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.     

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.     

Predictive accuracy of risk factors and markers: a simulation study of the effect of novel markers on different performance measures for logistic regression models

The change in c‐statistic is frequently used to summarize the change in predictive accuracy when a novel risk factor is added to an existing logistic regression model. We explored the relationship between the absolute change in the c‐statistic, Brier score, generalized R², and the discrimination slope when a risk factor was added to an existing model in an extensive set of Monte Carlo simulations. The increase in model accuracy due to the inclusion of a novel marker was proportional to both the prevalence of the marker and to the odds ratio relating the marker to the outcome but inversely proportional to the accuracy of the logistic regression model with the marker omitted. We observed greater improvements in model accuracy when the novel risk factor or marker was uncorrelated with the existing predictor variable compared with when the risk factor has a positive correlation with the existing predictor variable. We illustrated these findings by using a study on mortality prediction in patients hospitalized with heart failure. In conclusion, the increase in predictive accuracy by adding a marker should be considered in the context of the accuracy of the initial model. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

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)     

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.     

Multivariate analysis for matched case-control studies

A multivariate method based on the linear logistic model is presented for the analysis of case-control studies with pairwise matching. This technique enables one to investigate the effect of several variables simultaneously in the analysis while allowing for the matched design. The odds ratio is used as the basic measure of risk. One is able to control for variables which are not matching variables while investigating the odds ratio for a particular factor, and to estimate the change in the odds ratio as the level of one or more interval variables changes. The computing methods used for obtaining maximum conditional likelihood estimates of the parameters of interest are modifications of standard programs for logit regression.     

Loss of a child and the risk of amyotrophic lateral sclerosis

Between 1987 and 2005, the authors conducted a case-control study nested within the entire Swedish population to investigate whether loss of a child due to death is associated with the risk of amyotrophic lateral sclerosis (ALS). The study comprised 2,694 incident ALS cases and five controls per case individually matched by year of birth, gender, and parity. Odds ratios and their corresponding 95% confidence intervals for ALS were estimated by using conditional logistic regression models. Compared with that for parents who never lost a child, the overall odds ratio of ALS for bereaved parents was 0.7 (95% confidence interval (CI): 0.6, 0.8) and decreased to 0.4 (95% CI: 0.2, 0.8) 11-15 years after the loss. The risk reduction was also modified by parental age at the time of loss, with the lowest odds ratio of 0.4 (95% CI: 0.2, 0.9) for parents older than age 75 years. Loss of a child due to malignancy appeared to confer a lower risk of ALS (odds ratio = 0.5, 95% CI: 0.3, 0.8) than loss due to other causes. These data indicate that the risk of developing ALS decreases following the severe stress of parental bereavement. Further studies are needed to explore potential underlying mechanisms.     

Prevalence of nickel sensitization and urinary nickel content of children are increased by nickel in ambient air

In a cross-sectional study performed in 2000, an unexpected positive association between nickel (Ni) in ambient air, urinary Ni content and the prevalence of Ni sensitization in a subgroup of 6-yr-old children living near a steel mill was observed. Between 2005 and 2006, in a different and larger study population, we examined if Ni from ambient air or urinary Ni concentration was related to Ni sensitization in children living next to Ni-emitting steel mills.We studied 749 school beginners living in four Ni-polluted industrial areas of North Rhine-Westphalia, Germany. We assessed Ni in ambient air, Ni in urine from children and mothers, and Ni in tap water, conducted patch tests in children (including the NiSO₄-dilution test) and collected questionnaire data. Statistics were done by linear and logistic regression analyses, adjusted for covariates.At increased Ni concentration in ambient air (unit of increase: 10 ng/m³), urinary Ni concentrations rose in both mothers (9.1%; 95% CI: 6.8–11.4%) and children (2.4%; 95% CI: 0.4–4.4%). The prevalence of Ni sensitization in children was associated with increased Ni from ambient air (unit of increase: 18 ng/m³; odds ratio 1.28; 95% CI: 1.25–1.32) and urinary Ni concentration (unit of increase: 7.1 μg/L; odds ratio 2.4; 95% CI: 1.20–4.48). Ni in ambient air of areas with Ni-emitting factories contributes to internal Ni exposure in residents via inhalation and, furthermore, is a risk factor for the development of Ni sensitization in children.     

