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PubMed1995

Some unknown episodes in the history of population dynamics concerning the logistic law: the lesson of the great epidemiologists.

The author discusses the work of "the two great epidemiologists R. Ross and A. G. McKendrick. This talk is basically devoted to their 'lesson' about logistic theory and it is organised as follows: section 2 is devoted to McKendrick's [1911] contribution to the growth of microorganism populations, a paper which was not widely acclaimed, but which marks the first experimental laboratory verification of the logistic law. Section 3 is also devoted to McKendrick and to the way he repeatedly derived the logistic equation as a model for 'simple epidemics'. Finally, in section 4, we present Ross's 'A priori pathometry' theory that really emphasizes...the role played by the logistic law as the 'first principle' of the mathematical epidemiology."

Demography↗
PubMed1997

Comparison of a genetic algorithm neural network with logistic regression for predicting outcome after surgery for patients with nonsmall cell lung carcinoma.

BACKGROUND: Neural networks have been used to predict outcome in cancer patients. Their accuracy compared with standard statistical methods has not been fully assessed. METHODS: In this study, the authors examined the ability to predict the outcome of surgery in 620 patients with nonsmall cell lung carcinoma (NSCLC) by a genetic algorithm neural network (GANN) using Bayes' theorem compared with logistic regression, and the predictive value of tumor volume measures in addition to standard indices such as histologic type and stage. Predictive methods were compared by examining accuracy of classifying target outcome of patients living or dead at 6, 12, 18, and 24 months after surgery. RESULTS: GANN was a significantly better predictor of outcome than logistic regression at all time points (McNemar, P < 0.01). Measures of tumor volume produced significant improvement in the prediction of 12-, 18-, and 24-month time points with GANN, and at 18- and 24-month time points with logistic regression (Wilcoxon matched pairs signed rank test, P < 0.02). CONCLUSIONS: In this study of surgically treated NSCLC patients, outcome predictions were significantly improved by including measures of tumor volume. For predicting individual patient outcome, GANN was found to be highly accurate and significantly better than logistic regression.

Aged↗
PubMed1998

Empirical comparisons of proportional hazards, poisson, and logistic regression modeling of occupational cohort data.

This research was conducted to examine the effect of model choice on the epidemiologic interpretation of occupational cohort data. Three multiplicative models commonly employed in the analysis of occupational cohort studies--proportional hazards. Poisson, and logistic regression--were used to analyze data from an historical cohort study of workers exposed to formaldehyde. Samples were taken from this dataset to create a number of predetermined scenarios for comparing the models, varying study size, outcome frequency, strength of risk factors, and follow-up length. The Poisson and proportional hazards models yielded nearly identical relative risk estimates and confidence intervals in all situations except when confounding by age could not be closely controlled in the Poisson analysis. Logistic regression findings were more variable, with risk estimates differing most from the proportional hazards results when there was a common outcome or strong relative risk. The logistic model also provided less precise estimates than the other two. Thus, although logistic was the easiest model to implement, it should be used only in occupational cohort studies when the outcome is rare (5% or less), and the relative risk is less than approximately 2. Even then, the proportional hazards and Poisson models are better choices. Selecting between these two can be based on convenience in most circumstances.

Cohort Studies↗
PubMed2003

Evaluation of Cox's model and logistic regression for matched case-control data with time-dependent covariates: a simulation study.

Case-control studies are typically analysed using the conventional logistic model, which does not directly account for changes in the covariate values over time. Yet, many exposures may vary over time. The most natural alternative to handle such exposures would be to use the Cox model with time-dependent covariates. However, its application to case-control data opens the question of how to manipulate the risk sets. Through a simulation study, we investigate how the accuracy of the estimates of Cox's model depends on the operational definition of risk sets and/or on some aspects of the time-varying exposure. We also assess the estimates obtained from conventional logistic regression. The lifetime experience of a hypothetical population is first generated, and a matched case-control study is then simulated from this population. We control the frequency, the age at initiation, and the total duration of exposure, as well as the strengths of their effects. All models considered include a fixed-in-time covariate and one or two time-dependent covariate(s): the indicator of current exposure and/or the exposure duration. Simulation results show that none of the models always performs well. The discrepancies between the odds ratios yielded by logistic regression and the 'true' hazard ratio depend on both the type of the covariate and the strength of its effect. In addition, it seems that logistic regression has difficulty separating the effects of inter-correlated time-dependent covariates. By contrast, each of the two versions of Cox's model systematically induces either a serious under-estimation or a moderate over-estimation bias. The magnitude of the latter bias is proportional to the true effect, suggesting that an improved manipulation of the risk sets may eliminate, or at least reduce, the bias.

