A simple approach to power and sample size calculations in logistic regression and Cox regression models

For a given regression problem it is possible to identify a suitably defined equivalent two-sample problem such that the power or sample size obtained for the two-sample problem also applies to the regression problem. For a standard linear regression model the equivalent two-sample problem is easily identified, but for generalized linear models and for Cox regression models the situation is more complicated. An approximately equivalent two-sample problem may, however, also be identified here. In particular, we show that for logistic regression and Cox regression models the equivalent two-sample problem is obtained by selecting two equally sized samples for which the parameters differ by a value equal to the slope times twice the standard deviation of the independent variable and further requiring that the overall expected number of events is unchanged. In a simulation study we examine the validity of this approach to power calculations in logistic regression and Cox regression models. Several different covariate distributions are considered for selected values of the overall response probability and a range of alternatives. For the Cox regression model we consider both constant and non-constant hazard rates. The results show that in general the approach is remarkably accurate even in relatively small samples. Some discrepancies are, however, found in small samples with few events and a highly skewed covariate distribution. Comparison with results based on alternative methods for logistic regression models with a single continuous covariate indicates that the proposed method is at least as good as its competitors. The method is easy to implement and therefore provides a simple way to extend the range of problems that can be covered by the usual formulas for power and sample size determination.     

Logistic regression models with missing covariate values for complex survey data

Maximum likelihood methods are used to incorporate partially observed covariate values in fitting logistic regression models. We extend these methods to data collected through complex surveys using the pseudo-likelihood approach. One can obtain parameter estimates of the logistic regression model using standard statistical software and their standard errors by Taylor series expansion or the jackknife method. We apply the approach to data from a two-phase survey screening for dementia in a community sample of African Americans age 65 and older living in Indianapolis. The binary response variable is dementia and the covariate with missing values is a daily functioning score collected from interviews with a relative of the study subject. © 1997 John Wiley & Sons, Ltd.     

Comparison of methods incorporating quantitative covariates into affected sib pair linkage analysis

For complex traits, it may be possible to increase the power to detect linkage if one takes advantage of covariate information. Several statistics have been proposed that incorporate quantitative covariate information into affected sib pair (ASP) linkage analysis. However, it is not clear how these statistics perform under different gene-environment (G x E) interactions. We compare representative statistics to each other on simulated data under three biologically-plausible G x E models. We also compared their performance with a model-free method and with quantitative trait locus (QTL) linkage approaches. The statistics considered here are: (1) mixture model; (2) general conditional-logistic model (LODPAL); (3) multinomial logistic regression models (MLRM); (4) extension of the maximum-likelihood-binomial approach (MLB); (5) ordered-subset analysis (OSA); and (6) logistic regression modeling (COVLINK). In all three G x E models, most of these six statistics perform better when using the covariate C1 associated with a G x E interaction effect than when using the environmental risk factor C2 or the random noise covariate C3. Compared with a model-free method without covariates (S(all)), the mixture model performs the best when using C1, with the high-to-low OSA method also performing quite well. Generally, MLB is the least sensitive to covariate choice. However, most of these statistics do not provide better power than S(all). Thus, while inclusion of the "correct" covariate can lead to increased power, careful selection of appropriate covariates is vital for success.     

Value of multinomial model in epidemiology: application to the comparison of risk factors for severely and moderately preterm births

