In Berkson's paradox, is $\beta_1 = 0$ or $\ne 0$?

Question

There is "a general phenomenon known as Berkson’s paradox (Berkson, 1946), whereby observations on a common consequence of two independent causes render those causes dependent. For example, the outcomes of two independent coins are rendered dependent by the testimony that at least one of them is a tail" (Pearl, 2009).

$$ y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \epsilon $$

If (a) we make a linear regression model as above in which $x_2$ is a common consequence of independent causes $y$ and $x_1$ and (b) these latter two are rendered dependent by observations on $x_2$:

• Is $\hat\beta_1 \ne 0$? Why?
  • "Yes, because that is what Pearl must mean by "rendered dependent."
• Is the true value of $\beta_1 = 0$? Why?
  • "Yes, because that is what Pearl must mean by "independent causes."

*Or, do we not have enough information?

*Or, is Berkson's paradox not applicable (e.g., to a linear regression model with a continuous $y$)?

Berkson, J. (1946). Limitations of the application of fourfold table analysis to hospital data. Biometrics Bulletin 2, 47–53.

Answer 1

Your question is quite similar to this one Which OLS assumptions are colliders violating? I suggest you to see that one too and links therein.

My first point is that if you want to follow Pearl's Theory you have to use related rules. Probably confusion come from that. Therefore, first of all, you must distinguish clearly structural equations/parameters from regression one.

The structural causal model (SCM) you are interested in seems me \begin{aligned} Y&=u_Y \\ X_2&=\beta_1X_1+\beta_2Y+u_{X_2} \\ X_1&=u_{X_1} \end{aligned} so that $X_2$ is a collider: $X_1\rightarrow X_2\leftarrow Y$.

structural errors can be considered the exogenous variables in the system and we assume them as zero mean rvs independent each others. Note that one implication of that is: $E[u_{X_2}|X_1,Y]=0$. Note that, in general, the SCM encode (explicitly) all causal assumptions made by the researcher. In this case the SCM is consistent with Berkson's paradox.

Now your question is as follow: if you perform this regression

$ Y=\theta_0 + \theta_1X_1+\theta_2X_2+\epsilon $

what you have to expect as value for $\theta_1$?

Here $\theta_1$ stay for your $\hat\beta_1$. I do not consider hat symbol for $\theta$ because we can avoid sample/population difference and my notational distinction help us.

Now the answer: you have to expect $\theta_1 \ne 0$

It is so because $X_1$ and $Y$ are marginally independent but become dependent conditioning on collider ($X_2$). Note that $\theta_1$ do not have any causal role, do not identify any causal parameter. This is why, in general, to control for collider is not a good idea.

Indeed if you compute

$ Y=\theta_3 + \theta_4X_1+\epsilon_2 $

$\theta_4$ identify the causal effect of $X_1$ on $Y$, in fact $\theta_4=0$