Canada↗
PubMed2005

A permutation test for inference in logistic regression with small- and moderate-sized data sets.

Inference based on large sample results can be highly inaccurate if applied to logistic regression with small data sets. Furthermore, maximum likelihood estimates for the regression parameters will on occasion not exist, and large sample results will be invalid. Exact conditional logistic regression is an alternative that can be used whether or not maximum likelihood estimates exist, but can be overly conservative. This approach also requires grouping the values of continuous variables corresponding to nuisance parameters, and inference can depend on how this is done. A simple permutation test of the hypothesis that a regression parameter is zero can overcome these limitations. The variable of interest is replaced by the residuals from a linear regression of it on all other independent variables. Logistic regressions are then done for permutations of these residuals, and a p-value is computed by comparing the resulting likelihood ratio statistics to the original observed value. Simulations of binary outcome data with two independent variables that have binary or lognormal distributions yield the following results: (a) in small data sets consisting of 20 observations, type I error is well-controlled by the permutation test, but poorly controlled by the asymptotic likelihood ratio test; (b) in large data sets consisting of 1000 observations, performance of the permutation test appears equivalent to that of the asymptotic test; and (c) in small data sets, the p-value for the permutation test is usually similar to the mid-p-value for exact conditional logistic regression.

Clinical Trials as Topic↗
PubMed2007

Confidence intervals for multinomial logistic regression in sparse data.

Logistic regression is one of the most widely used regression models in practice, but alternatives to conventional maximum likelihood estimation methods may be more appropriate for small or sparse samples. Modification of the logistic regression score function to remove first-order bias is equivalent to penalizing the likelihood by the Jeffreys prior, and yields penalized maximum likelihood estimates (PLEs) that always exist, even in samples in which maximum likelihood estimates (MLEs) are infinite. PLEs are an attractive alternative in small-to-moderate-sized samples, and are preferred to exact conditional MLEs when there are continuous covariates. We present methods to construct confidence intervals (CI) in the penalized multinomial logistic regression model, and compare CI coverage and length for the PLE-based methods to that of conventional MLE-based methods in trinomial logistic regressions with both binary and continuous covariates. Based on simulation studies in sparse data sets, we recommend profile CIs over asymptotic Wald-type intervals for the PLEs in all cases. Furthermore, when finite sample bias and data separation are likely to occur, we prefer PLE profile CIs over MLE methods.

Aspartate Aminotransferases↗
PubMed2007

A comparison of regression trees, logistic regression, generalized additive models, and multivariate adaptive regression splines for predicting AMI mortality.

Clinicians and health service researchers are frequently interested in predicting patient-specific probabilities of adverse events (e.g. death, disease recurrence, post-operative complications, hospital readmission). There is an increasing interest in the use of classification and regression trees (CART) for predicting outcomes in clinical studies. We compared the predictive accuracy of logistic regression with that of regression trees for predicting mortality after hospitalization with an acute myocardial infarction (AMI). We also examined the predictive ability of two other types of data-driven models: generalized additive models (GAMs) and multivariate adaptive regression splines (MARS). We used data on 9484 patients admitted to hospital with an AMI in Ontario. We used repeated split-sample validation: the data were randomly divided into derivation and validation samples. Predictive models were estimated using the derivation sample and the predictive accuracy of the resultant model was assessed using the area under the receiver operating characteristic (ROC) curve in the validation sample. This process was repeated 1000 times-the initial data set was randomly divided into derivation and validation samples 1000 times, and the predictive accuracy of each method was assessed each time. The mean ROC curve area for the regression tree models in the 1000 derivation samples was 0.762, while the mean ROC curve area of a simple logistic regression model was 0.845. The mean ROC curve areas for the other methods ranged from a low of 0.831 to a high of 0.851. Our study shows that regression trees do not perform as well as logistic regression for predicting mortality following AMI. However, the logistic regression model had performance comparable to that of more flexible, data-driven models such as GAMs and MARS.