BACKGROUND: The multinomial logistic regression model is employed to model the relationship between an outcome variable with more than two categories and a set of covariates. This model is not widely used in epidemiology. We discuss the value of the multinomial model by comparing it with the binary logistic model, and we present a statistical comparison of odds ratios (OR) using the multinomial model. We studied the associations between obstetric history and very (< 33 weeks of amenorrhea) and moderate (33-36 weeks) preterm births. METHODS: Parameters (lnOR) of very and moderate preterm births, associated with the severity of obstetric history (none=0, moderate=1, severe=2), were estimated using two logistic binary models (moderate preterm births vs full-term births (>=37 weeks), and very preterm births vs full-term births) and one logistic multinomial model which compared very and moderate preterm births to full-term births. These analyses were performed before and after adjustment for a covariate: the country of survey. Parameters of very preterm birth and moderate preterm birth, estimated from multinomial model, were compared using Wald test. These analyses were performed using data from a large case-control survey in Europe, the EUROPOP survey; 1 675 very preterm births, 3 652 moderate preterm births and 7 965 full-term births were included. RESULTS: Crude parameters of very and moderate preterm births were similar, regardless the logistic regression model, binary or multinomial. The estimated parameters slightly differ after adjustment for the covariate, but lower variance estimates were obtained using multinomial logistic regression model. Parameters of very preterm birth associated with moderate obstetric history, B(gp)=0.5040, and severe obstetric history, B(gp)'=1.545, differ significantly from those of moderate preterm birth, B(pm)=0.4434 and B(pm)'=1.223 respectively (p < 0.001). CONCLUSION: Parameters obtained in separate logistic binary models are close to those obtained in a multinomial model. The multinomial model is useful for testing the heterogeneity of risk factors for distinct health problems.     

A semiparametric imputation approach for regression with censored covariate with application to an AMD progression study

This research is motivated by studying the progression of age‐related macular degeneration where both a covariate and the response variable are subject to censoring. We develop a general framework to handle regression with censored covariate where the response can be different types and the censoring can be random or subject to (constant) detection limits. Multiple imputation is a popular technique to handle missing data that requires compatibility between the imputation model and the substantive model to obtain valid estimates. With censored covariate, we propose a novel multiple imputation‐based approach, namely, the semiparametric two‐step importance sampling imputation (STISI) method, to impute the censored covariate. Specifically, STISI imputes the missing covariate from a semiparametric accelerated failure time model conditional on fully observed covariates (Step 1) with the acceptance probability derived from the substantive model (Step 2). The 2‐step procedure automatically ensures compatibility and takes full advantage of the relaxed semiparametric assumption in the imputation. Extensive simulations demonstrate that the STISI method yields valid estimates in all scenarios and outperforms some existing methods that are commonly used in practice. We apply STISI on data from the Age‐related Eye Disease Study, to investigate the association between the progression time of the less severe eye and that of the more severe eye. We also illustrate the method by analyzing the urine arsenic data for patients from National Health and Nutrition Examination Survey (2003‐2004) where the response is binary and 1 covariate is subject to detection limit.     

Sample size calculations for the two-sample problem using the multiplicative intensity model

In this paper we propose formulae for calculating the expected number of events or, alternatively, the required trial duration, for clinical trials involving two treatment groups in which patients may potentially experience multiple events and the data will be analysed using a multiplicative intensity (MI) model. We use a partial likelihood-based approach and examine in detail two MI models: one that includes a binary treatment variable as the only covariate and a three-state Markov process model in which a binary time-varying covariate is added to the previous model. For the simpler model, our formula coincides with those derived by Cook using full likelihood methods. We present applications of the derived formulae to chronic granulomatous disease and breast cancer data sets.     

A new approach to modelling interactions between treatment and continuous covariates in clinical trials by using fractional polynomials

