The Treatment of Missing Values in Logistic Regression

The efficiencies of the estimators in the linear logistic regression model are examined using simulations under six missing value treatments. These treatments use either the maximum likelihood or the discriminant function approach in the estimation of the regression coefficients. Missing values are assumed to occur at random. The cases of multivariate normal and dichotomous independent variables are both considered. We found that in general, there is no uniformly best method. However, mean substitution and discriminant function estimation using existing pairs of values for correlations turn out to be favourable for the cases considered.     

Generalized Walker-Duncan Procedure for Obtaining Maximum Likelihood Estimates of Parameters of Logistic Discriminant Function

A recursive method of obtaining the maximum likelihood estimates of the parameters of the quadratic logistic discriminant function is presented. This method is an extension of the Walker and Duncan procedure (1967) proposed for the linear logistic discriminant function in a dichotomous case. A generalization of the method to the problem of discrimination between several populations is also given in the paper. It works for both linear and quadratic logistic discriminant function. After an estimation of the parameters of the logistic function a classification can be performed. An example of application of the method to automatic diagnosis of some respiratory diseases is presented. Comparison with the standard procedures used for the estimation is done by a short simulation study.     

Analysis of Schizophrenia Data Using A Nonlinear Threshold Index Logistic Model

Genetic information, such as single nucleotide polymorphism (SNP) data, has been widely recognized as useful in prediction of disease risk. However, how to model the genetic data that is often categorical in disease class prediction is complex and challenging. In this paper, we propose a novel class of nonlinear threshold index logistic models to deal with the complex, nonlinear effects of categorical/discrete SNP covariates for Schizophrenia class prediction. A maximum likelihood methodology is suggested to estimate the unknown parameters in the models. Simulation studies demonstrate that the proposed methodology works viably well for moderate-size samples. The suggested approach is therefore applied to the analysis of the Schizophrenia classification by using a real set of SNP data from Western Australian Family Study of Schizophrenia (WAFSS). Our empirical findings provide evidence that the proposed nonlinear models well outperform the widely used linear and tree based logistic regression models in class prediction of schizophrenia risk with SNP data in terms of both Types I/II error rates and ROC curves.     

Logistic distributed activation energy model--part 2: application to cellulose pyrolysis

The pyrolysis behavior of cellulose has been investigated by using thermogravimetric analysis (TGA). The non-isothermal TGA data obtained at different heating rates have been analyzed simultaneously. Pattern Search Method has been proposed for the estimation of the model parameter values. Predicted values from the logistic distributed activation energy model have been compared with the experimental data and the results have indicated that the model describes the kinetic behavior of cellulose pyrolysis very well. The mean value and standard deviation of the logistic activation energy distribution for cellulose pyrolysis are found to be 258.5718 kJ mol(-1) and 2.6601 kJ mol(-1), the reaction order is 1.1101 and the k(0) is 1.6218×10(17) s(-1).     

Choice of Scale and Asymmetric Logistic Models

In recent years, a number of authors have proposed generalizations to the binomial logistic model. These proposals were motivated, in part, by the claimed inability of the logistic model to fit asymmetric data, that is, data that does not follow a symmetric S-shaped curve. In this note, it is demonstrated that by changing the scale of the independent variable(s), the logistic model can fit asymmetric data. Logistic models are fit to data from BLISS (1935) that had been used as an example by three of the authors; the logistic models fit Bliss' data as well as the proposed alternative models. In a general sense, these data serve as an example of the need to consider the appropriate choice of scale of the independent variables in logistic analysis.     

Linear Logistic Latent Class Analysis

In the present paper the linear logistic extension of latent class analysis is described. Thereby it is assumed that the item latent probabilities as well as the class sizes can be attributed to some explanatory variables. The basic equations of the model state the decomposition of the log-odds of the item latent probabilities and of the class sizes into weighted sums of basic parameters representing the effects of the predictor variables. Further, the maximum likelihood equations for these effect parameters and statistical tests for goodness-of-fit are given. Finally, an example illustrates the practical application of the model and the interpretation of the model parameters.     

Comparison of Logistic Regression and Linear Regression in Modeling Percentage Data

Percentage is widely used to describe different results in food microbiology, e.g., probability of microbial growth, percent inactivated, and percent of positive samples. Four sets of percentage data, percent-growth-positive, germination extent, probability for one cell to grow, and maximum fraction of positive tubes, were obtained from our own experiments and the literature. These data were modeled using linear and logistic regression. Five methods were used to compare the goodness of fit of the two models: percentage of predictions closer to observations, range of the differences (predicted value minus observed value), deviation of the model, linear regression between the observed and predicted values, and bias and accuracy factors. Logistic regression was a better predictor of at least 78% of the observations in all four data sets. In all cases, the deviation of logistic models was much smaller. The linear correlation between observations and logistic predictions was always stronger. Validation (accomplished using part of one data set) also demonstrated that the logistic model was more accurate in predicting new data points. Bias and accuracy factors were found to be less informative when evaluating models developed for percentage data, since neither of these indices can compare predictions at zero. Model simplification for the logistic model was demonstrated with one data set. The simplified model was as powerful in making predictions as the full linear model, and it also gave clearer insight in determining the key experimental factors.     

