Daniel Westreich

Invited Commentary: Positivity in Practice

Positivity, or the experimental treatment assignment assumption, requires that there be both exposed and unexposed participants at every combination of the values of the observed confounders in the population under study. Positivity is essential for inference but is often overlooked in practice by epidemiologists. This issue of the Journal includes 2 articles featuring discussions related to positivity. Here the authors define positivity, distinguish between deterministic and random positivity, and discuss the 2 relevant papers in this issue. In addition, the commentators illustrate positivity in simple 2 x 2 tables, as well as detail some ways in which epidemiologists may examine their data for nonpositivity and deal with violations of positivity in practice.

Berkson’s bias, selection bias, and missing data

Although Berksons bias is widely recognized in the epidemiologic literature, it remains underappreciated as a model of both selection bias and bias due to missing data. Simple causal diagrams and 2 × 2 tables illustrate how Berksons bias connects to collider bias and selection bias more generally, and show the strong analogies between Berksonian selection bias and bias due to missing data. In some situations, considerations of whether data are missing at random or missing not at random are less important than the causal structure of the missing data process. Although dealing with missing data always relies on strong assumptions about unobserved variables, the intuitions built with simple examples can provide a better understanding of approaches to missing data in real-world situations.

Splines for trend analysis and continuous confounder control

Spline regression often represents a less biased and more efficient alternative to standard linear, curvilinear, or categorical analyses of continuous exposures and confounders. Benefits of restricted cubic and quadratic splines have been described in the epidemiologic and biomedical literature.1-2 Analogous to the SAS (SAS Institute, Inc., Cary, North Carolina) code provided by Harrell3 for estimating restricted cubic splines, we present straightforward SAS code for estimating restricted quadratic splines. Using data from the HIV clinical cohort at the University of North Carolina Center for AIDS Research,4 we illustrate use of restricted quadratic splines in regression modeling for trend analysis and control of a continuous confounder. Details regarding the functional form of the restricted quadratic splines as well as SAS code for estimating restricted quadratic spline functions are provided in the eAppendix (http://links.lww.com). The data and SAS code used to generate the results included in this paper are also in the eAppendix. n nFirst, we illustrate the use of a restricted quadratic splines when estimating the association between log10 HIV-1 viral load centered at 2.301 log10 copies/ml and mortality. Figures S1 A-1C show the unadjusted association between centered log10 HIV-1 viral load at therapy initiation and the relative hazard of death estimated from several Cox proportional hazards models that (A) assume a log-linear relationship, (B) use indicators corresponding to quartiles of centered log10 HIV-1 viral load, or (C) include restricted quadratic splines with 4 equal knots based on the case distribution. n n n nFigure 1 n nUnadjusted associations between centered log10 HIV-1 viral load at therapy initiation and relative hazard of death among 557 male participants in the University of North Carolina Center for AIDS Research HIV clinical cohort, 1999-2010. HIV-1 viral load ... n n n nBased on the Akaike information criterion (AIC),5 presented in Figure 1, the restricted quadratic splines model provides the best fit to the data. The P- value for a joint Wald test of the three restricted quadratic splines basis functions included in the model was 0.010. The restricted quadratic splines model suggests a non-log-linear relationship between centered log10 HIV-1 viral load at therapy initiation and the relative hazard of death. n nSecond, we illustrate the use of restricted quadratic splines when controlling for centered log10 HIV-1 viral load as a confounder using a Cox model. The table shows the hazard ratios for the association between an indicator of CD4 cell count ≤350 cells/mm3 at therapy initiation and hazard of death, both unadjusted and adjusted for confounding by viral load at therapy initiation. Adjusting for viral load using a log-linear term attenuated the point estimate corresponding to the CD4 cell count indicator by 26%. Adjustment using restricted quadratic splines with 4 equal knots based on the case distribution attenuated the point estimate by 30%. Attenuation upon control for viral load is expected given that higher viral load was associated with lower CD4 cell count (http://links.lww.com), and an elevated risk of subsequent mortality. Similar results were observed when restricted cubic splines was used instead of a restricted quadratic splines with the same degrees of freedom and comparable knot locations (http://links.lww.com). n n n nTable n nHazard ratio for association between CD4 cell count less than or equal to 350 cells/mm3 versus greater than 350 cells/mm3 at therapy initiation and death among 557 male participants in the University of North Carolina Center for AIDS Research HIV clinical ... n n n nFor the first example, use of restricted quadratic splines rather than linear terms or indicators provided a better fit, revealing non-linear relationships that otherwise may have not been apparent. In the second example, use of a restricted quadratic spline resulted in stronger attenuation of a crude association, which likely represents better control of confounding by viral load. n nThe macro presented here offers users a straightforward SAS option for implementing restricted quadratic splines regression. This code is intended to aid in model selection as well as assessing robustness of inferences when comparing various modeling strategies.3,6-7 Furthermore, we hope the examples and code will facilitate the use of splines among researchers hesitant to employ less intuitive but largely equivalent modeling strategies,3,7 and in turn broaden the use of splines in applied epidemiologic research.

