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PubMed2002

Goodness-of-fit processes for logistic regression: simulation results.

In this paper we use simulations to compare the performance of new goodness-of-fit tests based on weighted statistical processes to three currently available tests: the Hosmer-Lemeshow decile-of-risk test; the Pearson chi-square, and the unweighted sum-of-squares tests. The simulations demonstrate that all tests have the correct size. The power for all tests to detect lack-of-fit due to an omitted quadratic term with a sample of size 100 is close to or exceeds 50 per cent to detect moderate departures from linearity and is over 90 per cent for these same alternatives for sample size 500. All tests have low power with sample size 100 to detect lack-of-fit due to an omitted interaction between a dichotomous and continuous covariate, while the power exceeds 80 per cent to detect extreme interaction with a sample size of 500. The power is low to detect any alternative link function with sample size 100 and for most alternative links for sample size 500. Only in the case of sample size 500 and an extremely asymmetric link function is the power over 80 per cent. The results from these simulations show that no single test, new or current, performs best in detecting lack-of-fit due to an omitted covariate or incorrect link function. However, one of the new weighted tests has power comparable to other tests in all settings simulated and had the highest power in the difficult case of an omitted interaction term. We illustrate the tests within the context of a model for factors associated with abstinence from drug use in a randomized trial of residential treatment programmes. We conclude the paper with a summary and specific recommendations for practice.

Computer Simulation↗
PubMed2004

Inflation of the type I error rate when a continuous confounding variable is categorized in logistic regression analyses.

This paper demonstrates an inflation of the type I error rate that occurs when testing the statistical significance of a continuous risk factor after adjusting for a correlated continuous confounding variable that has been divided into a categorical variable. We used Monte Carlo simulation methods to assess the inflation of the type I error rate when testing the statistical significance of a risk factor after adjusting for a continuous confounding variable that has been divided into categories. We found that the inflation of the type I error rate increases with increasing sample size, as the correlation between the risk factor and the confounding variable increases, and with a decrease in the number of categories into which the confounder is divided. Even when the confounder is divided in a five-level categorical variable, the inflation of the type I error rate remained high when both the sample size and the correlation between the risk factor and the confounder were high.

Cohort Studies↗
PubMed2005

A comparison of goodness of fit tests for the logistic GEE model.

Generalized estimating equations have become a popular regression method for analysing clustered binary data. Methods to assess the goodness of fit of the fitted models have recently been developed. However, evaluations and comparisons of these methods are limited. We discuss these methods and develop two additional statistics to evaluate goodness of fit. We evaluate the performance of each of the statistics with respect to type I error rates and power in a simulation study. Guidance is provided regarding appropriate use of the statistics under various scenarios.

Biometry↗
PubMed2005

Estimating treatment efficacy over time: a logistic regression model for binary longitudinal outcomes.

This paper presents a case study in longitudinal data analysis where the goal is to estimate the efficacy of a new drug for treatment of a severe chronic constipation. Data consist of long sequences of binary outcomes (relief/no relief) on each of a large number of patients randomized to treatment (low and high dose) or placebo. Data characteristics indicate: (1) the treatment effects vary non-linearly with time; (2) there is substantial heterogeneity across subjects in their responses to treatment; and (3) there is a high proportion of subjects who never experience any relief (the non-responders). To overcome these challenges, we develop a hierarchical model for binary longitudinal data with a mixture distribution on the probability of response to account for the high frequency of non-responders. While the model is specified conditionally on subject-specific latent variables, we also draw inferences on key population-average parameters for the assessment of the treatments' efficacy in a population. In addition we employ a model-checking method to compare the goodness-of-fit for our model against simpler modelling approaches for aggregated counts, such as the zero-inflated Poisson and zero-inflated negative binomial models. We estimate subject-specific and population-average rate ratios of relief for the treatment with respect to the placebo as functions of time (RR(t)), and compare them with the rate ratios estimated from the models for aggregated counts. We find that: (1) the treatment is effective with respect to the placebo with higher efficacy at the beginning of the study; (2) the estimated rate ratios from the models for aggregated counts appear to be similar to the average across time of the population-average rate ratios estimated under our model; and (3) model-checking suggests that the hierarchical and zero-inflated negative binomial model fit the data best. If we are mainly interested to establish the overall efficacy (or safety) of a new drug, it is appropriate to aggregate the longitudinal data over time and analyse the count data by use of standard statistical methods. However, the models for aggregated counts cannot capture time trend of treatment such as the initial treatment benefit or the development of tolerance during the early stage of the treatment which may be important information to physicians to predict the treatment effects for their patients.

Biometry↗
PubMed2001

Extinction and quasi-stationarity in the Verhulst logistic model.

