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PubMed2005

Comparison of artificial neural network and logistic regression models for prediction of mortality in head trauma based on initial clinical data.

BACKGROUND: In recent years, outcome prediction models using artificial neural network and multivariable logistic regression analysis have been developed in many areas of health care research. Both these methods have advantages and disadvantages. In this study we have compared the performance of artificial neural network and multivariable logistic regression models, in prediction of outcomes in head trauma and studied the reproducibility of the findings. METHODS: 1000 Logistic regression and ANN models based on initial clinical data related to the GCS, tracheal intubation status, age, systolic blood pressure, respiratory rate, pulse rate, injury severity score and the outcome of 1271 mainly head injured patients were compared in this study. For each of one thousand pairs of ANN and logistic models, the area under the receiver operating characteristic (ROC) curves, Hosmer-Lemeshow (HL) statistics and accuracy rate were calculated and compared using paired T-tests. RESULTS: ANN significantly outperformed logistic models in both fields of discrimination and calibration but under performed in accuracy. In 77.8% of cases the area under the ROC curves and in 56.4% of cases the HL statistics for the neural network model were superior to that for the logistic model. In 68% of cases the accuracy of the logistic model was superior to the neural network model. CONCLUSIONS: ANN significantly outperformed the logistic models in both fields of discrimination and calibration but lagged behind in accuracy. This study clearly showed that any single comparison between these two models might not reliably represent the true end results. External validation of the designed models, using larger databases with different rates of outcomes is necessary to get an accurate measure of performance outside the development population.

Adolescent↗

PubMed2005

Additive vs. logistic risk models for cardiac surgery mortality.

OBJECTIVE: Logistic regression is most often used to produce a cardiac operative risk model. But the logistic equation requires a computer to solve. Thus, simple additive models have been derived from logistic models by adding the odds ratios or modified coefficients. However, this simplification has no statistical justification, and the additive scores do not equal the original logistic probabilities. METHODS: The EuroSCORE risk model is a very successful and widely used cardiac surgery risk model and it comes in both an additive and a full logistic version. We applied the EuroSCORE model to the 28,337 cardiac surgeries in the Providence Health System Cardiovascular Study Group database. The discrimination of the models was assessed by the c index. The comparison of the mortality predictions of the logistic and the additive model are mostly descriptive and graphical. RESULTS: Theoretical considerations would predict that the additive model greatly underestimates the risk for the higher risk patients, and clinical data confirm this fact. For the 23,463 (83%) cases with complete data, the predicted mortality was 8.3% by the logistic model and 5.4% by the additive model. The discrimination (c index) of the additive (0.794) and logistic (0.791) models was equally good. A modified additive score is proposed (the mean of the logistic predicted mortality for each original additive score) which could be provided as a look-up table along with the scoring sheet. CONCLUSIONS: The additive EuroSCORE gives excellent discrimination, as good as the logistic risk model, but it greatly underestimates the risk of high-risk patients, compared to the logistic. The logistic equation should be used to predicate the mortality when possible. If this is not feasible, a modified additive score could be employed at the bedside. But the logistic should always be used for comparison of providers and for research publications.

Cardiac Surgical Procedures↗

PubMed1987

Equilibrium and local stability in a logistic matrix model for age-structured populations.

A logistic matrix model for age-structured population dynamics is constructed. This model discretizes a continuous, density-dependent model with age structure, i.e. it is an extension of the logistic model to the case of age-dependence. We prove the existence and uniqueness of its equilibrium and give a necessary and sufficient condition for the local stability of the equilibrium.

Aging↗

PubMed1999

Computing measures of explained variation for logistic regression models.

The proportion of explained variation (R2) is frequently used in the general linear model but in logistic regression no standard definition of R2 exists. We present a SAS macro which calculates two R2-measures based on Pearson and on deviance residuals for logistic regression. Also, adjusted versions for both measures are given, which should prevent the inflation of R2 in small samples.

Computer Simulation↗

PubMed1997

Regression models for ordinal responses: a review of methods and applications.