An Application of Logistic Models to the Analysis of Ordinal Responses

Logistic probability models—models linear in the log odds of the outcome event—have found extensive application in modelling of unordered categorical responses. This paper illustrates some extensions of logistic models to the modelling of probabilities of ordinal responses. The extensions arise naturally from discrete probability models for the conditional distribution of the ordinal response, as well as from linear modelling of the log odds of response. Methods of estimation and examination of fit developed for the binary logistic model extend in a straightforward manner to the ordinal models. The models and methods are illustrated in an analysis of the dependence of chronic obstructive respiratory disease prevalence on smoking and age.     

Expected Error Rates for Logistic Regression Versus Normal Discriminant Analysis

The expected error rates associated with using the allocation rule based on logistic regression are derived in the context of two multivariate normal populations with a common covariance matrix and compared with the corresponding error rates of the classical rule based on this normality assumption. It is shown in terms of the actual sizes of the asymptotic expected error rates that the performance of the logistic procedure does not fall far short of the normality based method, even for widely separated populations. This latter result is not obvious from previously available work on the asymptotic relative efficiency of the logistic procedure.     

Estimation of Relative Risk in Case Control Studies Through Logistic Regression Analysis

Multiple logistic regression analysis is used to estimate the relative risk in case control studies. The estimators obtained are valid when disease is rare. In this paper an estimator of relative risk in a case control study has been proposed using logistic regression results when the incidence of disease is not small. The bias of the usual estimator through logistic regression as compared to the new estimator has been worked out. The expression of Mean Square Error of proposed estimator has been derived in situations when the incidence of disease is known exactly as well as when estimated through an independent survey. It has been observed that there is a significant bias using the conventional estimator of relative risk when incidence of disease is high. In such situations the proposed estimator can be used with advantage.     

One‐Sided Test to Assess Correlation in Linear Logistic Models using Estimating Equations

A score‐type test is proposed for testing the hypothesis of independent binary random variables against positive correlation in linear logistic models with sparse data and cluster specific covariates. The test is developed for univariate and multivariate one‐sided alternatives. The main advantage of using score test is that it requires estimation of the model only under the null hypothesis, that in this case corresponds to the binomial maximum likelihood fit. The score‐type test is developed from a class of estimating equations with block‐diagonal structure in which the coefficients of the linear logistic model are estimated simultaneously with the correlation. The simplicity of the score test is illustrated in two particular examples.     

Use of Polytomous Logistic Model in Matched Case-Control Studies

Binary logistic model has been found useful for estimating odds ratio in case of dichotomous exposure variable under matched case-control retrospective design. We describe the use of polytomous logistic model for estimating odds ratios when the exposure of prime interests, relative to disease incidence, has more than two levels. An illustrative example is presented and discussed.     

A Generalized Logistic Model for Quantal Response Bioassay

In bioassay, where different levels of the stimulus may represent different doses of a drug, the binary response is the death or survival of an individual receiving a specified dose. In such applications, it is common to model the probability of a positive response P at the stimulus level x by P = F(x′β), where F is a cumulative distribution function and β is a vector of unknown parameters which characterize the response function. The two most popular models used for modelling binary response bioassay involve the probit model [BLISS (1935), FINNEY (1978)], and the logistic model [BERKSON (1944), BROWN (1982)]. However, these models have some limitations. The use of the probit model involves the inverse of the standard normal distribution function, making it rather intractable. The logistic model has a simple form and a closed expression for the inverse distribution function, however, neither the logistic nor the probit can provide a good fit to response functions which are not symmetric or are symmetric but have a steeper or gentler incline in the central probability region. In this paper we introduce a more realistic model for the analysis of quantal response bioassay. The proposed model, which we refer to it as the generalized logistic model, is a family of response curves indexed by shape parameters m₁ and m₂. This family is rich enough to include the probit and logistic models as well as many others as special cases or limiting distributions. In particular, we consider the generalized logistic three parameter model where we assume that m₁ = m, m is a positive real number, and m₂ = 1. We apply this model to various sets of data, comparing the fit results to those obtained previously by other dose-response curves such as the logistic and probit, and showing that the fit can be improved by using the generalized logistic.     

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.     

Efficient quadratic regularization for expression arrays

Gene expression arrays typically have 50 to 100 samples and 1000 to 20,000 variables (genes). There have been many attempts to adapt statistical models for regression and classification to these data, and in many cases these attempts have challenged the computational resources. In this article we expose a class of techniques based on quadratic regularization of linear models, including regularized (ridge) regression, logistic and multinomial regression, linear and mixture discriminant analysis, the Cox model and neural networks. For all of these models, we show that dramatic computational savings are possible over naive implementations, using standard transformations in numerical linear algebra.     

