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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↗
PubMed1998

Increased efficiency of analyses: cumulative logistic regression vs ordinary logistic regression.

The common practice of collapsing inherently continuous or ordinal variables into two categories causes information loss that may potentially weaken power to detect effects of explanatory variables and result in Type II errors in statistical inference. The purpose of this investigation was to illustrate, using a substantive example, the potential increase in power gained from an ordinal instead of a dichotomous specification for an inherently continuous response. Ordinary (OLR) and cumulative logistic regression (CLR) modeling were used to test the hypothesis that the risk of alveolar bone loss over 2 years is greater for subjects with poorer control of non-insulin-dependent diabetes mellitus (NIDDM) than for those who do not have diabetes or have better controlled NIDDM. There were 359 subjects; 21 of whom had NIDDM. Analysis of main effects using OLR for the dichotomous outcome (no change in radiographic bone loss vs any change) produced parameter estimates for better control and poorer control that were not statistically significant. CLR analysis of main effects using a 4-category ordinal specification for radiographic bone loss also produced a parameter estimate for better control that was not statistically significant, but which estimated poorer control to have a significant effect. The fit of this CLR model was significantly better at P < 0.05 than that for the OLR. While an OLR model testing the interaction between age and control status did not converge after 100 iterations, the CLR interaction model converged without difficulty and estimated a significant effect for interaction between age and poorer control. Results from the CLR analysis, in contrast to the OLR model, would lead one to conclude that the risk for more severe bone loss progression after 2 years is greater in subjects with poorer controlled NIDDM and that subjects with better controlled NIDDM may not have greater risk of bone loss progression than those without diabetes. The use of an ordinal instead of a dichotomous specification for an inherently continuous response provided increased power, more precise parameter estimates, and a significantly better fitting model. In estimating parameter estimates for odds ratios or risks, it is important to consider using ordinal logistic regression where the response is inherently continuous or ordinal.

Adolescent↗
PubMed1982

Survival models based upon the logistic and log--logistic distributions.

LOGC and LOGLOGC are interactive FORTRAN programs which fit the logistic and log--logistic regression models to censored survival data. Maximum likelihood estimates of parameters and estimates of their standard errors are given together with the corresponding log-likelihood. Survivorship functions, median remaining lifetimes and estimates of confidence intervals for these functions are given. Residual plotting is incorporated to assess the goodness of fit and can be plotted as can the survivorship function and median remaining lifetime estimates together with their corresponding confidence limits.

Computers↗
PubMed1993

Clinical chemical diagnosis of diseases assisted by logistic regression illustrated by diagnosis of canine primary and secondary hepatobiliary diseases.

OBJECTIVE: The purpose of the present study was to demonstrate the use of logistic regression models in the prediction of diseases using the prediction of canine primary and secondary hepatobiliary diseases as an example. Briefly, in a logistic regression model independent variables (i.e. the analytical results) are combined in a linear equation that is used to estimate the logarithm of the odds (logit) of an event (i.e. having primary or secondary hepatobiliary disease). From the estimated logit given by the logistic regression model, a conditional probability of the event (i.e. having primary or secondary hepatobiliary disease) can be calculated. STUDY DESIGN: Twenty-six dogs with verified primary and secondary hepatobiliary diseases and 19 dogs, initially suspected to have hepatobiliary diseases, but with apparently other diseases, were included in the study. The following clinical chemical parameters were measured: alanine aminotransferase (ALAT), aspartate aminotransferase (ASAT), alkaline phosphatase (AP), bilirubin(Total) (TB), urea, glucose, retention of bromosulphthalein (BSP), fasting and postprandial total serum bile acid concentration (FSBA and PSBA). Logistic regression analysis, using the CATMOD procedure in SAS, was used to select which of the measured parameters should be included in the model, and to derive a logistic regression model using the selected parameters. To observe more closely the potential of the logistic regression model, the model was also used to classify a test group consisting of 13 dogs (6 dogs with hepatobiliary diseases and 7 dogs with other diseases). RESULTS: By logistic regression analysis, ASAT and PSBA were selected to be included in the final model, and the final logistic regression model was Y = -3.194 + 0.044.PSBA + 3.251.ASAT. The logistic regression model classified correctly 38 (84%) of 45 dogs in the present study. Specifically, 21 (81%) of 26 dogs with verified primary or secondary hepatobiliary diseases and 17 (90%) of 19 dogs with various other diseases were correctly classified by the logistic regression model. When the model was used on the test group, 5 (83%) of 6 dogs with hepatobiliary diseases and 7 dogs (100%) of 7 dogs with other diseases were correctly classified. CONCLUSIONS: Even though the logistic regression model derived in the present study only serves as an example, thus reducing the practical usefulness of the derived logistic regression model, the present study indicates a great potential of logistic models for the diagnosis of diseases.

