Causal graphs and SEM

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R Causal mediation

Glymour, Scheines

G. Glymour, R. Scheines, P. Spirtes, C. Meek. 1994 Regression and Causation CMU Philosophy Methodology Logic. Pittsburgh.

Very clear as to problem with regression.

Effective connectivity

Guillaume Marrelec, Barry Horwitz, Jieun Kim, Mélanie Pélégrini-Issac, Habib Benali, Julien Doyon. n.d. Using Partial Correlation to Enhance Structural Equation Modeling of Functional MRI Data. Abstract. In functional magnetic resonance imaging (fMRI) data analysis, effective connectivity investigates the influence that brain regions exert on one another. Structural equation modeling (SEM) has been the main approach to examine effective connectivity. In this paper, we propose a method that, given a set of regions, performs partial correlation analysis. This method provides an approach to effective conne tivity that is data-driven, in the sense that it does not require any prior information regarding the anatomical or functional connections. To demonstrate the practical relevance of partial correlation analysis for effective connectivity investigation, we re-analyzed data previously published Bullmore, Horwitz, Honey, Brammer, Williams, Sharma, 2000. How good is good enough in path analysis of fMRI data? NeuroImage 11, 289–301. Specifically, we show that partial correlation analysis can serve several purposes. In a pre-processing step, it can hint at which effective connections are structuring the interactions and which have little influence on the pattern of connectivity. As a post-processing step, it can be used both as a simple and visual way to check the validity of SEM optimization algorithms and to show which assumptions made by the model are valid, and which ones should be further modified to better fit the data.

Wikipedia:Partial correlation#Using matrix inversion

Neuroimage. 2000 Apr;11(4):289-301. How good is good enough in path analysis of fMRI data? Bullmore E, Horwitz B, Honey G, Brammer M, Williams S, Sharma T.
Abstract. This paper is concerned with the problem of evaluating goodness-of-fit of a path analytic model to an interregional correlation matrix derived from functional magnetic resonance imaging (fMRI) data. We argue that model evaluation based on testing the null hypothesis that the correlation matrix predicted by the model equals the population correlation matrix is problematic because P values are conditional on asymptotic distributional results (which may not be valid for fMRI data acquired in less than 10 min), as well as arbitrary specification of residual variances and effective degrees of freedom in each regional fMRI time series. We introduce an alternative approach based on an algorithm for automatic identification of the best fitting model that can be found to account for the data. The algorithm starts from the null model, in which all path coefficients are zero, and iteratively unconstrains the coefficient which has the largest Lagrangian multiplier at each step until a model is identified which has maximum goodness by a parsimonious fit index. Repeating this process after bootstrapping the data generates a confidence interval for goodness-of-fit of the best model. If the goodness of the theoretically preferred model is within this confidence interval we can empirically say that the theoretical model could be the best model. This relativistic and data-based strategy for model evaluation is illustrated by analysis of functional MR images acquired from 20 normal volunteers during periodic performance (for 5 min) of a task demanding semantic decision and subvocal rehearsal. A model including unidirectional connections from frontal to parietal cortex, designed to represent sequential engagement of rehearsal and monitoring components of the articulatory loop, is found to be irrefutable by hypothesis-testing and within confidence limits for the best model that could be fitted to the data.

Copyright 2000 Academic Press.

Paragraph from an NSF proposal due Jan 15 2010

From 2SLS to Pearl’s Causal Graphs. The 2SLS approach stands on its own, provding important results in their own right,. Our regression model variable selection process, however, also involves one of the key elements of Bayesian statistical approaches to causal modeling: do not accept a model where the predicted variance from an independent to a dependent variable involves an implausibility (Bayesian constraint) such as “mud predicts rain” which cannot be causal. Causality in an indep X→c dep Y variable pair with a coefficient c means that a unit change in X will bring about c units of change in E(Y), the expected value of Y. "Change" means that if the physical means exists of fixing Y at some constant Y1, and of changing that constant from Y1 to Y2, then the observed change in the expected value E(Z) will be c(Y2-Y1). In a causal graph (Pearl 1987, 2000, 2009) pairs of nodes (for variables) may be causally asymmetric (X→Y) or mutually related through a latent common cause (X↔Y). No pair of variables represented by two nodes are reciprocally causal because this is a contradiction: Causality is temporally ordered. X and Y may be connected by a directed causal chain (partial ancestral graph, e.g., X→W→Y) but not a cycle. In a causal chain such as X→W→Y there is the possible additional causality, X→Y, or not, to be determined statistically. Letting d stand for dependence, X and Y are d-separated (Pearl 1987) if, with respect to the set Z of variables connecting X to Y in a chain, knowing X and something about a W in Z, where X→W→Y, gives you no additional knowledge (statistical prediction) about Y (Y is independent conditional of Z) and there is no possible direct causality X→Y consistent with a X→W→Y chain. It may be that no set Z of nodes d-separates X and Y, written X◦―◦Y. If X and Y are d-separated conditional on (“statistically independent” of) other variables, and X◦→Y (X is not an ancestor of Y), then for a no-implausibility regression relation indep X→b dep Y the coefficient b is an unbiased estimate of coefficient c in a causal relation X→c Y. All this is proven mathematically. Because a causal graph that can have any of these six possible relations between any X and Y, there is an equivalent structural equation model (SEM), which derives from the statistical technique of causal path analysis pioneered by Sewall Wright (1921). SEMs also test and estimate causal relationships using a combination of statistical data and qualitative causal assumptions. 2SLS indep X→b dep Y regression coefficients can be tested as qualifying or not for causality given d-separability computations(FN1) and give the same results as SEM. Algorithms such as TETRAD (Ramsey 2009), based on Heckerman, Meek and Cooper (1999), are capable of tracing causal graphs statistically from d-separation, starting from regression models with no contrafactuals. Ramsey (2001, 2009) provides algorithms (Scheines et al. 2009) at solve the triads problem in general, and similarly for longer chains.(FN2) These methods, then, provide our project with a means of moving from 2SLS results, valuable in their own right, to causal models.

  • FN1 Practical results can be checked with Pearl’s (2009:107-108) do(x) operator where Y(do(X)) simulates physical interventions of how an experimental or practical treatment fixing X to x0 transforms the distribution Y. “For example, if X represents a treatment variable, Y a response variable, and Z some covariate that affects the amount of treatment received, then the distribution P(z, y|do(x0)) gives the proportion of individuals that would attain response level Y = y and covariate level Z = z under the hypothetical situation in which treatment X = x0 is administered uniformly to the population.”
  • FN2 White et al. (1983) also provide an algorithm to solve the statistical d-separation question for triads of dichotomous variables.

Written by D.R. White re:Judea Pearl

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