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Identification Analysis

Overview

Identification analysis answers whether a causal parameter can be written as a function of the observed data distribution, given a Graphical Causal Model (GCM) and its assumptions. In causalinf, identification analysis is conducted with the method identification~analysis~() of a gcm.DAG object.

The analysis is conditional on the graph and on the assumptions discussed in Check Assumptions. It does not determine whether the graph is true. Instead, it answers a conditional question: if the GCM is a defensible representation of the data-generating process, which causal effects are identified and by which strategy?

The current GCM identification analysis checks the following strategies:

Strategy Abbreviation What it searches for
Selection on Observables SoO Adjustment sets for total and controlled direct effects.
Instrumental Variable IV Instruments, and any required adjustment variables, for the average causal effect.
do-calculus do A causal probability expression for the average causal effect.

The main causal parameters are:

Parameter Meaning
ACE Average Causal Effect
ACDE Average Controlled Direct Effect
cACE Conditional Average Causal Effect
cACDE Conditional Average Controlled Direct Effect

Basic analysis

Consider the front-door example included with the package:

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from causalinf import gcm
from causalinf import options as opt

opt.set_options(print_assumptions=False, print_assumptions_verbose=False)

G = gcm.examples("Front-door")
print(G)

Graph:
U -> Y
Z -> Y
Z2 -> D
D -> Z
U -> D
Z2 -> Y
Observed: Z, Z2
Exposure: D
Outcome: Y
Latent: U

The method identification_analysis() runs the identification search. If verbose=True (the default), the result is printed.

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G.identification_analysis()

Searching for identification by adjustment variables for ACE...done!
Searching for identification by adjustment variables for ACDE...done!
Searching for identification by instrumental variables...done!
Searching for identification by do-calculus...done!

Exposure: D
Outcome: Y        


Average Causal Effect (ACE)
---------------------            
Method: Selection on Observables (SoO)
Identified: False
Not identifiable by adjustment.

Method: do-calculus (do)
Identified: True
Causal probability: p(Y | do(D)) = sum_{Z,Z2} p(Z|D)sum_{D} p(D,Z2)p(Y|Z,D,Z2)

Method: Instrumental Variable (IV)
Identified: False
No instrument available in the DAG.                

Average Controlled Direct Effect (ACDE)
--------------------------------            
Method: Selection on Observables (SoO)
Identified: False
Not identifiable by adjustment.

The output reports the exposure, outcome, causal parameters, identification strategies, and whether each strategy identifies the parameter.

Concise output

After running identification_analysis(), the result is stored in the GCM object. Use G.identification() or G.print(what="identification") to print the stored result again.

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G.identification(print="concise")

Exposure: D
Outcome: Y        

==============================================================================
                                              Identified?
Causal Effects*                           SoO  IV   do-calculus 
==============================================================================
Averave Causal Effect (ACE)              No    No        Yes
Averave Controlled Direct Effect (ACDE)  No    NA/NC**    NA/NC**
==============================================================================
Notes:
*  For path effects (indirect effects), use SEM estimation
** NA/NC: Identification analysis not available or not conducted

The concise output is useful for quickly checking which causal parameters are identified by each strategy.

Detailed output

For more information about a specific strategy and parameter, use content="detailed" through G.print(). The argument strategy selects the identification strategy, and parameter selects the causal parameter.

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G.print(
    what="identification",
    identification={
        "content": "detailed",
        "strategy": "do",
        "parameter": "ACE",
    },
)

Exposure: D
Outcome: Y        

Identification method: do
---------------------            
Parameter: Average Causal Effect (ACE)
  tau_{ACE}(d, d') = E[Yi(d) - Yi(d')]
Identification:
  p(Y | do(D)) = sum_{Z,Z2} p(Z|D)sum_{D} p(D,Z2)p(Y|Z,D,Z2)
Details:
  tau_{ACE}(d, d') = E[Yi | do(Di)=d] - E[Yi | do(Di)=d']
  where
   Yi: Outcome (Y)
   Di: Exposure or treatment (D)
Models:
  Non-parametric: p(Y | do(D)) = sum_{Z,Z2} p(Z|D)sum_{D} p(D,Z2)p(Y|Z,D,Z2)

When the effect is identified by do-calculus, the detailed output includes the causal probability expression. When the effect is identified by adjustment or by an instrumental variable strategy, the detailed output includes the corresponding adjustment, formula, and parameter information.