Animals↗
PubMed1981

[Analysis of RIA standard curve by log-logistic and cubic log-logit models (author's transl)].

In order to improve goodness-of-fit in RIA standard analysis, programs for computing log-logistic and cubic log-logit were written in BASIC using personal computer P-6060 (Olivetti). Iterative least square method of Taylor series was applied for non-linear estimation of logistic and log-logistic. Here "log-logistic" represents Y = (a--d)/(1+log(X)/c)b)+d As weights either 1, 1/var(Y) or 1/sigma 2 were used in logistic or log-logistic and either Y2(1--Y)2, Y2(1-Y)2/var(Y), or Y2(1--Y)2/sigma 2 were used in logistic or log-logistic and either Y2(1--Y)2, Y2(1--Y)2/var(Y), or Y2(1--Y)2/sigma 2 were used in quadratic or cubic log-logit. The term var(Y) represents squares of pure error and sigma 2 represents estimated variance calculated using a following equation log(sigma 2 + 1) = log(A)+J log(y). As indicators for goodness-of-fit, MSL/Se2, CMD% and WRV (see text) were used. Better regression was obtained in case of alpha-fetoprotein by log-logistic than by logistic. Cortisol standard curve was much better fitted with cubic log-logit than quadratic log-logit. Predicted precision of AFP standard curve was below 5% in log-logistic instead of 8% in logistic analysis. Predicted precision obtained using cubic log-logit was about five times lower than that with quadratic log-logit. Importance of selecting good models in RIA data processing was stressed in conjunction with intrinsic precision of radioimmunoassay system indicated by predicted precision.

Computers↗
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↗
PubMedDate not supplied

Tailored logistics: the next advantage.

How many top executives have ever visited with managers who move materials from the factory to the store? How many still reduce the costs of logistics to the rent of warehouses and the fees charged by common carriers? To judge by hours of senior management attention, logistics problems do not rank high. But logistics have the potential to become the next governing element of strategy. Whether they know it or not, senior managers of every retail store and diversified manufacturing company compete in logistically distinct businesses. Customer needs vary, and companies can tailor their logistics systems to serve their customers better and more profitably. Companies do not create value for customers and sustainable advantage for themselves merely by offering varieties of goods. Rather, they offer goods in distinct ways. A particular can of Coca-Cola, for example, might be a can of Coca-Cola going to a vending machine, or a can of Coca-Cola that comes with billing services. There is a fortune buried in this distinction. The goal of logistics strategy is building distinct approaches to distinct groups of customers. The first step is organizing a cross-functional team to proceed through the following steps: segmenting customers according to purchase criteria, establishing different standards of service for different customer segments, tailoring logistics pipelines to support each segment, and creating economics of scale to determine which assets can be shared among various pipelines. The goal of establishing logistically distinct businesses is familiar: improved knowledge of customers and improved means of satisfying them.

Commerce↗
PubMed1999

EMS incident management: emergency medical logistics.

If you had to get x amount of supplies to point A or point B, or both, in 10 minutes, how would you do it? The answer lies in the following steps: 1. Develop a logistics plan. 2. Use emergency management as a partner agency for developing your logistics plan. 3. Implement a push logistics system by determining what supplies/medications and equipment are important. 4. Place mass casualty/disaster caches at key locations for rapid deployment. Have medication/fluid caches available at local hospitals. 5. Develop and implement command caches for key supervisors and managers. 6. Anticipate the logistics requirements of a terrorism/tactical violence event based on a community threat assessment. 7. Educate the public about preparing a BLS family disaster kit. 8. Test logistics capabilities at disaster exercises. 9. Budget for logistics needs. 10. Never underestimate the importance of logistics. When logistics support fails, the EMS system fails.

Decision Making, Organizational↗
PubMed1999

Logistic regression models in obstetrics and gynecology literature.

