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PubMed1990

Using logistic regression in perinatal epidemiology: an introduction for clinical researchers. Part 2: The logistic regression equation.

In Part 1 basic concepts were introduced as a preparation for an introductory explanation of logistic regression. Logistic regression is a statistical modelling technique, designed for the estimation of the simultaneous effects of predictors on the risk of a certain dichotomous outcome variable where each effect is estimated while adjusting for the effect of the other factors considered. The basic concepts--odds, odds ratio, confounding and interaction--were introduced in such a way that they naturally lead to the concept of logistic regression. In Part 2 the concepts are 'translated' into simple equations. By studying these equations the equivalence between such mathematical expressions and the underlying clinical assessment of risk will become clear.

Female↗

PubMed1989

Empirical comparisons of proportional hazards and logistic regression models.

We compare parameter estimates from the proportional hazards model, the cumulative logistic model and a new modified logistic model (referred to as the person-time logistic model), with the use of simulated data sets and with the following quantities varied: disease incidence, risk factor strength, length of follow-up, the proportion censored, non-proportional hazards, and sample size. Parameter estimates from the person-time logistic regression model closely approximated those from the Cox model when the survival time distribution was close to exponential, but could differ substantially in other situations. We found parameter estimates from the cumulative logistic model similar to those from the Cox and person-time logistic models when the disease was rare, the risk factor moderate, and censoring rates similar across the covariates. We also compare the models with analysis of a real data set that involves the relationship of age, race, sex, blood pressure, and smoking to subsequent mortality. In this example, the length of follow-up among survivors varied from 5 to 14 years and the Cox and person-time logistic approaches gave nearly identical results. The cumulative logistic results had somewhat larger p-values but were substantively similar for all but one coefficient (the age-race interaction). The latter difference reflects differential censoring rates by age, race and sex.

Adolescent↗

PubMed1989

Logistic patterning of brain activity.

Brain activity is a set of electrochemical events in the neural tissue during a specified period of time. Neural networks are the functionally connected and dynamically organized population of neurons and neuroglial cells, serving as a unified interface between environmental input and behavioral output at a specified time. They are patterned dynamically in neural space, time and intensity. It is proposed that 1. the electrophysiological patterning of brain activity follows the laws of logistic growth, decay and impulse response function and 2. that this process of logistic transform (LOGIT) of electrochemical processes is itself the general code of information transfer in the brain. A mathematical model of logistic transformation and new electrophysiological data based on the topographic mapping of event-related potentials (ERP) during selective auditory attentional tasks from nine healthy adults is presented. The N100-P200 and the N200-P300 segments of ERP's were analyzed at 20 ms intervals. Logarithmic latencies and voltages were highly correlated. Significant correlation was found between the mathematically estimated and actual values. The interpeak latencies also followed the logistic growth pattern and could be predicted by the model. A general equation of logistic transformation of brain activity, including variables of cortical voltage and frequency is proposed.

Adult↗

PubMed1991

The importance of assessing the fit of logistic regression models: a case study.

BACKGROUND: The logistic regression model is being used with increasing frequency in all areas of public health research. In the calendar year 1989, over 30% of the articles published in the American Journal of Public Health employed some form of logistic regression modeling. In spite of this increase, there has been no commensurate increase in the use of commonly available methods for assessing model adequacy. METHODS: We review the current status of the use of logistic regression modeling in the American Journal of Public Health. We present a brief overview of currently available and easily used methods for assessing the adequacy of a fitted logistic regression model. RESULTS: An example is used to demonstrate the methods as well as a few of the adverse consequences of failing to assess the fit of the model. One important adverse consequence illustrated in the example is the inclusion of variables in the model as a result of the influence of one subject. CONCLUSIONS: Failure to address model adequacy may lead to misleading or incorrect inferences. Recommendations are made for the use of methods for assessing model adequacy and for future editorial policy in regard to the review of articles using logistic regression.

Bias↗

PubMed1991

Age-dependent logistic regression model and its application.

In the present paper we introduce the theory and algorithm of the unconditional and conditional age-dependent logistic regression model, which combines logistic regression analysis of case-control study with survival analysis of cases in the data, thus facilitating simultaneous comparison analysis between cases and controls and among cases with different ages of disease onset under study. In age-dependent logistic regression analysis, estimated compound relative risk (CRR) and compound attributable risk (CAR) comprise the variance contributions of risk factors to disease occurrence and the time of disease onset, thereby the role played by various risk factors in etiology and etiopathology can be objectively evaluated. The current logistic regression model is only a particular case of age-dependent logistic regression theory neglecting the variations in onset age of diseases.

Age Factors↗

PubMed1990

Robust logistic discriminant functions in diagnosing chronic obstructive airways disease.

The paper gives a comparison of the classical logistic discriminant function, the alpha-trimmed logistic discriminant function and the L1-logistic discriminant function as used for assistance of medical diagnosis in chronic obstructive airways disease. The robustified logistic discriminant functions enable one to obtain higher rates of correctly classified individuals, especially the L1-logistic discriminant function should be recommended.

