Not all there

One of the most common problems with data is that it’s not there. Families don’t answer their phones, over-worked nurses miss some forms, and even tireless electronic recorders have power failures.

There’s a large field of statistical research devoted to ways of fixing the missing-data problem. None of them work — that’s not my cynical opinion, that’s a mathematical theorem — but many of them are more likely to make things better than worse.  The best ways to handle data you don’t have depends on what sort of data and why you don’t have it, but even the best ways can confuse people who aren’t paying attention.

Just ignoring the missing data problem and treating the data you have as all the data is effectively assuming the missing data look just like the observed data. This is often very implausible. For example, in a weight-loss study it is much more likely that people who aren’t losing weight will drop out. If you just analyse data from people who stay in the study and follow all your instructions, unless this is nearly everyone, they will probably have lost weight (on average) even if your treatment is just staring at a container of felt-tip pens.

That’s why it is often sensible to treat missing observations as if they were bad. The Ministry of Health drinking water standards do this.  For example, they say that only 96.7% of New Zealand received water complying with the bacteriological standards. That sounds serious. Of the 3.3% failures, however, more than half (2.0%) were just failures to monitor thoroughly enough, and only 0.1% had E. coli transgression that were not followed up by immediate corrective action.

From a regulatory point of view, lumping these together makes sense. The Ministry doesn’t want to create incentives for data to ‘accidentally’ go missing whenever there’s a problem. From a public health point of view, though, you can get badly confused if you just look at the headline compliance figure and don’t read down to page 18.

The Ministry takes a similarly conservative approach to the other standards, and the detailed explanations are more reassuring than the headline compliance figures. There are a small number of water supplies with worrying levels of arsenic — enough to increase lifetime cancer risk by a tenth of a percentage point or so — but in general the biggest problem is inadequate fluoride concentrations in drinking water for nearly half of Kiwi kids.