OBJECTIVE: To evaluate the reporting of multivariable logistic regression analyses and assess variations in quality over time in the obstetrics and gynecology literature. METHODS: Methodologic criteria for reporting logistic regression analyses were developed to identify problems affecting accuracy, precision, and interpretation of this approach to multivariable statistical analysis. These criteria were applied to 193 articles that reported multivariable logistic regression in the issues of four generic obstetrics and gynecology journals in 1985, 1990, and 1995. Rates of compliance with the methodologic criteria and their time trends were analyzed. RESULTS: The proportion of articles using logistic regression analysis increased over time: 1.7% in 1985, 2.8% in 1990, and 6.5% in 1995 (P < .001 for trend). Violations and omissions of methodologic criteria for reporting logistic models were common. The research question, in terms of dependent and independent variables, was not clearly reported in 32.1%. The process of variable selection was inadequately described in 51.8% of the articles. Among articles with ranked independent variables, 85.1% did not report assessment of conformity to linear gradient. Tests for goodness of fit were not given in 93.2% of articles. The contribution of the independent variables could not be evaluated in 36.2% of the articles because of a lack of coding of the variables. Interactions between variables were not assessed in 86.4% of articles. Analysis of variations in the quality of logistic regression analyses over time showed no increase in reporting of the criteria concerning variable selection and goodness of fit. However, the proportion of articles reporting one quality criterion concerning interpretation of the substantive significance of independent variables showed a trend toward improvement: 42.3% in 1985, 73.6% in 1990, and 75.4% in 1995 (P = .004 for trend). CONCLUSION: The reporting of multivariable logistic regression models in the obstetrics and gynecology literature is poor, and the time trends of improvement in quality of reporting are not particularly encouraging.

Gynecology↗
PubMed1988

Estimation of probabilities using the logistic model in retrospective studies.

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.

Computer Simulation↗
PubMed1997

Hybrid logistic characterization of isometric twitch force-time curve of intact blood-perfused canine right ventricular papillary muscle.

We previously found that a ventricular isovolumic pressure-time curve could be well fitted by the difference between two S-shaped logistic curves for the pressure rising and falling components, and called it "hybrid logistic" function: P(t)=A/[1+exp[-(4B/A)(t-C)]]-D/[1+exp[-(4E/D)(t-F)]]+G. We reported that the parameters of this hybrid logistic function are useful to characterize left ventricular contraction and relaxation comprehensively. In this study, we investigated how well this hybrid logistic function could fit the isometric twitch force-time curves of cross-circulated right ventricular papillary muscles of 7 dogs. This function precisely fitted the isometric force curves with correlation coefficients above 0.9996, much better than another fitting function (F(t)=C(t/A)(B)exp[1-(t/A)(B)]) proposed by Nwasokwa. The present results indicate that our hybrid logistic function can also reasonably express the canine right ventricular papillary muscle isometric twitch force-time curve. We suggest the possibility that the parameters of this hybrid logistic function are also useful to comprehensively characterize right ventricular papillary muscle twitch contraction and relaxation.

Animals↗
PubMed1998

Inference using conditional logistic regression with missing covariates.

When there are many nuisance parameters in a logistic regression model, a popular method for eliminating these nuisance parameters is conditional logistic regression. Unfortunately, another common problem in a logistic regression analysis is missing covariate data. With many nuisance parameters to eliminate and missing covariates, many investigators exclude any subject with missing covariates and then use conditional logistic regression, often called a complete-case analysis. In this article, we derive a modified conditional logistic regression that is appropriate with covariates that are missing at random. Performing a conditional logistic regression with only the complete cases is convenient with existing statistical packages, but it may give bias if missingness is not completely at random.

Bias↗
PubMed1998

Two-parameter logistic and Weibull equations provide better fits to survival data from isogenic populations of Caenorhabditis elegans in axenic culture than does the Gompertz model.

We have fitted Gompertz, Weibull, and two- and three-parameter logistic equations to survival data obtained from 77 cohorts of Caenorhabditis elegans in axenic culture. Statistical analysis showed that the fitting ability was in the order: three-parameter logistic > two-parameter logistic = Weibull > Gompertz. Pooled data were better fit by the logistic equations, which tended to perform equally well as population size increased, suggesting that the third parameter is likely to be biologically irrelevant. Considering restraints imposed by the small population sizes used, we simply conclude that the two-parameter logistic and Weibull mortality models for axenically grown C. elegans generally provided good fits to the data, whereas the Gompertz model was inappropriate in many cases. The survival curves of several short- and long-lived mutant strains could be predicted by adjusting only the logistic curve parameter that defines mean life span. We conclude that life expectancy is genetically determined; the life span-altering mutations reported in this study define a novel mean life span, but do not appear to fundamentally alter the aging process.

Animals↗
PubMed1997

Evaluation of logistic versus linear regression models for predicting pulmonary hypertension syndrome (ascites) using cold exposure or pulmonary artery clamp models in broilers.

