What is invariance and how to test it.

1. What is invariance and
how to test it
Federica Russo
Center Leo Apostel, VrijeUniversiteitBrussel

2. Overview
Invariance
The concept inherited from early econometrics
Invariance across changes
Of the effect-variable due to interventions in the cause-variable
Of the environment, i.e. across appropriate partitions of the dataset
Invariance of what?
Of joint-variations of variables
Of regular associations
Is it worthextendingthe concept?
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3. Invariance in early econometrics
The beginning of contemporary causal modelling:
Modelling and testing economic structures
Marschak, Frisch, Haavelmo:
Learning invariance properties of the model or economic
system from data
Test whether hypothetical variations (theoretical level) in
economic structure are actually invariant (empirical level)
Invariance of the parametrisation of the system
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4. Invariance across changes
Experimental contexts
Changes in the effect-variable
due to interventions in the cause-variable
Observational contexts
Changes across environments,
i.e. across appropriate partitions of the data set
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5. Changes in the effect-variable
P V = n R T
Pressure
Volume
Number
of moles
Gas
constant
Temperature
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6. Woodward’s conditions
Interventions I on cause-variable X:
Change in X (cause) is totally due to I
Change in Y (effect) totally due to change in X
I is not correlated with other possible causes of Y
If all this holds – and that’s a big if –
empirical generalisations are causal
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7. Changes in the environment
Environments are
appropriate partitions of the population of reference
Age groups
Socio-economic conditions
Exposures
…
Invariance is a test for stability across environments
Of the causal structure (arrangements)
Of the parametrisation (numerical values)
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8. What are the causes of self-rated health
in the Baltic countries in the ‘90s? 8

9. What environments?
1994 and 1999 data sets
Estonian Males; Estonian Females
Latvian Males; Latvia Females
Lithuanian Males; Lithuanian Females
Age groups (18–29, 30–44, 45–59, 60+)
Autochthons and other (mainly Russians)
Background knowledge looms large …
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10. Stability of the parametrisation
Baltic study: parametrisation stable for most
environments:
Time-frames, Gender, Age, Ethnical groups
Impact of alcohol consumption on self-rated health
Sign of parameter and numerical value are stable
Female Male
Estonia −0.094 −0.039
Latvia −0.181 −0.054
Lithuania −0.157 −0.068
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11. 11

12. Parametrisation of
the causal model is
stable for each
environment
The structure (= arrangement of variables)
is the same for
3 Baltic countries studied
Men and women
Ethnic groups
…
Stability of the causal structure
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13. Parametrisation and causal structure
Clearly not independent
If stability of parametrisationfails,
we are led to rethink causal structure
for at least some sub-populations
Homogeneity and heterogeneity
of population of reference
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14. What has to be invariant?
Detect joint-variations within and
between variables
Woodward: change-relating relations
How regular are joint variations?
Dependencies Strength of association
How stable are dependencies?
Invariance across changes
Variation
Regularity
Invariance
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15. Variation
• Visibility
No
variation, no
statistical
analysis
Regularity
• How strong
Frequency of
occurrence
Invariance
• How stable
Invariance
across changes
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16. Why extending
the concept of invariance?
Do not conflate
experimental and policy interventions
Interventions to get causal knowledge vs interventions based
on available causal knowledge
Avoid gold standards
Compare comparable methods
Preserving diversity of causal methods
Observational and experimental methods
need different tests
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17. For example, the authors note that some association appears between smoking and lung cancer in every well-
designed study on sufficiently large and representative populations with which they are familiar. There is
evidence of a higher frequency of lung cancer among smokers than among
nonsmokers, when potentially confounding variables are controlled for, among both men and
women, among people of different genetic backgrounds, across different diets, different
environments, and different socioeconomic conditions[…]. The precise level and quantitative
details of the association do vary, for example, the incidence of lung cancer among smokers is higher in lower
socioeconomic groups, but the fact that there is some association or other is stable or robust across a wide variety
or different groups and background circumstances. […] Thus, although Cornfield et al. do not exhibit a precise
deterministic or probabilistic generalization that is invariant across different circumstances [meaning: across
interventions] the cumulative impact of their evidence is to show that the relationship between
smoking and lung cancer is relatively invariant in the weak sense described above.
(Woodward 2003, p.312, emphasis and brackets added)
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