Logistic model for soldier production in aphids

Summary The number of aphid soldiers in a colony is positively correlated with colony size. This positive correlation has been repeatedly confirmed for the bamboo aphid Pseudoregma bambucicola. To explain this, we present a simple model assuming logistic growth of the aphid colony. The model predicts that a first soldier is more readily produced in large colonies than in small colonies. This is because the productivity of each reproductive decreases as the colony size increases, and also because the efficiency of defense increases as the number of reproductives defended by the soldier increases. The latter effect disappears when the number of reproductives exceeds N_p, the critical number of reproductives that a predator could damage. This argument holds for a second, a third, and an ith soldier in general. Although we assume logistic growth of the aphid colony, the model is applicable to any form of colony growth with minor changes of some premises.Received 12 February 2003; revised 28 March 2003; accepted 22 April 2003.     

Applications of Logistic Model to Medical Research

The estimation of odd ratio from contingency tables is one of the main problems faced by medical researchers. The purpose of this paper is to review and present applications of logistic model and its connection to odd ratio. Numerical examples are given illustrating the use of logistic model for estimating odd ratio.     

Logistic Proliferation of Cells in Scratch Assays is Delayed

Scratch assays are used to study how a population of cells re-colonises a vacant region on a two-dimensional substrate after a cell monolayer is scratched. These experiments are used in many applications including drug design for the treatment of cancer and chronic wounds. To provide insights into the mechanisms that drive scratch assays, solutions of continuum reaction–diffusion models have been calibrated to data from scratch assays. These models typically include a logistic source term to describe carrying capacity-limited proliferation; however, the choice of using a logistic source term is often made without examining whether it is valid. Here we study the proliferation of PC-3 prostate cancer cells in a scratch assay. All experimental results for the scratch assay are compared with equivalent results from a proliferation assay where the cell monolayer is not scratched. Visual inspection of the time evolution of the cell density away from the location of the scratch reveals a series of sigmoid curves that could be naively calibrated to the solution of the logistic growth model. However, careful analysis of the per capita growth rate as a function of density reveals several key differences between the proliferation of cells in scratch and proliferation assays. Our findings suggest that the logistic growth model is valid for the entire duration of the proliferation assay. On the other hand, guided by data, we suggest that there are two phases of proliferation in a scratch assay; at short time, we have a disturbance phase where proliferation is not logistic, and this is followed by a growth phase where proliferation appears to be logistic. These two phases are observed across a large number of experiments performed at different initial cell densities. Overall our study shows that simply calibrating the solution of a continuum model to a scratch assay might produce misleading parameter estimates, and this issue can be resolved by making a distinction between the disturbance and growth phases. Repeating our procedure for other scratch assays will provide insight into the roles of the disturbance and growth phases for different cell lines and scratch assays performed on different substrates.     

Sexing skulls using discriminant function analysis of visually assessed traits

The accuracy of sex determinations based on visual assessments of the mental eminence, orbital margin, glabellar area, nuchal area, and mastoid process was tested on a series of 304 skulls of known age and sex from people of European American, African American, and English ancestry as well as on an ancient Native American sample of 156 individuals whose sex could be reliably determined based on pelvic morphology. Ordinal scores of these sexually dimorphic traits were used to compute sex determination discriminant functions. Linear, kth-nearest-neighbor, logistic, and quadratic discriminant analysis models were evaluated based on their capacity to minimize both misclassifications and sex biases in classification errors. Logistic regression discriminant analysis produced the best results: a logistic model containing all five cranial trait scores correctly classified 88% of the modern skulls with a negligible sex bias of 0.1%. Adding age at death, birth year, and population affinity to the model did not appreciably improve its performance. For the ancient Native American sample, the best logistic regression model assigned the correct pelvic sex to 78% of the individuals with a sex bias of only 0.2%. Similar cranial trait frequency distributions were found in same-sex comparisons of the modern African American, European American, and English samples. The sexual dimorphism of these modern people contrasts markedly with that of the ancient Native Americans. Because of such population differences, discriminant functions like those presented in this paper should be used with caution on populations other than those for which they were developed.     

Taxonomy ofHordeum caespitosum,H. jubatum,andH. lechleri (Poaceae: Triticeae)

Hordeum caespitosum Scribner,H. jubatum L., andH. lechleri (Steudel)Schenck are very similar in appearance and therefore until recently were mostly not recognized as separate entities. The first two are tetraploid and natives to North America, but the second occurs naturally in eastern Siberia and has been introduced in Europe and South America and may become a cosmopolitan weed. The third is hexaploid and South American. This study analyses their morphological diversity by means of selected multivariate techniques in order to determine if there is justification to recognize them as three separate morphological species. Logistic discrimination, although based on a reduced set of characters, yielded the highest percent of correct assignments. A linear discriminant function is provided and validated by 100 bootstrap repeats. Canonical discriminant analysis indicated three groups. It is subsequently concluded that the three are separate morphological species. Although a linear discriminant function is given, a traditional identification key is provided based on the palea length and triad (the group of three spikelets at each rachis node) length.