Discriminant Analysis↗

PubMed1990

Association of logistic and Poisson models of infection with some physical characteristics of a single component plant virus.

A logistic model was recently formulated to describe the relationship between concentration of a single component plant virus and infections produced by inoculation to a local lesion host. In this paper the logistic is combined with a Poisson model. The logistic makes accurate fitting possible for a variety of infection-dilution series; and the Poisson acts as a base line, indicating whether lesion numbers are compatible with the hypothesis that random infection of similar infection sites has occurred. A logarithmic form of the logistic equation gives a straight line with negative slope (logit slope) which is useful in characterizing dilution series to which the logistic is fitted. A modified Poisson equation can also be fitted to a range of dilution series; it provides an independent estimate of slope for curves not widely divergent from the standard Poisson. Models have also been developed to define the limits of concentration within which single virions are likely to be randomly dispersed in inoculum without immediate contact with other virions, and are therefore more likely to enter inoculated tissue independently and cause random infections. Models are formulated for aggregation of tobacco mosaic virus in monolayers, crystals, and lenticular aggregates. Published and unpublished data are fitted and analyzed using some of these models.

Models, Statistical↗

PubMed1978

A critical analysis of the double and triple logistic growth curves.

Recently a model of human growth from the age of one year to maturity, based on two logistic terms, has been proposed. The originators of the model claim that it provides biologically meaningful parameters, and allows total growth to be partitioned into a pre-pubertal and an adolescent component. More recently, they have suggested an improved model, with three logistic terms, which gives a better fit. While the double logistic model gives an adequate fit to the observed height curve, differentiating it leads to a height velocity curve which differs considerably from the observed velocity curve. The triple logistic model gives an excellent fit to both the attained height and height velocity curves. Both models, however, imply a considerable time interval during which both the pre-pubertal and adolescent components are simultaneously contributing to growth, a situation that is difficult to justify biologically. The double model should therefore be discarded, and the triple logistic model considered to be of descriptive, rather than of interpretative, value.

Adolescent↗

PubMed1975

Comparison of logistic equations for population growth.

Two different forms of the logistic equation for population growth appear in the ecological literature. In the form of the logistic equation that appears in recent ecology textbooks the parameters are the instantaneous rate of natural increase per individual and the carrying capacity of the environment. In the form of the logistic equation that appears in some older literature the parameters are the instantaneous birth rate per individual and the carrying capacity. The decision whether to use one form or the other depends on which form of the equation is biologically more realistic. In this study the form of the logistic equation in which the instantaneous birth rate per individual is a parameter is shown to be more realistic in terms of the birth and death processes of population growth. Application of the logistic equation to calculate yield from an exploited fish population also shows that the parameters must be the instantaneous birth rate per individual and the carrying capacity.

Mathematics↗

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↗

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↗

PubMed1990

Using logistic regression in perinatal epidemiology: an introduction for clinical researchers. Part 1: Basic concepts.

Logistic regression is a statistical modelling technique which may be applied to estimate the simultaneous effect of a set of predictors (e.g. gestational age, birthweight) on the risk of a certain outcome variable (e.g. neonatal death) which can take either one of two possible values (yes/no, alive/dead) or in the situation where one wants to estimate the effect of a particular risk factor (e.g. sex) while adjusting (correcting) for the effect of other risk factors (e.g. gestational age). Since this situation often occurs both in medical or epidemiological research and in daily practice it is important to have a flexible and readily interpretable technique to predict risk of mortality and morbidity. Since the logistic regression technique is a powerful and widely applicable tool which is appearing more and more often in the epidemiological literature, a basic understanding of this technique becomes necessary for the clinical researcher. In this paper we explain logistic regression to medical researchers who do not have any particular statistical background. Part 1 covers the basic concepts. Part 2 will describe the actual representation of the basic concepts in a logistic framework.

Epidemiologic Methods↗

PubMed1989

[Study of logistic curve fitting for primary liver cancer death rate in Chengdu].

Logistic curve was used for fitting primary liver cancer (PLC) age-specific death rate in Chengdu. The results showed that the age-specific death rate from 1981 to 1986 in Chengdu (population about 4,000,000) was satisfactorily fitted by logistic curve in male, female and total (P less than 0.01). The R2s were 0.9885, 0.9912 and 0.9974, respectively. Velocity analysis of death rate showed that the increasing peak was in the group of age 40-60 for male and 50-65 for female. Male's velocity and slope of death rate were higher and steeper than female's. These results are consistent with the feature of logistic curve and parameters obtained from logistic curve fitting in this study. The age-specific death rate by sex for every year from 1981 to 1986 was fitted very well (P less than 0.01). R2s for male and female were 0.9415 +/- 0.044 and 0.9056 +/- 0.048, respectively.

Age Factors↗

PubMed1990

Assessing proportionality in the proportional odds model for ordinal logistic regression.

