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We study the large time behaviour of a nonlinear population model with a general logistic term. It is proved that every solution must have a limit when time becomes infinite. We present conditions that guarantee the boundedness of the solution. Furthermore, we prove that in general no oscillation is possible for the total number of population. This is in sharp contrast to the linear case.
Data from a case-control study of breast cancer in 441 cases and matched controls aged less than or equal to 38 in Los Angeles is fitted to two models of carcinogenesis: the two-stage model of Moolgavkar and Knudson; and a multistage adaptation of the log/log model proposed by Pike et al. In the two-stage model, risk factors (here age at menarche, age at first pregnancy, abortion, regularity of cycling, benign breast disease, and use of oral contraceptives (OCs] are postulated to act either by increasing the rate of mutation of normal to intermediate or of intermediate to malignant stage cells, or by increasing the proliferation rates of such cells. In the multistage model, it is postulated that all transition rates are equally determined by the rate of cell turnover, which is in turn influenced by risk factors. In both models, positive family history is modelled in two ways: (1) all cells may have started in an intermediate stage at birth, with probability depending on the number and degree of affected family members; or (2) as an event rate modifier in the same way as other covariates. With only menarche or menarche and family history included, the multistage and two-stage models both produced likelihoods very similar to those from a simple logistic model, though both fit better than the logistic model as more covariates were added to the model. The two stage models offer greater flexibility in modelling the time at which each factor is most effective. We found irregular menstrual cycling to reduce the rate of proliferation of normal cells or their mutation rate to intermediate cells by 50% (p less than 0.025) and use of OCs to increase these rates by 3.25 fold (p less than 0.01). Having completed a pregnancy of more than 26 weeks gestation appeared to reduce the rate of intermediate cell proliferation by 5% (p less than 0.05) relative to a baseline rate of 14% per year. Benign breast disease was associated with a 1.60 fold increased rate of the second mutation (p less than 0.025). In all models, family history was by far the strongest risk factor (RR = 4.44 from the logistic model, p less than 0.0001). Covariates showed similar effects in the multistage model, though their magnitudes were slightly different.
The clinical utility of conventional logistic regression models based on left ventricular ejection fraction (LVEF) for the prediction of cardiac events (death or recurrent infarction) was assessed in 646 postinfarction patients undergoing radionuclide ventriculography at rest and during exercise. The discriminant power of 2 different models (LVEF at rest alone vs LVEF at rest plus LVEF at peak exercise) was quantified in terms of the area under receiver-operating characteristic curves based on knowledge of patient outcome in the year after testing and the logistic probability of that outcome. Although LVEF at rest provided a significant amount of prognostic information (receiver-operating characteristic curve area = 62 +/- 4%, p less than 0.001), several limitations were observed: (1) powerful predictors of risk were uncommon (32% of patients with an LVEF at rest less than 0.20 had a cardiac event, but only 3% of the population had such extreme values); (2) the accuracy of predictions for high risk patients was less than for low risk patients (28 vs 98%, p less than 0.001); (3) addition of exercise LVEF to the model did not improve the accuracy of prediction (receiver-operating characteristic curve area = 68 +/- 4%, p = 0.11); and (4) predictions for individual patients were very imprecise (the 95% confidence interval of percent risk for an LVEF at rest of 0.20 [11 to 36%] overlapped that for an LVEF at rest of 0.60 [0 to 14%]).(ABSTRACT TRUNCATED AT 250 WORDS)
Logistic regression models are commonly used to study the association between a binary response variable and an exposure variable. Besides the exposure of interest, other covariates are frequently included in the fitted model in order to control for their effects on outcome. Unfortunately, misspecification of the main exposure variable and the other covariates is not uncommon, and this can adversely affect tests of the association between the exposure and response. We allow the term "misspecification" to cover a broad range of modeling errors including measurement errors, discretizing continuous explanatory variables, and completely excluding covariates from the model. This paper reviews some recent results on the consequences of model misspecification on the large sample properties of likelihood score tests of association between exposure and response.