We consider modelling interaction between a categoric covariate T and a continuous covariate Z in a regression model. Here T represents the two treatment arms in a parallel-group clinical trial and Z is a prognostic factor which may influence response to treatment (known as a predictive factor). Generalization to more than two treatments is straightforward. The usual approach to analysis is to categorize Z into groups according to cutpoint(s) and to analyse the interaction in a model with main effects and multiplicative terms. The cutpoint approach raises several well-known and difficult issues for the analyst. We propose an alternative approach based on fractional polynomial (FP) modelling of Z in all patients and at each level of T. Other prognostic variables can also be incorporated by first constructing a multivariable adjustment model which may contain binary covariates and FP transformations of continuous covariates other than Z. The main step involves FP modelling of Z and testing equality of regression coefficients between treatment groups in an interaction model adjusted for other covariates. Extensive experience suggests that a two-term fractional polynomial (FP2) function may describe the effect of a prognostic factor on a survival outcome quite well. In a controlled trial, this FP2 function describes the prognostic effect averaged over the treatment groups. We refit this function in each treatment group to see if there are substantial differences between groups. Allowing different parameter values for the chosen FP2 function is flexible enough to detect such differences. Within the same algorithm we can also deal with the conceptually different cases of a predefined hypothesis of interaction or searching for interactions. We demonstrate the ability of the approach to detect and display treatment/covariate interactions in two examples from controlled trials in cancer.     

倾向指数匹配法的模拟研究

目的 采用SAS程序模拟来研究倾向指数匹配法在处理非随机化试验数据中的效果.方法 利用蒙特-卡罗（Monte Carlo）模拟法产生有3个协变量（连续性变量X1,二分类变量X2,X3）的2组随机样本,以分组变量为因变量,以协变量为自变量建立logistic回归模型,并计算研究对象的倾向指数,然后按照倾向指数做组间无放回的卡钳匹配,得到一个各协变量均衡的处理组与对照组样本.用假设检验法和标准差异法分别评价匹配前后2组之间协变量的均衡性,并估计匹配前后2组间的处理效应.结果 假设检验法评价组间均衡性的结果为：匹配之前,协变量X1,X2,X3在2组间均有统计学差异,表明协变量X1,X2,X3在2组间不均衡;匹配之后,协变量X1,X2,X3在2组间均无统计学差异,表明协变量在2组间均衡.标准差异法评价组间均衡性的结果为：匹配之前,X1,X2,X3标准差异的均值分别为1 967.03%,117.29%,63.74%,均远远大于10%,表明协变量X1,X2,X3在处理组和对照组间都不均衡;匹配之后,X1,X2,X3标准差异的均值分别为19.46%,7.37%,6.85%,表明协变量X1,X2,X3在匹配后都基本变的均衡.可见使用基于倾向指数的卡钳匹配法对非随机化数据进行处理,协变量间不均衡的2个处理组在匹配以后达到了均衡.对处理效应的估计结果为：匹配之前,2组间的处理效应有统计学差异,但在匹配之后,2组间的处理效应变得没有统计学差异,表明匹配之前2组间的统计学差异是由协变量的不平衡引起的.结论 倾向指数法是一种有效的处理非随机化试验数据的方法,具有重要的应用价值.     

Evaluation of Cox's model and logistic regression for matched case-control data with time-dependent covariates: a simulation study

Case-control studies are typically analysed using the conventional logistic model, which does not directly account for changes in the covariate values over time. Yet, many exposures may vary over time. The most natural alternative to handle such exposures would be to use the Cox model with time-dependent covariates. However, its application to case-control data opens the question of how to manipulate the risk sets. Through a simulation study, we investigate how the accuracy of the estimates of Cox's model depends on the operational definition of risk sets and/or on some aspects of the time-varying exposure. We also assess the estimates obtained from conventional logistic regression. The lifetime experience of a hypothetical population is first generated, and a matched case-control study is then simulated from this population. We control the frequency, the age at initiation, and the total duration of exposure, as well as the strengths of their effects. All models considered include a fixed-in-time covariate and one or two time-dependent covariate(s): the indicator of current exposure and/or the exposure duration. Simulation results show that none of the models always performs well. The discrepancies between the odds ratios yielded by logistic regression and the 'true' hazard ratio depend on both the type of the covariate and the strength of its effect. In addition, it seems that logistic regression has difficulty separating the effects of inter-correlated time-dependent covariates. By contrast, each of the two versions of Cox's model systematically induces either a serious under-estimation or a moderate over-estimation bias. The magnitude of the latter bias is proportional to the true effect, suggesting that an improved manipulation of the risk sets may eliminate, or at least reduce, the bias.     

