A new u-statistic with superior design sensitivity in matched observational studies

Summary In an observational or nonrandomized study of treatment effects, a sensitivity analysis indicates the magnitude of bias from unmeasured covariates that would need to be present to alter the conclusions of a naïve analysis that presumes adjustments for observed covariates suffice to remove all bias. The power of sensitivity analysis is the probability that it will reject a false hypothesis about treatment effects allowing for a departure from random assignment of a specified magnitude; in particular, if this specified magnitude is “no departure” then this is the same as the power of a randomization test in a randomized experiment. A new family of u‐statistics is proposed that includes Wilcoxon's signed rank statistic but also includes other statistics with substantially higher power when a sensitivity analysis is performed in an observational study. Wilcoxon's statistic has high power to detect small effects in large randomized experiments—that is, it often has good Pitman efficiency—but small effects are invariably sensitive to small unobserved biases. Members of this family of u‐statistics that emphasize medium to large effects can have substantially higher power in a sensitivity analysis. For example, in one situation with 250 pair differences that are Normal with expectation 1/2 and variance 1, the power of a sensitivity analysis that uses Wilcoxon's statistic is 0.08 while the power of another member of the family of u‐statistics is 0.66. The topic is examined by performing a sensitivity analysis in three observational studies, using an asymptotic measure called the design sensitivity, and by simulating power in finite samples. The three examples are drawn from epidemiology, clinical medicine, and genetic toxicology.     

Bayesian semiparametric modeling for matched case-control studies with multiple disease states

We present a Bayesian approach to analyze matched "case-control" data with multiple disease states. The probability of disease development is described by a multinomial logistic regression model. The exposure distribution depends on the disease state and could vary across strata. In such a model, the number of stratum effect parameters grows in direct proportion to the sample size leading to inconsistent MLEs for the parameters of interest even when one uses a retrospective conditional likelihood. We adopt a semiparametric Bayesian framework instead, assuming a Dirichlet process prior with a mixing normal distribution on the distribution of the stratum effects. We also account for possible missingness in the exposure variable in our model. The actual estimation is carried out through a Markov chain Monte Carlo numerical integration scheme. The proposed methodology is illustrated through simulation and an example of a matched study on low birth weight of newborns (Hosmer, D. A. and Lemeshow, S., 2000, Applied Logistic Regression) with two possible disease groups matched with a control group.     

Confidence intervals for uncommon but dramatic responses to treatment

A small literature discusses locally most powerful rank tests when only a fraction of treated subjects respond to treatment. The ranks used in these tests are very different from conventional ranks, being relatively flat for low responses and then rising steeply, and the associated tests are much more powerful than conventional rank tests when, indeed, only a small fraction of treated subjects exhibit dramatic responses. Because the tests are derived from considerations of local power, they do not yield a plausible family of models for effect, and therefore they do not yield confidence intervals for the magnitude of effect formed by inverting the tests. There is a similarity between these tests and another family of tests, originally motivated by different considerations involving peak performance in small subsets. Exploiting this similarity, a method for obtaining confidence statements is proposed. In the case of observational studies, sensitivity to unobserved bias from nonrandom assignment of treatments is also examined. Two examples are used as illustrations: (i) a study of smoking during pregnancy and its effects on birth weight, in which smokers are matched to six controls, and (ii) a matched pair study of damage to DNA among workers at aluminum production plants.     

Statistical analysis for haplotype-based matched case-control studies

Using unphased genotype data, we studied statistical inference for association between a disease and a haplotype in matched case-control studies. Statistical inference for haplotype data is complicated due to ambiguity of genotype phases. An estimating equation-based method is developed for estimating odds ratios and testing disease-haplotype association. The method potentially can also be applied to testing haplotype-environment interaction. Simulation studies show that the proposed method has good performance. The performance of the method in the presence of departures from Hardy-Weinberg equilibrium is also studied.     