BACKGROUND: Epidemiologists are often interested in estimating the risk of several related diseases as well as adverse outcomes, which have a natural ordering of severity or certainty. While most investigators choose to model several dichotomous outcomes (such as very low birthweight versus normal and moderately low birthweight versus normal), this approach does not fully utilize the available information. Several statistical models for ordinal responses have been proposed, but have been underutilized. In this paper, we describe statistical methods for modelling ordinal response data, and illustrate the fit of these models to a large database from a perinatal health programme. METHODS: Models considered here include (1) the cumulative logit model, (2) continuation-ratio model, (3) constrained and unconstrained partial proportional odds models, (4) adjacent-category logit model, (5) polytomous logistic model, and (6) stereotype logistic model. We illustrate and compare the fit of these models on a perinatal database, to study the impact of midline episiotomy procedure on perineal lacerations during labour and delivery. Finally, we provide a discussion on graphical methods for the assessment of model assumptions and model constraints, and conclude with a discussion on the choice of an ordinal model. The primary focus in this paper is the formulation of ordinal models, interpretation of model parameters, and their implications for epidemiological research. CONCLUSIONS: This paper presents a synthesized review of generalized linear regression models for analysing ordered responses. We recommend that the analyst performs (i) goodness-of-fit tests and an analysis of residuals, (ii) sensitivity analysis by fitting and comparing different models, and (iii) by graphically examining the model assumptions.

Epidemiologic Methods↗

PubMed1999

Logistic regression models in obstetrics and gynecology literature.

OBJECTIVE: To evaluate the reporting of multivariable logistic regression analyses and assess variations in quality over time in the obstetrics and gynecology literature. METHODS: Methodologic criteria for reporting logistic regression analyses were developed to identify problems affecting accuracy, precision, and interpretation of this approach to multivariable statistical analysis. These criteria were applied to 193 articles that reported multivariable logistic regression in the issues of four generic obstetrics and gynecology journals in 1985, 1990, and 1995. Rates of compliance with the methodologic criteria and their time trends were analyzed. RESULTS: The proportion of articles using logistic regression analysis increased over time: 1.7% in 1985, 2.8% in 1990, and 6.5% in 1995 (P < .001 for trend). Violations and omissions of methodologic criteria for reporting logistic models were common. The research question, in terms of dependent and independent variables, was not clearly reported in 32.1%. The process of variable selection was inadequately described in 51.8% of the articles. Among articles with ranked independent variables, 85.1% did not report assessment of conformity to linear gradient. Tests for goodness of fit were not given in 93.2% of articles. The contribution of the independent variables could not be evaluated in 36.2% of the articles because of a lack of coding of the variables. Interactions between variables were not assessed in 86.4% of articles. Analysis of variations in the quality of logistic regression analyses over time showed no increase in reporting of the criteria concerning variable selection and goodness of fit. However, the proportion of articles reporting one quality criterion concerning interpretation of the substantive significance of independent variables showed a trend toward improvement: 42.3% in 1985, 73.6% in 1990, and 75.4% in 1995 (P = .004 for trend). CONCLUSION: The reporting of multivariable logistic regression models in the obstetrics and gynecology literature is poor, and the time trends of improvement in quality of reporting are not particularly encouraging.

Gynecology↗

PubMed2001

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↗

PubMed2003

A mixed-effects multinomial logistic regression model.

A mixed-effects multinomial logistic regression model is described for analysis of clustered or longitudinal nominal or ordinal response data. The model is parameterized to allow flexibility in the choice of contrasts used to represent comparisons across the response categories. Estimation is achieved using a maximum marginal likelihood (MML) solution that uses quadrature to numerically integrate over the distribution of random effects. An analysis of a psychiatric data set, in which homeless adults with serious mental illness are repeatedly classified in terms of their living arrangement, is used to illustrate features of the model.

Adult↗

PubMed2004

Demonstration of the applicability of the Weibull-log-logistic survival model to the isothermal and nonisothermal inactivation of Escherichia coli K-12 MG1655.

