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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↗
PubMed1987

Logistic regression for dependent binary observations.

The likelihood of a set of binary dependent outcomes, with or without explanatory variables, is expressed as a product of conditional probabilities each of which is assumed to be logistic. The models are called regressive logistic models. They provide a simple but relatively unknown parametrization of the multivariate distribution. They have the theoretical and practical advantage that they can be analyzed and fitted as in logistic regression for independent outcomes, and with the same computer programs. The paper is largely expository and is intended to motivate the development and usage of the regressive logistic models. The discussion includes serially dependent outcomes, equally predictive outcomes, more specialized patterns of dependence, multidimensional tables, and three examples.

Models, Theoretical↗
PubMed1986

Logistic regression methods for retrospective case-control studies using complex sampling procedures.

There are a number of possible designs for case-control studies. The simplest uses two separate simple random samples, but an actual study may use more complex sampling procedures. Typically, stratification is used to control for the effects of one or more risk factors in which we are interested. It has been shown (Anderson, 1972, Biometrika 59, 19-35; Prentice and Pyke, 1979, Biometrika 66, 403-411) that the unconditional logistic regression estimators apply under stratified sampling, so long as the logistic model includes a term for each stratum. We consider the case-control problem with stratified samples and assume a logistic model that does not include terms for strata, i.e., for fixed covariates the (prospective) probability of disease does not depend on stratum. We assume knowledge of the proportion sampled in each stratum as well as the total number in the stratum. We use this knowledge to obtain the maximum likelihood estimators for all parameters in the logistic model including those for variables completely associated with strata. The approach may also be applied to obtain estimators under probability sampling.

Clinical Trials as Topic↗
PubMed1985

[Practical application of logistic function to the growth of experimental tumors].

The properties of the logistic law of growth (Verhulst-Pearl) and a simple method for computing of statistical approximation are described. Two parameters are estimated: the generation rate c and the mortality rate c0, although the real biological processes are of greater complexity. The logistic law is modified by an additional term concerning the prefinal decline in the last life span. To prove the curve fitting to tumour growth by the modified logistic function the total number of Ehrlich ascites tumour cells was measured in 144 mice at 12 different times after inoculation. The accuracy of the curve fitting proved to be very good. Therefore the logistic function modified by an additional term for the final stages, is particularly suited for the characterization of Ehrlich ascites tumour growth and its changes.

Animals↗
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↗
PubMed1995

Truncated logistic regression.

Truncated binary data occurs when a group of individuals, who each have a binary response, are observed only if one or more of the individuals has a positive response. In this paper the group will be taken to be a motor vehicle accident and the binary response taken to be survival or death. We compare two regression techniques that can be used for truncated binary data. The first procedure, conditional logistic regression (Breslow and Day, 1980, Statistical Methods in Cancer Research. 1: The Analysis of Case-Control Studies. No. 32. Lyon: IARC) conditions on the actual number of deaths, and has been previously used for this type of data. The second procedure, truncated logistic regression, conditions on there being at least one death. It is computationally simpler than conditional logistic for groups of size greater than two and can be considerably more efficient. A major difference between the two methods is that only truncated logistic regression requires a knowledge of group level covariates and allows estimation of group level effects.

Accidents, Traffic↗
PubMed1984

[Variations on the theme of logistic function].

"The use of the logistic function in demographic techniques and analyses has increased significantly during recent years. Nevertheless, not too much attention has been given to the function of the constants nor to the consequences of the implicit assumptions when the function is used for estimating some demographic parameters. In this article, the author underlines the meaning of the constants in the logistic function when applied in some demographic cases, and gives some examples of the assumptions involved in the application of the logistic to the trends of urban/rural populations, life expectancies, and the logit transformation." (summary in ENG)

Demography↗
PubMed1996

Explained variation for logistic regression.

Different measures of the proportion of variation in a dependent variable explained by covariates are reported by different standard programs for logistic regression. We review twelve measures that have been suggested or might be useful to measure explained variation in logistic regression models. The definitions and properties of these measures are discussed and their performance is compared in an empirical study. Two of the measures (squared Pearson correlation between the binary outcome and the predictor, and the proportional reduction of squared Pearson residuals by the use of covariates) give almost identical results, agree very well with the multiple R2 of the general linear model, have an intuitively clear interpretation and perform satisfactorily in our study. For all measures the explained variation for the given sample and also the one expected in future samples can be obtained easily. For small samples an adjustment analogous to Radj2 in the general linear model is suggested. We discuss some aspects of application and recommend the routine use of a suitable measure of explained variation for logistic models.