Data Interpretation, Statistical↗
PubMed1989

Correction of logistic regression relative risk estimates and confidence intervals for systematic within-person measurement error.

Errors in the measurement of exposure that are independent of disease status tend to bias relative risk estimates and other measures of effect in epidemiologic studies toward the null value. Two methods are provided to correct relative risk estimates obtained from logistic regression models for measurement errors in continuous exposures within cohort studies that may be due to either random (unbiased) within-person variation or to systematic errors for individual subjects. These methods require a separate validation study to estimate the regression coefficient lambda relating the surrogate measure to true exposure. In the linear approximation method, the true logistic regression coefficient beta* is estimated by beta/lambda, where beta is the observed logistic regression coefficient based on the surrogate measure. In the likelihood approximation method, a second-order Taylor series expansion is used to approximate the logistic function, enabling closed-form likelihood estimation of beta*. Confidence intervals for the corrected relative risks are provided that include a component representing error in the estimation of lambda. Based on simulation studies, both methods perform well for true odds ratios up to 3.0; for higher odds ratios the likelihood approximation method was superior with respect to both bias and coverage probability. An example is provided based on data from a prospective study of dietary fat intake and risk of breast cancer and a validation study of the questionnaire used to assess dietary fat intake.

Bias↗
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

On the cumulants of population size for the stochastic power law logistic model.

The deterministic power law logistic model is used to describe density-dependent population growth in cases where the ordinary logistic model is insufficient. This paper investigates an analogous stochastic power law logistic model. The exact (unconditional) population size distributions and the cumulant functions for this stochastic model are intractable for large population sizes. Approximating cumulant functions are derived for populations of any size, and are illustrated with examples of assumed Africanized honey bee population dynamics. Outstanding among the findings is that the approximations for the cumulant functions are very accurate for these examples. The stochastic power law logistic model is very general and may be applied to describe the growth of many other natural populations.

Animals↗
PubMed1994

The analysis of survival (mortality) data: fitting Gompertz, Weibull, and logistic functions.

Survival functions are fitted to survival data from several large populations. The Gompertz survival function corresponds to exponential mortality rate increases with time. The Weibull survival function corresponds to mortality rates that increase as a power function of time. A two-parameter, logistic survival function is introduced, and corresponds to mortality rates that increase, and then decrease, with time. A three-parameter logistic-mortality function also is examined. It reflects mortality rates that rise, and then plateau, with age. Data are from published studies of medflies, Drosophila, house flies, flour beetles, and humans. Some survival data are better fit by a logistic survival function than by the more traditionally used Gompertz or Weibull functions. Gompertz, Weibull, or logistic survival functions often fit the survival of 95+% of a population, and the 'tails' of the survival curves usually appear to fall between the values predicted by the three functions. For some populations, such 'tails' appear to be too complex to be fit well by any simple function. Survival data for males and females in some populations are best fit by different functions. Populations of 100 or more are needed to distinguish among the functions. When testing effects of environmental or genetic manipulations on survival, it has been common to determine the changes in parameter values for a given function, such as Gompertz. It may be equally important to determine whether the best-fit function has changed as well.

Animals↗
PubMed1992

The logistic modeling of interobserver agreement.

An approach to the logistic modeling of interobserver agreement is described that allows for the estimation of a commonly employed measure of agreement. The dependent variable is defined to be 1 if the two raters agree, and 0 otherwise. Covariates may be included in the regression equation in order to obtain adjusted or subgroup-specific estimates of percent agreement. As an empirical example, logistic models were fitted to data from a validation study of the agreement between interview information and physician records on the history of post-menopausal estrogen use, from a case-control study of breast cancer conducted on Oahu, Hawaii. Variables found to be related to agreement in previous univariate analyses were examined as covariates in the logistic model. The directly calculated estimates of percent agreement agreed well with the modeled estimates derived from the regression coefficients. Thus, the logistic model may provide a useful alternative to existing methods for the description of interobserver agreement.

