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Simple polynomial multiplication algorithms for exact conditional tests of linearity in a logistic model.

The linear logistic model is often employed in the analysis of binary response data. The well-known asymptotic chi-square and likelihood ratio tests are usually used to detect the assumption of linearity in such a model. For small, sparse, or skewed data, the asymptotic theory is however dubious and exact conditional chi-square and likelihood ratio tests may provide reliable alternatives. In this article, we propose efficient polynomial multiplication algorithms to compute exact significance levels as well as exact powers of these tests. Two options, namely the cell- and stage-wise approaches, in implementing these algorithms will be discussed. When sample sizes are large, we propose an efficient Monte Carlo method for estimating the exact significance levels and exact powers. Real data are used to demonstrate the performance with an application of the proposed algorithms.

Algorithms↗

Stepwise logistic regression analysis of tumor contour features for breast ultrasound diagnosis.

To assist the ultrasound (US) differential diagnosis of solid breast tumors by using stepwise logistic regression (SLR) analysis of tumor contour features, we retrospectively reviewed 111 medical records of digitized US images of breast pathologies. They were pathologically proved benign breast tumors from 40 patients (i.e., 40 fibroadenomas) and malignant breast tumors from 71 patients (i.e., 71 infiltrative ductal carcinomas). Radiologists, before analysis by the computer-aided diagnosis (CAD) system, segmented the tumors manually. The contour features were calculated by measuring the radial length of tumor boundaries. The features selection process was accomplished using a stepwise analysis procedure. Then, an SLR model with contour features was used to classify tumors as benign or malignant. In this experiment, cases were sampled with "leave-one-out" test methods to evaluate the SLR performance using a receiver operating characteristic (ROC) curve. The accuracy of our SLR model with contour features for classifying malignancies was 91.0% (101 of 111 tumors), the sensitivity was 97.2% (69 of 71), the specificity was 80.0% (32 of 40), the positive predictive value was 89.6% (69 of 77), and the negative predictive value was 94.1% (32 of 34). The CAD system using SLR can differentiate solid breast nodules with relatively high accuracy and its high negative predictive value could potentially help inexperienced operators to avoid misdiagnoses. Because the SLR model is trainable, it could be optimized if a larger set of tumor images were supplied.

Adolescent↗

A note on R2 measures for Poisson and logistic regression models when both models are applicable.

The aim of many epidemiological studies is the regression of a dichotomous outcome (e.g., death or affection by a certain disease) on prognostic covariables. Thereby the Poisson regression model is often used alternatively to the logistic regression model. Modelling the number of events and individual outcomes, respectively, both models lead to nearly the same results concerning the parameter estimates and their significances. However, when calculating the proportion of explained variation, quantified by an R2 measure, a large difference between both models usually occurs. We illustrate this difference by an example and explain it with theoretical arguments. We conclude, the R2 measure of the Poisson regression quantifies the predictability of event rates, but it is not adequate to quantify the predictability of the outcome of individual observations.

Adult↗

Logistic regression in the medical literature: standards for use and reporting, with particular attention to one medical domain.

Logistic regression (LR) is a widely used multivariable method for modeling dichotomous outcomes. This article examines use and reporting of LR in the medical literature by comprehensively assessing its use in a selected area of medical study. Medline, followed by bibliography searches, identified 15 peer-reviewed English-language articles with original data, employing LR, published between 1985 and 1999, pertaining to patient interest in genetic testing for cancer susceptibility. Articles were examined for each of 10 criteria for proper use and reporting of LR models. Substantial shortcomings were found in both use of LR and reporting of results. For many studies, the ratio of the number of outcome events to predictor variables (events per variable) was sufficiently small to call into question the accuracy of the regression model. Additionally, no studies reported validation analysis, regression diagnostics, or goodness-of-fit measures. It is recommended that authors, reviewers, and editors pay greater attention to guidelines concerning the use and reporting of LR models.