Published isothermal semilogarithmic survival curves of Escherichia coli K-12 MG1655, in the range of 49.8 to 60.6 degrees C, all had noticeable downward concavity. They could be described by the model log S(t) = -b(T)t n, where S(t) = N(t)/N0, N(t) and N0 being the momentary and initial number of organisms, respectively; b(T) is a temperature-dependent rate parameter; and n is a constant found to be about 1.5. The temperature dependence of b(T) could be described by the log-logistic model, b(T) = ln[1 + exp[k(T - Tc)]], which had an almost perfect fit, with k = 0.88 degrees C(-1) and Tc = 60.5 degrees C. The constants, n, k, and Tc were considered the organism's survival parameters in the particular medium. They were incorporated into a rate equation on the assumption that in nonisothermal heating, the momentary inactivation rate is the isothermal rate at the momentary temperature at a time that corresponds to the momentary survival ratio. This model's estimates matched the actual survival curves obtained in the same work under two different nonisothermal heating profiles, lending support to the notion that the Weibull-log-logistic model combination can be used not only to describe isothermal inactivation mathematically, but also to predict survival patterns under nonisothermal conditions.

Colony Count, Microbial↗

PubMed1999

Validation of logistic regression models in small samples: application to calvarial lesions diagnosis.

We have used the leave-one-out (LOO) method and the area under the receiver operating characteristic (ROC) curve to validate logistic models with a sample of 167 patients with calvarial lesions. Seven logistic regression models were developed from 12 clinical and radiological variables to predict the most common diagnoses separately. The LOO method was used to test the validity of the equations. The discriminant power of every model was assessed by means of the area under the ROC curve (Az). The model with the greatest discrimination ability for the whole data set was the osteoma equation (Az = 0.951). The discriminatory ability of the statistical models decreased significantly with the LOO procedure, having the malignancy model the highest value (Az = 0.931). The LOO method can obtain a high benefit from small samples in order to validate prediction rules. In studies with small samples, resampling techniques such as the LOO should be routinely used in predictive modeling. This method may improve the forecast of infrequent diseases, such as calvarial lesions.

Bone Neoplasms↗

PubMed1994

Corrections for exposure measurement error in logistic regression models with an application to nutritional data.

Two correction methods are considered for multiple logistic regression models with some covariates measured with error. Both methods are based on approximating the complicated regression model between the response and the observed covariates with simpler models. The first model is the logistic approximation proposed by Rosner et al., and the second is a second-order extension of this model. Only the mean and covariance matrix of the true values of the covariates given the observed values have to be specified, but no distributional assumptions about the measurement error are made. The parameters related to the conditional moments are estimated from a separate validation data set. The correction methods considered here are compared to other methods proposed in the literature. They are also applied to a multiple logistic model describing the effect of nutrient intakes on the ratio of serum HDL cholesterol. The data constitute baseline data from an epidemiological cohort study, in which a separate pilot study has been carried out to obtain validation information. In the example the corrected parameter estimates from the two approximate models are very similar. Both differ considerably from the naive logistic estimates, indicating a large effect of the measurement error. The various assumptions required by the correction methods are also discussed.

Aged↗

PubMed1998

Using binary logistic regression models for ordinal data with non-proportional odds.

The proportional odds model (POM) is the most popular logistic regression model for analyzing ordinal response variables. However, violation of the main model assumption can lead to invalid results. This is demonstrated by application of this method to data of a study investigating the effect of smoking on diabetic retinopathy. Since the proportional odds assumption is not fulfilled, separate binary logistic regression models are used for dichotomized response variables based upon cumulative probabilities. This approach is compared with polytomous logistic regression and the partial proportional odds model. The separate binary logistic regression approach is slightly less efficient than a joint model for the ordinal response. However, model building, investigating goodness-of-fit, and interpretation of the results is much easier for binary responses. The careful application of separate binary logistic regressions represents a simple and adequate tool to analyze ordinal data with non-proportional odds.

