S. Hispanic population. These educational needs include tobacco use prevention and smoking cessation. However, there is a lack of information on Hispanic physicians' tobacco intervention practices, their level of awareness and use of cessation protocols, and the type of programs that would best address their tobacco training needs. The purpose of this study was to assess the tobacco intervention practices and training needs of Hispanic physicians. Methods Data was collected through a validated survey instrument among a cross-sectional sample of self-reported Hispanic physicians. Data analyses included frequencies, descriptive statistics, and factorial analyses of variance. Results The response rate was 55.5%. The majority of respondents (73.3%) were middle-age males. Less than half of respondents routinely performed the most basic intervention: asking patients about smoking status (44.4%) and advising smoking patients to quit (42.2%). Twenty-five percent assisted smoking patients by talking to them about the health risks of smoking, providing education materials or referring them to cessation programs. Only 4.4% routinely arranged follow-up visits or phone calls for smoking patients. The majority of respondents (64.4%) indicated that they prescribe cessation treatments to less than 20% of smoking patients. A few (4.4%) routinely used behavioral change techniques or programs. A minority (15.6%) indicated that they routinely ask their patients about exposure to tobacco smoke, and 6.7% assisted patients exposed to secondhand smoke in understanding the health risks associated with environmental tobacco smoke (ETS). The most frequently encountered barriers preventing respondents from intervening with patients who smoke included: time, lack of training, lack of receptivity by patients, and lack of reimbursement by third party payers. There was no significant main effect of type of physician, nor was there an interaction effect (gender by type of physician), on tobacco-related practices. Conclusion The results indicate that Hispanic physicians, similarly to U.S. physicians in general, do not meet the level of intervention recommended by health care agencies. The results presented will assist in the development of tobacco training initiatives for Hispanic physicians.

Extended kalman filter for estimation of parameters in nonlinear state-space models of biochemical networks.

It is system dynamics that determines the function of cells, tissues and organisms. To develop mathematical models and estimate their parameters are an essential issue for studying dynamic behaviors of biological systems which include metabolic networks, genetic regulatory networks and signal transduction pathways, under perturbation of external stimuli. In general, biological dynamic systems are partially observed. Therefore, a natural way to model dynamic biological systems is to employ nonlinear state-space equations. Although statistical methods for parameter estimation of linear models in biological dynamic systems have been developed intensively in the recent years, the estimation of both states and parameters of nonlinear dynamic systems remains a challenging task. In this report, we apply extended Kalman Filter (EKF) to the estimation of both states and parameters of nonlinear state-space models. To evaluate the performance of the EKF for parameter estimation, we apply the EKF to a simulation dataset and two real datasets: JAK-STAT signal transduction pathway and Ras/Raf/MEK/ERK signaling transduction pathways datasets. The preliminary results show that EKF can accurately estimate the parameters and predict states in nonlinear state-space equations for modeling dynamic biochemical networks.