Two-stage instrumental variable methods for estimating the causal odds ratio: analysis of bias

We present closed-form expressions of asymptotic bias for the causal odds ratio from two estimation approaches of instrumental variable logistic regression: (i) the two-stage predictor substitution (2SPS) method and (ii) the two-stage residual inclusion (2SRI) approach. Under the 2SPS approach, the first stage model yields the predicted value of treatment as a function of an instrument and covariates, and in the second stage model for the outcome, this predicted value replaces the observed value of treatment as a covariate. Under the 2SRI approach, the first stage is the same, but the residual term of the first stage regression is included in the second stage regression, retaining the observed treatment as a covariate. Our bias assessment is for a different context from that of Terza (J. Health Econ. 2008; 27(3):531-543), who focused on the causal odds ratio conditional on the unmeasured confounder, whereas we focus on the causal odds ratio among compliers under the principal stratification framework. Our closed-form bias results show that the 2SPS logistic regression generates asymptotically biased estimates of this causal odds ratio when there is no unmeasured confounding and that this bias increases with increasing unmeasured confounding. The 2SRI logistic regression is asymptotically unbiased when there is no unmeasured confounding, but when there is unmeasured confounding, there is bias and it increases with increasing unmeasured confounding. The closed-form bias results provide guidance for using these IV logistic regression methods. Our simulation results are consistent with our closed-form analytic results under different combinations of parameter settings.     

Model specification and unmeasured confounders in partially ecologic analyses based on group proportions of exposed

OBJECTIVES: The aim of this study was to quantify bias from a partially ecologic analysis due to (i) model misspecification and (ii) an unmeasured confounder, considering various scenarios that may occur in occupational and environmental epidemiology. A study with an aggregate exposure variable, PE, but with outcome, group membership, and covariates assessed individually is partially ecologic. In this paper, PE is the proportion exposed; PE can vary across geographic areas or occupational groups. METHODS: Several hypothetical scenarios were considered, varying the baseline risk, the exposure effect, the exposure distribution across groups, the impact of the (unmeasured) confounder, and the confounder distribution across groups. First, confounding within groups was introduced. Thereafter, confounding between groups was introduced by co-varying PE and the confounder prevalence across the groups. The expected odds ratio (exposed versus unexposed) was calculated in two alternative models, the logistic regression and linear odds models, both with PE as the independent variable. Moreover, empirical data on noise exposure and sleeping disturbances were analyzed. RESULTS: Compared with the logistic regression model, the linear odds model yielded a markedly less biased odds ratio (OR) when the outcome was rare (< or = 5% baseline risk). Confounding within groups resulted in marginal bias, whereas confounding between groups resulted in more pronounced bias. CONCLUSIONS: A logistic regression analysis, with PE as an independent variable, can yield substantial model misspecification bias. By contrast, the linear odds model is valid when the outcome is rare. Confounding between groups should be of more concern than confounding within groups in partially ecologic analyses.     

休哈特和累积和联合控制图提高临床检验质量控制水平

目的 本文描述了改编的决定限累积和方法应用于临床检验质量控制。方法 累积和方法的累积和是根据数字限而不使用V型模板来进行解释。该方法在计算机化质量控制系统或手工控制图上易于完成。本研究主要是手工应用并证明如何在已有的Shewhart(休哈持)质控图上执行这一方法。结果 在单一质控制图上同时使用累积和及Shewhart质控规则，也可简化累积和方法的计算。计算机模拟研究几种单个累积和规则以及与Shewhart联合规则的性能特征。研究表明通过累积和方法和Shewhart控制图的联合应用能改善存在的质量控制系统。结论 在实验室手工质量控制系统上引入累积和方法具有其优点。     

Alternative parameterization of polychotomous models: theory and application to matched case-control studies

A method is proposed for transforming a class of models having an outcome variable with more than two levels into an equivalent binary model. The polychotomous logistic model is used to demonstrate the method. The equivalency to a simple logistic regression model after some data transformation (augmentation) is shown. The method is applied to the data from two case-control studies each with two control groups, and further applications are indicated.     

