Intermediate and advanced topics in multilevel logistic regression analysis

Multilevel data occur frequently in health services, population and public health, and epidemiologic research. In such research, binary outcomes are common. Multilevel logistic regression models allow one to account for the clustering of subjects within clusters of higher‐level units when estimating the effect of subject and cluster characteristics on subject outcomes. A search of the PubMed database demonstrated that the use of multilevel or hierarchical regression models is increasing rapidly. However, our impression is that many analysts simply use multilevel regression models to account for the nuisance of within‐cluster homogeneity that is induced by clustering. In this article, we describe a suite of analyses that can complement the fitting of multilevel logistic regression models. These ancillary analyses permit analysts to estimate the marginal or population‐average effect of covariates measured at the subject and cluster level, in contrast to the within‐cluster or cluster‐specific effects arising from the original multilevel logistic regression model. We describe the interval odds ratio and the proportion of opposed odds ratios, which are summary measures of effect for cluster‐level covariates. We describe the variance partition coefficient and the median odds ratio which are measures of components of variance and heterogeneity in outcomes. These measures allow one to quantify the magnitude of the general contextual effect. We describe an R² measure that allows analysts to quantify the proportion of variation explained by different multilevel logistic regression models. We illustrate the application and interpretation of these measures by analyzing mortality in patients hospitalized with a diagnosis of acute myocardial infarction. © 2017 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.     

A weighting approach to causal effects and additive interaction in case-control studies: marginal structural linear odds models

Estimates of additive interaction from case-control data are often obtained by logistic regression; such models can also be used to adjust for covariates. This approach to estimating additive interaction has come under some criticism because of possible misspecification of the logistic model: If the underlying model is linear, the logistic model will be misspecified. The authors propose an inverse probability of treatment weighting approach to causal effects and additive interaction in case-control studies. Under the assumption of no unmeasured confounding, the approach amounts to fitting a marginal structural linear odds model. The approach allows for the estimation of measures of additive interaction between dichotomous exposures, such as the relative excess risk due to interaction, using case-control data without having to rely on modeling assumptions for the outcome conditional on the exposures and covariates. Rather than using conditional models for the outcome, models are instead specified for the exposures conditional on the covariates. The approach is illustrated by assessing additive interaction between genetic and environmental factors using data from a case-control study.     

The median hazard ratio: a useful measure of variance and general contextual effects in multilevel survival analysis

Multilevel data occurs frequently in many research areas like health services research and epidemiology. A suitable way to analyze such data is through the use of multilevel regression models (MLRM). MLRM incorporate cluster‐specific random effects which allow one to partition the total individual variance into between‐cluster variation and between‐individual variation. Statistically, MLRM account for the dependency of the data within clusters and provide correct estimates of uncertainty around regression coefficients. Substantively, the magnitude of the effect of clustering provides a measure of the General Contextual Effect (GCE). When outcomes are binary, the GCE can also be quantified by measures of heterogeneity like the Median Odds Ratio (MOR) calculated from a multilevel logistic regression model. Time‐to‐event outcomes within a multilevel structure occur commonly in epidemiological and medical research. However, the Median Hazard Ratio (MHR) that corresponds to the MOR in multilevel (i.e., ‘frailty’) Cox proportional hazards regression is rarely used. Analogously to the MOR, the MHR is the median relative change in the hazard of the occurrence of the outcome when comparing identical subjects from two randomly selected different clusters that are ordered by risk. We illustrate the application and interpretation of the MHR in a case study analyzing the hazard of mortality in patients hospitalized for acute myocardial infarction at hospitals in Ontario, Canada. We provide R code for computing the MHR. The MHR is a useful and intuitive measure for expressing cluster heterogeneity in the outcome and, thereby, estimating general contextual effects in multilevel survival analysis. © 2016 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.     