Epidemiologic Studies↗

Comparing performance of multinomial logistic regression and discriminant analysis for monitoring access to care for acute myocardial infarction.

One way to monitor patient access to emergent health care services is to use patient characteristics to predict arrival time at the hospital after onset of symptoms. This predicted arrival time can then be compared with actual arrival time to allow monitoring of access to services. Predicted arrival time could also be used to estimate potential effects of changes in health care service availability, such as closure of an emergency department or an acute care hospital. Our goal was to determine the best statistical method for prediction of arrival intervals for patients with acute myocardial infarction (AMI) symptoms. We compared the performance of multinomial logistic regression (MLR) and discriminant analysis (DA) models. Models for MLR and DA were developed using a dataset of 3,566 male veterans hospitalized with AMI in 81 VA Medical Centers in 1994-1995 throughout the United States. The dataset was randomly divided into a training set (n = 1,846) and a test set (n = 1,720). Arrival times were grouped into three intervals on the basis of treatment considerations: <6 hours, 6-12 hours, and >12 hours. One model for MLR and two models for DA were developed using the training dataset. One DA model had equal prior probabilities, and one DA model had proportional prior probabilities. Predictive performance of the models was compared using the test (n = 1,720) dataset. Using the test dataset, the proportions of patients in the three arrival time groups were 60.9% for <6 hours, 10.3% for 6-12 hours, and 28.8% for >12 hours after symptom onset. Whereas the overall predictive performance by MLR and DA with proportional priors was higher, the DA models with equal priors performed much better in the smaller groups. Correct classifications were 62.6% by MLR, 62.4% by DA using proportional prior probabilities, and 48.1% using equal prior probabilities of the groups. The misclassifications by MLR for the three groups were 9.5%, 100.0%, 74.2% for each time interval, respectively. Misclassifications by DA models were 9.8%, 100.0%, and 74.4% for the model with proportional priors and 47.6%, 79.5%, and 51.0% for the model with equal priors. The choice of MLR or DA with proportional priors, or DA with equal priors for monitoring time intervals of predicted hospital arrival time for a population should depend on the consequences of misclassification errors.

Aged↗

The clustering of neonatal deaths in triplet pregnancies: application of response conditional multivariate logistic regression models.

BACKGROUND/OBJECTIVES: A population-based retrospective cohort study of triplet pregnancies was conducted to estimate individual probabilities of neonatal mortality (death within 28 days of birth) conditional on the number of neonatal deaths experienced by other infants in the triplet set. METHODS: Data on 4,697 triplet sets (14,091 births) were derived from the U.S. 1995-1997 matched multiple birth file assembled by the National Center for Health Statistics. Response conditional multivariate logistic regression was used to model the association of neonatal mortality among cotriplets. To account for the correlation of the outcomes among cotriplets, regression parameters were estimated by the methodology of generalized estimating equations with robust variance estimates. RESULTS: Compared with a triplet where both cotriplets survived the neonatal period, the adjusted odds ratio and 95% confidence interval (CI) for a neonatal death associated with one and two cotriplet neonatal deaths were 1.80 (95% CI 1.06, 3.04), and 13.41 (95% CI 2.31, 77.7), respectively, after adjusting for birthweight and gestational age. CONCLUSIONS: These results show strong evidence of clustering of neonatal deaths in triplet pregnancies.

Humans↗

A logistic regression model when some events precede treatment: the effect of thrombolytic therapy for acute myocardial infarction on the risk of cardiac arrest.

When outcomes occur in clinical trials before treatment can be given, neither intent-to-treat nor according-to-protocol analyses give optimal estimates of the treatment effect. A better approach employs a time-dependent variable for treatment. Intent-to-treat analyses are conservative, biasing against treatment; according-to-protocol analyses bias in favor of treatment. We show how to measure the effect of a time-dependent variable in a logistic regression using person-time intervals as units of measurement and describe appropriate methods for reporting model performance. The method is applied to develop a model to predict the probability that a patient with a myocardial infarction will have a sudden cardiac arrest within 48 hours of presentation to emergency medical services both when treated with thrombolysis and when not treated. We use a time-dependent treatment variable because many patients went into cardiac arrest while awaiting treatment. This technique has been programmed into an electrocardiograph for real-time use in an emergency department.

