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This paper reviews concepts of multivariate statistical modeling via the logistic regression model, which has become very popular for modeling the relationship between a positive clinical outcome and a variety of predictor variables. The process is illustrated using a composite of data from three large prostate specific antigen based screening studies of prostate cancer.
Nuisance parameters are parameters that are not of immediate interest to the experimenter. For log-linear and logistic models the null distribution of (most) statistics of interest depends on such parameters. Traditionally, nuisance parameters were eliminated by performing inference with respect to the chi-squared limiting distribution of common test statistics. An alternative solution is to eliminate the nuisance parameters by conditioning on their minimal sufficient statistics. The support of the resulting conditional distribution is often intractable, making null probability calculations challenging. An often feasible way to avoid complete enumeration of this support is to approximate conditional probabilities using Monte Carlo methods. In this article we survey recent developments in Monte Carlo conditional analysis for log-linear and logistic models focusing on the algorithms proposed by Booth and Butler, Diaconis and Sturmfels, Smith et al., and Mehta et al. We illustrate these algorithms with simple motivating examples.
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.
"We revive the logistic model, which was tested and found wanting in early-20th-century studies of aggregate human populations, and apply it instead to life expectancy (death) and fertility (birth)....For death...the logistic portrays the situation crisply. Human life expectancy is reaching the culmination of a two-hundred year-process that forestalls death until about 80 for men and the mid 80s for women. No breakthroughs in longevity are in sight unless genetic engineering comes to help. For birth, the logistic covers quantitatively its actual morphology. However, because we have not been able to model this essential parameter in a predictive way over long periods, we cannot say whether the future of human population is runaway growth or slow implosion... From a niche point of view, resources are the limits to numbers, and access to resources depends on technologies. The logistic makes clear that for homo faber, the limits to numbers keep shifting. These moving edges may most confound forecasting the long-run size of humanity."
The purpose of this study was to determine factors predictive of clinical outcome for pediatric patients with maxillofacial infections. Using linear logistic regression, four important variables, age, admission temperature, admission white blood cell count, and source of infection, were identified. Relevant study variables were abstracted from the records of all children less than 15 years old admitted to San Francisco General Hospital (SFGH) with facial infections between 1982 and 1986 (n = 105). An unfavorable clinical outcome was defined as a length of hospital stay (LOS) greater than or equal to 4 days and/or the need for an operation to resolve the infection. A favorable outcome was a LOS less than 4 days and no operation. To develop and validate the linear logistic model, the original group of patients (n = 105) was divided into index and validation sets. The index set was created by randomly selecting 80 of the original patients. The model was then applied to a validation set of the 25 remaining children. The model predicted that 13.95 of the patients in the validation set would have an unfavorable outcome. The actual number of unfavorable outcomes was 16. To further test the model's validity, a third data set was collected. It was composed of pediatric patients admitted to SFGH between January 1, 1987 and June 30, 1989 (n = 24). The model predicted that 15.99 of these patients would have an unfavorable outcome; 17 patients actually did have an unfavorable outcome.(ABSTRACT TRUNCATED AT 250 WORDS)
The linear-logistic regression model and Cox's proportional hazard model are widely used in epidemiology. Their successful application leaves no doubt that they are accurate reflections of observed disease processes and their associated risks or incidence rates. In spite of their prominence, it is not a priori evident why these models work. This article presents a derivation of the two models from the framework of canonical modeling. It begins with a general description of the dynamics between risk sources and disease development, formulates this description in the canonical representation of an S-system, and shows how the linear-logistic model and Cox's proportional hazard model follow naturally from this representation. The article interprets the model parameters in terms of epidemiological concepts as well as in terms of general systems theory and explains the assumptions and limitations generally accepted in the application of these epidemiological models.
Using linear logistic regression, six factors were identified as important predictors of risk of DST sensitization in a group of 195 patients. Factors increasing the risk were: percent panel reactive antibody (PRA), previous transplants, and pregnancy; those decreasing the risk were HLA antigens matched, third-party blood transfusions, and Imuran administration. From this analysis, the magnitude of the effect of each factor on the risk of sensitization was obtained. An equation was then obtained that can be used to compute an estimated probability of sensitization (PS) for each patient. As a test of predictive ability of the model, the PS was calculated for 66 patients in an independent patient group. These observations were arranged according to the estimated probability and then divided into intervals of risk. Overall, for each interval, a very high level of agreement was found between the predicted and actual number of sensitized patients. A total of 16.13 patients were predicted to become sensitized and 17 actually did.