A new exact and more powerful unconditional test of no treatment effect from binary matched pairs

Summary .   We consider the problem of testing for a difference in the probability of success from matched binary pairs. Starting with three standard inexact tests, the nuisance parameter is first estimated and then the residual dependence is eliminated by maximization, producing what I call an E+M P-value. The E+M P-value based on McNemar's statistic is shown numerically to dominate previous suggestions, including partially maximized P-values as described in Berger and Sidik (2003, Statistical Methods in Medical Research 12, 91–108). The latter method, however, may have computational advantages for large samples.     

Does a dose-response relationship reduce sensitivity to hidden bias?

It is often said that an important consideration in judging whether an association between treatment and response is causal is the presence or absence of a dose-response relationship, that is, larger ostensible treatment effects when doses of treatment are larger. This criterion is widely discussed in textbooks and is often mentioned in empirical papers. At the same time, it is well known through both important examples and elementary theory that a treatment may cause dramatic effects with no dose-response relationship, and hidden biases may produce a dose-response relationship when the treatment is without effect. What does a dose-response relationship say about causality? It is observed here that a dose-response relationship may or may not reduce sensitivity to hidden bias, and whether it has or has not can be determined by a suitable analysis using the data at hand. Moreover, a study without a dose-response relationship may or may not be less sensitive to hidden bias than another study with such a relationship, and this, too, can be determined from the data at hand. An example concerning cytogenetic damage among professional painters is used to illustrate.     

Sensitivity analysis: distributional assumptions and confounding assumptions

Summary .   In a presentation of various methods for assessing the sensitivity of regression results to unmeasured confounding, Lin, Psaty, and Kronmal (1998, Biometrics 54 , 948–963) use a conditional independence assumption to derive algebraic relationships between the true exposure effect and the apparent exposure effect in a reduced model that does not control for the unmeasured confounding variable. However, Hernán and Robins (1999, Biometrics 55 , 1316–1317) have noted that if the measured covariates and the unmeasured confounder both affect the exposure of interest then the principal conditional independence assumption that is used to derive these algebraic relationships cannot hold. One particular result of Lin et al. does not rely on the conditional independence assumption but only on assumptions concerning additivity. It can be shown that this assumption is satisfied for an entire family of distributions even if both the measured covariates and the unmeasured confounder affect the exposure of interest. These considerations clarify the appropriate contexts in which relevant sensitivity analysis techniques can be applied.     

Sensitivity analysis for causal inference using inverse probability weighting

Evaluation of impact of potential uncontrolled confounding is an important component for causal inference based on observational studies. In this article, we introduce a general framework of sensitivity analysis that is based on inverse probability weighting. We propose a general methodology that allows both non‐parametric and parametric analyses, which are driven by two parameters that govern the magnitude of the variation of the multiplicative errors of the propensity score and their correlations with the potential outcomes. We also introduce a specific parametric model that offers a mechanistic view on how the uncontrolled confounding may bias the inference through these parameters. Our method can be readily applied to both binary and continuous outcomes and depends on the covariates only through the propensity score that can be estimated by any parametric or non‐parametric method. We illustrate our method with two medical data sets.     

A powerful and robust test statistic for randomization inference in group-randomized trials with matched pairs of groups

For group-randomized trials, randomization inference based on rank statistics provides robust, exact inference against nonnormal distributions. However, in a matched-pair design, the currently available rank-based statistics lose significant power compared to normal linear mixed model (LMM) test statistics when the LMM is true. In this article, we investigate and develop an optimal test statistic over all statistics in the form of the weighted sum of signed Mann-Whitney-Wilcoxon statistics under certain assumptions. This test is almost as powerful as the LMM even when the LMM is true, but it is much more powerful for heavy tailed distributions. A simulation study is conducted to examine the power.     