Syndromes such as ascites (pulmonary hypertension syndrome) present difficulties both in the interpretation of associated physiological observations and in their analyses. The ability to predict which physiological variables have the greatest influence on survival or, more importantly, which individuals are most susceptible or resistant to ascites would be very useful selection tools. When addressed in this manner, ascites data become binary data sets (healthy or affected). Binary data can be problematic in that they do not meet all of the assumptions necessary for more traditional analyses such as ANOVA and linear regression. Binary data are discrete and do not have normally distributed errors, which violates a fundamental assumption of linear models. The predictive abilities of linear and logistic regression were evaluated in two replicated experiments using two methods to induce ascites, cold exposure (COLD) and surgical clamping of one pulmonary artery (PAC). The logistic and linear predictive models were derived using the same data and variables. The first data set from PAC and COLD were used to develop the predictive models and the replicate data sets of PAC and COLD were used as "test data sets" for the prediction of ascites. The linear models developed were complex, using four or five variables and requiring up to seven different measurements. On average, the linear models predicted ascites correctly 87.6% of the time. The logistic models were simple (single variable) models that predicted ascites correctly 92.0% of the time. The variables used in the logistic models were derivations of the ratio of right ventricular weight to total ventricular weight, either corrected for age or the body weight of the bird. Although linear regression predicted the incidence of ascites almost as well as logistic regression did, logistic regression is the more appropriate test statistic to use.

Analysis of Variance↗
PubMed1998

A simple method of sample size calculation for linear and logistic regression.

A sample size calculation for logistic regression involves complicated formulae. This paper suggests use of sample size formulae for comparing means or for comparing proportions in order to calculate the required sample size for a simple logistic regression model. One can then adjust the required sample size for a multiple logistic regression model by a variance inflation factor. This method requires no assumption of low response probability in the logistic model as in a previous publication. One can similarly calculate the sample size for linear regression models. This paper also compares the accuracy of some existing sample-size software for logistic regression with computer power simulations. An example illustrates the methods.

Humans↗
PubMed1998

Incorporation of family history in logistic regression models.

For diseases with a genetic component, logistic regression models are presented that incorporate family history in a quantitative way. In the largest model, every type of relative has their own regression coefficient. The other two models are submodels, which incorporate family history either by the number of cases in the family minus its expectation or by a weighted number of cases in the family minus its expectation. For various genetic effects, namely polygenic and autosomal dominant effects, the performance of these simple logistic models is studied. First, the predictive values of the logistic and true genetic models are computed and compared. Secondly, a simulation study is carried out to investigate the effects of estimation of the parameters in a small data set. Thirdly, the logistic models are fitted to a data set of Von Willebrand Factor responses of target individuals and their families; in these models, family history has a significant effect. The conclusion is that for the genetic effects considered the logistic models perform well.

Female↗
PubMed1997

Value of logistic discriminant analysis for interpreting initial visual field defects.

PURPOSE: The authors evaluate logistic discriminant analysis as a method for interpreting visual field results in initial stages of several ophthalmic diseases. METHODS: The authors retrospectively studied the visual field defects of 96 patients with early glaucomatous damage and prospectively studied 84 subjects with normal eyes (n = 28), cataracts (n = 27), diabetic retinopathy (n = 14), or hypertensive retinopathy (n = 15). The visual fields were examined at least twice with program G1 of Octopus 500 (Interzeag AG, Schlieren, Switzerland). Only one eye per patient was randomly selected and included in the study. The authors created a database with all visual field information provided by Octopus and applied logistic discriminant analysis (SAS Logistic Procedures, SAS Institute, Cary, NC) to obtain classification rules capable of qualifying visual field defects as glaucomatous or nonglaucomatous. The rules were tested with an independent sample. RESULTS: There were significant differences between the groups in the distribution of visual field defects tested by program G1. Logistic discriminant analysis correctly identified the glaucomatous or nonglaucomatous origin of the defects with a sensitivity of 65% to 85% and a specificity of 60% to 92%. CONCLUSIONS: Logistic discriminant analysis is a useful tool to aid in the interpretation of early glaucomatous and nonglaucomatous visual field defects.

Data Interpretation, Statistical↗
PubMed1998

Using binary logistic regression models for ordinal data with non-proportional odds.

The proportional odds model (POM) is the most popular logistic regression model for analyzing ordinal response variables. However, violation of the main model assumption can lead to invalid results. This is demonstrated by application of this method to data of a study investigating the effect of smoking on diabetic retinopathy. Since the proportional odds assumption is not fulfilled, separate binary logistic regression models are used for dichotomized response variables based upon cumulative probabilities. This approach is compared with polytomous logistic regression and the partial proportional odds model. The separate binary logistic regression approach is slightly less efficient than a joint model for the ordinal response. However, model building, investigating goodness-of-fit, and interpretation of the results is much easier for binary responses. The careful application of separate binary logistic regressions represents a simple and adequate tool to analyze ordinal data with non-proportional odds.

Bias↗