Doubly robust conditional logistic regression

Epidemiologic research often aims to estimate the association between a binary exposure and a binary outcome, while adjusting for a set of covariates (eg, confounders). When data are clustered, as in, for instance, matched case-control studies and co-twin-control studies, it is common to use conditional logistic regression. In this model, all cluster-constant covariates are absorbed into a cluster-specific intercept, whereas cluster-varying covariates are adjusted for by explicitly adding these as explanatory variables to the model. In this paper, we propose a doubly robust estimator of the exposure-outcome odds ratio in conditional logistic regression models. This estimator protects against bias in the odds ratio estimator due to misspecification of the part of the model that contains the cluster-varying covariates. The doubly robust estimator uses two conditional logistic regression models for the odds ratio, one prospective and one retrospective, and is consistent for the exposure-outcome odds ratio if at least one of these models is correctly specified, not necessarily both. We demonstrate the properties of the proposed method by simulations and by re-analyzing a publicly available dataset from a matched case-control study on induced abortion and infertility.     

Measures of clustering and heterogeneity in multilevel Poisson regression analyses of rates/count data

Multilevel data occur frequently in many research areas like health services research and epidemiology. A suitable way to analyze such data is through the use of multilevel regression models. These models incorporate cluster‐specific random effects that allow one to partition the total variation in the outcome into between‐cluster variation and between‐individual variation. The magnitude of the effect of clustering provides a measure of the general contextual effect. When outcomes are binary or time‐to‐event in nature, the general contextual effect can be quantified by measures of heterogeneity like the median odds ratio or the median hazard ratio, respectively, which can be calculated from a multilevel regression model. Outcomes that are integer counts denoting the number of times that an event occurred are common in epidemiological and medical research. The median (incidence) rate ratio in multilevel Poisson regression for counts that corresponds to the median odds ratio or median hazard ratio for binary or time‐to‐event outcomes respectively is relatively unknown and is rarely used. The median rate ratio is the median relative change in the rate of the occurrence of the event when comparing identical subjects from 2 randomly selected different clusters that are ordered by rate. We also describe how the variance partition coefficient, which denotes the proportion of the variation in the outcome that is attributable to between‐cluster differences, can be computed with count outcomes. We illustrate the application and interpretation of these measures in a case study analyzing the rate of hospital readmission in patients discharged from hospital with a diagnosis of heart failure.     

Interaction as departure from additivity in case-control studies: a cautionary note

It has been argued that assessment of interaction should be based on departures from additive rates or risks. The corresponding fundamental interaction parameter cannot generally be estimated from case-control studies. Thus, surrogate measures of interaction based on relative risks from logistic models have been proposed, such as the relative excess risk due to interaction (RERI), the attributable proportion due to interaction (AP), and the synergy index (S). In practice, it is usually necessary to include covariates such as age and gender to control for confounding. The author uncovers two problems associated with surrogate interaction measures in this case: First, RERI and AP vary across strata defined by the covariates, whereas the fundamental interaction parameter is unvarying. S does not vary across strata, which suggests that it is the measure of choice. Second, a misspecification problem implies that measures based on logistic regression only approximate the true measures. This problem can be rectified by using a linear odds model, which also enables investigators to test whether the fundamental interaction parameter is zero. A simulation study reveals that coverage is much improved by using the linear odds model, but bias may be a concern regardless of whether logistic regression or the linear odds model is used.     

log-binomial模型估计的患病比及其应用

[导读]探讨有和无连续协变量时的log-binomial模型估计患病比的统计方法及其应用.文中使用禁烟立法态度与吸烟关联的研究实例,比较log-binomial模型估计的现患比(PR)与logistic回归模型估计的优势比(OR).当模型中无连续协变量时,采用最大似然估计拟合log-binomial模型;当因含有连续协变量导致模型不收敛时,则采用COPY方法估计PR.分别估计男、女禁烟立法态度与吸烟的关联.由于女性吸烟率低,用PR与OR所估计的关联结果相似.而男性吸烟率较高,OR值明显大于PR.当年龄作为连续协变量纳入模型时,导致log-binomial模型不收敛,采用COPY方法解决此问题.所有分析均在SAS软件中实现.结论:当患病率较高时,PR比OR更好地测量了暴露与疾病的关联.文中给出log-binomial回归模型和COPY方法估计PR的SAS程序.     

