Sequences of regressions and their independences

Ordered sequences of univariate or multivariate regressions provide statistical models for analysing data from randomized, possibly sequential interventions, from cohort or multi-wave panel studies, but also from cross-sectional or retrospective studies. Conditional independences are captured by what we name regression graphs, provided the generated distribution shares some properties with a joint Gaussian distribution. Regression graphs extend purely directed, acyclic graphs by two types of undirected graph, one type for components of joint responses and the other for components of the context vector variable. We review the special features and the history of regression graphs, derive criteria to read all implied independences of a regression graph and prove criteria for Markov equivalence that is to judge whether two different graphs imply the same set of independence statements. Knowledge of Markov equivalence provides alternative interpretations of a given sequence of regressions, is essential for machine learning strategies and permits to use the simple graphical criteria of regression graphs on graphs for which the corresponding criteria are in general more complex. Under the known conditions that a Markov equivalent directed acyclic graph exists for any given regression graph, we give a polynomial time algorithm to find one such graph.Comment: 43 pages with 17 figures The manuscript is to appear as an invited discussion paper in the journal TES

Pairwise Markov properties for regression graphs

With a sequence of regressions, one may generate joint probability distributions. One starts with a joint, marginal distribution of context variables having possibly a concentration graph structure and continues with an ordered sequence of conditional distributions, named regressions in joint responses. The involved random variables may be discrete, continuous or of both types. Such a generating process specifies for each response a conditioning set that contains just its regressor variables, and it leads to at least one valid ordering of all nodes in the corresponding regression graph that has three types of edge: one for undirected dependences among context variables, another for undirected dependences among joint responses and one for any directed dependence of a response on a regressor variable. For this regression graph, there are several definitions of pairwise Markov properties, where each interprets the conditional independence associated with a missing edge in the graph in a different way. We explain how these properties arise, prove their equivalence for compositional graphoids and point at the equivalence of each one of them to the global Markov property.Work of the first author was partially supported by grant #FA9550-14-1-0141 from the US Air Force Office of Scientific Research (AFOSR) and the Defense Advanced Research Projects Agency (DARPA)

Single-shot imaging of OH radicals and simultaneous OH radical/acetone imaging with a tunable Nd : YAG laser

Single shot imaging capability for OH radical distributions in various atmospheric pressure methane flames upon excitation with a tunable frequency-quadrupled Nd : YAG laser is demonstrated. The laser wavelength can be tuned with an intra-cavity etalon to produce laser-induced fluorescence (LIF) signals from OH via absorption in the OH A–X (2,0) P 1 (10) line. Simultaneous single-shot imaging of the burnt and unburnt zones in laminar nonpremixed, premixed and turbulent flames is presented. The unburnt areas are visualized with LIF of acetone that is seeded to the methane fuel. Acetone levels are set to match signal intensities to that of the OH signals to allow imaging on a single intensified CCD camera.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47048/1/340_2004_Article_1545.pd

Eliciting a directed acyclic graph for a multivariate time series of vehicle counts in a traffic network

The problem of modelling multivariate time series of vehicle counts in traffic networks is considered. It is proposed to use a model called the linear multiregression dynamic model (LMDM). The LMDM is a multivariate Bayesian dynamic model which uses any conditional independence and causal structure across the time series to break down the complex multivariate model into simpler univariate dynamic linear models. The conditional independence and causal structure in the time series can be represented by a directed acyclic graph (DAG). The DAG not only gives a useful pictorial representation of the multivariate structure, but it is also used to build the LMDM. Therefore, eliciting a DAG which gives a realistic representation of the series is a crucial part of the modelling process. A DAG is elicited for the multivariate time series of hourly vehicle counts at the junction of three major roads in the UK. A flow diagram is introduced to give a pictorial representation of the possible vehicle routes through the network. It is shown how this flow diagram, together with a map of the network, can suggest a DAG for the time series suitable for use with an LMDM

Discovering a junction tree behind a Markov network by a greedy algorithm

In an earlier paper we introduced a special kind of k-width junction tree, called k-th order t-cherry junction tree in order to approximate a joint probability distribution. The approximation is the best if the Kullback-Leibler divergence between the true joint probability distribution and the approximating one is minimal. Finding the best approximating k-width junction tree is NP-complete if k>2. In our earlier paper we also proved that the best approximating k-width junction tree can be embedded into a k-th order t-cherry junction tree. We introduce a greedy algorithm resulting very good approximations in reasonable computing time. In this paper we prove that if the Markov network underlying fullfills some requirements then our greedy algorithm is able to find the true probability distribution or its best approximation in the family of the k-th order t-cherry tree probability distributions. Our algorithm uses just the k-th order marginal probability distributions as input. We compare the results of the greedy algorithm proposed in this paper with the greedy algorithm proposed by Malvestuto in 1991.Comment: The paper was presented at VOCAL 2010 in Veszprem, Hungar