Likelihood Functions↗
PubMed1996

Univariate analysis of dichotomous or ordinal data from twin pairs: a simulation study comparing structural equation modeling and logistic regression.

The univariate analysis of categorical twin data can be performed using either structural equation modeling (SEM) or logistic regression. This paper presents a comparison between these two methods using a simulation study. Dichotomous and ordinal (three category) twin data are simulated under two different sample sizes (1,000 and 2,000 twin pairs) and according to different additive genetic and common environmental models of phenotypic variation. The two methods are found to be generally comparable in their ability to detect a "correct" model under the specifications of the simulation. Both methods lack power to detect the right model for dichotomous data when the additive genetic effect is low (between 10 and 20%) or medium (between 30 and 40%); the ordinal data simulations produce similar results except for the additive genetic model with medium or high heritability. Neither method could adequately detect a correct model that included a modest common environmental effect (20%) even when the additive genetic effect was large and the sample size included 2,000 twin pairs. The SEM method was found to have better power than logistic regression when there is a medium (30%) or high (50%) additive genetic effect and a modest common environmental effect. Conversely, logistic regression performed better than SEM in correctly detecting additive genetic effects with simulated ordinal data (for both 1,000 and 2,000 pairs) that did not contain modest common environmental effects; in this case the SEM method incorrectly detected a common environmental effect that was not present.

Analysis of Variance↗
PubMed2001

Efficient regression calibration for logistic regression in main study/internal validation study designs with an imperfect reference instrument.

An extension to the version of the regression calibration estimator proposed by Rosner et al. for logistic and other generalized linear regression models is given for main study/internal validation study designs. This estimator combines the information about the parameter of interest contained in the internal validation study with Rosner et al.'s regression calibration estimate, using a generalized inverse-variance weighted average. It is shown that the validation study selection model can be ignored as long as this model is jointly independent of the outcome and the incompletely observed covariates, conditional, at most, upon the surrogates and other completely observed covariates. In an extensive simulation study designed to follow a complex, multivariate setting in nutritional epidemiology, it is shown that with validation study sizes of 340 or more, this estimator appears to be asymptotically optimal in the sense that it is nearly unbiased and nearly as efficient as a properly specified maximum likelihood estimator. A modification to the regression calibration variance estimator which replaces the standard uncorrected logistic regression coefficient variance with the sandwich estimator to account for the possible misspecification of the logistic regression fit to the surrogate covariates in the main study, was also studied in this same simulation experiment. In this study, the alternative variance formula yielded results virtually identical to the original formula. A version of the proposed estimator is also derived for the case where the reference instrument, available only in the validation study, is imperfect but unbiased at the individual level and contains error that is uncorrelated with other covariates and with error in the surrogate instrument. Replicate measures are obtained in a subset of study participants. In this case it is shown that the validation study selection model can be ignored when sampling into the validation study depends, at most, only upon perfectly measured covariates. Two data sets, a study of fever in relation to occupational exposure to antineoplastics among hospital pharmacists and a study of breast cancer incidence in relation to dietary intakes of alcohol and vitamin A, adjusted for total energy intake, from the Nurses' Health Study, were analysed using these new methods. In these data, because the validation studies contained less than 200 observations and the events of interest were relatively rare, as is typical, the potential improvements offered by this new estimator were not apparent.

Adult↗
PubMed1994

The Common Alcohol Logistic-Revised scale (CAL-R): a revised alcoholism scale for the MMPI and MMPI-2.

The Common Alcohol Logistic-Revised (CAL-R) scale was developed for use with MMPI or MMPI-2 with medical patients. It was derived from the Common Alcohol Logistic (CAL) scale and developed by (1) dropping the six CAL items deleted from the MMPI during construction of the MMPI-2 and recomputing item weights (using logistic regression); (2) calculating norms; and (3) repeating the validation procedure used in developing CAL. We used the same criterion group (1,221 alcoholics) and same contrast sample (7,621 nonalcoholic medical patients) used for CAL. Comparison of receiver operating characteristic curves, positive predictive value, and negative predictive value for CAL and CAL-R indicates that the latter has the same favorable ability to screen medical patients for alcoholism as the former.