An insight on the use of multiple logistic regression analysis to estimate association between risk factor and disease occurrence

Multiple logistic regression is an accepted statistical method for assessing association between an anticedant characteristic (risk factor) and a quantal outcome (probability of disease occurrence), statistically adjusting for potential confounding effects of other covariates. Yet the method has potential drawbacks which are not generally recognized. This article considers one important drawback of logistic regression. Specifically the so-called main effect logistic model assumes that the probability of developing disease is linearly and additively related to the risk factors on the logistic scale. This assumption stipulates that for each risk factor, the odds ratio is constant over all reference exposure levels, and that the odds ratio exposed to two or more factors is equal to the product of individual risk factor odds ratios. If the observed odds ratios in the data follow this pattern, the model-predicted odds ratios will be accurate, and the meaning of the odds ratio for each risk factor will be straightforward. But if the observed odds ratios deviate from the model assumption, the model will not fit the data accurately, and the model-predicted odds ratios will not reflect those in the data. Although satisfactory fit can always be achieved by adding to the model polynomial and product terms derived from the original risk factors, the odds ratios estimated by such an interaction logistic model are difficult to interpret, viz., the odds ratio for each risk factor depends not only on the reference exposure levels of that factor, but also on the exposure level in other factors.(ABSTRACT TRUNCATED AT 250 WORDS)     

Absolute risk reductions,relative risks,relative risk reductions,and numbers needed to treat can be obtained from a logistic regression model

ObjectiveLogistic regression models are frequently used in cohort studies to determine the association between treatment and dichotomous outcomes in the presence of confounding variables. In a logistic regression model, the association between exposure and outcome is measured using the odds ratio (OR). The OR can be difficult to interpret and only approximates the relative risk (RR) in certain restrictive settings. Several authors have suggested that for dichotomous outcomes, RRs, RR reductions, absolute risk reductions, and the number needed to treat (NNT) are more clinically meaningful measures of treatment effect.Study Design and SettingWe describe a method for deriving clinically meaningful measures of treatment effect from a logistic regression model. This method involves determining the probability of the outcome if each subject in the cohort was treated and if each subject was untreated. These probabilities are then averaged across the study cohort to determine the average probability of the outcome in the population if all subjects were treated and if they were untreated.ResultsRisk differences, RRs, and NNTs were derived using a logistic regression model.ConclusionsClinically meaningful measures of effect can be derived from a logistic regression model in a cohort study. These methods can also be used in randomized controlled trials when logistic regression is used to adjust for possible imbalance in prognostically important baseline covariates.     

A brief conceptual tutorial of multilevel analysis in social epidemiology: using measures of clustering in multilevel logistic regression to investigate contextual phenomena

STUDY OBJECTIVE: In social epidemiology, it is easy to compute and interpret measures of variation in multilevel linear regression, but technical difficulties exist in the case of logistic regression. The aim of this study was to present measures of variation appropriate for the logistic case in a didactic rather than a mathematical way. Design and PARTICIPANTS: Data were used from the health survey conducted in 2000 in the county of Scania, Sweden, that comprised 10 723 persons aged 18-80 years living in 60 areas. Conducting multilevel logistic regression different techniques were applied to investigate whether the individual propensity to consult private physicians was statistically dependent on the area of residence (that is, intraclass correlation (ICC), median odds ratio (MOR)), the 80% interval odds ratio (IOR-80), and the sorting out index). RESULTS: The MOR provided more interpretable information than the ICC on the relevance of the residential area for understanding the individual propensity of consulting private physicians. The MOR showed that the unexplained heterogeneity between areas was of greater relevance than the individual variables considered in the analysis (age, sex, and education) for understanding the individual propensity of visiting private physicians. Residing in a high education area increased the probability of visiting a private physician. However, the IOR showed that the unexplained variability between areas did not allow to clearly distinguishing low from high propensity areas with the area educational level. The sorting out index was equal to 82%. CONCLUSION: Measures of variation in logistic regression should be promoted in social epidemiological and public health research as efficient means of quantifying the importance of the context of residence for understanding disparities in health and health related behaviour.     