Bayesian adjustment for covariate measurement errors: A flexible parametric approach

In most epidemiological investigations, the study units are people, the outcome variable (or the response) is a health‐related event, and the explanatory variables are usually environmental and/or socio‐demographic factors. The fundamental task in such investigations is to quantify the association between the explanatory variables (covariates/exposures) and the outcome variable through a suitable regression model. The accuracy of such quantification depends on how precisely the relevant covariates are measured. In many instances, we cannot measure some of the covariates accurately. Rather, we can measure noisy (mismeasured) versions of them. In statistical terminology, mismeasurement in continuous covariates is known as measurement errors or errors‐in‐variables. Regression analyses based on mismeasured covariates lead to biased inference about the true underlying response–covariate associations. In this paper, we suggest a flexible parametric approach for avoiding this bias when estimating the response–covariate relationship through a logistic regression model. More specifically, we consider the flexible generalized skew‐normal and the flexible generalized skew‐t distributions for modeling the unobserved true exposure. For inference and computational purposes, we use Bayesian Markov chain Monte Carlo techniques. We investigate the performance of the proposed flexible parametric approach in comparison with a common flexible parametric approach through extensive simulation studies. We also compare the proposed method with the competing flexible parametric method on a real‐life data set. Though emphasis is put on the logistic regression model, the proposed method is unified and is applicable to the other generalized linear models, and to other types of non‐linear regression models as well. Copyright © 2009 John Wiley & Sons, Ltd.     

Bayesian analysis of logistic regression with an unknown change point and covariate measurement error

We discuss Bayesian estimation of a logistic regression model with an unknown threshold limiting value (TLV). In these models it is assumed that there is no effect of a covariate on the response under a certain unknown TLV. The estimation of these models in a Bayesian context by Markov chain Monte Carlo (MCMC) methods is considered with focus on the TLV. We extend the model by accounting for measurement error in the covariate. The Bayesian solution is compared with the likelihood solution proposed by Küchenhoff and Carroll using a data set concerning the relationship between dust concentration in the working place and the occurrence of chronic bronchitis.     

Understanding the effects of conditional dependence in research studies involving imperfect diagnostic tests

When two imperfect diagnostic tests are carried out on the same subject, their results may be correlated even after conditioning on the true disease status. While past work has focused on the consequences of ignoring conditional dependence, the degree to which conditional dependence can be induced has not been systematically studied. We examine this issue in detail by introducing a hypothetical missing covariate that affects the sensitivities of two imperfect dichotomous tests. We consider four forms for this covariate, normal, uniform, dichotomous and trichotomous. In the case of a dichotomous covariate, we derive an expression showing that the conditional covariance is a function of the product of the changes in test sensitivities (or specificities) between the subgroups defined by the covariate. The maximum possible covariance is induced by a dichotomous covariate with a very strong effect on both tests. Through simulations, we evaluate the extent to which fitting a latent class model ignoring each type of covariate but including a general covariance term can adjust for the correlation induced by the covariate. We compare the results to when the conditional dependence is ignored. We find that the bias because of ignoring conditional dependence is generally small even for moderate covariate effects, and when bias is present, a model including a covariance term works well. We illustrate our methods by analyzing data from a childhood tuberculosis study. Copyright © 2016 John Wiley & Sons, Ltd.     