Models of DNA sequence evolution

Models of DNA sequence evolution and methods for estimating evolutionary distances are needed for studying the rate and pattern of molecular evolution and for inferring the evolutionary relationships of organisms or genes. In this dissertation, several new models and methods are developed.^ The rate variation among nucleotide sites: To obtain unbiased estimates of evolutionary distances, the rate heterogeneity among nucleotide sites of a gene should be considered. Commonly, it is assumed that the substitution rate varies among sites according to a gamma distribution (gamma model) or, more generally, an invariant+gamma model which includes some invariable sites. A maximum likelihood (ML) approach was developed for estimating the shape parameter of the gamma distribution $(\alpha)$ and/or the proportion of invariable sites $(\theta).$ Computer simulation showed that (1) under the gamma model, $\alpha$ can be well estimated from 3 or 4 sequences if the sequence length is long; and (2) the distance estimate is unbiased and robust against violations of the assumptions of the invariant+gamma model.^ However, this ML method requires a huge amount of computational time and is useful only for less than 6 sequences. Therefore, I developed a fast method for estimating $\alpha,$ which is easy to implement and requires no knowledge of tree. A computer program was developed for estimating $\alpha$ and evolutionary distances, which can handle the number of sequences as large as 30.^ Evolutionary distances under the stationary, time-reversible (SR) model: The SR model is a general model of nucleotide substitution, which assumes (i) stationary nucleotide frequencies and (ii) time-reversibility. It can be extended to SRV model which allows rate variation among sites. I developed a method for estimating the distance under the SR or SRV model, as well as the variance-covariance matrix of distances. Computer simulation showed that the SR method is better than a simpler method when the sequence length $L>1,000$ bp and is robust against deviations from time-reversibility. As expected, when the rate varies among sites, the SRV method is much better than the SR method.^ The evolutionary distances under nonstationary nucleotide frequencies: The statistical properties of the paralinear and LogDet distances under nonstationary nucleotide frequencies were studied. First, I developed formulas for correcting the estimation biases of the paralinear and LogDet distances. The performances of these formulas and the formulas for sampling variances were examined by computer simulation. Second, I developed a method for estimating the variance-covariance matrix of the paralinear distance, so that statistical tests of phylogenies can be conducted when the nucleotide frequencies are nonstationary. Third, a new method for testing the molecular clock hypothesis was developed in the nonstationary case. ^

A comparison of Markov and generalized linear models of hospital mortality

This paper reports a comparison of three modeling strategies for the analysis of hospital mortality in a sample of general medicine inpatients in a Department of Veterans Affairs medical center. Logistic regression, a Markov chain model, and longitudinal logistic regression were evaluated on predictive performance as measured by the c-index and on accuracy of expected numbers of deaths compared to observed. The logistic regression used patient information collected at admission; the Markov model was comprised of two absorbing states for discharge and death and three transient states reflecting increasing severity of illness as measured by laboratory data collected during the hospital stay; longitudinal regression employed Generalized Estimating Equations (GEE) to model covariance structure for the repeated binary outcome. Results showed that the logistic regression predicted hospital mortality as well as the alternative methods but was limited in scope of application. The Markov chain provides insights into how day to day changes of illness severity lead to discharge or death. The longitudinal logistic regression showed that increasing illness trajectory is associated with hospital mortality. The conclusion is reached that for standard applications in modeling hospital mortality, logistic regression is adequate, but for new challenges facing health services research today, alternative methods are equally predictive, practical, and can provide new insights. ^

Bayesian generalized linear models for meta-analysis of diagnostic tests

With the recognition of the importance of evidence-based medicine, there is an emerging need for methods to systematically synthesize available data. Specifically, methods to provide accurate estimates of test characteristics for diagnostic tests are needed to help physicians make better clinical decisions. To provide more flexible approaches for meta-analysis of diagnostic tests, we developed three Bayesian generalized linear models. Two of these models, a bivariate normal and a binomial model, analyzed pairs of sensitivity and specificity values while incorporating the correlation between these two outcome variables. Noninformative independent uniform priors were used for the variance of sensitivity, specificity and correlation. We also applied an inverse Wishart prior to check the sensitivity of the results. The third model was a multinomial model where the test results were modeled as multinomial random variables. All three models can include specific imaging techniques as covariates in order to compare performance. Vague normal priors were assigned to the coefficients of the covariates. The computations were carried out using the 'Bayesian inference using Gibbs sampling' implementation of Markov chain Monte Carlo techniques. We investigated the properties of the three proposed models through extensive simulation studies. We also applied these models to a previously published meta-analysis dataset on cervical cancer as well as to an unpublished melanoma dataset. In general, our findings show that the point estimates of sensitivity and specificity were consistent among Bayesian and frequentist bivariate normal and binomial models. However, in the simulation studies, the estimates of the correlation coefficient from Bayesian bivariate models are not as good as those obtained from frequentist estimation regardless of which prior distribution was used for the covariance matrix. The Bayesian multinomial model consistently underestimated the sensitivity and specificity regardless of the sample size and correlation coefficient. In conclusion, the Bayesian bivariate binomial model provides the most flexible framework for future applications because of its following strengths: (1) it facilitates direct comparison between different tests; (2) it captures the variability in both sensitivity and specificity simultaneously as well as the intercorrelation between the two; and (3) it can be directly applied to sparse data without ad hoc correction. ^