Adult↗
PubMed2003

Regressive logistic and proportional hazards disease models for within-family analyses of measured genotypes, with application to a CYP17 polymorphism and breast cancer.

Various statistical methods have been proposed to evaluate associations between measured genetic variants and disease, including some using family designs. For breast cancer and rare variants, we applied a modified segregation analysis method that uses the population cancer incidence and population-based case families in which a mutation is known to be segregating. Here we extend the method to a common polymorphism, and use a regressive logistic approach to model familial aggregation by conditioning each individual on their mother's breast cancer history. We considered three models: 1) class A regressive logistic model; 2) age-of-onset regressive logistic model; and 3) proportional hazards familial model. Maximum likelihood estimates were calculated using the software MENDEL. We applied these methods to data from the Australian Breast Cancer Family Study on the CYP17 5'UTR T-->C MspA1 polymorphism measured for 1,447 case probands, 787 controls, and 213 relatives of case probands found to have the CC genotype. Breast cancer data for first- and second-degree relatives of case probands were used. The three methods gave consistent estimates. The best-fitting model involved a recessive inheritance, with homozygotes being at an increased risk of 47% (95% CI, 28-68%). The cumulative risk of the disease up to age 70 years was estimated to be 10% or 22% for a CYP17 homozygote whose mother was unaffected or affected, respectively. This analytical approach is well-suited to the data that arise from population-based case-control-family studies, in which cases, controls and relatives are studied, and genotype is measured for some but not all subjects.

Adult↗
PubMed2003

Logistic regression of family data from retrospective study designs.

We wish to study the effects of genetic and environmental factors on disease risk, using data from families ascertained because they contain multiple cases of the disease. To do so, we must account for the way participants were ascertained, and for within-family correlations in both disease occurrences and covariates. We model the joint probability distribution of the covariates of ascertained family members, given family disease occurrence and pedigree structure. We describe two such covariate models: the random effects model and the marginal model. Both models assume a logistic form for the distribution of one person's covariates that involves a vector beta of regression parameters. The components of beta in the two models have different interpretations, and they differ in magnitude when the covariates are correlated within families. We describe ascertainment assumptions needed to estimate consistently the parameters beta(RE) in the random effects model and the parameters beta(M) in the marginal model. Under the ascertainment assumptions for the random effects model, we show that conditional logistic regression (CLR) of matched family data gives a consistent estimate beta(RE) for beta(RE) and a consistent estimate for the covariance matrix of beta(RE). Under the ascertainment assumptions for the marginal model, we show that unconditional logistic regression (ULR) gives a consistent estimate for beta(M), and we give a consistent estimator for its covariance matrix. The random effects/CLR approach is simple to use and to interpret, but it can use data only from families containing both affected and unaffected members. The marginal/ULR approach uses data from all individuals, but its variance estimates require special computations. A C program to compute these variance estimates is available at http://www.stanford.edu/dept/HRP/epidemiology. We illustrate these pros and cons by application to data on the effects of parity on ovarian cancer risk in mother/daughter pairs, and use simulations to study the performance of the estimates.

Algorithms↗
PubMed2003

Causal logistic models for non-compliance under randomized treatment with univariate binary response.

We propose a method for estimating the marginal causal log-odds ratio for binary outcomes under treatment non-compliance in placebo-randomized trials. This estimation method is a marginal alternative to the causal logistic approach by Nagelkerke et al. (2000) that conditions on partially unknown compliance (that is, adherence to treatment) status, and also differs from previous approaches that estimate risk differences or ratios in subgroups defined by compliance status. The marginal causal method proposed in this paper is based on an extension of Robins' G-estimation approach for fitting linear or log-linear structural nested models to a logistic model. Comparing the marginal and conditional causal log-odds ratio estimates provides a way of assessing the magnitude of unmeasured confounding of the treatment effect due to treatment non-adherence. More specifically, we show through simulations that under weak confounding, the conditional and marginal procedures yield similar estimates, whereas under stronger confounding, they behave differently in terms of bias and confidence interval coverage. The parametric structures that represent such confounding are not identifiable. Hence, the proof of consistency of causal estimators and corresponding simulations are based on two different models that fully identify the causal effects being estimated. These models differ in the way that compliance is related to potential outcomes, and thus differ in the way that the causal effect is identified. The simulations also show that the proposed marginal causal estimation approach performs well in terms of bias under the different levels of confounding due to non-adherence and under different causal logistic models. We also provide results from the analyses of two data sets further showing how a comparison of the marginal and conditional estimators can help evaluate the magnitude of confounding due to non-adherence.