The logistic regression model has been in use in statistical analysis for many years. The paper introduces a spline model to remove the linear restriction on logit function. By considering knot locations as free variables, spline approximation of data is improved. The number of knots and the degree of the spline functions can still be determined by using a model selection procedure. Moreover, a knot, seen as a free parameter for a piecewise linear spline, represents a break point in the logit function which may be interpreted as a threshold value. This method is applied to a clinical trial for an in vitro fertilization program.
We propose an algorithm for constructing minimax D-optimal designs for the logistic model when only the ranges of the values for both parameters are assumed known. Properties of these designs are studied and compared with optimal Bayesian designs and Sitter's (1992, Biometrics, 48, 1145-1155) minimax D-optimal kk-designs. Examples of minimax D-optimal designs are presented for the logistic and power logistic models, including a dose-response design for rheumatoid arthritis patients.
The quasi-stationary distribution of the stochastic logistic model is studied in the parameter region where its body is approximately normal. Improved asymptotic approximations of its first three cumulants are derived. It is shown that the same results can be derived with the aid of the moment closure method. This indicates that the moment closure method leads to expressions for the cumulants that are asymptotic approximations of the cumulants of the quasi-stationary distribution.
Three models of simple logistical growth were used to describe volumetric growth in heterogeneous tumours. Two clonal subpopulations (designated as clone A and clone D) originally obtained from a human colon adenocarcinoma were used to produce solid xenograft tumours in nude mice. Volumetric growth of tumours produced from pure cells alone was compared to that produced from 50% A:50% D, 88% A:12% D, and 9% A:91% D admixtures. Gompertzian analysis of the in vivo growth data indicated significant differences in both the initial growth rates and final asymptotic limiting volumes of the pure versus the admixed tumours. Verhulstian and modified Verhulstian models were also used to derive regression curves from the same data. The fit of the curves was compared with each other using standard (Akaike, 1974; Schwartz, 1978) information criteria. In four of the five tumour populations the Gompertz equation fitted best. Only in the 88% A:12% D tumours did the modified Verhulst model fit best. The deviations from the regression curves, the residuals, for all three models were systematically distributed. These systematic errors are likely to be the result of using simplified logistical models to describe the growth kinetics of interacting populations in heterogeneous tumours.
We discuss the use of the trichotomous logistic model to discriminate between patients with gastrointestinal (GI) cancer, patients with benign GI disease and 'normal' subjects, using symptoms and the concentrations of some serum proteins that are potentially indicative of malignancy as covariates. A parsimonious model can be obtained by invoking an indistinguishability hypothesis which is appropriate when a covariate is considered to have no predictive value between categories. It is shown that the polychotomous model can be re-parameterised under the null hypothesis to give a 'reduced form', which can be fitted by maximum likelihood. The validity of the use of the same methods for retrospective sampling is discussed. The approach is illustrated by the development of a logistic model to identify symptomatic and asymptomatic subjects with a high risk of GI cancer.
Methods for estimating the parameters of the logistic regression model when the data are collected using a case-control (retrospective) scheme are compared. The regression coefficients are estimated by maximum likelihood methodology. This leaves the constant term parameter to be estimated. Four methods for estimating this parameter are proposed. The comparison of the four estimators is in two parts. First, they are compared for large samples. This is accomplished via the asymptotic distribution of the estimators. Second, the estimators are compared for small samples. This is conducted via stimulation using 11 logistic models. The estimation of the posterior probability of the response variable being a success (Px), as given by the logistic regression model, when the constant parameter is estimated by each of the four proposed methods is the main focus of this paper. A third concern is the comparison of the logistic discriminant procedures when each of the four methods of estimating the constant parameters is used. In addition, the linear discriminant function procedure is included. This comparison is executed only for small samples via simulation. It was found that when estimating Px, method 1 (which is essentially the MLE) minimizes the expected mean square error. The results were not as clear when the parameter of interest was the constant term itself. The results from the classification comparisons implied that when the logistic model contains mostly (or all) binary regression variables the logistic discriminant procedure using method 1 to estimate the constant term gives minimum expected error rate; otherwise the linear discriminant function gives minimum expected error rate. In the latter case the logistic discriminant procedure (method 1 estimator of the constant term) is approximately as good.