Breast Neoplasms↗
PubMed2004

Comparison of logistic regression and neural network analysis applied to predicting living setting after hip fracture.

PURPOSE: Describe and compare the characteristics of artificial neural networks and logistic regression to develop prediction models in epidemiological research. METHODS: The sample included 3708 persons with hip fracture from 46 different states included in the Uniform Data System for Medical Rehabilitation. Mean age was 75.5 years (sd=14.2), 73.7% of patients were female, and 82% were non-Hispanic white. Average length of stay was 17.0 days (sd=10.6). The primary outcome measure was living setting (at home vs. not at home) at 80 to 180 days after discharge. RESULTS: Statistically significant variables (p <.05) in the logistic model included follow-up therapy, sphincter control, self-care ability, marital status, age, and length of stay. Areas under the receiver operating characteristic curves were 0.67 for logistic regression and 0.73 for neural network analysis. Calibration curves indicated a slightly better fit for the neural network model. CONCLUSIONS: Follow-up therapy and independent bowel and/or bladder function were strong predictors of living at home up to 6 months after hospitalization for hip fracture. No practical differences were found between the predictive ability of logistic regression and neural network analysis in this sample.

Aged↗
PubMed1997

Identification of early pregnancy landmarks by transvaginal sonography: analysis by logistic regression.

OBJECTIVE: To assess the feasibility of logistic regression analysis for determining the gestational ages at which detection of early pregnancy landmarks first can be observed. DESIGN: Retrospective analysis. SETTING: University-based tertiary care clinic. PATIENT(S): Eighty-two women with viable singleton pregnancies in whom ovulation had been achieved by an injection of hCG. INTERVENTION(S): Two hundred fifteen transvaginal sonographic scans. MAIN OUTCOME MEASURE(S): Logistic regression was used to estimate the probability of detection of sonographic findings as a function of gestational age. RESULT(S): We found that the likelihood of visualization of a gestational sac or fetal heart motion could be represented accurately by logistic equations. Gestational age at which there was 95% probability of visualization was 35.5 days for the gestational sac and 44.5 days for fetal cardiac activity. The probability of detecting fetal cardiac activity was 95% when the mean gestational sac diameter was 1.6 cm and was 99% at 1.9 cm. CONCLUSION(S): The sonographic appearances of developmental landmarks in early pregnancy occurs within well-defined gestational time periods, and the probabilities for visualization can be closely approximated using a logistic model. Our results suggest that the number of sonographic examinations required to document infertility treatment success can be minimized by surveillance at standardized gestational ages.

Adult↗
PubMed1996

A simulation study of the number of events per variable in logistic regression analysis.

We performed a Monte Carlo study to evaluate the effect of the number of events per variable (EPV) analyzed in logistic regression analysis. The simulations were based on data from a cardiac trial of 673 patients in which 252 deaths occurred and seven variables were cogent predictors of mortality; the number of events per predictive variable was (252/7 =) 36 for the full sample. For the simulations, at values of EPV = 2, 5, 10, 15, 20, and 25, we randomly generated 500 samples of the 673 patients, chosen with replacement, according to a logistic model derived from the full sample. Simulation results for the regression coefficients for each variable in each group of 500 samples were compared for bias, precision, and significance testing against the results of the model fitted to the original sample. For EPV values of 10 or greater, no major problems occurred. For EPV values less than 10, however, the regression coefficients were biased in both positive and negative directions; the large sample variance estimates from the logistic model both overestimated and underestimated the sample variance of the regression coefficients; the 90% confidence limits about the estimated values did not have proper coverage; the Wald statistic was conservative under the null hypothesis; and paradoxical associations (significance in the wrong direction) were increased. Although other factors (such as the total number of events, or sample size) may influence the validity of the logistic model, our findings indicate that low EPV can lead to major problems.

Bias↗
PubMed2001

Comparison of 'pattern recognition' and logistic regression models for discrimination between benign and malignant pelvic masses: a prospective cross validation.

