Logistic regression with incompletely observed binary

Logistic regression is one of the most important tools in the analysis of epidemiological and clinical data. Such data often contain missing values for one or more variables. Common practice is to eliminate all individuals for whom any information is missing. This deletion approach does not make efficient use of available information and often introduces bias.^ Two methods were developed to estimate logistic regression coefficients for mixed dichotomous and continuous covariates including partially observed binary covariates. The data were assumed missing at random (MAR). One method (PD) used predictive distribution as weight to calculate the average of the logistic regressions performing on all possible values of missing observations, and the second method (RS) used a variant of resampling technique. Additional seven methods were compared with these two approaches in a simulation study. They are: (1) Analysis based on only the complete cases, (2) Substituting the mean of the observed values for the missing value, (3) An imputation technique based on the proportions of observed data, (4) Regressing the partially observed covariates on the remaining continuous covariates, (5) Regressing the partially observed covariates on the remaining continuous covariates conditional on response variable, (6) Regressing the partially observed covariates on the remaining continuous covariates and response variable, and (7) EM algorithm. Both proposed methods showed smaller standard errors (s.e.) for the coefficient involving the partially observed covariate and for the other coefficients as well. However, both methods, especially PD, are computationally demanding; thus for analysis of large data sets with partially observed covariates, further refinement of these approaches is needed. ^

Logistic regression with Markov chains as covariates

The tobacco-specific nitrosamine 4-(methylnitrosamino)-1-(3-pyridyl)-1-butanone (NNK) is an obvious carcinogen for lung cancer. Since CBMN (Cytokinesis-blocked micronucleus) has been found to be extremely sensitive to NNK-induced genetic damage, it is a potential important factor to predict the lung cancer risk. However, the association between lung cancer and NNK-induced genetic damage measured by CBMN assay has not been rigorously examined. ^ This research develops a methodology to model the chromosomal changes under NNK-induced genetic damage in a logistic regression framework in order to predict the occurrence of lung cancer. Since these chromosomal changes were usually not observed very long due to laboratory cost and time, a resampling technique was applied to generate the Markov chain of the normal and the damaged cell for each individual. A joint likelihood between the resampled Markov chains and the logistic regression model including transition probabilities of this chain as covariates was established. The Maximum likelihood estimation was applied to carry on the statistical test for comparison. The ability of this approach to increase discriminating power to predict lung cancer was compared to a baseline "non-genetic" model. ^ Our method offered an option to understand the association between the dynamic cell information and lung cancer. Our study indicated the extent of DNA damage/non-damage using the CBMN assay provides critical information that impacts public health studies of lung cancer risk. This novel statistical method could simultaneously estimate the process of DNA damage/non-damage and its relationship with lung cancer for each individual.^

Age -period -cohort analysis: An extension of Cox's lifetable regression with time -dependent covariates

It is well known that an identification problem exists in the analysis of age-period-cohort data because of the relationship among the three factors (date of birth + age at death = date of death). There are numerous suggestions about how to analyze the data. No one solution has been satisfactory. The purpose of this study is to provide another analytic method by extending the Cox's lifetable regression model with time-dependent covariates. The new approach contains the following features: (1) It is based on the conditional maximum likelihood procedure using a proportional hazard function described by Cox (1972), treating the age factor as the underlying hazard to estimate the parameters for the cohort and period factors. (2) The model is flexible so that both the cohort and period factors can be treated as dummy or continuous variables, and the parameter estimations can be obtained for numerous combinations of variables as in a regression analysis. (3) The model is applicable even when the time period is unequally spaced.^ Two specific models are considered to illustrate the new approach and applied to the U.S. prostate cancer data. We find that there are significant differences between all cohorts and there is a significant period effect for both whites and nonwhites. The underlying hazard increases exponentially with age indicating that old people have much higher risk than young people. A log transformation of relative risk shows that the prostate cancer risk declined in recent cohorts for both models. However, prostate cancer risk declined 5 cohorts (25 years) earlier for whites than for nonwhites under the period factor model (0 0 0 1 1 1 1). These latter results are similar to the previous study by Holford (1983).^ The new approach offers a general method to analyze the age-period-cohort data without using any arbitrary constraint in the model. ^