Programmatic results

The printed output is convenient for reports, but the identification results can also be retrieved as Python objects.

The full result is available in G.identification_dict.

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from pprint import pprint

pprint(G.identification_dict)

{'IV': {'ACE': {'adjusted': False,
                'conditional on': None,
                'conducted': True,
                'exposure': ['D'],
                'formulas': {'non-parametric': '', 'parametric': ''},
                'identified': False,
                'latex': {'Non-parametric': '',
                          'Parametric tau': '',
                          'Parametric(*)': '\n'
                                           '(*) Under linearity and no '
                                           'interaction'},
                'outcome': 'Y',
                'result': {},
                'result str': 'No instrument available in the DAG.',
                'text': {'Non-parametric': '',
                         'Parametric tau': '',
                         'Parametric(*)': '\n'
                                          '(*) Under linearity and no '
                                          'interaction'},
                'variables': None,
                'where': ''}},
 'SoO': {'ACDE': {'adjusted': False,
                  'conditional on': None,
                  'conducted': True,
                  'exposure': ['D'],
                  'formulas': {'non-parametric': '', 'parametric': ''},
                  'identified': False,
                  'latex': {'Non-parametric': '',
                            'Parametric tau': '',
                            'Parametric(*)': '\n'
                                             '(*) Under linearity and no '
                                             'interaction'},
                  'outcome': 'Y',
                  'result': None,
                  'result str': 'Not identifiable by adjustment.',
                  'text': {'Non-parametric': '',
                           'Parametric tau': '',
                           'Parametric(*)': '\n'
                                            '(*) Under linearity and no '
                                            'interaction'},
                  'variables': None,
                  'where': ''},
         'ACE': {'adjusted': False,
                 'conditional on': None,
                 'conducted': True,
                 'exposure': ['D'],
                 'formulas': {'non-parametric': '', 'parametric': ''},
                 'identified': False,
                 'latex': {'Non-parametric': '',
                           'Parametric tau': '',
                           'Parametric(*)': '\n'
                                            '(*) Under linearity and no '
                                            'interaction'},
                 'outcome': 'Y',
                 'result': None,
                 'result str': 'Not identifiable by adjustment.',
                 'text': {'Non-parametric': '',
                          'Parametric tau': '',
                          'Parametric(*)': '\n'
                                           '(*) Under linearity and no '
                                           'interaction'},
                 'variables': None,
                 'where': ''}},
 'do': {'ACE': {'adjusted': False,
                'conditional on': None,
                'conducted': True,
                'exposure': ['D'],
                'formulas': {'non-parametric': 'p(Y | do(D)) = sum_{Z,Z2} '
                                               'p(Z|D)sum_{D} '
                                               'p(D,Z2)p(Y|Z,D,Z2)',
                             'parametric': 'p(Y | do(D)) = sum_{Z,Z2} '
                                           'p(Z|D)sum_{D} p(D,Z2)p(Y|Z,D,Z2)'},
                'identified': True,
                'latex': {'Non-parametric': '$p(Y | do(D)) = '
                                            '\\sum_{Z,Z2}\\left(p(Z|D)\\sum_{D}\\left(p(D,Z2)p(Y|Z,D,Z2)\\right)\\right)$',
                          'Parametric tau': '',
                          'Parametric(*)': '$p(Y | do(D)) = '
                                           '\\sum_{Z,Z2}\\left(p(Z|D)\\sum_{D}\\left(p(D,Z2)p(Y|Z,D,Z2)\\right)\\right)$'},
                'outcome': 'Y',
                'result': 'p(Y | do(D)) = sum_{Z,Z2} p(Z|D)sum_{D} '
                          'p(D,Z2)p(Y|Z,D,Z2)',
                'result str': 'Causal probability: p(Y | do(D)) = sum_{Z,Z2} '
                              'p(Z|D)sum_{D} p(D,Z2)p(Y|Z,D,Z2)',
                'text': {'Non-parametric': 'p(Y | do(D)) = sum_{Z,Z2} '
                                           'p(Z|D)sum_{D} p(D,Z2)p(Y|Z,D,Z2)',
                         'Parametric tau': '',
                         'Parametric(*)': 'p(Y | do(D)) = sum_{Z,Z2} '
                                          'p(Z|D)sum_{D} p(D,Z2)p(Y|Z,D,Z2)'},
                'variables': {'Di': ['D'], 'Yi': 'Y'},
                'where': 'where\n'
                         '  Yi: Outcome (Y)\n'
                         '  Di: Exposure or treatment (D)'}}}

To retrieve only the strategies that identify each parameter, use G.get_identified().

