Richard Scheines

Bayesian estimation and testing of structural equation models

The Gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model (SEM) given covariance data and a prior distribution over the parameters. Point estimates, standard deviations and interval estimates for the parameters can be computed from these samples. If the prior distribution over the parameters is uninformative, the posterior is proportional to the likelihood, and asymptotically the inferences based on the Gibbs sample are the same as those based on the maximum likelihood solution, for example, output from LISREL or EQS. In small samples, however, the likelihood surface is not Gaussian and in some cases contains local maxima. Nevertheless, the Gibbs sample comes from the correct posterior distribution over the parameters regardless of the sample size and the shape of the likelihood surface. With an informative prior distribution over the parameters, the posterior can be used to make inferences about the parameters underidentified models, as we illustrate on a simple errors-in-variables model.

Multidimensionality and Structural Coefficient Bias in Structural Equation Modeling: A Bifactor Perspective

In this study, the authors consider several indices to indicate whether multidimensional data are “unidimensional enough” to fit with a unidimensional measurement model, especially when the goal is to avoid excessive bias in structural parameter estimates. They examine two factor strength indices (the explained common variance and omega hierarchical) and several model fit indices (root mean square error of approximation, comparative fit index, and standardized root mean square residual). These statistics are compared in population correlation matrices determined by known bifactor structures that vary on the (a) relative strength of general and group factor loadings, (b) number of group factors, and (c) number of items or indicators. When fit with a unidimensional measurement model, the degree of structural coefficient bias depends strongly and inversely on explained common variance, but its effects are moderated by the percentage of correlations uncontaminated by multidimensionality, a statistic that rises combinatorially with the number of group factors. When the percentage of uncontaminated correlations is high, structural coefficients are relatively unbiased even when general factor strength is low relative to group factor strength. On the other hand, popular structural equation modeling fit indices such as comparative fit index or standardized root mean square residual routinely reject unidimensional measurement models even in contexts in which the structural coefficient bias is low. In general, such statistics cannot be used to predict the magnitude of structural coefficient bias.

The TETRAD Project: Constraint Based Aids to Causal Model Specification

The statistical community has brought logical rigor and mathematical precision to the problem of using data to make inferences about a models parameter values. The TETRAD project, and related work in computer science and statistics, aims to apply those standards to the problem of using data and background knowledge to make inferences about a models specification. We begin by drawing the analogy between parameter estimation and model specification search. We then describe how the specification of a structural equation model entails familiar constraints on the covariance matrix for all admissible values of its parameters; we survey results on the equivalence of structural equation models, and we discuss search strategies for model specification. We end by presenting several algorithms that are implemented in the TETRAD I1 program.

Using Path Diagrams as a Structural Equation Modeling Tool

A linear structural equation model (SEM) without free parameters has two parts: a probability distribution and an associated path diagram corresponding to the causal relations among variables specified by the structural equations and the correlations among the error terms. This article shows how path diagrams can be used to solve a number of important problems in structural equation modeling; for example, How much do sample data underdetermine the correct model specification? Given that there are equivalent models, is it possible to extract the features common to those models? When a modeler draws conclusions about coefficients in an unknown underlying SEM from a multivariate regression, precisely what assumptions are being made about the SEM? The authors explain how the path diagram provides much more than heuristics for special cases; the theory of path diagrams helps to clarify several of the issues just noted.

Interventions and Causal Inference

The literature on causal discovery has focused on interventions that involve randomly assigning values to a single variable. But such a randomized intervention is not the only possibility, nor is it always optimal. In some cases it is impossible or it would be unethical to perform such an intervention. We provide an account of ‘hard’ and ‘soft’ interventions and discuss what they can contribute to causal discovery. We also describe how the choice of the optimal intervention(s) depends heavily on the particular experimental setup and the assumptions that can be made.

