Bias in casecrossover analyses of environmental time series data

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Bias in case-crossover analyses of environmental
time series data

• Paddy Farrington,
• Heather Whitaker and Mounia Hocine
• The Open University, UK

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The message

• Case-crossover methods are frequently used in
  environmental epidemiology.
• Some of them are biased, owing to incorrect use
  of the case-control paradigm.
• This can be fixed by using case series methods.
• However there is little point in doing so as time
  series methods are more flexible.

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Environmental time series

• Data xt on an environmental exposure (eg
  temperature, pollution level) in a given area are
  available at times t 1, 2, , T.
• Counts of events nt (eg vascular events, asthma
  hospitalizations) are documented at each time
  point.
• Question is there an association between the
  counts and the exposures?

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(No Transcript)
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Case-crossover methods applied to environmental
time series

• For an event at time t, select a window Wt
  determined by t and containing t (eg times t ?1,
  t and t 1).
• Treat the exposures in Wt as a matched
  case-control set, xt being the case exposure.
• Do a conditional logistic regression as if this
  were a matched case-control study.

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t 1 t t 1
Control Case Control
Exposures xt-1 xt
xt1 Count nt
Likelihood contribution of events at t
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Overlap bias

• There is a large industry on how to choose the
  referent window Wt.
• It has been shown that this method of analysis
  sometimes produces biased estimates.
• This bias has been called overlap bias.
• But up till now the source of the bias has not
  been clear.

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Exchangeability of the exposure distribution
within matched sets

• Matched case-control studies require the
  distribution of exposures in each matched set to
  be exchangeable
• given the unordered set of exposures E X0, X1,
  , XM), for any permutation ? of the indices,
• P(X0,,XME) P(X?(0) ,,X?(M)E)

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The (full) case-control likelihood

• Consider a matched set of size M 1 (one case
  and M controls). For each label r 0, 1, , M let

The likelihood contribution for this matched set
is then
provided that exposures are exchangeable.
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Exchangeability of environmental exposures?

• For environmental time series, sampling of the
  reference windows Wt determines the exposures.
• Available orderings of exposures across reference
  windows are determined by the time series (and
  form a finite population).
• The exchangeability condition may not be met if
  not this results in overlap bias.

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Binary exposures - Example

• Consider the exposure series 01011 and case
  positions 2, 3 and 4.
• Exposure vectors
• Case position 2 (0,1,0)
• Case position 3 (1,0,1)
• Case position 4 (0,1,1)
• Exposures are not exchangeable e.g. the vectors
  (1,0,0), (0,0,1) do not appear.
• This will result in overlap bias.

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Case series methods

• Overlap bias can be avoided by dropping the
  case-control paradigm.
• Instead, partition the time series into
  non-overlapping windows Wt and treat event times
  within each window as random.
• This is a cohort approach, which is not subject
  to overlap bias.
• It has recently been extended to allow for the
  residual effects of seasonality.

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Is this approach worth rescuing?

• The key assumptions of the method when applied to
  such data are
• The counts are Poisson, and
• There is no underlying time trend in event rates
  within each window Wt.
• Failure of either assumption invalidates the
  method.
• Time series methods make no such assumptions.

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Example Relative risk of RSV per 10oC
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Conclusions

• For modelling environmental time series, the
  case-crossover method is biased as exposures are
  usually not exchangeable.
• Case series methods are unbiased.
• They are equivalent to Poisson time series models
  with piecewise constant rate.
• Time series regression models are far more
  flexible and should be preferred.
• Whitaker et al, Environmetrics 18 157 (2007)