Appropriate assessment of neighborhood effects on individual health: integrating random and fixed effects in multilevel logistic regression

The logistic regression model is frequently used in epidemiologic studies, yielding odds ratio or relative risk interpretations. Inspired by the theory of linear normal models, the logistic regression model has been extended to allow for correlated responses by introducing random effects. However, the model does not inherit the interpretational features of the normal model. In this paper, the authors argue that the existing measures are unsatisfactory (and some of them are even improper) when quantifying results from multilevel logistic regression analyses. The authors suggest a measure of heterogeneity, the median odds ratio, that quantifies cluster heterogeneity and facilitates a direct comparison between covariate effects and the magnitude of heterogeneity in terms of well-known odds ratios. Quantifying cluster-level covariates in a meaningful way is a challenge in multilevel logistic regression. For this purpose, the authors propose an odds ratio measure, the interval odds ratio, that takes these difficulties into account. The authors demonstrate the two measures by investigating heterogeneity between neighborhoods and effects of neighborhood-level covariates in two examples--public physician visits and ischemic heart disease hospitalizations--using 1999 data on 11,312 men aged 45-85 years in Malmo, Sweden.     

Fitting stratified proportional odds models by amalgamating conditional likelihoods

Classical methods for fitting a varying intercept logistic regression model to stratified data are based on the conditional likelihood principle to eliminate the stratum-specific nuisance parameters. When the outcome variable has multiple ordered categories, a natural choice for the outcome model is a stratified proportional odds or cumulative logit model. However, classical conditioning techniques do not apply to the general K-category cumulative logit model (K>2) with varying stratum-specific intercepts as there is no reduction due to sufficiency; the nuisance parameters remain in the conditional likelihood. We propose a methodology to fit stratified proportional odds model by amalgamating conditional likelihoods obtained from all possible binary collapsings of the ordinal scale. The method allows for categorical and continuous covariates in a general regression framework. We provide a robust sandwich estimate of the variance of the proposed estimator. For binary exposures, we show equivalence of our approach to the estimators already proposed in the literature. The proposed recipe can be implemented very easily in standard software. We illustrate the methods via three real data examples related to biomedical research. Simulation results comparing the proposed method with a random effects model on the stratification parameters are also furnished.     

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.     

Weighted estimation for confounded binary outcomes subject to misclassification

In the presence of confounding, the consistency assumption required for identification of causal effects may be violated due to misclassification of the outcome variable. We introduce an inverse probability weighted approach to rebalance covariates across treatment groups while mitigating the influence of differential misclassification bias. First, using a simplified example taken from an administrative health care dataset, we introduce the approach for estimation of the marginal causal odds ratio in a simple setting with the use of internal validation information. We then extend this to the presence of additional covariates and use simulated data to investigate the finite sample properties of the proposed weighted estimators. Estimation of the weights is done using logistic regression with misclassified outcomes, and a bootstrap approach is used for variance estimation.     

Body mass index and risk of infections: a Mendelian randomization study of 101,447 individuals

Body mass index (BMI) has been related to risk of infections. The aim of this study was to assess the shape of the association between BMI and risk of infections and to evaluate whether such associations represent causality. We included 101,447 individuals from The Copenhagen General Population Study who had BMI measured. Outcome was hospital contacts related to infections. The shape of the association between BMI and risk of infections was examined using restricted cubic spline Cox regression. To evaluate causality, we used Mendelian randomization, an epidemiological method that counteracts confounding and reverse causality by using genetic variation as instrumental variables. We created a genetic risk score based on five genetic variants causing lifelong higher BMI and used this score in instrumental variable analysis. During median follow-up of 8.8 years, 10,263 hospital contacts related to infections were recorded. We found a U-shaped association between BMI and risk of any infection and pneumonia, and a linear association between BMI and risk of skin infection, urinary tract infection, and sepsis. In instrumental variable analyses, higher BMI was associated with increased risk of skin infection: odds ratio 1.12 (95% CI 1.03–1.22) for a genetically induced 1 unit increase in BMI. Observationally, low as well as high BMI was associated with increased risk of any infection and pneumonia, whereas only high BMI was associated with increased risk of skin infection, urinary tract infection, and sepsis. High BMI was causally associated with increased risk of skin infection.     

Comparing Standard Regression, Propensity Score Matching, and Instrumental Variables Methods for Determining the Influence of Mammography on Stage of Diagnosis

In situations where randomized trials are not feasible, analysis of observational data must be used instead. However, when using observational data, there is often selection bias for which we must account in order to adjust for pre-treatment differences between groups in their baseline characteristics. As an example of this, we used the Linked Medicare-Tumor Registry Database created by the National Cancer Institute and the Centers for Medicare and Medicaid Services to look at screening with mammography in older women to determine its effectiveness in detecting cancer at an earlier stage. The standard regression method and two methods of adjusting for selection bias are compared. We start with the standard analysis, a logistic regression predicting stage at diagnosis that includes as independent variables a set of covariates to adjust for differences in baseline risk plus an indicator variable for whether the woman used screening. Next, we employ propensity score matching, which evens out the distribution of measured baseline characteristics across groups, and is more robust to model mis-specification than the standard analysis. Lastly, we conduct an instrumental variable analysis, which addresses unmeasured differences between the users and non-users. This article compares these methods and discusses issues of which researchers and analysts should be aware. It is important to look beyond the standard analysis and to consider propensity score matching when there is concern about group differences in measured covariates and instrumental variable analysis when there is concern about differences in unmeasured covariates.     

