Simpson’s Paradox

Signal-to-noise ratio, 71, 154 Simpson s paradox, 231 Slope covariance, 2491 Spare receptors, 21 Specific binding, 60 Spiropiperidines, 155f Spongo thymidine, 148 Spongouridine, 148 SSq, 239 SSqc, 241... 

This phenomenon, known as Simpson s Paradox, illustrates the dangers of simple pooling. A meta-analysis for the data in the example would be based on the separate tables and compute a treatment difference for those patients with small stones and a treatment difference for patients with larger stones. These two differences would then be averaged to give a valid combined treatment difference. 

Julious SA and MuUeeMA (1994) Confounding and Simpson s Paradox British Medical Journal, 309, 1480-1481... 

Averaging of proportionate responses from non-homogenous treatment groups, also known as Simpson s paradox (see Spilker, 1991)... 

Simpson s paradox is a phenomenon whereby an overall treatment effect in one direction is reversed in every stratum. It is named after a paper of 19 51 in which it was extensively examined by E.H. Simpson (Simpson, 1951), although a related phenomenon had been discussed at least as early as 1899 by Karl Pearson (Pearson et al., 1899). (See Aldrich (1995) for discussion.) A hypothetical example is given in Table 9.1, in which a trial has been run comparing two treatments in two strata of patients moderately ill and... 

Of course, in a clinical trial we would never see imbalances like this at baseline and would therefore very rarely expect to be faced with Simpson s paradox. Its importance is far greater in an epidemiological context. For example, we might suspect when a new drug is launched on the market that patients with the most refractory disease might be more likely to take it. In the example there were 600 severe patients who took A and only 60 who took B, the numbers being reversed for moderate disease, so this might be a possible pattern if A were a newer treatment. Disease severity would then be a confounder for which it was appropriate to adjust. 

In other words, what is a main effect in logistic regression terms becomes an interaction in log-linear model terms. However, it is really the log-linear model that is the odd one out among generalized linear models as regards use of interactions and, in more conventional terms, Simpson s paradox does not involve interactions. 

See Garrett (2006) for further discussion of Simpson s paradox and Aldrich (1995) for a historical account. 

An advantage of a randomized clinical trial is that on average over all randomizations any covariates are balanced at baseline. This implies that a phenomenon such as Simpson s paradox is rather unlikely to arise in practice. For certain types of model it also implies that the expected value over all randomizations of a treatment effect that does not include a given covariate in the model is the same as the expected value that does include the covariate. For example, this applies to Normal linear models. 

This section presents the two most widely used statistical methods for meta-analysis, namely, the fixed effecfs model and the random effects model. In addition, we want to emphasize that an analysis based on crude pooling of adverse event numbers across different studies to compare treatment groups should be avoided, as the analysis is vulnerable to the mischief of Simpson s paradox (Chuang-Stein and Beltangady, 2011). 

Notice that in comparisons such as these sometimes slight inconsistencies in the results can be obtained. In two cases A was considered better than B, and B better than C, yet C was judged superior to A This inconsistency or non-transitivity is known as Simpson s or de Condorcet s paradox. In this particular case it can perfectly well be attributed to random variation. Assessors who are not sure about their conclusion are forced to make a choice, which then can only be a random guess. It is possible, however, to obtain results which are conflicting and statistically significant at the same time A < B and B < C, but C < A. This situation may occur when the attribute to be assessed in the comparisons is open to different interpretations. Actually, this is a case of multicriteria decision making (see Chapter 26) and it may be impossible to rank the three products unambiguously... 

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