page_title: Causal graph
section_title: Example
hierarchical_section_title: Causal graph / Example
caption_attribution_description: English: Figure 3: Identified model with latent variables ( Q 1 {\displaystyle Q_{1}} and Q 2 {\displaystyle Q_{2}} ) shown explicitly
context_page_description: In statistics, econometrics, epidemiology, genetics and related disciplines, causal graphs are probabilistic graphical models used to encode assumptions about the data-generating process. They can also be viewed as a blueprint of the algorithm by which Nature assigns values to the variables in the domain of interest.
  Causal graphs can be used for communication and for inference. As communication devices, the graphs provide formal and transparent representation of the causal assumptions that researchers may wish to convey and defend. As inference tools, the graphs enable researchers to estimate effect sizes from non-experimental data, derive testable implications of the assumptions encoded, test for external validity, and manage missing data and selection bias.
  Causal graphs were first used by the geneticist Sewall Wright under the rubric "path diagrams". They were later adopted by social scientists and, to a lesser extent, by economists. These models were initially confined to linear equations with fixed parameters.
context_section_description: Suppose we wish to estimate the effect of attending an elite college on future earnings.  Simply regressing earnings on college rating will not give an unbiased estimate of the target effect because elite colleges are highly selective, and students attending them are likely to have qualifications for high-earning jobs prior to attending the school.  Assuming that the causal relationships are linear, this background knowledge can be expressed in the following structural equation model (SEM) specification.
  Model 1
  where  represents the individual's qualifications prior to college,  represents qualifications after college,  contains attributes representing the quality of the college attended, and  the individual's salary.
  Figure 1 is a causal graph that represents this model specification.  Each variable in the model has a corresponding node or vertex in the graph.  Additionally, for each equation, arrows are drawn from the independent variables to the dependent variables.  These arrows reflect the direction of causation.  In some cases, we may label the arrow with its corresponding structural coefficient as in Figure 1.
  If  and  are unobserved or latent variables their influence on  and  can be attributed to their error terms.  By removing them, we obtain the following model specification:
  Model 2
  The background information specified by Model 1 imply that the error term of , , is correlated with C's error term, .  As a result, we add a bidirected arc between S and C, as in Figure 2.
  Since  is correlated with  and, therefore, ,  is endogenous and  is not identified in Model 2. However, if we include the strength of an individual's college application, , as shown in Figure 3, we obtain the following model:
  Model 3
  By removing the latent variables from the model specification we obtain:
  Model 4
  with  correlated with .
  Now,  is identified and can be estimated using the regression of  on  and . This can be verified using the single-door criterion, a necessary and sufficient graphical condition for the identification of a structural coefficients, like , using regression.
caption: Figure 3: Identified model with latent variables (
    
      
        
          
            Q
            
              1
            
          
        
      
      {\displaystyle Q_{1}}
    
   and 
    
      
        
          
            Q
            
              2
            
          
        
      
      {\displaystyle Q_{2}}
    
  ) shown explicitly
exif: {}