Suppose we study the effect of random variable $X$ on $Y$ and we suspect a third variable $C$ to be a confounder.

Is there a sound way to highlight this suspicion ?

I can imagine showing the correlation between $C$ and $X$, and between $C$ and $Y$, but is this sufficient ?

Also, can I prove anything without a causal study ?

  • $\begingroup$ Numeric or categorical variables? And, if categorical, how many categories? $\endgroup$ – Creosote Sep 16 '15 at 15:38
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    $\begingroup$ One way would be to show that the relationship between $X$ and $Y$ conditioned on $C$ is weaker than the relationship between $X$ and $Y$ without conditioning. $\endgroup$ – Michael K Sep 16 '15 at 18:57
  • $\begingroup$ @Creosote continuous, but let's assume they're categorical with 10's of categories. $\endgroup$ – oDDsKooL Sep 17 '15 at 6:45
  • $\begingroup$ @MichaelK do you mean e.g. $corr(X,Y | C) < corr(X,Y)$ ? $\endgroup$ – oDDsKooL Sep 17 '15 at 6:46
  • $\begingroup$ just compute covariance to know its value. answer to your 2nd question is no. Third part of your question is too vague $\endgroup$ – Subhash C. Davar Sep 17 '15 at 14:45

I'm going to assume that $X$ is marginally dependent on $Y$, and $C$ is marginally dependent on both $X$ and $Y$. Furthermore, I'm going to assume that though $X$ may or may not affect $Y$, there's no situation in which $Y$ causes $X$. Given these dependences and the Causal Markov Assumption, there are only eight Directed Acyclic Graphs (DAGs) that can describe the causal relationships between these variables, and they fall into only four Markov equivalence classes. Let's talk about the patterns of correlation and partial correlation* you would see in these eight possibilities (shown below).

[*] You mentioned correlation in the question, so I'll assume the data are multivariate Gaussian, and just talk about correlation rather than dependence in general.

equivalence classes compatible with assumptions

In the first two equivalence classes (graphs 1a, 2a and 2b), you would see that $\rho_{XY} = \rho_{XY.C}$, that is, conditioning on $C$ does not change the strength of the correlation between $X$ and $Y$. $C$ is not a confounder in any of these cases.

In the third equivalence class (graphs 3a and 3b), $X$ and $Y$ are correlated ($\rho_{XY} \neq 0$) but they have zero partial correlation on $C$ ($\rho_{XY.C} = 0$). However, in graph 3b $C$ is a confounder, whereas in graph 3a, $C$ is a mediating variable: $X$ affects $Y$ through $C$. If the true graph is 3a, then the true effect of $X$ on $Y$ is given by $\rho_{XY}$, whereas if the true graph is 3b, the true effect of $X$ on $Y$ is given by $\rho_{XY.C} = 0$. But the patterns of correlations are identical between the two models. Correlations can't distinguish between these two possibilities.

The fourth equivalence class is similar. If the true graph is 4a, then the total causal effect of $X$ on $Y$ is given by $\rho_{XY}$, and the direct causal effect (i.e. the effect that doesn't go through the mediating variable, $C$) is given by $\rho_{XY.C}$. So both the marginal correlation and the partial correlation give you causal quantities of interest. However, if the true graph is model 4b, then only $\rho_{XY.C}$ gives you the effect of $X$ on $Y$; the marginal correlation $\rho_{XY}$ will be confounded by $C$. Whereas if the true graph is 4c, then only $\rho_{XY}$ gives you the effect of $X$ on $Y$. If you were to condition on $C$ you would induce "collider bias," so $\rho_{XY.C}$ would give a biased estimate of the effect of $X$ on $Y$. Again, the three models cannot be distinguished by correlations or partial correlations alone.

The upshot is: you need more than correlations to decide whether $C$ is a confounder or not. The options are:

  1. Causal background knowledge: Maybe you think it's very implausible that $X$ should influence $C$. Then you can rule out some of these eight options.
  2. Controlled experiment: If you manipulate $X$ you will learn the true effect on $Y$.
  3. Non-Gaussian data or non-linear relationships between variables: If your data is of the right form, there are causal structure learning algorithms that can distinguish between models that fall into the same Markov Equivalence Class.
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  • $\begingroup$ excellent answer, thanks! do you have any pointers on causal structure learning algorithms ? $\endgroup$ – oDDsKooL Oct 20 '15 at 8:48

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