The role of the c-statistic in variable selection for propensity score models

The applied literature on propensity scores has often cited the c‐statistic as a measure of the ability of the propensity score to control confounding. However, a high c‐statistic in the propensity model is neither necessary nor sufficient for control of confounding. Moreover, use of the c‐statistic as a guide in constructing propensity scores may result in less overlap in propensity scores between treated and untreated subjects; this may require the analyst to restrict populations for inference. Such restrictions may reduce precision of estimates and change the population to which the estimate applies. Variable selection based on prior subject matter knowledge, empirical observation, and sensitivity analysis is preferable and avoids many of these problems. Copyright

The parametric g-formula to estimate the effect of highly active antiretroviral therapy on incident AIDS or death

The parametric g-formula can be used to contrast the distribution of potential outcomes under arbitrary treatment regimes. Like g-estimation of structural nested models and inverse probability weighting of marginal structural models, the parametric g-formula can appropriately adjust for measured time-varying confounders that are affected by prior treatment. However, there have been few implementations of the parametric g-formula to date. Here, we apply the parametric g-formula to assess the impact of highly active antiretroviral therapy on time to acquired immune deficiency syndrome (AIDS) or death in two US-based human immunodeficiency virus cohorts including 1498 participants. These participants contributed approximately 7300 person-years of follow-up (49% exposed to highly active antiretroviral therapy) during which 382 events occurred and 259 participants were censored because of dropout. Using the parametric g-formula, we estimated that antiretroviral therapy substantially reduces the hazard of AIDS or death (hazard ratiou2009=u20090.55; 95% confidence limits [CL]: 0.42, 0.71). This estimate was similar to one previously reported using a marginal structural model, 0.54 (95% CL: 0.38, 0.78). The 6.5-year difference in risk of AIDS or death was 13% (95% CL: 8%, 18%). Results were robust to assumptions about temporal ordering, and extent of history modeled, for time-varying covariates. The parametric g-formula is a viable alternative to inverse probability weighting of marginal structural models and g-estimation of structural nested models for the analysis of complex longitudinal data.

Early life soy exposure and age at menarche

This study examines the timing of menarche in relation to infant-feeding methods, specifically addressing the potential effects of soy isoflavone exposure through soy-based infant feeding. Subjects were participants in the Avon Longitudinal Study of Parents and Children (ALSPAC). Mothers were enrolled during pregnancy and their children have been followed prospectively. Early-life feeding regimes, categorised as primarily breast, early formula, early soy and late soy, were defined using infant-feeding questionnaires administered during infancy. For this analysis, age at menarche was assessed using questionnaires administered approximately annually between ages 8 and 14.5. Eligible subjects were limited to term, singleton, White females. We used Kaplan-Meier survival curves and Cox proportional hazards models to assess age at menarche and risk of menarche over the study period. The present analysis included 2920 girls. Approximately 2% of mothers reported that soy products were introduced into the infant diet at or before 4 months of age (early soy). The median age at menarche [interquartile range (IQR)] in the study sample was 153 months [144-163], approximately 12.8 years. The median age at menarche among early soy-fed girls was 149 months (12.4 years) [IQR, 140-159]. Compared with girls fed non-soy-based infant formula or milk (early formula), early soy-fed girls were at 25% higher risk of menarche throughout the course of follow-up (hazard ratio 1.25 [95% confidence interval 0.92, 1.71]). Our results also suggest that girls fed soy products in early infancy may have an increased risk of menarche specifically in early adolescence. These findings may be the observable manifestation of mild endocrine-disrupting effects of soy isoflavone exposure. However, our study is limited by few soy-exposed subjects and is not designed to assess biological mechanisms. Because soy formula use is common in some populations, this subtle association with menarche warrants more in-depth evaluation in future studies.

Time Scale and Adjusted Survival Curves for Marginal Structural Cox Models

Typical applications of marginal structural time-to-event (e.g., Cox) models have used time on study as the time scale. Here, the authors illustrate use of time on treatment as an alternative time scale. In addition, a method is provided for estimating Kaplan-Meier-type survival curves for marginal structural models. For illustration, the authors estimate the total effect of highly active antiretroviral therapy on time to acquired immunodeficiency syndrome (AIDS) or death in 1,498 US men and women infected with human immunodeficiency virus and followed for 6,556 person-years between 1995 and 2002; 323 incident cases of clinical AIDS and 59 deaths occurred. Of the remaining 1,116 participants, 77% were still under observation at the end of follow-up. By using time on study, the hazard ratio for AIDS or death comparing always with never using highly active antiretroviral therapy from the marginal structural model was 0.52 (95% confidence interval: 0.35, 0.76). By using time on treatment, the analogous hazard ratio was 0.44 (95% confidence interval: 0.32, 0.60). In time-to-event analyses, the choice of time scale may have a meaningful impact on estimates of association and precision. In the present example, use of time on treatment yielded a hazard ratio further from the null and more precise than use of time on study as the time scale.

