Lord's paradox
[4] Judea Pearl used these examples to illustrate how graphical causal models resolve the issue of when control for baseline status is appropriate.
Lord imagines two statisticians who use different common statistical methods but reach opposite conclusions.
The first statistician claims no significant difference between genders: "[A]s far as these data are concerned, there is no evidence of any interesting effect of diet (or of anything else) on student weights.
Visually, the second statistician fits a regression model (green dotted lines), finds that the intercept differs for boys vs girls, and concludes that the new diet had a larger impact for males.
Lord concluded: "there simply is no logical or statistical procedure that can be counted on to make proper allowance for uncontrolled preexisting differences between groups."
While initially framed as a paradox, later authors have used the example to clarify the importance of untestable assumptions in causal inference.
However, unless he too makes further assumptions, his descriptive statement is completely irrelevant to the campus dietician's interest in the effect of the dining hall diet."
The latter is unobservable in the real world, a fact that Holland & Rubin term "the fundamental problem of causal inference" (pg.
Pearl (2016)[5] agrees with Lord’s conclusion that the answer cannot be found in the data, but he finds Holland and Rubin’s account to be incomplete.
In his views, a complete resolution of the Paradox should provide an answer to Lord’s essential question: "How to allow for preexisting differences between groups?"
To this end, Pearl used a simplified version of Lord’s Paradox, proposed by Wainer and Brown,[8] in which gender differences are not considered.
One intuition claims that, to get the needed effect, we must make “proper allowances” for uncontrolled preexisting differences between groups” (i.e. initial weights).
[fn 1] This can also be seen from Figure 2(b), which allows D to causally affect Y while, simultaneously, be statistically independent of it (due to path cancelations).
Broadly, the "fundamental problem of causal inference"[4] and related aggregation concepts Simpson's paradox play major roles in applied statistics.
Lord's Paradox and associated analyses provide a powerful teaching tool to understand these fundamental statistical concepts.
Unless there is experimental evidence to support the notion that there are indeed different paths of direct and indirect effects from birth weight to BP, we are cautious of using such terminology to label the results from multiple regression, as with model 3.