A statistical problem for inference to regulatory structure from associations of gene expression measurements with microarrays

MOTIVATION One approach to inferring genetic regulatory structure from microarray measurements of mRNA transcript hybridization is to estimate the associations of gene expression levels measured in repeated samples. The associations may be estimated by correlation coefficients or by conditional frequencies (for discretized measurements) or by some other statistic. Although these procedures have been successfully applied to other areas, their validity when applied to microarray measurements has yet to be tested. RESULTS This paper describes an elementary statistical difficulty for all such procedures, no matter whether based on Bayesian updating, conditional independence testing, or other machine learning procedures such as simulated annealing or neural net pruning. The difficulty obtains if a number of cells from a common population are aggregated in a measurement of expression levels. Although there are special cases where the conditional associations are preserved under aggregation, in general inference of genetic regulatory structure based on conditional association is unwarranted

Simulation Studies of the Reliability of Computer-Aided Model Specification Using the TETRAD II, EQS, and LISREL Programs

TETRAD II, a fully automated successor to the TETRAD program, is intended to aid in the respecification of underspecified linear causal models, or structural equation models. The performance of TETRAD II is compared with the automatic respecification procedures in the EQS and LISREL VI programs using 360 simulated data sets from nine different linear models containing “latent” or unmeasured variables. LISREL VI and EQS each output a single suggested model; TETRAD II outputs a small list of such models. For these cases, we find that the TETRAD II program, which uses graph algorithms and heuristic search techniques, is more reliable (although less precise) than either EQS or LISREL VI, which use numerical algorithms and beam search techniques. A detailed analysis of the reasons for these differences is offered. Contrary to those who dismiss automated search techniques as unreliable “ransacking” or “data mining,” TETRAD II provides correct information about the true model for 95% of the large sample data sets. The need for further simulation tests and the prospects for the development of automated techniques to aid in the initial specification of causal models for nonexperimental data are discussed.

Replacing Lecture with Web-Based Course Materials

In a series of 5 experiments in 2000 and 2001, several hundred students at two different universities with three different professors and six different teaching assistants took a semester long course on causal and statistical reasoning in either traditional lecture/recitation or online/recitation format. In this article we compare the pre-post test gains of these students, we identify features of the online experience that were helpful and features that were not, and we identify student learning strategies that were effective and those that were not. Students who entirely replaced going to lecture with doing online modules did as well and usually better than those who went to lecture. Simple strategies like incorporating frequent interactive comprehension checks into the online material (something that is difficult to do in lecture) proved effective, but online students attended face-to-face recitations less often than lecture students and suffered because of it. Supporting the idea that small, interactive recitations are more effective than large, passive lectures, recitation attendance was three times as important as lecture attendance for predicting pre-test to post-test gains. For the online student, embracing the online environment as opposed to trying to convert it into a traditional print-based one was an important strategy, but simple diligence in attempting “voluntary” exercises was by far the most important factor in student success.

Computer environments for proof construction

Abstract Does the presentation and use of the search space matter for complex problem solving tasks? We address these questions for the construction of proofs in sentential logic. Using a fully computerized logic course, we isolated crucial features of computer environments and assessed their relative pedagogical effectiveness. After being given a pretest for logical aptitude, students were divided into three matched groups, each of which used a distinct computerized environment to construct proofs. All students were presented with identical course material on sentential logic for approximately five weeks. Students completed more than one hundred exercises during those five weeks and took a midterm at the end of the period. The group using the most informative and most flexible interface performed substantially better on the midterm— the difference was particularly striking for hard problems. In two follow‐up experiments we added strategic problem solving help; student performance improved again (entirely...

Actual causation: a stone soup essay

AbstractWe argue that current discussions of criteria for actual causation are ill-posed in several respects. (1) The methodology of current discussions is by induction from intuitions about an infinitesimal fraction of the possible examples and counterexamples; (2) cases with larger numbers of causes generate novel puzzles; (3) “neuron” and causal Bayes net diagrams are, as deployed in discussions of actual causation, almost always ambiguous; (4) actual causation is (intuitively) relative to an initial system state since state changes are relevant, but most current accounts ignore state changes through time; (5) more generally, there is no reason to think that philosophical judgements about these sorts of cases are normative; but (6) there is a dearth of relevant psychological research that bears on whether various philosophical accounts are descriptive. Our skepticism is not directed towards the possibility of a correct account of actual causation; rather, we argue that standard methods will not lead to such an account. A different approach is required. Once upon a time a hungry wanderer came into a village. He filled an iron cauldron with water, built a fire under it, and dropped a stone into the water. “I do like a tasty stone soup” he announced. Soon a villager added a cabbage to the pot, another added some salt and others added potatoes, onions, carrots, mushrooms, and so on, until there was a meal for all.