Aged↗

Statistical evaluation of the role of Helicobacter pylori in stress gastritis: applications of splines and bootstrapping to the logistic model.

Stress gastritis is a serious problem in the intensive care unit population. The recent discovery of the causal nature of Helicobacter pylori (H. pylori) in the development of gastric ulcers led us to examine its relationship with stress gastritis. We investigated this relationship in 874 veterans admitted to intensive care units who were tested for the presence of H. pylori and followed for 6 weeks for the development of stress gastritis. We fit spline models to assess functional relationships and used the logistic model to determine the association between H. pylori and stress gastritis. The predictive ability of the model was assessed with receiver operating characteristic (ROC) curve analysis and validated with the bootstrapping technique. Increased anti-H. pylori immunoglobulin A concentrations were found to be an important predictor of stress gastritis independent of other known risk factors.

Aged↗

Stepwise selection in small data sets: a simulation study of bias in logistic regression analysis.

Stepwise selection methods are widely applied to identify covariables for inclusion in regression models. One of the problems of stepwise selection is biased estimation of the regression coefficients. We illustrate this "selection bias" with logistic regression in the GUSTO-I trial (40,830 patients with an acute myocardial infarction). Random samples were drawn that included 3, 5, 10, 20, or 40 events per variable (EPV). Backward stepwise selection was applied in models containing 8 or 16 pre-specified predictors of 30-day mortality. We found a considerable overestimation of regression coefficients of selected covariables. The selection bias decreased with increasing EPV. For EPV 3, 10, or 40, the bias exceeded 25% for 7, 3, and 1 in the 8-predictor model respectively, when a conventional selection criterion was used (alpha = 0.05). For these EPV values, the bias was less than 20% for all covariables when no selection was applied. We conclude that stepwise selection may result in a substantial bias of estimated regression coefficients.

Aged↗

Concept formation vs. logistic regression: predicting death in trauma patients.

This study compares two classification models used to predict survival of injured patients entering the emergency department. Concept formation is a machine learning technique that summarizes known examples cases in the form of a tree. After the tree is constructed, it can then be used to predict the classification of new cases. Logistic regression, on the other hand, is a statistical model that allows for a quantitative relationship for a dichotomous event with several independent variables. The outcome (dependent) variable must have only two choices, e.g. does or does not occur, alive or dead, etc. The result of this model is an equation which is then used to predict the probability of class membership of a new case. The two models were evaluated on a trauma registry database composed of information on all trauma patients admitted in 1992 to a Level I trauma center. A total of 2155 records. representing all trauma patients admitted for more than 24 h or who died in the Emergency Department, were grouped into two databases as follows: (1) discharge status of 'died' (containing 151 records), and (2) any discharge status other than 'died' (containing 2004 records). Both databases contained the same variables.

Artificial Intelligence↗

Qualitative diagnosis of calvarial metastasis by neural network and logistic regression.

RATIONALE AND OBJECTIVES: To simplify the diagnostic features used by an artificial neural network compared with logistic regression (LR) in the diagnosis of calvarial metastasis with computed tomography and analyze their accuracy. MATERIALS AND METHODS: Twenty-one of 167 patients with calvarial lesions were found to have metastasis. Clinical and computed tomography data were used for LR and neural network models. Both models were tested with the leave-one-out method. The final results of each model were compared using the area under receiver operating characteristic curve (Az). RESULTS: The neural network identified metastasis significantly more successfully than LR with an Az of 0.9324 +/- 0.0386 versus 0.9192 +/- 0.0373, P = .01. The most important features selected by the LR and neural network were age and edge definition. CONCLUSION: Neural networks offer wide possibilities over statistics for the study of calvarial metastases other than their minimum clinical and radiologic features for diagnosis.