An ordinal logistic regression model with misclassification of the outcome variable and categorical covariate

Ordinal outcomes are frequently employed in diagnosis and clinical trials. Clinical trials of Alzheimer's disease (AD) treatments are a case in point using the status of mild, moderate or severe disease as outcome measures. As in many other outcome oriented studies, the disease status may be misclassified. This study estimates the extent of misclassification in an ordinal outcome such as disease status. Also, this study estimates the extent of misclassification of a predictor variable such as genotype status. An ordinal logistic regression model is commonly used to model the relationship between disease status, the effect of treatment, and other predictive factors. A simulation study was done. First, data based on a set of hypothetical parameters and hypothetical rates of misclassification was created. Next, the maximum likelihood method was employed to generate likelihood equations accounting for misclassification. The Nelder-Mead Simplex method was used to solve for the misclassification and model parameters. Finally, this method was applied to an AD dataset to detect the amount of misclassification present. The estimates of the ordinal regression model parameters were close to the hypothetical parameters. β1 was hypothesized at 0.50 and the mean estimate was 0.488, β2 was hypothesized at 0.04 and the mean of the estimates was 0.04. Although the estimates for the rates of misclassification of X1 were not as close as β1 and β2, they validate this method. X 1 0-1 misclassification was hypothesized as 2.98% and the mean of the simulated estimates was 1.54% and, in the best case, the misclassification of k from high to medium was hypothesized at 4.87% and had a sample mean of 3.62%. In the AD dataset, the estimate for the odds ratio of X 1 of having both copies of the APOE 4 allele changed from an estimate of 1.377 to an estimate 1.418, demonstrating that the estimates of the odds ratio changed when the analysis includes adjustment for misclassification. ^

POLYCHOTOMOUS LOGISTIC REGRESSION ANALYSIS (HYPERTENSION, CORONARY HEART DISEASE)

The history of the logistic function since its introduction in 1838 is reviewed, and the logistic model for a polychotomous response variable is presented with a discussion of the assumptions involved in its derivation and use. Following this, the maximum likelihood estimators for the model parameters are derived along with a Newton-Raphson iterative procedure for evaluation. A rigorous mathematical derivation of the limiting distribution of the maximum likelihood estimators is then presented using a characteristic function approach. An appendix with theorems on the asymptotic normality of sample sums when the observations are not identically distributed, with proofs, supports the presentation on asymptotic properties of the maximum likelihood estimators. Finally, two applications of the model are presented using data from the Hypertension Detection and Follow-up Program, a prospective, population-based, randomized trial of treatment for hypertension. The first application compares the risk of five-year mortality from cardiovascular causes with that from noncardiovascular causes; the second application compares risk factors for fatal or nonfatal coronary heart disease with those for fatal or nonfatal stroke. ^

Evaluation of the Hosmer -Lemeshow global goodness -of-fit statistic for mixed variable logistic regression models

