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I read from statistics books that numerical analysis is not in important role in statistics. Are there any examples in history where numerical approximation has been caused wrong decisions in statistics, like $p$ value has been too close to $0.05$?

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There was an example that ended up in the New York Times in 2002. There were two problems: one with numerical analysis and one with computationally-feasible standard error estimation.

The topic was the short-term health effects of air pollution, where a lot of analyses had been done using gam() from S-PLUS. The models being fitted were different from a lot of past uses of gam(), with many more degrees of freedom in the smoothing terms to model seasonal effects over a period of years.

Shortly after the deadline for evidence to be taken into account for the new particulate air pollution standards, a research team at Johns Hopkins realised that

  • the convergence tolerance in gam() was too loose for these models
  • the standard error approximation wasn't all that good when the fitted smooth term had high correlation with some predictors.

Fortunately, the qualitative results were not affected, but there was a certain amount of wailing and gnashing of teeth at the time.

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