The proportional odds model for ordinal logistic regression provides a useful extension of the binary logistic model to situations where the response variable takes on values in a set of ordered categories. The model may be represented by a series of logistic regressions for dependent binary variables, with common regression parameters reflecting the proportional odds assumption. Key to the valid application of the model is the assessment of the proportionality assumption. An approach is described arising from comparisons of the separate (correlated) fits to the binary logistic models underlying the overall model. Based on asymptotic distributional results, formal goodness-of-fit measures are constructed to supplement informal comparisons of the different fits. A number of proposals, including application of bootstrap simulation, are discussed and illustrated with a data example.

Biometry↗

PubMed1990

Variance calculations and confidence intervals for estimates of the attributable risk based on logistic models.

The attributable risk (AR), defined as AR = [Pr(disease) - Pr(disease/no exposure)]/Pr(disease), measures the proportion of disease risk that is attributable to an exposure. Recently Bruzzi et al. (1985, American Journal of Epidemiology 122, 904-914) presented point estimates of AR based on logistic models for case-control data to allow for confounding factors and secondary exposures. To produce confidence intervals, we derived variance estimates for AR under the logistic model and for various designs for sampling controls. Calculations for discrete exposure and confounding factors require covariances between estimates of the risk parameters of the logistic model and the proportions of cases with given levels of exposure and confounding factors. These covariances are estimated from Taylor series expansions applied to implicit functions. Similar calculations for continuous exposures are derived using influence functions. Simulations indicate that those asymptotic procedures yield reliable variance estimates and confidence intervals with near nominal coverage. An example illustrates the usefulness of variance calculations in selecting a logistic model that is neither so simplified as to exhibit systematic lack of fit nor so complicated as to inflate the variance of the estimate of AR.

Alcohol Drinking↗

PubMed1978

Application of the four-parameter logistic model to bioassay: comparison with slope ratio and parallel line models.

Bioassays with a quantitative response showing a sigmoid log-dose relationship can be analysed by fitting a non-linear dose-response model directly to the data. It is demonstrated that the four-parameter logistic model, previously applied to immunoassay (Healy 1972), is applicable to the free fat cell bioassay of insulin (Moody, Stan, Stan and Gliemann 1974). It is shown that the standard slope ratio and parallel line models for bioassay can be considered as approximations to the logistic in the extreme dose regions, while the parallel line model can be expected to fit in the middle region. The full statistical analysis of the four-parameter logistic model applied to a general assay design is described. An APL computer program has been developed to facilitate the calculations, which include non-linear curve-fitting, tests of goodness of fit and parallelity, as well as point and interval estimates of the relative potency. Examples of free fat cell bioassays of insulin that have been analysed according to these methods are given. Efficient estimation of the potency calls for concentrating the doses in the region with the steepest slope of the dose-response curve. With respect to testing the parallelity and to allow for assay-to-assay variability and unpredictable potencies, it may be preferable to use an assay design with doses distributed over a wide range and to apply a dose-response model which, like the four-parameter logistic, is capable of fitting over the whole feasible dose range.

Biological Assay↗

PubMed1978

[Logistic law of growth and its implications].

The "morphological" (i.e. structural and quantitative) properties of VERHULST's "Logistic Law of Growth" in its versions as differential equations and as analytical functions will be discussed. It follows the attempts of generalizations of the logistic law of growth by parameterization or by changing its structure due to adding parameters. Such an additional parameter is the constant term paying regard to the level (in y-direction) on which the (growth) process may start. The second manner of introducing additional parameters is the substitution of the independent variable in its linear form by a polynomial of degree k. These generalizations will be called "generalized logistic (growth-) function". Its "morphology" will be discussed. Special points there are the use of this function as empirical expression for smoothing and quantitative description of courses of measured values (of growth variables), and the genesis of the function type as solution of a first order differential equation. "Philosophy of the generalized law of growth" means a detailed discussion of properties which could be interpreted as "time structure", and of the modelling relevancy of the differential equations resp. of the analytical function expressions which represent the versions of the generalized logistic law.

Humans↗

PubMed1990

On the use of a logistic risk score in predicting risk of coronary heart disease.

Many studies over the last 20 years have used logistic regression to model the relationship between the risk of developing coronary heart disease (CHD) and the levels of risk factors such as high blood pressure, high serum cholesterol, and cigarette smoking. Subsequently, several investigators have proposed the use of some of the published estimated logistic risk functions to predict risk in new populations. Because of great variation in definition of event, duration of follow-up, population characteristics, definition of risk variables, and selection of other variables in the logistic functions, direct use of such established functions would generally not have validity for the prediction of absolute risk levels. A review of fifteen of these studies indicates on the one hand generally similar results in direction and order of magnitude of effects of the major risk factors, confirming the importance of these risk factors of CHD. On the other hand the reviews indicate sufficient variation to suggest that extrapolation to new populations even to predict relative risk is not justified.

Age Factors↗