Maximum Likelihood Estimation in Gaussian Chain Graph Models under the Alternative Markov Property

The AMP Markov property is a recently proposed alternative Markov property for chain graphs. In the case of continuous variables with a joint multivariate Gaussian distribution, it is the AMP rather than the earlier introduced LWF Markov property that is coherent with data-generation by natural block-recursive regressions. In this paper, we show that maximum likelihood estimates in Gaussian AMP chain graph models can be obtained by combining generalized least squares and iterative proportional fitting to an iterative algorithm. In an appendix, we give useful convergence results for iterative partial maximization algorithms that apply in particular to the described algorithm.Comment: 15 pages, article will appear in Scandinavian Journal of Statistic

Pitfalls of using the risk ratio in meta‐analysis

For meta-analysis of studies that report outcomes as binomial proportions, the most popular measure of effect is the odds ratio (OR), usually analyzed as log(OR). Many meta-analyses use the risk ratio (RR) and its logarithm, because of its simpler interpretation. Although log(OR) and log(RR) are both unbounded, use of log(RR) must ensure that estimates are compatible with study-level event rates in the interval (0, 1). These complications pose a particular challenge for random-effects models, both in applications and in generating data for simulations. As background we review the conventional random-effects model and then binomial generalized linear mixed models (GLMMs) with the logit link function, which do not have these complications. We then focus on log-binomial models and explore implications of using them; theoretical calculations and simulation show evidence of biases. The main competitors to the binomial GLMMs use the beta-binomial (BB) distribution, either in BB regression or by maximizing a BB likelihood; a simulation produces mixed results. Two examples and an examination of Cochrane meta-analyses that used RR suggest bias in the results from the conventional inverse-variance-weighted approach. Finally, we comment on other measures of effect that have range restrictions, including risk difference, and outline further research

Qualitative evaluation of a preventive intervention for the offspring of parents with a history of depression

Background: Meta-analyses of randomised controlled trials suggest that psychological interventions to reduce children’s risk of depression are effective. Nevertheless, these effects are modest and diminish over time. The Medical Research Council recommends a mixed-methods approach to the evaluation of complex interventions. By gaining a more thorough understanding of participants’ perspectives, qualitative evaluations of preventive interventions could improve their efficacy, longevity and transfer into clinical practice. Methods: 18 parents and 22 children who had received a 12-session family- and group-based cognitivebehavioural intervention to prevent youth depression as part of a randomised controlled trial took part in semistructured interviews or a focus group about aspects which had been perceived as helpful, elements they were still using after the intervention had ended, and suggestions they had for improving the intervention. Results: The chance to openly share and discuss their experiences of depression within and between families was considered helpful by both children and parents. Children benefitted the most from learning coping strategies for dealing with stress and many still used them in everyday life. Parents profited mostly from increasing positive family time, but noted that maintaining new routines after the end of the intervention proved difficult. Participants were generally content with the intervention but commented on how tiring and time consuming it was. Conclusions: Managing parents’ expectations of family-based interventions in terms of their own mental health needs (versus those of their children) and leaving more room for open discussions may result in interventions which are more appealing to participating families. Increasing intervals between sessions may be one means of improving the longevity of interventions. Trial registration: The original RCT this evaluation is a part of was registered under NCT02115880

Total Positivity in Markov Structures

We discuss properties of distributions that are multivariate totally positive of order two (MTP$_{2}$) related to conditional independence. In particular, we show that any independence model generated by an MTP$_{2}$ distribution is a compositional semigraphoid which is upward-stable and singleton-transitive. In addition, we prove that any MTP$_{2}$ distribution satisfying an appropriate support condition is faithful to its concentration graph. Finally, we analyze factorization properties of MTP$_{2}$ distributions and discuss ways of constructing MTP$_{2}$ distributions; in particular we give conditions on the log-linear parameters of a discrete distribution which ensure MTP$_{2}$ and characterize conditional Gaussian distributions which satisfy MTP$_{2}$.SF was supported in part by an NSERC Discovery Research Grant, KS by grant #FA9550-12-1-0392 from the U.S. Air Force Office of Scientic Research (AFOSR) and the Defense Advanced Research Projects Agency (DARPA), CU by the Austrian Science Fund (FWF) Y 903-N35, and PZ by the European Union Seventh Framework Programme PIOF-GA-2011-300975