Adolescent↗

Mechanisms of cuticular uptake of xenobiotics into living plants: evaluation of a logistic-kinetic penetration model.

The objective of this study was to determine whether a logistic-kinetic penetration model could be applied to whole plant uptake. Uptake over 24 h was determined for three model compounds, applied in the presence and absence of surfactants, into the leaves of two plant species. Data for two time intervals were used in the model to predict uptake at intermediate intervals and compared with experimental results. Overall, the model fit the whole plant uptake data well. The study confirmed that an increase (or decrease) in active ingredient (ai) concentration or an increase in contact area will have no effect on the penetration rate factor, q, within the normal working concentration range. This enabled uptake to be predicted at different times for concentrations of ai not already studied, having first derived q for one concentration of the formulation of interest and having 24 h (maximum) uptake results for all formulations and concentrations of interest. The advantages of the models and equations described are that few variables are required, and they are simple to measure.

Carbon Radioisotopes↗

Sequential logistic models for 30 days mortality after CABG: pre-operative, intra-operative and post-operative experience--The Israeli CABG study (ISCAB). Three models for early mortality after CABG.

OBJECTIVES: The goal of this paper was to examine the added effect of operative and post-operative variables on 30 days mortality, in addition to patients' case-mix factors. SETTING AND DESIGN: A prospective study of 4835 patients, 95% of all Israeli patients who underwent coronary artery bypass grafting (CABG) in 1994. Information related to risk of death was collected at admission to hospital (preceding the operation), at time of the operation and in the immediate post-operative period. Deaths were independently ascertained. METHOD: Data collectors followed every patient from admission to discharge. Sequential logistic models were constructed for the 'case-mix', 'operative' and the 'post-operative' periods in chronological order. Each model incorporated and adjusted for the risk estimated at the previous point in time, by forcing individual risk scores. RESULTS: Significant pre-operative risk factors for 30 days mortality, in the case-mix model included mainly severity of illness characteristics, such as, left ventricular dysfunction and emergency admission, (c-statistic 78.8%). Model 2 (the 'operation' model) included in addition to the case-mix score, excessive duration of the operation per graft, bleeding, etc. (c-statistic 85.3%). The post-operative model showed the added effect of the post-operative factors such as low haemoglobin, additional surgery, and excessive time on respirator, (c-statistic 92.4%). CONCLUSIONS: The sequential analysis was an efficient method for updating patients' risk over time, where the number of events was small, relative to the number of risk factors. The addition of peri-operative factors increased significantly the predictive power of the model, adding clinical insights to the role of the hospital experience on 30 days mortality.

Adult↗

Clarithromycin resistance, but not CYP2C-19 polymorphism, has a major impact on treatment success in 7-day treatment regimen for cure of H. pylori infection: a multiple logistic regression analysis.

Mutations in the gene encoding the CYP2C-19 enzyme for PPI metabolism have been shown to enhance the chance for a cure in a H. pylori-positive patients using a two-week dual-therapy regimen involving omeprazole and amoxicillin. However, the impact of CYP2C-19 genetic polymorphism on eradication rates of a one-week triple-therapy regimen has not been examined. In this cohort study, 156 H. pylori-positive peptic ulcer or NUD patients who presented to our university hospital were recruited. They were treated by one-week omeprazole-amoxicillin-clarithromycin therapy. Host and bacterial predictive factors including H. pylori susceptibility and CYP2C-19 genotyping, as well as cure rate for H. pylori infection, were studied. Cure rate was 85.9% (95% CI: 79-91%) on an intent to treat (ITT) basis. By multiple logistic regression analysis, only clarithromycin resistance had a significant impact on treatment success (odds ratio 28.7: 95% CI: 6-172). CYP2C-19 genetic polymorphism was not associated with a significant change in cure rate. These observations indicate only clarithromycin susceptibility, not CYP2C-19 polymorphism, has a major impact on the treatment success when using a seven-day OAC H. pylori treatment regimen.