Attributable effects in case2-studies

Summary In an effort to determine whether a particular treatment causes a particular outcome event, data are obtained from a database system that records events when they occur, and for such events, the system records exposure to the treatment. That is, the system records information about cases. The system provides no information about events that might have occurred but did not, that is, about units which are not cases. Roughly speaking, we know the number of successes for two proportions, treated and control, but not the numbers of trials or units for these proportions; indeed, the concept of a “trial” may be somewhat vague. With no further information, the situation is quite hopeless. However, an interesting strategy that is sometimes used entails identifying two types of cases whose origin is entirely different so that it is known the cases of the second type were definitely not affected by the treatment under study. This strategy—the case–case or case²‐study—seems to have been reinvented independently many times, and has recently been offered as a general strategy for infectious disease epidemiology by McCarthy and Giesecke (1999, International Journal of Epidemiology 28, 764–768). Can this strategy permit estimation of the number of cases caused by the treatment? Using attributable effects in a new way, a method of exact inference is proposed, along with a large sample approximation. Two examples are discussed: one concerning the effects of daytime running lights (DRLs) on the risk of multivehicle accidents; the other concerning the origin of a Salmonella infection. A counterexample with superficially similar appearance is also discussed concerning suicide rates following the publication of Final Exit; here, the treatment may alter the outcome, or it may alter the type, and the attributable effect cannot be estimated.     

Reduced sensitivity to hidden bias at upper quantiles in observational studies with dilated treatment effects

When a treatment has a dilated effect, with larger effects when responses are higher, there can be much less sensitivity to bias at upper quantiles than at lower quantiles; i.e., small, plausible hidden biases might explain the ostensible effect of the treatment for many subjects, and yet only quite large hidden biases could explain the effect on a few subjects having dramatically elevated responses. An example concerning kidney function of cadmium workers is discussed in detail. In that example, the treatment effect is far from additive: It is plausibly zero at the lower quartile of responses to control, and it is large and fairly insensitive to bias at the upper quartile.     

Causal inference for non-mortality outcomes in the presence of death

Evaluation of the causal effect of a baseline exposure on a morbidity outcome at a fixed time point is often complicated when study participants die before morbidity outcomes are measured. In this setting, the causal effect is only well defined for the principal stratum of subjects who would live regardless of the exposure. Motivated by gerontologic researchers interested in understanding the causal effect of vision loss on emotional distress in a population with a high mortality rate, we investigate the effect among those who would live both with and without vision loss. Since this subpopulation is not readily identifiable from the data and vision loss is not randomized, we introduce a set of scientifically driven assumptions to identify the causal effect. Since these assumptions are not empirically verifiable, we embed our methodology within a sensitivity analysis framework. We apply our method using the first three rounds of survey data from the Salisbury Eye Evaluation, a population-based cohort study of older adults. We also present a simulation study that validates our method.     

Conditional likelihood methods for haplotype-based association analysis using matched case-control data

Genetic epidemiologists routinely assess disease susceptibility in relation to haplotypes, that is, combinations of alleles on a single chromosome. We study statistical methods for inferring haplotype-related disease risk using single nucleotide polymorphism (SNP) genotype data from matched case-control studies, where controls are individually matched to cases on some selected factors. Assuming a logistic regression model for haplotype-disease association, we propose two conditional likelihood approaches that address the issue that haplotypes cannot be inferred with certainty from SNP genotype data (phase ambiguity). One approach is based on the likelihood of disease status conditioned on the total number of cases, genotypes, and other covariates within each matching stratum, and the other is based on the joint likelihood of disease status and genotypes conditioned only on the total number of cases and other covariates. The joint-likelihood approach is generally more efficient, particularly for assessing haplotype-environment interactions. Simulation studies demonstrated that the first approach was more robust to model assumptions on the diplotype distribution conditioned on environmental risk variables and matching factors in the control population. We applied the two methods to analyze a matched case-control study of prostate cancer.     

Bayesian Inference for the Causal Effect of Mediation

Summary We propose a nonparametric Bayesian approach to estimate the natural direct and indirect effects through a mediator in the setting of a continuous mediator and a binary response. Several conditional independence assumptions are introduced (with corresponding sensitivity parameters) to make these effects identifiable from the observed data. We suggest strategies for eliciting sensitivity parameters and conduct simulations to assess violations to the assumptions. This approach is used to assess mediation in a recent weight management clinical trial.