Bias↗

PubMed2003

Evaluation of the American College of Cardiology/American Heart Association and the Society for Coronary Angiography and Interventions lesion classification system in the current "stent era" of coronary interventions (from the ACC-National Cardiovascular Data Registry).

In 1988 American College of Cardiology (ACC)/American Heart Association (AHA) Guidelines for Coronary Angioplasty proposed a lesion classification system to stratify lesions by difficulty and risk to better understand the outcomes of coronary interventions. It was a 3-level (A, B, and C) classification based on 11 lesion characteristics. A modification, dividing the intermediate B category into B1 and B2, is also in common use. Recently, a simplification of this classification was evaluated using the large Society for Cardiac Angiography and Interventions (SCAI) Registry (SCAI I = non-C/patent; SCAI II = C/patent; SCAI III = non-C/occluded; SCAI IV = C/occluded). The lesion classification systems were evaluated in 61,926 patients from the ACC National Cardiovascular Data Registry who underwent single-vessel percutaneous coronary intervention between January 1998 and September 2000. Stents were placed in 74.5% of patients. Logistic models for lesion success and complications were constructed and compared. The c statistic for success using the ACC/AHA original classification system was 0.69, 0.71 for the modified ACC/AHA system, and 0.75 for the SCAI classification. The range of complication and success rates was greater using the SCAI models, and the logistic models for success and complication were more robust for the SCAI system. Thus, in the large ACC-National Cardiovascular Data Registry, with a high percentage of stent usage, the simpler SCAI lesion classification provided better discrimination for success and complications than the more complex ACC/AHA lesion classification system-original or modified.

Angioplasty, Balloon, Coronary↗

PubMed2006

Evaluating mortality in intensive care units: contribution of competing risks analyses.

INTRODUCTION: Kaplan-Meier curves and logistic models are widely used to describe and explain the variability of survival in intensive care unit (ICU) patients. The Kaplan-Meier approach considers that patients discharged alive from hospital are 'non-informatively' censored (for instance, representative of all other individuals who have survived to that time but are still in hospital); this is probably wrong. Logistic models are adapted to this so-called 'competing risks' setting but fail to take into account censoring and differences in exposure time. To address these issues, we exemplified the usefulness of standard competing risks methods; namely, cumulative incidence function (CIF) curves and the Fine and Gray model. METHODS: We studied 203 mechanically ventilated cancer patients with acute respiratory failure consecutively admitted over a five-year period to a teaching hospital medical ICU. Among these patients, 97 died before hospital discharge. After estimating the CIF of hospital death, we used Fine and Gray models and logistic models to explain variability hospital mortality. RESULTS: The CIF of hospital death was 35.5% on day 14 and was 47.8% on day 60 (97/203); there were no further deaths. Univariate models, either the Fine and Gray model or the logistic model, selected the same eight variables as carrying independent information on hospital mortality at the 5% level. Results of multivariate were close, with four variables selected by both models: autologous stem cell transplantation, absence of congestive heart failure, neurological impairment, and acute respiratory distress syndrome. Two additional variables, clinically documented pneumonia and the logistic organ dysfunction, were selected by the Fine and Gray model. CONCLUSION: The Fine and Gray model appears of interest when predicting mortality in ICU patients. It is closely related to the logistic model, through direct modeling of times to death, and can be easily extended to model non-fatal outcomes.

Adult↗

PubMed2004

A Bayesian approach to estimate and validate the false negative fraction in a two-stage multiple screening test.