Environmental tobacco smoke exposure, respiratory and cardiovascular health in restaurant and bar workers in Mexico

Environmental tobacco smoke (ETS) is a well established health hazard, being causally associated to lung cancer and cardiovascular disease. ETS regulations have been developed worldwide to reduce or eliminate exposure in most public places. Restaurants and bars constitute an exception. Restaurants and bar workers experience the highest ETS exposure levels across several occupations, with correspondingly increased health risks. In Mexico, previous exposure assessment in restaurants and bars showed concentrations in bars and restaurants to be the highest across different public and workplaces. Recently, Mexico developed at the federal level the General Law for Tobacco Control restricting indoors smoking to separated areas. AT the local level Mexico City developed the Law for the Protection of Non-smokers Health, completely banning smoking in restaurants and bars. Studies to assess ETS exposure in restaurants and bars, along with potential health effects were required to evaluate the impact of these legislative changes and to set a baseline measurement for future evaluations.^ A large cross-sectional study conducted in restaurants and bars from four Mexican cities was conducted from July to October 2008, to evaluate the following aims: Aim 1) Explore the potential impact of the Mexico City ban on ETS concentrations through comparison of Mexico City with other cities. Aim 2). Explore the association between ETS exposure, respiratory function indicators and respiratory symptoms. Aim 3). Explore the association between ETS exposure and blood pressure and heart rate.^ Three cities with no smoking ban were selected: Colima (11.5% smoking prevalence), Cuernavaca (21.5% smoking prevalence) and Toluca (27.8% smoking prevalence). Mexico City (27.9% smoking prevalence), the only city with a ban at the time of the study, was also selected. Restaurants and bars were randomly selected from municipal records. A goal of 26 restaurants and 26 bars per city was set, 50% of them under 100 m2. Each establishment was visited during the highest occupancy shift, and managers and workers answered to a questionnaire. Vapor-phase nicotine was measured using passive monitors, that were activated at the beginning and deactivated at the end of the shift. Also, workers participated at the beginning and end of the shift in a short physical evaluation, comprising the measurement of Forced Expiratory Volume in the first second (FEV1) and Peak Expiratory Flow (PEF), as well as blood pressure and heart rate.^ A total of 371 establishments were invited, 219 agreed to participate for a 60.1% participation rate. In them, 828 workers were invited, 633 agreed to participate for a 76% participation rate. Mexico City had at least 4 times less nicotine compared to any of the other cities. Differences between Mexico City and other cities were not explained by establishment characteristics, such as ventilation or air extraction. However, differences between cities disappeared when ban mechanisms, such as policy towards costumer's smoking, were considered in the models. An association between ETS exposure and respiratory symptoms (cough OR=1.27, 95%CI=1.04, 1.55) and respiratory illness (asthma OR=1.97, 95%CI=1.20, 3.24; respiratory illness OR=1.79, 95%CI=1.10, 2.94) was observed. No association between ETS and phlegm, wheezing or respiratory infections was observed. No association between ETS and any of the spirometric indicators was observed. An association between ETS exposure and increased systolic and diastolic blood pressure at the end of the shift was observed among non-smokers (systolic blood pressure beta=1.51, 95%CI=0.44, 2.58; diastolic blood pressure beta=1.50, 95%CI=0.72, 2.28). The opposite effect was observed in heavy smokers, were increased ETS exposure was associated with lower blood pressure at the end of the shift (systolic blood pressure beta=1.90, 95%CI=-3.57, -0.23; diastolic blood pressure beta=-1.46, 95%CI=-2.72, -0.02). No association in light smokers was observed. No association for heart rate was observed. ^ Results from this dissertation suggest Mexico City's smoking ban has had a larger impact on ETS exposure. Ventilation or air extraction, mechanisms of ETS control suggested frequently by tobacco companies to avoid smoking bans were not associated with ETS exposure. This dissertation suggests ETS exposure could be linked to changes in blood pressure and to increased respiratory symptoms. Evidence derived from this dissertation points to the potential negative health effects of ETS exposure in restaurants and bars, and provides support for the development of total smoking bans in this economic sector. ^