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pprint(G.get_identified())

{'ACE': ['do']}

Use by="strategy" to group the result by identification strategy instead of by causal parameter.

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pprint(G.get_identified(by="strategy"))

{'do': ['ACE']}

By default, get_identified() omits strategies that do not identify a parameter. Use include_all=True when the non-identifying strategies should also be shown.

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pprint(G.get_identified(include_all=True))

{'ACDE': ['SoO'], 'ACE': ['SoO', 'do', 'IV']}

Identification by adjustment

For graphs in which adjustment is sufficient, identification_analysis() reports the valid adjustment set for the ACE and ACDE.

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G_adj = gcm.examples("Two confounders")
G_adj.identification_analysis()

Searching for identification by adjustment variables for ACE...done!
Searching for identification by adjustment variables for ACDE...done!
Searching for identification by instrumental variables skipped.
Searching for identification by do-calculus skipped.

Exposure: D
Outcome: Y        


Average Causal Effect (ACE)
---------------------            
Method: Selection on Observables (SoO)
Identified: True
Adjustments: {Z1, Z2}

Method: do-calculus (do)
Identified: False
Not conducted. Identification by adjustment or instrumental variable available.

Method: Instrumental Variable (IV)
Identified: False
Not conducted. Identification by adjustment available.                

Average Controlled Direct Effect (ACDE)
--------------------------------            
Method: Selection on Observables (SoO)
Identified: True
Adjustments: {Z1, Z2}

If identification by adjustment succeeds, instrumental-variable and do-calculus searches are skipped by default when they are not needed. To force these searches, use iv="always" and causal_probability="always".

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G_adj.identification_analysis(iv="always", causal_probability="always")

Searching for identification by adjustment variables for ACE...done!
Searching for identification by adjustment variables for ACDE...done!
Searching for identification by instrumental variables...done!
Searching for identification by do-calculus...done!

Exposure: D
Outcome: Y        


Average Causal Effect (ACE)
---------------------            
Method: Selection on Observables (SoO)
Identified: True
Adjustments: {Z1, Z2}

Method: do-calculus (do)
Identified: True
Causal probability: p(Y | do(D)) = sum_{Z1,Z2} p(Z1,Z2)p(Y|D,Z1,Z2)

Method: Instrumental Variable (IV)
Identified: False
No instrument available in the DAG.                

Average Controlled Direct Effect (ACDE)
--------------------------------            
Method: Selection on Observables (SoO)
Identified: True
Adjustments: {Z1, Z2}

Identification by instrumental variables

Instrumental-variable identification is searched for the ACE. It is especially useful when the exposure and outcome are confounded but the graph contains a valid instrument.

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G_iv = gcm.examples("IV with 1 instrument")
G_iv.identification_analysis()

Searching for identification by adjustment variables for ACE...done!
Searching for identification by adjustment variables for ACDE...done!
Searching for identification by instrumental variables...done!
Searching for identification by do-calculus skipped.

Exposure: X
Outcome: Y        


Average Causal Effect (ACE)
---------------------            
Method: Selection on Observables (SoO)
Identified: False
Not identifiable by adjustment.

Method: do-calculus (do)
Identified: False
Not conducted. Identification by adjustment or instrumental variable available.

Method: Instrumental Variable (IV)
Identified: True
Instrument: Z                 

Average Controlled Direct Effect (ACDE)
--------------------------------            
Method: Selection on Observables (SoO)
Identified: False
Not identifiable by adjustment.

The IV output reports whether an instrument is available in the DAG. In examples with multiple instruments, the result can include instruments that require adjustment.

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G_iv_multi = gcm.examples("IV with 3 instruments")
G_iv_multi.identification_analysis()

Searching for identification by adjustment variables for ACE...done!
Searching for identification by adjustment variables for ACDE...done!
Searching for identification by instrumental variables...done!
Searching for identification by do-calculus skipped.

