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

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.

• Think you should specify the regression model even if your notation's fairly standard, to avoid confusion. – Scortchi - Reinstate Monica Mar 13 '15 at 14:00
• The misunderstanding on which this question is based is addressed in your subsequent question at stats.stackexchange.com/questions/141607/…. – whuber Mar 13 '15 at 14:43
• @whuber: Sorry I don't see my misunderstanding yet. My Berkson's paradox question seems to me a special case where I think "rendered dependent" means $\hat\beta_1 \ne 0$ but I want to know if CV community thinks it means $\beta_1 \ne 0$ or something else. Other question I hope is more general theory (of which Berkson's is a special case) and thus I hope they illuminate rather than duplicate each other. But perhaps someone has a simple answer to $\beta_1 = 0$ question that will make Berkson's question obvious. – jtd Mar 13 '15 at 15:57
• @whuber: Sorry, I hope latest edits are clear. I just want to be clear that Berkson's Paradox requires the true $\beta_1 = 0$. This is in part because I wondered if NeilG's answer here was in fact an example of Berkson's Paradox or something else (stats.stackexchange.com/questions/141309/…). If it is a BP example, then I know how to classify it; if not, I need to learn something from NeilG. – jtd Mar 13 '15 at 20:20
• You seem to be thinking that to say the true value of $\beta_1\neq0$ implies $x_2$ causes $y$. It doesn't; it implies only a correlation. If the true value of $\beta_1\neq0$ because $y$ & $x_1$ have causal effects on $x_2$ then $\hat\beta_1$ is a consistent estimator of the non-zero $\beta_1$ (it approaches it closer & closer as the sample size increases). – Scortchi - Reinstate Monica Mar 13 '15 at 23:49