OBJECTIVES: In estimating sensitivity and specificity of a diagnostic kit it is imperative that all study subjects are verified via a gold standard procedure. However the application of such a procedure to all the study subjects may not be feasible due to associated cost, risk and invasiveness. As a result only a part of the study subjects receive the definitive assessment. The accuracy of a diagnostic kit can also be expressed in terms of its error rates. Our first objective is to estimate the false negative fraction (FNF) under partial verification in a particular case of a two-stage multiple screening test using a beta-binomial model and a Bayesian logistic model. The second objective is to validate the two models in order to determine which fits the data better. METHODS: We estimate the FNF from the above mentioned models using Bayesian approach. The validation of the models is based on their out-of-sample predictive capabilities. RESULTS: For the bowel cancer data that was used in this study we found the median posterior estimate of the FNF, based on the beta-binomial model, to be 26.4% (95% credible interval: 0.123-0.650). The corresponding estimate based on the Bayesian logistic model was 23.3% (95% credible interval: 0.124-0.375). Validation results showed that the betabinomial model gave slightly better predictions compared to the Bayesian logistic model. CONCLUSIONS: Estimation of the FNF can be done by adopting the Bayesian approach. Models fitted can be validated by comparing their performance in terms of their out-of-sample predicitve potential.

Bayes Theorem↗

PubMed2005

Nomographic representation of logistic regression models: a case study using patient self-assessment data.

Logistic regression models are widely used in medicine, but difficult to apply without the aid of electronic devices. In this paper, we present a novel approach to represent logistic regression models as nomograms that can be evaluated by simple line drawings. As a case study, we show how data obtained from a questionnaire-based patient self-assessment study on the risks of developing melanoma can be used to first identify a subset of significant covariates, build a logistic regression model, and finally transform the model to a graphical format. The advantage of the nomogram is that it can easily be mass-produced, distributed and evaluated, while providing the same information as the logistic regression model it represents.

Algorithms↗

PubMed2006

Comparison of models to identify lame cows based on gait and lesion scores, and limb movement variables.

Bovine lameness results in pain and suffering in cattle and economic loss for producers. A system for automatically detecting lame cows was developed recently that measures vertical force components attributable to individual limbs. These measurements can be used to calculate a number of limb movement variables. The objective of this investigation was to explore whether gait scores, lesion scores, or combined gait and lesion scores were more effectively captured by a set of 5 limb movement variables. A set of 700 hind limb examinations was used to create gait-based, lesion-based, and combined (gait- and lesion-based) models. Logistic regression models were constructed using 1, 2, or 3 d of measurements. Resulting models were tested on cows not used in modeling. The accuracy of lesion-score models was superior to that of gait-score models; lesion-based models generated greater values of areas under the receiving operating characteristic curves (range 0.75 to 0.84) and lower mean-squared errors (0.13 to 0.16) compared with corresponding values for the gait-based models (0.63 to 0.73 and 0.26 to 0.31 for receiving operating characteristic and mean-squared errors, respectively). These results indicate that further model development and investigation could generate automated and objective methods of lameness detection in dairy cattle.

Animals↗

PubMed2004

Modeling the boundaries of growth of Salmonella Typhimurium in broth as a function of temperature, water activity, and pH.

The growth limits of a mixture of five strains of Salmonella Typhimurium in tryptic soy broth were examined at different environmental conditions. The response of the pathogen was monitored in a total of 350 combination treatments of temperature (10 to 35 degrees C), pH (3.76 to 6.44), and water activity (aw, 0.913 to 0.990) for 62 days. No growth/growth (turbidity) data were modeled by logistic polynomial regression. The concordance index of the logistic model was 99.8%, indicating a good fit to the observed data. The minimum pH and aw values that permitted growth were 3.94 and 0.942, respectively, and occurred in the temperature range of 25 to 35 degrees C. At temperatures below this range, the minimum pH and aw allowing growth increased as the temperature decreased. The results showed an abrupt change in the probability of growth close to the boundary with minor changes of the environmental factors. The probabilities predicted by the model were compared with published data on the actual response of Salmonella Typhimurium or other salmonellae serotypes in 50 cases of food products, including salad dressing, mayonnaise, meat, cheese, vegetables, and fruits. The model predicted successfully the response of the pathogen in 90% of the tested cases. The results of the study indicated that the developed model predicts satisfactorily the growth/no growth interface of Salmonella Typhimurium in foods and can provide useful quantitative data for the development of safer food products and processes.

Colony Count, Microbial↗