Simpsons paradox says that a relationship can __________ when an additional variable is considered.
The main issue is that you are equating one simple way to show the paradox as the paradox itself. The simple example of the contingency table is not the paradox per se. Simpson's paradox is about conflicting causal intuitions when comparing marginal and conditional associations, most often due to sign reversals (or extreme attenuations such as independence, as in the original example given by Simpson himself, in which there isn't a sign reversal). The paradox arises when you interpret both estimates causally, which could lead to different conclusions --- does the treatment help or hurt the patient? And which estimate should you use? Whether the paradoxical pattern shows up on a contingency table or in a regression, it doesn't matter. All variables can be continuous and the paradox could still happen --- for instance, you could have a case where $\frac{\partial E(Y|X)}{\partial X} > 0$ yet $\frac{\partial E(Y|X, C = c)}{\partial X} < 0, \forall c$.
This is incorrect! Simpson's paradox is not a particular instance of confounding error -- if it were just that, then there would be no paradox at all. After all, if you are sure some relationship is confounded you would not be surprised to see sign reversals or attenuations in contingency tables or regression coefficients --- maybe you would even expect that. So while Simpson's paradox refers to a reversal (or extreme attenuation) of "effects" when comparing marginal and conditional associations, this might not be due to confounding and a priori you can't know whether the marginal or the conditional table is the "correct" one to consult to answer your causal query. In order to do that, you need to know more about the causal structure of the problem. Consider these examples given in Pearl:  Imagine that you are interested in the total causal effect of $X$ on $Y$. The reversal of associations could happen in all of these graphs. In (a) and (d) we have confounding, and you would adjust for $Z$. In (b) there's no confounding, $Z$ is a mediator, and you should not adjust for $Z$. In (c) $Z$ is a collider and there's no confounding, so you should not adjust for $Z$ either. That is, in two of these examples (b and c) you could observe Simpson's paradox, yet, there's no confounding whatsoever and the correct answer for your causal query would be given by the unadjusted estimate. Pearl's explanation of why this was deemed a "paradox" and why it still puzzles people is very plausible. Take the simple case depicted in (a) for instance: causal effects can't simply reverse like that. Hence, if we are mistakenly assuming both estimates are causal (the marginal and the conditional), we would be surprised to see such a thing happening --- and humans seem to be wired to see causation in most associations. So back to your main (title) question:
In a sense, this is the current definition of Simpson's paradox. But obviously the conditioning variable is not hidden, it has to be observed otherwise you would not see the paradox happening. Most of the puzzling part of the paradox stems from causal considerations and this "hidden" variable is not necessarily a confounder. Contigency tables and regression As discussed in the comments, the algebraic identity of running a regression with binary data and computing the differences of proportions from the contingency tables might help understanding why the paradox showing up in regressions is of similar nature. Imagine your outcome is $y$, your treatment $x$ and your groups $z$, all variables binary. Then the overall difference in proportion is simply the regression coefficient of $y$ on $x$. Using your notation: $$ \frac{a+b}{c+d} - \frac{e+f}{g+h} = \frac{cov(y,x)}{var(x)} $$ And the same thing holds for each subgroup of $z$ if you run separate regressions, one for $z=1$: $$ \frac{a}{c} - \frac{e}{g} = \frac{cov(y,x|z =1)}{var(x|z=1)} $$ And another for $z =0$: $$ \frac{b}{d} - \frac{f}{h} = \frac{cov(y,x|z=0)}{var(x|z=0)} $$ Hence in terms of regression, the paradox corresponds to estimating the first coefficient $\left(\frac{cov(y,x)}{var(x)}\right)$ in one direction and the two coefficients of the subgroups $\left(\frac{cov(y,x|z)}{var(x|z)}\right)$ in a different direction than the coefficient for the whole population $\left(\frac{cov(y,x)}{var(x)}\right)$. What does Simpson's paradox show?Simpson's Paradox is a statistical phenomenon where an association between two variables in a population emerges, disappears or reverses when the population is divided into subpopulations.
What does Simpson's paradox teach us quizlet?-In probability and statistics, Simpson's paradox is a paradox in which a correlation present in different groups is reversed when the groups are combined. -An association in sub-populations may be reversed in the population.
What is Simpson's paradox in statistics quizlet?Simpson's paradox. occurs when an association or comparison that holds for all of several groups reverses direction when the data are combined to form a single group. confounding. occurs when effects of two explanatory variables on a response variable cannot be distinguished from each other. common response.
How does Simpson's paradox affect the outcome of a study?The effect of Simpson's paradox in experimental research is that a false association can lead to an incorrect conclusion. The effect of the incorrect conclusion is that a researcher may admit a wrong treatment and even, the researcher may continue to make a further study on the incorrect conclusion.
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