OBJECTIVES: To test prospectively the diagnostic performance of two logistic regression models for calculation of individual risk of malignancy in adnexal tumors (the 'Tailor model' and the 'Timmerman model'), and to compare them to that of 'pattern recognition' (subjective evaluation of the gray-scale ultrasound image and color Doppler ultrasound examination). DESIGN: Consecutive women with a pelvic mass judged clinically to be of adnexal origin underwent preoperative ultrasound examination including color and spectral Doppler examination. The same examination techniques and definitions as those used in the studies in which the logistic regression models had been created were used. The Tailor model was tested in 133 women (35 of whom hada malignancy) and the Timmerman model in 82 women (29 of whom had a malignancy). A subset of 79 women (28 of whom had a malignancy) was used to compare the performance of the Tailor model and the Timmerman model by calculating and comparing the areas under the receiver operating characteristics curves of the two models. Sensitivity and specificity with regard to malignancy were calculated for all three methods. RESULTS: Pattern recognition performed better than the two logistic regression models (sensitivity around 85%, specificity around 90%). Using a risk of malignancy of > 50% to indicate malignancy (as suggested in the original publications), the sensitivity of the Tailor model was 69% and the specificity 88% (n = 133). The corresponding values for the Timmerman model were 62% and 79% (n = 82). The receiver operating characteristics curves showed the two logistic regression models to have similar diagnostic properties (area under the curve, 0.87 vs. 0.84; P = 0.25; n = 79). The diagnostic performance of the mathematical models was much poorer in this study than in those in which the models had been created. CONCLUSION: The poor diagnostic performance of the mathematical models can probably be explained by subtle differences in definitions and examination technique and by differences between the original tumor populations and the study population. For mathematical models to be generally useful, they probably need to be created on the basis of a very large number of tumors, and the variables in the model must be unequivocally defined and the examination technique meticulously standardized.

Adenofibroma↗
PubMed1999

A logistic regression based extension of the TDT for continuous and categorical traits.

The transmission disequilibrium test (TDT), designed as a test of linkage in the presence of association (i.e. linkage disequilibrium), has received considerable attention in the recent statistical genetics literature due to its advantages over other within-family analytic methods. One limitation of the conventional TDT is its application solely to linkage disequilibrium between a genetic marker and a single categorical trait (e.g. presence or absence of a disease). In this paper, we present an extension of the TDT using logistic regression to examine the relation between a candidate gene or genetic marker and one or more continuous or categorical explanatory variables. This logistic regression extension of the TDT possesses all of the desirable features of the conventional TDT, as well as many advantages associated with traditional regression analysis. We describe the model and its properties, as well as a number of its possible applications, and apply it to examine linkage disequilibrium between the dopamine receptor D2 gene (DRD2) and symptoms of childhood attention deficit hyperactivity disorder (ADHD). We also briefly compare the logistic regression TDT to other quantitative TDTs that have been proposed in the literature, and highlight the advantages of a regression-based approach for examining the relation between a candidate gene and one or more continuous or categorical traits. Given its features, we regard the logistic regression extension of the TDT as a flexible new data analytic method with extensive potential applications to problems in medical, psychiatric, and behavioral genetics.

Attention Deficit Disorder with Hyperactivity↗
PubMed2004

The utility of structure-activity relationship (SAR) models for prediction and covariate selection in developmental toxicity: comparative analysis of logistic regression and decision tree models.

Structure-activity relationship (SAR) models can be used to predict the biological activity of potential developmental toxicants whose adverse effects include death, structural abnormalities, altered growth and functional deficiencies in the developing organism. Physico-chemical descriptors of spatial, electronic and lipophilic properties were used to derive SAR models by two modeling approaches, logistic regression and Classification and Regression Tree (CART), using a new developmental database of 293 chemicals (FDA/TERIS). Both single models and ensembles of models (termed bagging) were derived to predict toxicity. Assessment of the empirical distributions of the prediction measures was performed by repeated random partitioning of the data set. Results showed that both the decision tree and logistic regression derived developmental SAR models exhibited modest prediction accuracy. Bagging tended to enhance the prediction accuracy and reduced the variability of prediction measures compared to the single model for CART-based models but not consistently for logistic-based models. Prediction accuracy of single logistic-based models was higher than single CART-based models but bagged CART-based models were more predictive. Descriptor selection in SAR for the understanding of the developmental mechanism was highly dependent on the modeling approach. Although prediction accuracy was similar in the two modeling approaches, there was inconsistency in the model descriptors.

Animals↗