Exposure: D
Outcome: Y        


Average Causal Effect (ACE)
---------------------            
Method: Selection on Observables (SoO)
Identified: False
Not identifiable by adjustment.

Method: do-calculus (do)
Identified: False
Not conducted. Identification by adjustment or instrumental variable available.

Method: Instrumental Variable (IV)
Identified: True
Instrument: Z1 (if adjusted by Z2) or 
            Z3  or 
            Z4 (if adjusted by X2, Z2)                

Average Controlled Direct Effect (ACDE)
--------------------------------            
Method: Selection on Observables (SoO)
Identified: False
Not identifiable by adjustment.

Conditional effects

Conditional causal effects can be requested with the argument conditional. The argument can be a string for one conditioning variable or a list for more than one.

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G_cond = gcm.examples("Two confounders")
G_cond.identification_analysis(conditional="Z1")

Searching for identification by adjustment variables for cACE...done!
Searching for identification by adjustment variables for cACDE...done!
Searching for identification by instrumental variables skipped.
Searching for identification by do-calculus skipped.

For conditional effects, the reported parameters are prefixed with c, such as cACE and cACDE. Instrumental-variable identification is not available for conditional average effects in the current implementation.

Identification results are conditional on the assumptions encoded in the GCM. To print the identification result together with the identification assumptions, set print_assumptions=True.

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G.print(
    what="identification",
    identification={
        "print_assumptions": True,
        "print_assumptions_verbose": True,
    },
)

Exposure: D
Outcome: Y        


Average Causal Effect (ACE)
---------------------            
Method: Selection on Observables (SoO)
Identified: False
Not identifiable by adjustment.

Method: do-calculus (do)
Identified: True
Causal probability: p(Y | do(D)) = sum_{Z,Z2} p(Z|D)sum_{D} p(D,Z2)p(Y|Z,D,Z2)

Method: Instrumental Variable (IV)
Identified: False
No instrument available in the DAG.                

Average Controlled Direct Effect (ACDE)
--------------------------------            
Method: Selection on Observables (SoO)
Identified: False
Not identifiable by adjustment.                

Assumptions for identification:
------------------------------
1. Correct DAG
   - Definition: DAG structure matches the true causal relations: (a) A directed arrow from a variable A to a variable B means that there is a causal effect of A on B, which may or may not be zero; (b) Absence of an arrow from a variable C to a variable D implies certainty that C does not cause D; (c) A bidirected arrow between a variable E and a variable F means that they share a common unobserved or latent cause.
   - Scope: Connection between reality and the DAG model
   - Role: Ensures adjustment sets and do-calculus yield the correct identifiable causal effect.
   - Usage: identification
   - Violation: Biased or invalid causal effect estimates and incorrect adjustment sets.            
2. Causal Markov Condition (CMC)
   - Definition: Each variable is independent of its non-descendants given its parents
   - Scope: Connects the DAG and the conditional distribution of each variable
   - Role: Links d-separation to conditional independencies, grounding do-calculus.
   - Usage: identification, discovery
   - Violation: Graph–distribution link breaks and identification results may be incorrect.            
3. Positivity (Overlap)
   - Definition: Each treatment level has a positive probability of occurring, including at all relevant levels of the adjustment variables if they are used for identification.
   - Scope: Variables' distributions
   - Role: Required for the g-formula, IPW, and many identification and estimation strategies.
   - Usage: identification, estimation, inference
   - Violation: Effects are undefined or non-estimable in certain regions of the covariate space.

For a fuller discussion of the assumptions themselves, see Check Assumptions.

Practical workflow

A typical GCM identification workflow is:

  1. Build the graph with the exposure, outcome, observed variables, and latent variables.
  2. Check whether the graph represents the intended causal assumptions.
  3. Run G.identification_analysis().
  4. Inspect the default or concise output to see which parameters are identified.
  5. Use detailed output for the strategy and parameter that will be reported.
  6. Retrieve G.identification_dict or G.get_identified() when the result is needed programmatically.
  7. If an assumption is uncertain, compare the identification result with an alternative graph.

References

  • Ferrari, D. (forthcoming). The Identification of Causal Effects. Cambridge University Press.
  • Pearl, J. (2009). Causality: Models, Reasoning and Inference. Cambridge University Press.