We extend the random intercept logistic model to accommodate negative intracluster correlations for bivariate binary response data. This approach assumes a single random effect per cluster, but entails separate affine transformations of this random effect for the two responses of the pair. We show this approach works for two data sets and a simulation, whereas other mixed effects approaches fail. The two data sets are from a crossover trial and a developmental toxicity study of the effects of chemical exposure on malformation risk among rat pups. Comparisons are made with the conditional likelihood approach and with generalized estimating equations estimation of the population-averaged logit model. Simulations show the conditional likelihood approach does not perform well for moderate to strong negative correlations, as a positive intracluster correlation is assumed. The proposed mixed effects approach appears to be slightly more conservative than the population-averaged approach with respect to coverage of confidence intervals. Nonetheless, the statistical literature suggests that mixed effects models provide information in addition to that provided by population-averaged models under scientific contexts such as crossover trials. Extensions to trivariate and higher-dimensional responses also are addressed. However, such extensions require certain constraints on the correlation structure.
This article aims at modeling the impact of road characteristics and local spatial environment on road (un)safety. The study applies to Belgium where some 1,500 people are killed annually on the roads. This statistic corresponds to one of the highest risks in Europe. Road unsafety is expressed here as whether an hectometer of road belongs to a black zone; a black zone is defined as a segment of road where roads accidents are concentrated. Logistic modeling including spatial autocorrelation is used and compared to non-spatial regression. It is shown that a spatial model is needed to avoid biased estimated parameters. Results show that local environment and road infrastructure play a substantial role in the co-occurrence of road accidents. Hence, education and enforcement cannot be the only measures taken to reach a sustainable road safety. To attain their objectives of accident reduction, public authorities should also take their responsibilities in the matter of securing road infrastructure.
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.
The development of the regressive logistic model applicable to the analysis of codon frequencies of DNA sequences in terms of explanatory variables is presented. A codon is a triplet of nucleotides that code for an amino acid, and may be considered as a trivariate response (B1, B2, B3), where Bi (i = 1, 2, 3) is a categorical random variable with values A, C, G, T. The linear order of bases in the DNA and possible statistical dependence of the bases in a given codon make the regressive logistic model a suitable tool for the analysis of codon frequencies. A problem of structural zeros arises from the fact that the stopping codons (terminators) do not code for amino acids; this is solved by normalizing the likelihood function. Codon frequencies may also depend on the function of the gene and they are known to differ between genes of the same genome. Differences also occur between synonymous codons for the same amino acid. Thus, the use of covariates that differ between synonymous codons as well as covariates that are constant within codons of the same amino acid may be useful in explaining the frequencies. As an illustration, the method is applied to the human mitochondrial genome using the following as explanatory variables: (1) TSCORE, a measure of the number of single base mutations required for a given codon to become a terminator; (2) AARISK, an indicator of a codon's ability of changing by a single base substitution to triplets coding for amino acids with very different characteristics; (3) AVDIST, a measure of the typicality of the amino acid coded for by the triplets. The results indicate that models that incorporate dependency structure and covariates are to be preferred to either the models comprising covariates alone or dependency structure alone.
Multinomial responses frequently occur in dose level experiments. For example, in a study of the influence of gamma radiation on the emergence of house flies (Musca domestica L., 1758), three disjoint outcomes occurred: death before the pupae opened, death during emergence, and life after emergence. Although the flies are easy to breed, this sort of bioassay is, in general, very expensive since it requires the use of a gamma radiation source. Experiments therefore need to be designed to involve the minimum number of different doses. Here the theory of optimum experimental design is applied to provide efficient experiments to estimate the parameters of those multinomial logistic models that are a special case of the multivariate logistic models of Glonek and McCullagh (1995, Journal of the Royal Statistical Society, Series B 57, 533-546). The purpose is to reduce the overall experimental cost. The general equivalence theorem (Fedorov, 1972, Theory of Optimal Experiments) is adapted to this class of models, providing an effective method of generating and checking the optimality of designs. One example on flies demonstrates the method, which can be easily implemented.