Amoxicillin↗

Apolipoprotein E genotype frequency patterns in aged Danes as revealed by logistic regression models.

Although the ApoE gene has been intensively studied in aging research, most of the studies conducted so far have been based on the traditional case-control design with subjects consisting of young controls and long-lived cases. The genotype frequency pattern in and between the two age-groups has been rarely investigated due to limitations in either research design or data analytical method. In this study, we genotyped 748 individuals (including both twin pairs and unrelated individuals) aged from 73 to 95 with aim at examining the genotype frequency trajectory of ApoE gene at high ages. Binomial and multinomial logistic regression models have been applied to model the gene frequency as a function of age and to investigate the modes of gene function (dominant, recessive, additive). The generalized estimation equations (GEEs) are introduced to account for the intra-pair genotype correlation in the twin pairs included in the data. Both the observed and the fitted frequencies show a constantly declining pattern of ApoE epsilon4 allele as age advances indicating a significant and steadily deleterious effect of the dominant allele that increases the hazard of death at high ages.

Aged↗

A logistic regression model for measuring gene-longevity associations.

The logistic regression model is a popular model for data analysis in epidemiological research. In this paper, we use this model to analyze genetic data collected from gene-longevity association studies. This new approach models the probability of observing one genotype as a function of the age of investigated individuals. Applying the model to genotype data on the TH and 3'ApoB-VNTR loci collected from an Italian centenarian study, we show how it can be used to model the different ways that genes affect survival, including sex- and age-specific influences. We highlight the advantages of this application over other available models. The application of the model to empirical data indicates that it is an efficient and easily applicable approach for determining the influences of genes on human longevity.

Adolescent↗

Combining the transmission disequilibrium test and case-control methodology using generalized logistic regression.

To study the role of genetic factors in the etiology, susceptibility, or severity of disease, several methods are available. In a transmission disequilibrium test, genotypes of cases are compared to those of their parents to explore whether a specific allele, or marker, at a locus of interest appears to be transmitted in excess of what is expected on the basis of Mendelian inheritance. Such apparent excess transmission indicates that cases are being selected for that allele, thereby providing evidence that this allele is a risk factor for disease. In case-control studies, genotypes of cases are compared to those of controls from the same population to identify whether a specific allele is associated with disease. If so, either the allele at this locus or one in linkage disequilibrium with it may be causally related to the etiology of the disease. Here, we discuss the problem of combining a transmission disequilibrium test and a case-control comparison, in order to integrate all available information, and thereby increase statistical power. As the same cases are used in both approaches, the two results are not independent. However, parents of cases can be independently compared to controls. Both the issue of testing for a genetic effect and the estimation of relative risks under the multiplicative model using generalized logistic regression are discussed.

Case-Control Studies↗

Reactivity of six antibodies in effusions of mesothelioma, adenocarcinoma and mesotheliosis: stepwise logistic regression analysis.

Anti-CEA, anti-vimentin, CAM5.2, BerEp4, Leu-M1 and anti-EMA were applied to effusions from 36 mesotheliomas, 53 adenocarcinomas and 24 reactive mesothelial proliferations. Stepwise logistic regression analysis selected three criteria of major importance for distinguishing between adenocarcinoma and mesothelioma: BerEp4, CEA and EMA accentuated at the cell membrane (mEMA), these three being of similar diagnostic value. The pattern BerEp4-, CEA- and mEMA+ was fully predictive for mesothelioma (sensitivity 47%), whereas the opposite pattern was fully predictive for adenocarcinoma (sensitivity 80%). Only EMA seemed to distinguish between mesotheliosis and mesothelioma. Comparison of reactivity in cytological and histological material from the same mesotheliomas showed similar staining frequencies for CEA and CAM5.2, with some random variation for Leu-M1 and EMA, whereas vimentin and BerEp4 reactivity was more frequent in cytological specimens.

Adenocarcinoma↗