Hypogonadism in male cancer patients: A cross-sectional study

Introduction. Cancer is the second most common cause of death in the USA (2). Studies have shown a coexistence of cancer and hypogonadism (9,31,13). The majority of patients with cancer develop cachexia, which cannot be solely explained by anorexia seen in these patients. Testosterone is a male sex hormone which is known to increase muscle mass and strength, maintain cancellous bone mass, and increase cortical bone mass, in addition to improving libido, sexual desire, and fantasy (14). If a high prevalence of hypogonadism is detected in male cancer patients, and a significant difference exists in testosterone levels in cancer patients with cachexia versus those without cachexia, testosterone may be administered in future randomized trials to help alleviate cachexia. Study group and design The study group consisted of male cancer patients and non-cancer controls aged between 40 and 70 years. The primary study design was cross-sectional with a sample size of 135. The present data analysis is done on a subset convenience sample of 72 patients recruited between November 2006 and January 2010. ^ Methods. Patients aged 40-70 years with or without a diagnosis of cancer were recruited into the study. All patients with a BMI over 35, significant edema, non-melanomatous skin cancer, current alcohol or illicit drug abuse, concomitant usage of medications interfering with gonadal axis, and anabolic agents, patients on tube feeds or parenteral nutrition within 3 months prior to enrollment were excluded from the study. The study was approved by the Institutional Review Board of Baylor College of Medicine and is being conducted at the Michael E. DeBakey Veterans Affairs Medical Center at Houston. My thesis is a pilot data analysis that employs a smaller subset convenience sample of 72 patients determined by using the data available for the 72 patients (of the intended sample of 135 patients) recruited between November 2006 and January 2010. The primary aim of this analysis is to compare the proportion of patients with hypogonadism in the male cancer and non-cancer control groups, and to evaluate if a significant difference exists with respect to testosterone levels in male cancer patients with cachexia versus those without cachexia. The procedures of the study relevant to the current data analysis included blood collection to measure levels of testosterone and measurement of body weight to categorize cancer patients into cancer cachexia and cancer non-cachexia sub-groups. ^ Results. After logarithmic transformation of data of cancer and control groups, the unpaired t test with unequal variances was done. The proportion of patients with hypogonadism in the male cancer and non-cancer control groups was 47.5% and 22.7% with a Pearson chi2 statistic of 1.6036 and a p value of 0.205. Comparing the mean calculated Bioavailable testosterone in male cancer patients and non-cancer controls resulted in a t statistic of 21.83 and a p value less than 0.001. When the cancer group alone was taken, the mean free testosterone, calculated bioavailable testosterone and total testosterone levels in the cancer non-cachexia sub-group were 3.93, 5.09, 103.51 respectively and in the cancer cachexia sub-group were 3.58, 4.17, 84.08 respectively. The unpaired t test with equal variances showed that the two sub-groups had p values of 0.2015, 0.1842, and 0.4894 with respect to calculated bioavailable testosterone, free testosterone, and total testosterone respectively. ^ Conclusions. The small sample size of this exploratory study, resulting in a small power, does not allow us to draw definitive conclusions. For the given sub-sample, the proportion of patients with hypogonadism in the cancer group was not significantly different from that of patients with hypogonadism in the control group. Inferences on prevalence of hypogonadism in male cancer patients could not be made in this paper as the sub-sample is small and therefore not representative of the general population. However, there was a statistically significant difference in calculated Bioavailable testosterone levels in male cancer patients versus non-cancer controls. Analysis of cachectic and non-cachectic patients within the male cancer group showed no significant difference in testosterone levels (total, free, and calculated bioavailable testosterone) between both sub-groups. However, to re-iterate, this study is exploratory and the results may change once the complete dataset is obtained and analyzed. It however serves as a good template to guide further research and analysis.^