The early recognition of life-threatening injury is paramount to the prompt initiation of appropriate care. This study assesses the importance of multiple rib fractures as a marker of severe injury in children. We analyzed physiologic, etiologic, and injury data for 2,080 children with blunt or penetrating trauma aged 0-14 years consecutively admitted to a Level I pediatric trauma center. Analysis of variance, Student's t-test, and the Chi-square test of independence were used to test for differences between children with rib fractures and other children. Probability of survival was modeled using stepwise logistic regression. There were 14 deaths among 33 children with rib fractures, a mortality rate of 42%. Child abuse accounted for 63% of the injuries to children less than 3 years old, while pedestrian injuries predominated among older children. Children with rib fractures were significantly more severely injured than children with blunt or penetrating trauma but without rib fractures. When compared to children without rib fractures, children with rib fractures had a higher mortality rate, but no statistically significant difference in morbidity. The mortality rate for the 18 children with both rib fractures and head injury was 71%. A logistic model with variables measuring severity of head injury and number of ribs fractured correctly predicted survival in more than 85% of children with thoracic trauma. Although rib fractures are rare injuries in childhood, they are associated with a high risk of death. The risk of mortality increases with the number of ribs fractured. The combination of rib fractures and head injury was usually fatal.
A multivariate statistical analysis of 9 potential prognostic factors was made, using data gathered on 107 patients admitted between 1975 and 1987. After each variable was assessed separately for its prognostic importance by means of univariate analysis, a multivariate analysis using Cox's proportional hazards regression model and multivariate logistic model was done. These tests identified Gleason's score as the most important factor for all patients followed by M category and alkaline phosphatase (ALP), while Gleason's score alkaline phosphatase and age at diagnosis were most important for stage D patients. For short-term survival, the significant prognostic factors were Gleason's score and M category while for long term survival they were Gleason's score and age. T. category, stage, acid phosphatase, prostatic acid phosphatase, and treatment were found to be significantly related to survival time when examined individually; they were not found to be significant in the multivariate analysis.
Small sample size (n less than 10) and inappropriate analysis of multivariate data have hindered previous attempts to describe which physiologic and demographic variables are most important in determining how long humans can tolerate acceleration. Data from previous centrifuge studies conducted at NASA/Ames Research Center, utilizing a 7-14 d bed rest protocol to simulate weightlessness, were included in the current investigation. After review, data on 25 women and 22 men were available for analysis. Study variables included gender, age, weight, height, percent body fat, resting heart rate, mean arterial pressure, VO2max, and plasma volume. Since the dependent variable was time to greyout (failure), two contemporary biostatistical modeling procedures (proportional hazard and logistic discriminant function) were used to estimate risk, given a particular subject's profile. After adjusting for pre-bed-rest tolerance time, none of the profile variables remained in the risk equation for post-bed-rest tolerance greyout. However, prior to bed rest, risk of greyout could be predicted with 91% accuracy. All of the profile variables except weight, MAP, and those related to inherent aerobic capacity (VO2max, percent body fat, resting heart rate) entered the risk equation for pre-bed-rest greyout. A cross-validation using 24 new subjects indicated a very stable model for risk prediction, accurate within 5% of the original equation. The result for the inherent fitness variables is significant in that a consensus as to whether an increased aerobic capacity is beneficial or detrimental has not been satisfactorily established. We conclude that tolerance to +Gz acceleration before and after simulated weightlessness is independent of inherent aerobic fitness.
In a community based, prospective study to determine risk factors for falls, 465 women and 296 men 70 years and over were followed for 1 year and 507 falls were documented. A greater proportion of women (32.7%) than men (23.0%) experienced at least one fall in which there was no or minimal external contribution. Using unconditional logistic regression models we investigated the effect of physical and sociological variables on the sex difference in fall rate. Controlling for the variables age, use of psychotropic drugs, inability to rise from a chair without using arms, going outdoors less than daily and living alone decreased the relative risk of women falling compared to men from 2.02 (95% CI, 1.40-2.92) to 1.55 (95% CI 1.04-2.31). Some of the increased risk of falling associated with being a women was able to be explained and is potentially correctable. But even after controlling for the physical and social variables which we had assessed, women compared to men still had a significantly increased relative risk of falling.
