Inferring causal effects from observational and interventional data is a highly desirable but ambitious goal. Many of the computational and statistical methods are plagued by fundamental identifiability issues, instability, and unreliable performance, especially for largescale systems with many measured variables. We present software and provide some validation of a recently developed methodology based on an invariance principle, called invariant causal prediction (ICP). The ICP method quantifies confidence probabilities for inferring causal structures and thus leads to more reliable and confirmatory statements for causal relations and predictions of external intervention effects. We validate the ICP method and some other procedures using large-scale genome-wide gene perturbation experiments in Saccharomyces cerevisiae. The results suggest that prediction and prioritization of future experimental interventions, such as gene deletions, can be improved by using our statistical inference techniques.interventional-observational data | invariant causal prediction | genome database validation | graphical models I n this article, we discuss statistical methods for causal inference from perturbation experiments. As this is a rather general topic, we focus on the following problem: based on data from observational and perturbation settings, we want to predict the effect and outcome of an unseen and new intervention or perturbation. Taking applications in genomics as an example, a typical task is as follows: based on observational data from wild-type organisms and interventional data from gene knockout or knockdown experiments, we want to predict the effect of a new gene knockout or knockdown on a phenotype of interest. For example, the organism is the model plant Arabidopsis thaliana, the gene knockouts correspond to mutant plants, and the phenotype of interest is the time it takes until the plant is flowering (1).From a methodological viewpoint, the prediction of unseen future interventions belongs to the area of causal inference where one aims to quantify presence and strength of causal effects among various variables. Loosely speaking, a causal effect is the effect of an external intervention (or say the response to a "What if I do?" question). The corresponding theory, e.g., using Pearl's do-operator (2), provides a link between causal effects and perturbations or randomized experiments. We mostly assume here that all of the variables in the causal model (for inferring causal effects) are observed: the case with hidden variables is mentioned only briefly in a later section, although it is an important theme in causal inference (due to the problem of hidden confounding variables) (cf. refs. 2 and 3).A popular and powerful route for causal modeling is given by structural equation models (SEMs) (2, 4). We consider a set of random variables X 1 , . . . , X p , X p+1 , and we often denote by Y = X 1 , emphasizing that Y is our response variable of interest (e.g., a phenotype of interest). The main building blocks of a SEM...
SUMMARY1. Mechanical properties of permeabilized single fibres from rabbit psoas and soleus muscle were determined by measuring the length responses due to abrupt changes in load and the force responses due to isovelocity length changes at different phosphate and Ca21 concentrations.2. The length responses due to abrupt increases in load from psoas fibres showed a rapid lengthening during the change in load followed by a phase of lengthening during which the velocity gradually decreased. In soleus fibres an abrupt lengthening during the change in load was followed by a phase of lengthening during which the velocity remained constant or decreased slightly for increases in load to less than 1-45 of the isometric force (FO). For larger increases in load the velocity during this later phase first increased and thereafter decreased.3. The initial force-velocity curve, derived from the early part of the isotonic responses after the change in load, as well as the late force-velocity curve derived from the force level attained during isovelocity length changes, were sensitive to phosphate. Phosphate caused a shift of the absolute force-velocity curves of both psoas and soleus fibres towards lower values of force. In psoas fibres, the relative force-velocity curves derived by normalization of the force level to the force developed isometrically was shifted by phosphate to smaller velocities. In soleus fibres, the initial velocity at low and intermediate relative loads (< 1-75 FO) was increased by phosphate but at higher loads it decreased, while the late force-velocity curve showed an overall decrease in velocity.4. The force responses during isovelocity lengthening of psoas fibres showed an early rapid increase in force followed by a slow rise in force. The position of this break point in force was sensitive to the phosphate concentration. In soleus fibres, the force responses without phosphate showed an overshoot followed by a slow rise in force. The overshoot diminished with increasing phosphate concentration.5. Phosphate and Ca2+ affected the force responses in psoas and soleus fibres in different ways. When the isometric starting levels were the same, force during and after the length change at submaximal activation was always less than at maximal activation in the presence of 15 mM-phosphate.MS 8477 17 PHY 451 5. J. M. STIENEN AND OTHERS 6. The changes in the mechanical performance during lengthening caused by phosphate in psoas as well as in soleus fibres, are in agreement with a decrease in the average force per attached crossbridge. The results are compatible with a crossbridge model in which phosphate causes a shift of attached crossbridges from a high-forceproducing state to a low-or non-force-producing state. However, a decrease in the number of attached crossbridges might contribute to the kinetics of the phosphate effects.
We evaluated three automated systems for measurement of O2 uptake and compared them with the conventional Douglas bag method. One system (9000 IV Ergometric system) was only tested with respect to its oxygen kinetics, the other two (MMC-Horizon and EOS-Sprint) were tested during three exercise programs: (1) steady-state exercise at 50-W steps from 0 to 200 W, (2) progressive increasing exercise to maximal load, and (3) single-step exercise from 0 to 250 W. The regression lines of mean O2 uptake and load for six subjects were different for intercept (MMC-Horizon) or slope (EOS-Sprint) compared with the conventional method. The maximal O2 uptake values of six subjects were not significantly different for the two systems when compared with the Douglas bag method. The time constants of the exponential function describing oxygen kinetics during repeated (6 times) step changes in load in two subjects were different for the three systems. MMC-Horizon and 9000 IV Ergometric system had lower (51.8 s and 55.1 s, respectively, vs 62.5 s) and EOS-Sprint higher time constant (94.6 vs 47.7 s) than the conventional method. The automated systems were convenient and efficient for measurement of O2 uptake during steady-state and maximal exercise. When O2 uptake kinetics are essential, one has to take into account the response time of the system.
An important goal common to domain adaptation and causal inference is to make accurate predictions when the distributions for the source (or training) domain(s) and target (or test) domain(s) differ. In many cases, these different distributions can be modeled as different contexts of a single underlying system, in which each distribution corresponds to a different perturbation of the system, or in causal terms, an intervention. We focus on a class of such causal domain adaptation problems, where data for one or more source domains are given, and the task is to predict the distribution of a certain target variable from measurements of other variables in one or more target domains. We propose an approach for solving these problems that exploits causal inference and does not rely on prior knowledge of the causal graph, the type of interventions or the intervention targets. We demonstrate our approach by evaluating a possible implementation on simulated and real world data.
We study the performance of Local Causal Discovery (LCD) [5], a simple and efficient constraint-based method for causal discovery, in predicting causal effects in large-scale gene expression data. We construct practical estimators specific to the high-dimensional regime. Inspired by the ICP algorithm [13], we use an optional preselection method and two different statistical tests. Empirically, the resulting LCD estimator is seen to closely approach the accuracy of ICP, the state-of-the-art method, while it is algorithmically simpler and computationally more efficient.
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