COX-2 expression in operable triple negative breast cancer: A hospital based cross-sectional study

Objectives. Triple Negative Breast Cancer (TNBC) lack expression of estrogen receptors (ER), progesterone receptors (PR), and absence of Her2 gene amplification. Current literature has identified TNBC and over-expression of cyclo-oxygenase-2 (COX-2) protein in primary breast cancer to be independent markers of poor prognosis in terms of overall and distant disease free survival. The purpose of this study was to compare COX-2 over-expression in TNBC patients to those patients who expressed one or more of the three tumor markers (i.e. ER, and/or PR, and/or Her2).^ Methods. Using a secondary data analysis, a cross-sectional design was implemented to examine the association of interest. Data collected from two ongoing protocols titled "LAB04-0657: a model for COX-2 mediated bone metastasis (Specific aim 3)" and "LAB04-0698: correlation of circulating tumor cells and COX-2 expression in primary breast cancer metastasis" was used for analysis. A sample of 125 female patients was analyzed using Chi-square tests and logistic regression models. ^ Results. COX-2 over-expression was present in 33% (41/125) and 28% (35/124) patients were identified as having TNBC. TNBC status was associated with elevated COX-2 expression (OR= 3.34; 95% CI= 1.40–8.22) and high tumor grade (OR= 4.09; 95% CI= 1.58–10.82). In a multivariable analysis, TNBC status was an important predictor of COX-2 expression after adjusting for age, menopausal status, BMI, and lymph node status (OR= 3.31; 95% CI: 1.26–8.67; p=0.01).^ Conclusion. TNBC is associated with COX-2 expression—a known marker of poor prognosis in patients with operable breast cancer. Replication of these results in a study with a larger sample size, or a future randomized clinical trial demonstrating an improved prognosis with COX-2 suppression in these patients would support this hypothesis.^

Assessing time-by-covariate interactions in Cox proportional hazards regression models using cubic spline functions

This dissertation develops and explores the methodology for the use of cubic spline functions in assessing time-by-covariate interactions in Cox proportional hazards regression models. These interactions indicate violations of the proportional hazards assumption of the Cox model. Use of cubic spline functions allows for the investigation of the shape of a possible covariate time-dependence without having to specify a particular functional form. Cubic spline functions yield both a graphical method and a formal test for the proportional hazards assumption as well as a test of the nonlinearity of the time-by-covariate interaction. Five existing methods for assessing violations of the proportional hazards assumption are reviewed and applied along with cubic splines to three well known two-sample datasets. An additional dataset with three covariates is used to explore the use of cubic spline functions in a more general setting. ^

A STUDY ON THE EPIDEMIOLOGY OF POSITIVE PHYSICAL HEALTH (NHANES I)

HANES 1 detailed sample data were used to operationalize a definition of health in the absence of disease and to describe and compare the characteristics of the normal (healthy) group versus an abnormal (unhealthy) group.^ Parallel screening gave a 3.8 percent prevalence proportion of physical health, with a female:male ratio of 2:1 and younger ages in the healthy group. Statistically significant Mantel-Haenszel gender-age-adjusted odds ratios (MHOR) were estimated among abnormal non-migrants (1.53), skilled workers/unemployed (1.76), annual family incomes of less than $10,000 (1.56), having ever smoked (1.58), and started smoking before 18 years of age (1.58). Significant MHOR were also found for abnormals for health promoting measures: non-iodized salt use (1.94), needed dental care (1.91); and for fair to poor perceived health (4.28), perceiving health problems (2.52), and low energy level (1.68). Significant protective effects for much to moderate recreational exercise (MHOR 0.42) and very active to moderate non-recreational activity (MHOR 0.49) were also obtained. Covariance analysis additive models detected statistically significant higher mean values for abnormals than normals for serum magnesium, hemoglobin, hematocrit, urinary creatinine, and systolic and diastolic blood pressures, and lower values for abnormals than normals for serum iron. No difference was detected for serum cholesterol. Significant non-additive joint effects were found for body mass index.^ The results suggest positive physical health can be measured with cross-sectional survey data. Gender differentials, and associations between ecologic, socioeconomic, hazardous risk factors, health promoting activities and physical health are in general agreement with published findings on studies of morbidity. Longitudinal prospective studies are suggested to establish the direction of the associations and to enhance present knowledge of health and its promoting factors. ^