Confounding Factors, Epidemiologic↗
PubMed2003

Using lowess to remove systematic trends over time in predictor variables prior to logistic regression with quantile categories.

In case-control studies one may employ logistic regression to model the relationship between binary responses and continuous predictor variables that have been categorized by the empirical quartiles of the controls. Sometimes, however, systematic trends over time (or drifts) contaminate the laboratory measurements of predictor variables. In this paper we consider the use of locally weighted robust regression (lowess) to estimate and remove these systematic trends when the trends for the cases and controls have a common shape. One can then use the lowess adjusted data in the desired logistic regression model. We illustrate these methods with a case-control study that was designed to assess the risk of oesophageal cancer as a function of the quartile categories of sphinganine levels in the blood serum. Upon examination of the data, it was discovered that the sphinganine laboratory measurements were contaminated by a systematic trend, the magnitude of which depended only on the day of analysis. This trend needed to be removed before performing further analyses of the data. In addition, we present simulations to examine the use of lowess methods to estimate and remove various shapes of trends from contaminated predictor data before constructing logistic regression models with quartile categories. We found that using the trend-contaminated data tends to give attenuated parameter estimates and hence lower significance and power levels than using the uncontaminated data. Conversely, using appropriate lowess methods to adjust the data tends to give nearly unbiased parameter estimates, near nominal significance levels, and improved power.

Adult↗
PubMed2004

A simple approach to power and sample size calculations in logistic regression and Cox regression models.

For a given regression problem it is possible to identify a suitably defined equivalent two-sample problem such that the power or sample size obtained for the two-sample problem also applies to the regression problem. For a standard linear regression model the equivalent two-sample problem is easily identified, but for generalized linear models and for Cox regression models the situation is more complicated. An approximately equivalent two-sample problem may, however, also be identified here. In particular, we show that for logistic regression and Cox regression models the equivalent two-sample problem is obtained by selecting two equally sized samples for which the parameters differ by a value equal to the slope times twice the standard deviation of the independent variable and further requiring that the overall expected number of events is unchanged. In a simulation study we examine the validity of this approach to power calculations in logistic regression and Cox regression models. Several different covariate distributions are considered for selected values of the overall response probability and a range of alternatives. For the Cox regression model we consider both constant and non-constant hazard rates. The results show that in general the approach is remarkably accurate even in relatively small samples. Some discrepancies are, however, found in small samples with few events and a highly skewed covariate distribution. Comparison with results based on alternative methods for logistic regression models with a single continuous covariate indicates that the proposed method is at least as good as its competitors. The method is easy to implement and therefore provides a simple way to extend the range of problems that can be covered by the usual formulas for power and sample size determination.

Breast Neoplasms↗
PubMed2004

Comparison of Bayesian model averaging and stepwise methods for model selection in logistic regression.

Logistic regression is the standard method for assessing predictors of diseases. In logistic regression analyses, a stepwise strategy is often adopted to choose a subset of variables. Inference about the predictors is then made based on the chosen model constructed of only those variables retained in that model. This method subsequently ignores both the variables not selected by the procedure, and the uncertainty due to the variable selection procedure. This limitation may be addressed by adopting a Bayesian model averaging approach, which selects a number of all possible such models, and uses the posterior probabilities of these models to perform all inferences and predictions. This study compares the Bayesian model averaging approach with the stepwise procedures for selection of predictor variables in logistic regression using simulated data sets and the Framingham Heart Study data. The results show that in most cases Bayesian model averaging selects the correct model and out-performs stepwise approaches at predicting an event of interest.

Age Factors↗