The mobility effects hypothesis: Methods and applications.

We discuss hypotheses researchers have put forth to explain how outcomes of socially mobile and immobile individuals might differ and/or how mobility experiences are related to outcomes of interest. Next, we examine the methodological literature on this topic, culminating in the development of the diagonal mobility model (DMM, also called the diagonal reference model in some studies), the primary tool of use since the 1980's. We then discuss some of the many applications of the DMM. Although the model was proposed to examine the effects of social mobility on outcomes of interest, the estimated relationships between mobility and outcomes that researchers have called mobility effects are more appropriately regarded as partial associations. When mobility is not associated with outcomes, as is often found in empirical work, the outcomes of movers from origin o to destination d are a weighted average of the outcomes of individuals who remained in states o and d respectively, and the weights capture the relative salience of origins and destinations in the acculturation process. In light of this attractive feature of the model, we briefly develop several generalizations of the current DMM that future researchers should also find useful. Finally, we propose new estimands of mobility effects, based on the explicit notion that a unit effect of mobility is a comparison of an individual with herself under two conditions, one in which she is mobile, the other in which she is immobile, and we discuss some of the challenges in identifying such effects.

Cloak and DAG: a response to the comments on our comment.

Our original comment (Lindquist and Sobel, 2011) made explicit the types of assumptions neuroimaging researchers are making when directed graphical models (DGMs), which include certain types of structural equation models (SEMs), are used to estimate causal effects. When these assumptions, which many researchers are not aware of, are not met, parameters of these models should not be interpreted as effects. Thus it is imperative that neuroimaging researchers interested in issues involving causation, for example, effective connectivity, consider the plausibility of these assumptions for their particular problem before using SEMs. In cases where these additional assumptions are not met, researchers may be able to use other methods and/or design experimental studies where the use of unrealistic assumptions can be avoided. Pearl does not disagree with anything we stated. However, he takes exception to our use of potential outcomes' notation, which is the standard notation used in the statistical literature on causal inference, and his comment is devoted to promoting his alternative conventions. Glymour's comment is based on three claims that he inappropriately attributes to us. Glymour is also more optimistic than us about the potential of using directed graphical models (DGMs) to discover causal relations in neuroimaging research; we briefly address this issue toward the end of our rejoinder.

Compliance mixture modelling with a zero-effect complier class and missing data.

Randomized experiments are the gold standard for evaluating proposed treatments. The intent to treat estimand measures the effect of treatment assignment, but not the effect of treatment if subjects take treatments to which they are not assigned. The desire to estimate the efficacy of the treatment in this case has been the impetus for a substantial literature on compliance over the last 15 years. In papers dealing with this issue, it is typically assumed there are different types of subjects, for example, those who will follow treatment assignment (compliers), and those who will always take a particular treatment irrespective of treatment assignment. The estimands of primary interest are the complier proportion and the complier average treatment effect (CACE). To estimate CACE, researchers have used various methods, for example, instrumental variables and parametric mixture models, treating compliers as a single class. However, it is often unreasonable to believe all compliers will be affected. This article therefore treats compliers as a mixture of two types, those belonging to a zero-effect class, others to an effect class. Second, in most experiments, some subjects drop out or simply do not report the value of the outcome variable, and the failure to take into account missing data can lead to biased estimates of treatment effects. Recent work on compliance in randomized experiments has addressed this issue by assuming missing data are missing at random or latently ignorable. We extend this work to the case where compliers are a mixture of types and also examine alternative types of nonignorable missing data assumptions.

Graphical models, potential outcomes and causal inference: comment on Ramsey, Spirtes and Glymour.

Ramsey, Spirtes and Glymour (RSG) critique a method proposed by Neumann et al. (2010) for the discovery of functional networks from fMRI meta-analysis data. We concur with this critique, but are unconvinced that directed graphical models (DGMs) are generally useful for estimating causal effects. We express our reservations using the "potential outcomes" framework for causal inference widely used in statistics.

ESTIMATING CAUSAL EFFECTS IN STUDIES OF HUMAN BRAIN FUNCTION: NEW MODELS, METHODS AND ESTIMANDS.

Neuroscientists often use functional magnetic resonance imaging (fMRI) to infer effects of treatments on neural activity in brain regions. In a typical fMRI experiment, each subject is observed at several hundred time points. At each point, the blood oxygenation level dependent (BOLD) response is measured at 100,000 or more locations (voxels). Typically, these responses are modeled treating each voxel separately, and no rationale for interpreting associations as effects is given. Building on Sobel and Lindquist (J. Amer. Statist. Assoc. 109 (2014) 967-976), who used potential outcomes to define unit and average effects at each voxel and time point, we define and estimate both "point" and "cumulated" effects for brain regions. Second, we construct a multisubject, multivoxel, multirun whole brain causal model with explicit parameters for regions. We justify estimation using BOLD responses averaged over voxels within regions, making feasible estimation for all regions simultaneously, thereby also facilitating inferences about association between effects in different regions. We apply the model to a study of pain, finding effects in standard pain regions. We also observe more cerebellar activity than observed in previous studies using prevailing methods.

Semiparametric transformation models for causal inference in time to event studies with all-or-nothing compliance.

We consider causal inference in randomized survival studies with right censored outcomes and all-or-nothing compliance, using semiparametric transformation models to estimate the distribution of survival times in treatment and control groups, conditional on covariates and latent compliance type. Estimands depending on these distributions, for example, the complier average causal effect (CACE), the complier effect on survival beyond time t, and the complier quantile effect are then considered. Maximum likelihood is used to estimate the parameters of the transformation models, using a specially designed expectation-maximization (EM) algorithm to overcome the computational difficulties created by the mixture structure of the problem and the infinite dimensional parameter in the transformation models. The estimators are shown to be consistent, asymptotically normal, and semiparametrically efficient. Inferential procedures for the causal parameters are developed. A simulation study is conducted to evaluate the finite sample performance of the estimated causal parameters. We also apply our methodology to a randomized study conducted by the Health Insurance Plan of Greater New York to assess the reduction in breast cancer mortality due to screening.

Causal Inference for fMRI Time Series Data with Systematic Errors of Measurement in a Balanced On/Off Study of Social Evaluative Threat.

Functional magnetic resonance imaging (fMRI) has facilitated major advances in understanding human brain function. Neuroscientists are interested in using fMRI to study the effects of external stimuli on brain activity and causal relationships among brain regions, but have not stated what is meant by causation or defined the effects they purport to estimate. Building on Rubin's causal model, we construct a framework for causal inference using blood oxygenation level dependent (BOLD) fMRI time series data. In the usual statistical literature on causal inference, potential outcomes, assumed to be measured without systematic error, are used to define unit and average causal effects. However, in general the potential BOLD responses are measured with stimulus dependent systematic error. Thus we define unit and average causal effects that are free of systematic error. In contrast to the usual case of a randomized experiment where adjustment for intermediate outcomes leads to biased estimates of treatment effects (Rosenbaum, 1984), here the failure to adjust for task dependent systematic error leads to biased estimates. We therefore adjust for systematic error using measured "noise covariates" , using a linear mixed model to estimate the effects and the systematic error. Our results are important for neuroscientists, who typically do not adjust for systematic error. They should also prove useful to researchers in other areas where responses are measured with error and in fields where large amounts of data are collected on relatively few subjects. To illustrate our approach, we re-analyze data from a social evaluative threat task, comparing the findings with results that ignore systematic error.