A cross-sectional analysis of the association between polycyclic aromatic hydrocarbons and diesel particulate matter exposures and hypertension among individuals of Mexican origin residing in Houston, TX

Background. Research has shown that elevations of only 10 mmHg diastolic blood pressure (BP) and 5 mmHg systolic BP are associated with substantial (as large as 50%) increases in risks for cardiovascular disease, a leading cause of death, worldwide. Epidemiological studies have found that particulate matter (PM) increases blood pressure (BP) and many biological mechanisms which may suggest that the organic matter of PM contributes to the increase in BP. To understand components of PM which may contribute to the increase in BP, this study focuses on diesel particulate matter (DPM) and polycyclic aromatic hydrocarbons (PAHs). To our knowledge, there have been only four epidemiological studies on BP and DPM, and no epidemiological studies on BP and PAHs. ^ Objective. Our objective was to evaluate the association between prevalent hypertension and two ambient exposures: DPM and PAHs amongst the Mano a Mano cohort. ^ Methods. The Mano a Mano cohort which was established by the M.D. Anderson Cancer Center in 2001, is comprised of individuals of Mexican origin residing in Houston, TX. Using geographical information systems, we linked modeled annual estimates of PAHs and DPM at the census track level from the U.S. Environmental Protection Agency's National-Scale Air Toxics Assessment to residential addresses of cohort members. Mixed-effects logistic regression models were applied to determine associations between DPM and PAHs and hypertension while adjusting for confounders. ^ Results. Ambient levels of DPM, categorized into quartiles, were not statistically associated with hypertension and did not indicate a dose response relationship. Ambient levels of PAHs, categorized into quartiles, were not associated with hypertension, but did indicate a dose response relationship in multiple models (for example: Q2: OR = 0.98; 95% CI, 0.73–1.31, Q3: OR = 1.08; 95% CI, 0.82–1.41, Q4: OR = 1.26; 95% CI, 0.94–1.70). ^ Conclusion. This is the first assessment to analyze the relationship between ambient levels of PAHs and hypertension and it is amongst a few studies investigating the association between ambient levels of DPM and hypertension. Future analyses are warranted to explore the effects DPM and PAHs using different categorizations in order to clarify their relationships with hypertension.^

Analyzing left-truncated right-censored data with uncertain onset time with parametric models

Prevalent sampling is an efficient and focused approach to the study of the natural history of disease. Right-censored time-to-event data observed from prospective prevalent cohort studies are often subject to left-truncated sampling. Left-truncated samples are not randomly selected from the population of interest and have a selection bias. Extensive studies have focused on estimating the unbiased distribution given left-truncated samples. However, in many applications, the exact date of disease onset was not observed. For example, in an HIV infection study, the exact HIV infection time is not observable. However, it is known that the HIV infection date occurred between two observable dates. Meeting these challenges motivated our study. We propose parametric models to estimate the unbiased distribution of left-truncated, right-censored time-to-event data with uncertain onset times. We first consider data from a length-biased sampling, a specific case in left-truncated samplings. Then we extend the proposed method to general left-truncated sampling. With a parametric model, we construct the full likelihood, given a biased sample with unobservable onset of disease. The parameters are estimated through the maximization of the constructed likelihood by adjusting the selection bias and unobservable exact onset. Simulations are conducted to evaluate the finite sample performance of the proposed methods. We apply the proposed method to an HIV infection study, estimating the unbiased survival function and covariance coefficients. ^