The performance of the Hosmer-Lemeshow global goodness-of-fit statistic for logistic regression models was explored in a wide variety of conditions not previously fully investigated. Computer simulations, each consisting of 500 regression models, were run to assess the statistic in 23 different situations. The items which varied among the situations included the number of observations used in each regression, the number of covariates, the degree of dependence among the covariates, the combinations of continuous and discrete variables, and the generation of the values of the dependent variable for model fit or lack of fit.^ The study found that the $\rm\ C$g* statistic was adequate in tests of significance for most situations. However, when testing data which deviate from a logistic model, the statistic has low power to detect such deviation. Although grouping of the estimated probabilities into quantiles from 8 to 30 was studied, the deciles of risk approach was generally sufficient. Subdividing the estimated probabilities into more than 10 quantiles when there are many covariates in the model is not necessary, despite theoretical reasons which suggest otherwise. Because it does not follow a X$\sp2$ distribution, the statistic is not recommended for use in models containing only categorical variables with a limited number of covariate patterns.^ The statistic performed adequately when there were at least 10 observations per quantile. Large numbers of observations per quantile did not lead to incorrect conclusions that the model did not fit the data when it actually did. However, the statistic failed to detect lack of fit when it existed and should be supplemented with further tests for the influence of individual observations. Careful examination of the parameter estimates is also essential since the statistic did not perform as desired when there was moderate to severe collinearity among covariates.^ Two methods studied for handling tied values of the estimated probabilities made only a slight difference in conclusions about model fit. Neither method split observations with identical probabilities into different quantiles. Approaches which create equal size groups by separating ties should be avoided. ^

Logistic regression models for ordinal response: A study of self-efficacy in colorectal cancer screening

The ordinal logistic regression models are used to analyze the dependant variable with multiple outcomes that can be ranked, but have been underutilized. In this study, we describe four logistic regression models for analyzing the ordinal response variable. ^ In this methodological study, the four regression models are proposed. The first model uses the multinomial logistic model. The second is adjacent-category logit model. The third is the proportional odds model and the fourth model is the continuation-ratio model. We illustrate and compare the fit of these models using data from the survey designed by the University of Texas, School of Public Health research project PCCaSO (Promoting Colon Cancer Screening in people 50 and Over), to study the patient’s confidence in the completion colorectal cancer screening (CRCS). ^ The purpose of this study is two fold: first, to provide a synthesized review of models for analyzing data with ordinal response, and second, to evaluate their usefulness in epidemiological research, with particular emphasis on model formulation, interpretation of model coefficients, and their implications. Four ordinal logistic models that are used in this study include (1) Multinomial logistic model, (2) Adjacent-category logistic model [9], (3) Continuation-ratio logistic model [10], (4) Proportional logistic model [11]. We recommend that the analyst performs (1) goodness-of-fit tests, (2) sensitivity analysis by fitting and comparing different models.^

Predicting childhood sexual or physical abuse: a logistic regression geo-mapping approach to prevention.

This study investigates the degree to which gender, ethnicity, relationship to perpetrator, and geomapped socio-economic factors significantly predict the incidence of childhood sexual abuse, physical abuse and non- abuse. These variables are then linked to geographic identifiers using geographic information system (GIS) technology to develop a geo-mapping framework for child sexual and physical abuse prevention.

Postoperative renal function preservation with nonischemic femoral arterial cannulation for thoracoabdominal aortic repair.

BACKGROUND: Renal failure after thoracoabdominal aortic repair is a significant clinical problem. Distal aortic perfusion for organ and spinal cord protection requires cannulation of the left femoral artery. In 2006, we reported the finding that direct cannulation led to leg ischemia in some patients and was associated with increased renal failure. After this finding, we modified our perfusion technique to eliminate leg ischemia from cannulation. In this article, we present the effects of this change on postoperative renal function. METHODS: Between February 1991 and July 2008, we repaired 1464 thoracoabdominal aortic aneurysms. Distal aortic perfusion was used in 1088, and these were studied. Median patient age was 68 years, and 378 (35%) were women. In September 2006, we began to adopt a sidearm femoral cannulation technique that provides distal aortic perfusion while maintaining downstream flow to the leg. This was used in 167 patients (15%). We measured the joint effects of preoperative glomerular filtration rate (GFR) and cannulation technique on the highest postoperative creatinine level, postoperative renal failure, and death. Analysis was by multiple linear or logistic regression with interaction. RESULTS: The preoperative GFR was the strongest predictor of postoperative renal dysfunction and death. No significant main effects of sidearm cannulation were noted. For peak creatinine level and postoperative renal failure, however, strong interactions between preoperative GFR and sidearm cannulation were present, resulting in reductions of postoperative renal complications of 15% to 20% when GFR was <60 mL>/min/1.73 m(2). For normal GFR, the effect was negated or even reversed at very high levels of GFR. Mortality, although not significantly affected by sidearm cannulation, showed a similar trend to the renal outcomes. CONCLUSION: Use of sidearm cannulation is associated with a clinically important and highly statistically significant reduction in postoperative renal complications in patients with a low GFR. Reduced renal effect of skeletal muscle ischemia is the proposed mechanism. Effects among patients with good preoperative renal function are less clear. A randomized trial is needed.