Measuring variability between clusters by subgroup: An extension of the median odds ratio

Investigating clustered data requires consideration of the variation across clusters, including consideration of the component of the total individual variance that is at the cluster level. The median odds ratio and analogues are useful intuitive measures available to communicate variability in outcomes across clusters using the variance of random intercepts from a multilevel regression model. However, the median odds ratio cannot describe variability across clusters for different patient subgroups because the random intercepts do not vary by subgroup. To empower investigators interested in equity and other applications of this scenario, we describe an extension of the median odds ratio to multilevel regression models employing both random intercepts and random coefficients. By example, we conducted a retrospective cohort analysis of variation in care limitations (goals of care preferences) according to ethnicity in patients admitted to intensive care. Using mixed-effects logistic regression clustered by hospital, we demonstrated that patients of non-Caucasian ethnicity were less likely to have care limitations but experienced similar variability across hospitals. Limitations of the extended median odds ratio include the large sample sizes and computational power needed for models with random coefficients. This extension of the median odds ratio to multilevel regression models with random coefficients will provide insight into cluster-level variability for researchers interested in equity and other phenomena where variability by patient subgroup is important.     

A new logistic regression approach for the evaluation of diagnostic test results.

The value of a dichotomous diagnostic test is often described in terms of sensitivity, specificity, and likelihood ratios (LRs). Although it is known that these test characteristics vary between subgroups of patients, they are generally interpreted, on average, without considering information on patient characteristics, such as clinical signs and symptoms, or on previous test results. This article presents a reformulation of the logistic regression model that allows to calculate the LRs of diagnostic test results conditional on these covariates. The proposed method starts with estimating logistic regression models for the prior and posterior odds of disease. The regression model for the prior odds is based on patient characteristics, whereas the regression model for the posterior odds also includes the diagnostic test of interest. Following the Bayes theorem, the authors demontsrate that the regression model for the LR can be derived from taking the differences between the regression coefficients of the 2 models. In a clinical example, they demonstrate that the LRs of positive and negative test results and the sensitivity and specificity of the diagnostic test varied considerably between patients with different risk profiles, even when a constant odds ratio was assumed. The proposed logistic regression approach proves an efficient method to determine the performance of tests at the level of the individual patient risk profile and to examine the effect of patient characteristics on diagnostic test characteristics.     

Semiparametric regression models for detecting effect modification in matched case-crossover studies

In matched case-crossover studies, it is generally accepted that covariates on which a case and associated controls are matched cannot exert a confounding effect on independent predictors included in the conditional logistic regression model because any stratum effect is removed by the conditioning on the fixed number of sets of a case and controls in the stratum. Hence, the conditional logistic regression model is not able to detect any effects associated with the matching covariates by stratum. In addition, the matching covariates may be effect modification and the methods for assessing and characterizing effect modification by matching covariates are quite limited. In this article, we propose a unified approach in its ability to detect both parametric and nonparametric relationships between the predictor and the relative risk of disease or binary outcome, as well as potential effect modifications by matching covariates. Two methods are developed using two semiparametric models: (1) the regression spline varying coefficients model and (2) the regression spline interaction model. Simulation results show that the two approaches are comparable. These methods can be used in any matched case-control study and extend to multilevel effect modification studies. We demonstrate the advantage of our approach using an epidemiological example of a 1-4 bi-directional case-crossover study of childhood aseptic meningitis associated with drinking water turbidity.     