Logistic regression was used to analyze the clinical findings (attributes) which predicted coliform mastitis in 113 dairy cattle, 36 of which had coliforms cultured from milk. Weakness of the cow, swelling of the udder, decreased body temperature and watery consistency of the milk were selected for the model. An analysis was then done to find the attributes which clinicians used when predicting that a cow would have a coliform cultured. Clinicians appeared to use water consistency of the milk, shivering, firmness of the udder, pulse rate, elevated body temperature, and respiratory rate. In a final analysis the clinicians' predictions were forced into the model to determine which attributes might be used by clinicians to increase diagnostic accuracy. Inclusion of weakness of the cow, swelling of the udder, decreased temperature of the cow, and duration of mastitis of less than 24 hours increased accuracy over clinical prediction alone. Accuracy of cowside diagnosis might be increased if more attention were paid to these attributes when making a diagnosis of coliform mastitis.
Surgical risk is defined as the occurrence of complications arising in the individual as a result of surgical stress. The ability to forecast these consequences is an important factor in determining decision taken by surgeon. Several attempts have been made to quantify postsurgical prospects but up till now no overall solution has been found. This paper attempts to define a multifactorial risk index for adults subjected to surgery, with respect to immediate and early per- and post-surgical complications. 1182 adult patients, 14 yrs or more, surgically treated not for urgency during 1985 in six Italian centres, were prospectively studied in order to derive a multivariate prognostic index of after surgery mortality. Stepwise logistic regression model was applied to a set of preoperative and operative factors, five of which were found significantly correlate with death: nutritional status, renal failure, reintervention, bacterial contamination during surgery, age greater than 70 years. Thus, from regression coefficients, scores were derived for modalities of significant variables, allowing to build four classes of risk patients: low (less than 1%), medium (between 1% and 10%), high (between 10% and 50%), extremely high risk (greater than 50%).
One hundred and twelve new cases of Graves' disease were treated by tapazole for 6 months and followed up for another 12 months. The initial dose was 30 mg/d. Clinical and biochemical euthyroidism was achieved within 1 to 3 months, then a maintenance was given until cessation of drugs at 6 months. One hundred and eleven cases completed the study. Remission and relapse were defined at the end of follow-up for 1 year according to the presence or absence of clinical manifestation of hyperthyroidism and the levels of T3 and T4. The results of the 12-month follow-up showed that 46 of 111 cases were in remission, and the remaining 65 cases suffered relapse. A logistic regression model with 4 variables was established, which included thyroid suppression rate and goitre size by palpation at the end of drug treatment, the level of T3 before therapy, and the patients' age. The model had 81.5% sensitivity, 84.8% specificity and 82.9% (92/111) accuracy in predicting the outcome of Graves' disease after withdrawal of drug for 1 year. The results were much better than any other univariate analysis in this study.
Various measures for estimating the goodness-of-fit of the multiple logistic regression (MLR) model have been suggested, although there is no clear consensus as to which measure is most suitable. In this paper, a simple measure of the discriminatory power of the fitted MLR model, based on maximization of Youden's J index (J*), is proposed and compared with several goodness-of-fit statistics described previously. The relative effectiveness of the measure is illustrated using data from the Lipid Research Clinics Prevalence Study. It is suggested that J* may be a useful alternative index of goodness-of-fit of an MLR model, with the added advantage of having a simple practical interpretation.
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The assessment of continuous covariates singly as possible predictors in a multivariable logistic regression model is an important first step in the analysis. An approach to plotting which uses a cusum (cumulative sum) of the binary response variable is described. Extreme-deviation statistics associated with the cusum may be used to detect monotonic and non-monotonic trends. Probability plots of the covariate in the two groups defined by the response variable may help to determine the appropriate scale (transformation) of the covariate and to anticipate possible problems with the logistic fit. The ratio of the variances in the response/non-response groups is informative about the need for a quadratic term in the logistic model. Smoothed scatterplots of the response are valuable in displaying the observed and fitted values. The techniques are illustrated with two data sets.
Ordered logistic regression is an extension of binary logistic regression, and is particularly well suited to the analysis of many psychiatric scores. Its use is demonstrated in a pair of linked cross-sectional surveys of dementia in residents of old people's homes, first to model the association of dementia with demographic characteristics, and then to explore possible reasons for a rise in the prevalence of dementia in the homes over a four-year period.
This paper describes the dynamics of a continuously reproducing diploid population with two alleles at one locus. The dependent variables are allele frequency and population density. We modify the basic density-dependent logistic growth model by inserting three possible types of frequency dependence in the fitness functions. These models are analyzed and contrasted with the purely density-dependent situation. Examples are given of periodic fluctuations in allele frequency and population density, which would be impossible for purely density-dependent fitness functions.
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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.