Binomial regression in GLIM: estimating risk ratios and risk differences

Although an estimate of the odds ratio adjusted for other covariates can be obtained by logistic regression, until now there has been no simple way to estimate other interesting parameters such as the risk ratio and risk difference multivariately for prospective binomial data. These parameters can be estimated in the generalized linear model framework by choosing different link functions or transformations of binomial or binary data. Macros for use with the program GLIM provide a simple method to compute parameters other than the odds ratio while adjusting for confounding factors. A data set presented previously is used as an example.     

Lymphohematopoietic cancer in styrene-butadiene polymerization workers.

1,3-Butadiene and styrene are suspected carcinogens and common chemicals used in the synthesis of rubber. To investigate any potential human hazards from exposure to these chemicals, a case-control study of 59 lymphohematopoietic cancers was conducted within a cohort of male workers employed between 1943 and 1982 in eight North American styrene-butadiene rubber polymer-producing plants. A total of 193 controls were matched to the cases by plant, age, year of hire, duration worked, and survival to time of death of the case. Each job was assigned an estimated exposure rank, and each worker's cumulated rank score was calculated on the basis of the time spent in each job throughout his employment. "Exposure" as a dichotomous variable was defined as a log rank score above the mean of the log scores for the total population of cases and controls within a subtype of cancer. Matched-pair analysis identified a strong association between leukemia and butadiene, with an odds ratio of 9.36 (95% confidence interval 2.05-22.9) and an association between styrene and leukemia (odds ratio = 3.13, 95% confidence interval 0.84-11.2) that did not achieve statistical significance. When exposure to both styrene and butadiene was included in a conditional logistic regression model, the odds ratio for butadiene remained high (odds ratio = 7.39), but the estimated association of leukemia with styrene was small. The results of this study support the hypothesis that exposure to butadiene is associated with the risk of leukemia. There also appears to be an additional risk from work in specific subdivisions of the industry.     

Reducing attrition bias with an instrumental variable in a regression model: Results from a panel of rheumatoid arthritis patients

This study proposes an econometric technique to reduce attrition bias in panel data. In the simplest case, one estimates two regressions. The first is a probit regression based on sociodemographic and clinical characteristics measured at baseline. The probit regression estimates the probability that subjects stay or leave over the duration of the study. We insert the predicted probabilities from the probit regression into an inverse Mills ratio (IMR) or hazard rate to form an instrumental variable. We use this instrumental variable subsequently as an additional covariate in a second regression model that attempts to explain fluctuations in the dependent variable. The second regression, which is linear, includes only subjects who remained in the study. In alternative models, instrumental variables are created using predicted values from least squares and logit regressions estimating the probability that subjects stay or leave. The use of the instrumental variables reduces the effects of attrition bias in the linear regression model We applied the technique to a panel of patients with rheumatoid arthritis (RA) enrolled in 1981 and followed through 1990. We attempted to predict values for a measure of functional disability recorded in 1990 with use of covariates measured in 1981. The dependent variable was an index of disability in 1990 and the independent variables (covariates) included the disability index from 1981, the years of duration of RA, gender, martial status, education, and age in 1981. The correction technique suggested that ignoring attrition bias would underestimate the strength of associations between being female and the subsequent disability index, and overestimate the strength of associations between being married spouse present, age, and the initial disability index on the one hand and the subsequent disability index on the other.     

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.     

Sample size evaluation for a multiply matched case-control study using the score test from a conditional logistic (discrete Cox PH) regression model

The conditional logistic regression model (Biometrics 1982; 38:661-672) provides a convenient method for the assessment of qualitative or quantitative covariate effects on risk in a study with matched sets, each containing a possibly different number of cases and controls. The conditional logistic likelihood is identical to the stratified Cox proportional hazards model likelihood, with an adjustment for ties (J. R. Stat. Soc. B 1972; 34:187-220). This likelihood also applies to a nested case-control study with multiply matched cases and controls, selected from those at risk at selected event times. Herein the distribution of the score test for the effect of a covariate in the model is used to derive simple equations to describe the power of the test to detect a coefficient theta (log odds ratio or log hazard ratio) or the number of cases (or matched sets) and controls required to provide a desired level of power. Additional expressions are derived for a quantitative covariate as a function of the difference in the assumed mean covariate values among cases and controls and for a qualitative covariate in terms of the difference in the probabilities of exposure for cases and controls. Examples are presented for a nested case-control study and a multiply matched case-control study.