AN ILLNESS-DEATH PROCESS WITH TIME-DEPENDENT COVARIATES

A general model for the illness-death stochastic process with covariates has been developed for the analysis of survival data. This model incorporates important baseline and time-dependent covariates to make proper adjustment for the transition probabilities and survival probabilities. The follow-up period is subdivided into small intervals and a constant hazard is assumed for each interval. An approximation formula is derived to estimate the transition parameters when the exact transition time is unknown.^ The method developed is illustrated by using data from a study on the prevention of the recurrence of a myocardial infarction and subsequent mortality, the Beta-Blocker Heart Attack Trial (BHAT). This method provides an analytical approach which simultaneously includes provision for both fatal and nonfatal events in the model. According to this analysis, the effectiveness of the treatment can be compared between the Placebo and Propranolol treatment groups with respect to fatal and nonfatal events. ^

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.^

Radiomics of NSCLC: Quantitative CT Image Feature Characterization and Tumor Shrinkage Prediction

Radiomics is the high-throughput extraction and analysis of quantitative image features. For non-small cell lung cancer (NSCLC) patients, radiomics can be applied to standard of care computed tomography (CT) images to improve tumor diagnosis, staging, and response assessment. The first objective of this work was to show that CT image features extracted from pre-treatment NSCLC tumors could be used to predict tumor shrinkage in response to therapy. This is important since tumor shrinkage is an important cancer treatment endpoint that is correlated with probability of disease progression and overall survival. Accurate prediction of tumor shrinkage could also lead to individually customized treatment plans. To accomplish this objective, 64 stage NSCLC patients with similar treatments were all imaged using the same CT scanner and protocol. Quantitative image features were extracted and principal component regression with simulated annealing subset selection was used to predict shrinkage. Cross validation and permutation tests were used to validate the results. The optimal model gave a strong correlation between the observed and predicted shrinkages with . The second objective of this work was to identify sets of NSCLC CT image features that are reproducible, non-redundant, and informative across multiple machines. Feature sets with these qualities are needed for NSCLC radiomics models to be robust to machine variation and spurious correlation. To accomplish this objective, test-retest CT image pairs were obtained from 56 NSCLC patients imaged on three CT machines from two institutions. For each machine, quantitative image features with concordance correlation coefficient values greater than 0.90 were considered reproducible. Multi-machine reproducible feature sets were created by taking the intersection of individual machine reproducible feature sets. Redundant features were removed through hierarchical clustering. The findings showed that image feature reproducibility and redundancy depended on both the CT machine and the CT image type (average cine 4D-CT imaging vs. end-exhale cine 4D-CT imaging vs. helical inspiratory breath-hold 3D CT). For each image type, a set of cross-machine reproducible, non-redundant, and informative image features was identified. Compared to end-exhale 4D-CT and breath-hold 3D-CT, average 4D-CT derived image features showed superior multi-machine reproducibility and are the best candidates for clinical correlation.

Prevalence of gastro esophageal reflux disease following the Montreal Consensus and its association with Helicobacter pylori infection and obesity in a random sample of adults of Ciudad Juarez, Mexico, 2004.

