Sensitivity Analysis for Observational Studies with Recurrent Events

We conduct an observational study of the effect of sickle cell trait Haemoglobin AS (HbAS) on the hazard rate of malaria fevers in children. Assuming no unmeasured confounding, there is strong evidence that HbAS reduces the rate of malarial fevers. Since this is an observational study, however, the no unmeasured confounding assumption is strong. A sensitivity analysis considers how robust a conclusion is to a potential unmeasured confounder. We propose a new sensitivity analysis method for recurrent event data and apply it to the malaria study. We find that for the causal conclusion that HbAS is protective against malarial fevers to be overturned, the hypothesized unmeasured confounder must be as influential as all but one of the measured confounders.

A second evidence factor for a second control group

In an observational study of the effects caused by a treatment, a second control group is used in an effort to detect bias from unmeasured covariates, and the investigator is content if no evidence of bias is found. This strategy is not entirely satisfactory: two control groups may differ significantly, yet the difference may be too small to invalidate inferences about the treatment, or the control groups may not differ yet nonetheless fail to provide a tangible strengthening of the evidence of a treatment effect. Is a firmer conclusion possible? Is there a way to analyze a second control group such that the data might report measurably strengthened evidence of cause and effect, that is, insensitivity to larger unmeasured biases? Evidence factor analyses are not commonly used with a second control group: most analyses compare the treated group to each control group, but analyses of that kind are partially redundant; so, they do not constitute evidence factors. An alternative analysis is proposed here, one that does yield two evidence factors, and with a carefully designed test statistic, is capable of extracting strong evidence from the second factor. The new technical work here concerns the development of a test statistic with high design sensitivity and high Bahadur efficiency in a sensitivity analysis for the second factor. A study of binge drinking as a cause of high blood pressure is used as an illustration.

Sensitivity analyses informed by tests for bias in observational studies

In an observational study, the treatment received and the outcome exhibited may be associated in the absence of an effect caused by the treatment, even after controlling for observed covariates. Two tactics are common: (i) a test for unmeasured bias may be obtained using a secondary outcome for which the effect is known and (ii) a sensitivity analysis may explore the magnitude of unmeasured bias that would need to be present to explain the observed association as something other than an effect caused by the treatment. Can such a test for unmeasured bias inform the sensitivity analysis? If the test for bias does not discover evidence of unmeasured bias, then ask: Are conclusions therefore insensitive to larger unmeasured biases? Conversely, if the test for bias does find evidence of bias, then ask: What does that imply about sensitivity to biases? This problem is formulated in a new way as a convex quadratically constrained quadratic program and solved on a large scale using interior point methods by a modern solver. That is, a convex quadratic function of N variables is minimized subject to constraints on linear and convex quadratic functions of these variables. The quadratic function that is minimized is a statistic for the primary outcome that is a function of the unknown treatment assignment probabilities. The quadratic function that constrains this minimization is a statistic for subsidiary outcome that is also a function of these same unknown treatment assignment probabilities. In effect, the first statistic is minimized over a confidence set for the unknown treatment assignment probabilities supplied by the unaffected outcome. This process avoids the mistake of interpreting the failure to reject a hypothesis as support for the truth of that hypothesis. The method is illustrated by a study of the effects of light daily alcohol consumption on high-density lipoprotein (HDL) cholesterol levels. In this study, the method quickly optimizes a nonlinear function of N = 800 $N=800$ variables subject to linear and quadratic constraints. In the example, strong evidence of unmeasured bias is found using the subsidiary outcome, but, perhaps surprisingly, this finding makes the primary comparison insensitive to larger biases.

A statistic with demonstrated insensitivity to unmeasured bias for 2 × 2 × S tables in observational studies

Are weak associations between a treatment and a binary outcome always sensitive to small unmeasured biases in observational studies? This possibility is often discussed in epidemiology. The familiar Mantel-Haenszel test for a 2 × 2 × S $$ 2\times 2\times S $$ contingency table exaggerates sensitivity to unmeasured biases when the population odds ratios vary among the S $$ S $$ strata. A statistic built from several components, here from the S $$ S $$  strata, is said to have demonstrated insensitivity to bias if it uses only those components that provide indications of insensitivity to bias. Briefly, such a statistic is a d $$ d $$ -statistic. There are 2 S - 1 $$ {2}^S-1 $$ candidate statistics with S $$ S $$ strata, and a d $$ d $$ -statistic considers them all.  To have level α $$ \alpha $$ , a test based on a d $$ d $$ -statistic must pay a price for its double use of the data, but as the sample size increases, that price becomes small, while the gain may be large. The price is paid by conditioning on the limited information used to identify components that are insensitive to a bias of specified magnitude, basing the test result on the information that remains after conditioning. In large samples, the d $$ d $$ -statistic achieves the largest possible design sensitivity, so it does not exaggerate sensitivity to unmeasured bias. A simulation verifies that the large sample result has traction in samples of practical size. A study of sunlight as a cause of cataract is used to illustrate issues and methods. Several extensions of the method are discussed. An R package dstat2x2xk implements the method.

EVIDENCE FACTORS IN A CASE-CONTROL STUDY WITH APPLICATION TO THE EFFECT OF FLEXIBLE SIGMOIDOSCOPY SCREENING ON COLORECTAL CANCER

As in any observational study, in a case-control study a primary concern is potential unmeasured confounders. Bias, due to unmeasured confounders, can result in a false discovery of an apparent treatment effect when there is none. Replication of an observational study, which tries to provide multiple analyses of the data where the biases affecting each analysis are thought to be different, is one way to strengthen the evidence from an observational study. Evidence factors allow for internal replication by testing a hypothesis using multiple comparisons in a way that the comparisons yield independent evidence and differ in the sources of potential bias. We construct evidence factors in a case-control study in which there are two types of cases, "narrow" cases which are thought to be potentially more affected by the exposure and "marginal" cases which are thought to have more heterogeneous causes. We develop and study an inference procedure for using such evidence factors and apply it to a study of the effect of sigmoidoscopy screening on colorectal cancer.

Using Evidence Factors to Clarify Exposure Biomarkers

A study has 2 evidence factors if it permits 2 statistically independent inferences about 1 treatment effect such that each factor is immune to some bias that would invalidate the other factor. Because the 2 factors are statistically independent, the evidence they provide can be combined using methods associated with meta-analysis for independent studies, despite using the same data twice in different ways. We illustrate evidence factors, applying them in a new way in investigations that have both an exposure biomarker and a coarse external measure of exposure to a treatment. To illustrate, we consider the possible effects of cigarette smoking on homocysteine levels, with self-reported smoking and a cotinine biomarker. We examine joint sensitivity of 2 factors to bias from confounding, a central aspect of any observational study.