Identifying the odds ratio estimated by a two‐stage instrumental variable analysis with a logistic regression model

An adjustment for an uncorrelated covariate in a logistic regression changes the true value of an odds ratio for a unit increase in a risk factor. Even when there is no variation due to covariates, the odds ratio for a unit increase in a risk factor also depends on the distribution of the risk factor. We can use an instrumental variable to consistently estimate a causal effect in the presence of arbitrary confounding. With a logistic outcome model, we show that the simple ratio or two‐stage instrumental variable estimate is consistent for the odds ratio of an increase in the population distribution of the risk factor equal to the change due to a unit increase in the instrument divided by the average change in the risk factor due to the increase in the instrument. This odds ratio is conditional within the strata of the instrumental variable, but marginal across all other covariates, and is averaged across the population distribution of the risk factor. Where the proportion of variance in the risk factor explained by the instrument is small, this is similar to the odds ratio from a RCT without adjustment for any covariates, where the intervention corresponds to the effect of a change in the population distribution of the risk factor. This implies that the ratio or two‐stage instrumental variable method is not biased, as has been suggested, but estimates a different quantity to the conditional odds ratio from an adjusted multiple regression, a quantity that has arguably more relevance to an epidemiologist or a policy maker, especially in the context of Mendelian randomization. Copyright © 2013 John Wiley & Sons, Ltd.     

Multilevel logistic regression modelling with correlated random effects: application to the Smoking Cessation for Youth study

A multilevel logistic regression model is presented for the analysis of clustered and repeated binary response data. At the subject level, serial dependence is expected between repeated measures recorded on the same individual. At the cluster level, correlations of observations within the same subgroup are present due to the inherent hierarchical setting. Two random components are therefore incorporated explicitly within the linear predictor to account for the simultaneous heterogeneity and autoregressive structure. Application to analyse a set of longitudinal data from an adolescent smoking cessation intervention that motivated this study is illustrated.     

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.     

EXPLAINED VARIATION FOR LOGISTIC REGRESSION

Different measures of the proportion of variation in a dependent variable explained by covariates are reported by different standard programs for logistic regression. We review twelve measures that have been suggested or might be useful to measure explained variation in logistic regression models. The definitions and properties of these measures are discussed and their performance is compared in an empirical study. Two of the measures (squared Pearson correlation between the binary outcome and the predictor, and the proportional reduction of squared Pearson residuals by the use of covariates) give almost identical results, agree very well with the multiple R² of the general linear model, have an intuitively clear interpretation and perform satisfactorily in our study. For all measures the explained variation for the given sample and also the one expected in future samples can be obtained easily. For small samples an adjustment analogous to R²_adj in the general linear model is suggested. We discuss some aspects of application and recommend the routine use of a suitable measure of explained variation for logistic models.     

Random effects meta‐analysis of event outcome in the framework of the generalized linear mixed model with applications in sparse data

We consider random effects meta‐analysis where the outcome variable is the occurrence of some event of interest. The data structures handled are where one has one or more groups in each study, and in each group either the number of subjects with and without the event, or the number of events and the total duration of follow‐up is available. Traditionally, the meta‐analysis follows the summary measures approach based on the estimates of the outcome measure(s) and the corresponding standard error(s). This approach assumes an approximate normal within‐study likelihood and treats the standard errors as known. This approach has several potential disadvantages, such as not accounting for the standard errors being estimated, not accounting for correlation between the estimate and the standard error, the use of an (arbitrary) continuity correction in case of zero events, and the normal approximation being bad in studies with few events. We show that these problems can be overcome in most cases occurring in practice by replacing the approximate normal within‐study likelihood by the appropriate exact likelihood. This leads to a generalized linear mixed model that can be fitted in standard statistical software. For instance, in the case of odds ratio meta‐analysis, one can use the non‐central hypergeometric distribution likelihood leading to mixed‐effects conditional logistic regression. For incidence rate ratio meta‐analysis, it leads to random effects logistic regression with an offset variable. We also present bivariate and multivariate extensions. We present a number of examples, especially with rare events, among which an example of network meta‐analysis. Copyright © 2010 John Wiley & Sons, Ltd.     

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.