Gastroesophageal reflux disease is a common condition affecting 25 to 40% of the population and causes significant morbidity in the U.S., accounting for at least 9 million office visits to physicians with estimated annual costs of $10 billion. Previous research has not clearly established whether infection with Helicobacter pylori, a known cause of peptic ulcer, atrophic gastritis and non cardia adenocarcinoma of the stomach, is associated with gastroesophageal reflux disease. This study is a secondary analysis of data collected in a cross-sectional study of a random sample of adult residents of Ciudad Juarez, Mexico, that was conducted in 2004 (Prevalence and Determinants of Chronic Atrophic Gastritis Study or CAG study, Dr. Victor M. Cardenas, Principal Investigator). In this study, the presence of gastroesophageal reflux disease was based on responses to the previously validated Spanish Language Dyspepsia Questionnaire. Responses to this questionnaire indicating the presence of gastroesophageal reflux symptoms and disease were compared with the presence of H. pylori infection as measured by culture, histology and rapid urease test, and with findings of upper endoscopy (i.e., hiatus hernia and erosive and atrophic esophagitis). The prevalence ratio was calculated using bivariate, stratified and multivariate negative binomial logistic regression analyses in order to assess the relation between active H. pylori infection and the prevalence of gastroesophageal reflux typical syndrome and disease, while controlling for known risk factors of gastroesophageal reflux disease such as obesity. In a random sample of 174 adults 48 (27.6%) of the study participants had typical reflux syndrome and only 5% (or 9/174) had gastroesophageal reflux disease per se according to the Montreal consensus, which defines reflux syndromes and disease based on whether the symptoms are perceived as troublesome by the subject. There was no association between H. pylori infection and typical reflux syndrome or gastroesophageal reflux disease. However, we found that in this Northern Mexican population, there was a moderate association (Prevalence Ratio=2.5; 95% CI=1.3, 4.7) between obesity (≥30 kg/m2) and typical reflux syndrome. Management and prevention of obesity will significantly curb the growing numbers of persons affected by gastroesophageal reflux symptoms and disease in Northern Mexico. ^

A GENETIC MARKER STUDY OF NON - INSULIN DEPENDENT DIABETICS IN THE MEXICAN-AMERICAN COMMUNITY FROM STARR COUNTY, TEXAS (PLASMA GLUCOSE, HEMOGLOBIN ANALYSIS)

Diabetes mellitus occurs in two forms, insulin-dependent (IDDM, formerly called juvenile type) and non-insulin dependent (NIDDM, formerly called adult type). Prevalence figures from around the world for NIDDM, show that all societies and all races are affected; although uncommon in some populations (.4%), it is common (10%) or very common (40%) in others (Tables 1 and 2).^ In Mexican-Americans in particular, the prevalence rates (7-10%) are intermediate to those in Caucasians (1-2%) and Amerindians (35%). Information about the distribution of the disease and identification of high risk groups for developing glucose intolerance or its vascular manifestations by the study of genetic markers will help to clarify and solve some of the problems from the public health and the genetic point of view.^ This research was designed to examine two general areas in relation to NIDDM. The first aims to determine the prevalence of polymorphic genetic markers in two groups distinguished by the presence or absence of diabetes and to observe if there are any genetic marker-disease association (univariate analysis using two by two tables and logistic regression to study the individual and joint effects of the different variables). The second deals with the effect of genetic differences on the variation in fasting plasma glucose and percent glycosylated hemoglobin (HbAl) (analysis of Covariance for each marker, using age and sex as covariates).^ The results from the first analysis were not statistically significant at the corrected p value of 0.003 given the number of tests that were performed. From the analysis of covariance of all the markers studied, only Duffy and Phosphoglucomutase were statistically significant but poor predictors, given that the amount they explain in terms of variation in glycosylated hemoglobin is very small.^ Trying to determine the polygenic component of chronic disease is not an easy task. This study confirms the fact that a larger and random or representative sample is needed to be able to detect differences in the prevalence of a marker for association studies and in the genetic contribution to the variation in glucose and glycosylated hemoglobin. The importance that ethnic homogeneity in the groups studied and standardization in the methodology will have on the results has been stressed. ^