We formulate and analyse a stochastic version of the Verhulst deterministic model for density-dependent growth of a single population. Three parameter regions with qualitatively different behaviours are identified. Explicit approximations of the quasi-stationary distribution and of the expected time to extinction are presented in each of these regions. The quasi-stationary distribution is approximately normal, and the time to extinction is long, in one of these regions. Another region has a short time to extinction and a quasi-stationary distribution that is approximately truncated geometric. A third region is a transition region between these two. Here the time to extinction is moderately long and the quasi-stationary distribution has a more complicated behaviour. Numerical illustrations are given.

Animals↗
PubMed2006

Combining logistic models with multivariate methods for the rapid biological assessment of rivers using macroinvertebrates.

This work represents an attempt to define a simple method to classify the relative degree of disturbance of sites in lotic systems on the basis of comparison of their faunistic composition with reference sites. Two ecotypes were selected in northern Portugal where benthic invertebrates were sampled in reaches with different levels of contamination. As a first stage, previous Geographic Information System information was used to define reference sites in each ecotype. Afterwards, multivariate techniques and non linear estimation models were combined to assess biological quality. This method allowed us to quantify sites according to increasing levels of contamination, after the probabilities of occurrence of taxa along a gradient of contamination taking into account the reference condition. The results suggest that this method is sensitive to organic pollution, easy to interpret, namely the species tolerance, and could be a good framework to establish regional rankings depending on the ecological impact of river sites.

Altitude↗
PubMed2006

Modeling crash outcome probabilities at rural intersections: application of hierarchical binomial logistic models.

It is important to examine the nature of the relationships between roadway, environmental, and traffic factors and motor vehicle crashes, with the aim to improve the collective understanding of causal mechanisms involved in crashes and to better predict their occurrence. Statistical models of motor vehicle crashes are one path of inquiry often used to gain these initial insights. Recent efforts have focused on the estimation of negative binomial and Poisson regression models (and related deviants) due to their relatively good fit to crash data. Of course analysts constantly seek methods that offer greater consistency with the data generating mechanism (motor vehicle crashes in this case), provide better statistical fit, and provide insight into data structure that was previously unavailable. One such opportunity exists with some types of crash data, in particular crash-level data that are collected across roadway segments, intersections, etc. It is argued in this paper that some crash data possess hierarchical structure that has not routinely been exploited. This paper describes the application of binomial multilevel models of crash types using 548 motor vehicle crashes collected from 91 two-lane rural intersections in the state of Georgia. Crash prediction models are estimated for angle, rear-end, and sideswipe (both same direction and opposite direction) crashes. The contributions of the paper are the realization of hierarchical data structure and the application of a theoretically appealing and suitable analysis approach for multilevel data, yielding insights into intersection-related crashes by crash type.

Accidents, Traffic↗
PubMed2006

Initial biopsy outcome prediction--head-to-head comparison of a logistic regression-based nomogram versus artificial neural network.

OBJECTIVES: Nomograms and artificial neural networks (ANNs) represent alternative methodologic approaches to predict the probability of prostate cancer on initial biopsy. We hypothesized that, in a head-to-head comparison, one of the approaches might demonstrate better accuracy and performance characteristics than the other. METHODS: A previously published nomogram, which relies on age, digital rectal examination, serum prostate-specific antigen (PSA), and percent-free PSA, and an ANN, which relies on the same predictors plus prostate volume, were applied to a cohort of 3980 men, who were subjected to multicore systematic prostate biopsy. The accuracy and the performance characteristics were compared between these two approaches. RESULTS: The accuracy of the nomogram was 71% versus 67% for the ANN (p=0.0001). Graphical exploration of the performance characteristics demonstrated virtually perfect predictions for the nomogram. Conversely, the ANN underestimated the observed rate of prostate cancer. CONCLUSIONS: A 4% increase in predictive accuracy implies that the use of the nomogram instead of the ANN will result in 40 additional patients who will be correctly classified between benign and cancer.

Adult↗
PubMed2005

Use of a hierarchical logistic regression model to determine the impact of war injuries on clinical pregnancy rates in patients undergoing intracytoplasmic sperm injection.

OBJECTIVE: To determine the effects of war injuries, including spinal cord injuries, poisoning, and other injuries, on clinical pregnancy rates in infertile couples undergoing intracytoplasmic sperm injection (ICSI) and to compare the results with infertile couples who were not victims of the war. DESIGN: Retrospective study. SETTING: A large, national clinic in Tehran, Iran. PATIENT(S): Six hundred sixty-seven couples undergoing ICSI. INTERVENTION(S): Database analysis. MAIN OUTCOME MEASURE(S): Ongoing clinical pregnancy. RESULT(S): War victims with spinal cord injuries and those who had been poisoned had the same pregnancy rates as did infertile couples who were not victims of the war. The predictive model shows that the more embryos that are transferred, the more the chances for clinical pregnancies; in addition, the more cycles a couple undergoes, the fewer are the chances for clinical pregnancy in an additional cycle. Finally, younger females have more chances for clinical pregnancy than do older ones. CONCLUSION(S): Intracytoplasmic sperm injection is an appropriate option for the treatment of infertile war victim couples, especially for couples suffering from spinal cord injuries and poisoning.

Adult↗