Assessing the effect of hormonal contraception on HIV acquisition in observational data: challenges and recommended analytic approaches.

Introduction:Determining whether hormonal contraception, particularly the injectable contraceptive depot-medroxyprogesterone acetate (DMPA), increases a womans risk of HIV acquisition is a priority question for public health. However, assessing the relationship between various hormonal contraceptive methods and HIV acquisition with observational data involves substantial analytic design issues and challenges. Studies to date have used inconsistent approaches and generated a body of evidence that is complex and challenging to interpret. Methods:In January 2013, the United States Agency for International Development and FHI 360 supported a meeting of epidemiologists, statisticians, and content experts to develop recommendations for future observational analyses of hormonal contraception and HIV acquisition. Results:Meeting participants generated recommendations regarding careful definition of exposure groups; handling potential confounders, mediators, and effect modifiers; estimating and addressing the magnitude of measurement error; using multiple methods to account for pregnancy; and exploring the potential for differential exposure to HIV-infected partners. Advantages and disadvantages of various statistical approaches to account for time-varying confounding and estimating total and direct effects were also discussed. Conclusion:Implementing these recommendations in future observational hormonal contraception-HIV acquisition analyses will enhance interpretation of existing studies and strengthen the overall evidence base for this complex and important area.

An information criterion for marginal structural models

Marginal structural models were developed as a semiparametric alternative to the G-computation formula to estimate causal effects of exposures. In practice, these models are often specified using parametric regression models. As such, the usual conventions regarding regression model specification apply. This paper outlines strategies for marginal structural model specification and considerations for the functional form of the exposure metric in the final structural model. We propose a quasi-likelihood information criterion adapted from use in generalized estimating equations. We evaluate the properties of our proposed information criterion using a limited simulation study. We illustrate our approach using two empirical examples. In the first example, we use data from a randomized breastfeeding promotion trial to estimate the effect of breastfeeding duration on infant weight at 1u2009year. In the second example, we use data from two prospective cohorts studies to estimate the effect of highly active antiretroviral therapy on CD4 count in an observational cohort of HIV-infected men and women. The marginal structural model specified should reflect the scientific question being addressed but can also assist in exploration of other plausible and closely related questions. In marginal structural models, as in any regression setting, correct inference depends on correct model specification. Our proposed information criterion provides a formal method for comparing model fit for different specifications.

Effectiveness and safety of 30 mg versus 40 mg stavudine regimens: a cohort study among HIV-infected adults initiating HAART in South Africa

BackgroundAs stavudine remains an important and widely prescribed drug in resource-limited settings, the effect of a reduced dose of stavudine (from 40 mg to 30 mg) on outcomes of highly active antiretroviral therapy (HAART) remains an important public health question.MethodsWe analyzed prospectively collected data from the Themba Lethu Clinic in Johannesburg, South Africa. We assessed the relationship between stavudine dose and six- and/or 12-month outcomes of stavudine substitution, failure to suppress viral load to below 400 copies/ml, development of peripheral neuropathy, lipoatrophy and hyperlactatemia/lactic acidosis. Since individuals with a baseline weight of less than 60 kg were expected to have received the same dose of stavudine throughout the study period, analysis was restricted to individuals who weighed 60 kg or more at baseline. Data were analyzed using logistic regression.ResultsBetween 1 April 2004 and 30 September 2009, 3910 patients were initiated on antiretroviral therapy (ART) with a recorded stavudine dose and were included in the analysis. Of these, 2445 (62.5%) received a 40 mg stavudine dose while 1565 (37.5%) received 30 mg. In multivariate analysis, patients receiving a 40 mg dose were more likely to discontinue stavudine use (adjusted odds ratio, OR 1.71; 95% confidence limits, CI 1.13-2.57) than those receiving 30 mg by 12 months on ART. Additionally, patients receiving 40 mg doses of stavudine were more likely to report peripheral neuropathy (OR 3.12; 95% CI 1.86-5.25), lipoatrophy (OR 11.8; 95% CI 3.2-43.8) and hyperlactatemia/lactic acidosis (OR 8.37; 95% CI 3.83-18.29) in the same time period. Failure to suppress HIV viral load within 12 months of HAART initiation was somewhat more common among those given 40 mg doses (OR 1.62; 95% CI 0.88, 2.97) although this result lacked precision. Sensitivity analyses accounting for death and loss to follow up generally supported these estimates.ConclusionsLower stavudine dosage is associated with fewer reports of several stavudine-associated adverse events and also a lower risk of stavudine discontinuation within the first year on ART.