Robust extraction of covariate information to improve estimation efficiency in randomized trials

In randomized trials, investigators typically rely upon an unadjusted estimate of the mean outcome within each treatment arm to draw causal inferences. Statisticians have underscored the gain in efficiency that can be achieved from covariate adjustment in randomized trials with a focus on problems involving linear models. Despite recent theoretical advances, there has been a reluctance to adjust for covariates based on two primary reasons: (i) covariate-adjusted estimates based on conditional logistic regression models have been shown to be less precise and (ii) concern over the opportunity to manipulate the model selection process for covariate adjustments to obtain favorable results. In this paper, we address these two issues and summarize recent theoretical results on which is based a proposed general methodology for covariate adjustment under the framework of targeted maximum likelihood estimation in trials with two arms where the probability of treatment is 50%. The proposed methodology provides an estimate of the true causal parameter of interest representing the population-level treatment effect. It is compared with the estimates based on conditional logistic modeling, which only provide estimates of subgroup-level treatment effects rather than marginal (unconditional) treatment effects. We provide a clear criterion for determining whether a gain in efficiency can be achieved with covariate adjustment over the unadjusted method. We illustrate our strategy using a resampled clinical trial dataset from a placebo controlled phase 4 study. Results demonstrate that gains in efficiency can be achieved even with binary outcomes through covariate adjustment leading to increased statistical power.     

The Association Between Short Interpregnancy Interval and Preterm Birth in Louisiana: A Comparison of Methods

There is growing interest in the application of propensity scores (PS) in epidemiologic studies, especially within the field of reproductive epidemiology. This retrospective cohort study assesses the impact of a short interpregnancy interval (IPI) on preterm birth and compares the results of the conventional logistic regression analysis with analyses utilizing a PS. The study included 96,378 singleton infants from Louisiana birth certificate data (1995–2007). Five regression models designed for methods comparison are presented. Ten percent (10.17 %) of all births were preterm; 26.83 % of births were from a short IPI. The PS-adjusted model produced a more conservative estimate of the exposure variable compared to the conventional logistic regression method (β-coefficient: 0.21 vs. 0.43), as well as a smaller standard error (0.024 vs. 0.028), odds ratio and 95 % confidence intervals [1.15 (1.09, 1.20) vs. 1.23 (1.17, 1.30)]. The inclusion of more covariate and interaction terms in the PS did not change the estimates of the exposure variable. This analysis indicates that PS-adjusted regression may be appropriate for validation of conventional methods in a large dataset with a fairly common outcome. PS’s may be beneficial in producing more precise estimates, especially for models with many confounders and effect modifiers and where conventional adjustment with logistic regression is unsatisfactory. Short intervals between pregnancies are associated with preterm birth in this population, according to either technique. Birth spacing is an issue that women have some control over. Educational interventions, including birth control, should be applied during prenatal visits and following delivery.     

Regressive threshold model for familial analysis of complex diseases with variable age of onset

Few models for segregation (or combined segregation-linkage) analysis have been developed to account for variable age of onset. The unified model (UM) can only take into account age at examination. In the logistic hazard model (LHM), Abel and Bonney ([1990] Genet. Epidemiol. 7:391-407) incorporated survival analysis concepts into the regressive logistic model of Bonney ([1986] Am. J. Med. Genet. 18:731-749), but interpretation of familial dependence parameters is difficult. In this article, we extended the regressive threshold model (RTM) proposed by Demenais ([1991] Am. J. Hum. Genet. 49:773-785) to account for a variable age of onset of complex diseases. This model assumes an underlying liability to disease and is more general than the original logistic formulation, since the phenotypes of each individual's antecedents can be adjusted for their own genotypes and covariate effects. The variation of risk with age can be expressed as a general step function, and variants of the model have been proposed by imposing different types of constraints among the time-dependent thresholds. The performances of the three models (UM, LHM, and RTM) were compared in the context of segregation analysis of family data generated with variable age of onset. All analysis models were robust with respect to false conclusion of a major gene, and the best results were obtained under RTM. The power to detect the major gene was higher under LHM than RTM, but the best fit of the estimated cumulative age-dependent penetrance with respect to the true value was obtained under RTM. This new model may thus prove helpful in contributing to identification of genes underlying complex diseases, since it can easily include linked marker loci and linkage disequilibrium.