Brute Force: Missing Data (Part 4)

A Clockwork Orange, Relax by Frankie Goes To Hollywood and Nevermind the Bollocks. Which is the odd-one-out and why?

The answer is Nevermind the Bollocks because the other two were censored from cinema release in the former's case and from Radio 1 in the later. No-one will ever know for sure if Relax would have made it to the top of the pops if this hadn't happened. But it seems like Relax being absent on our radios actually helped it reach the dizzy heights in the charts.

In mathematics, when things go missing then theres work to be done with extra adjustment and ammendments to a full model needing to be made, often at some cost. Dealing with missingness can seriously complicate matters. Filling-in what isn't there needs to take into account what might have been there or how the missing parts affect all of the other parts too.

Removal of some component of a mathematical model can lead to involved and complex stuctures.

Now the problem is, this new missing-model with all of the extra bells and whistles is much harder to handle than the more well-behaved simpler model when we knew all the parts. Because of this situation, often the only solution to getting any work done is to resort to simulation.

Censored data is a common example of a missing data problem. Censored data is data where we do not necessarily observe the true value. In some instances instead of the true value of what ever it is that we're observing we only have the information that it is no bigger or smaller than some threshold value. In the case of time, for example, whats called right-censoring is when we stop observing something at some time so we only have information about it up to that point and not afterwards. For example, if people were monitored whilst in hospital for infection but not once they were discharged then the censored time would be at the time they leave hospital.


Lets suppose that we hadn't heard Relax on the radio upto a few days before the chart count down but then our radio broke, so it may have been played freely for all we know after that. Then if we wanted to make a guess about how well it was going to do in the charts we're missing a bit of important information (we'll leave whether bannning it was a good or bad thing for sales for the time being.)

Censoring means that if we want to calculate some statistics, say, using this data then we need to account for the fact that some of the data is censored. For example, if we wanted a mean average then this is not the simple mean as a sum of all the values divided by how many values there are. This is because some of those values we have are not actually the true value but just a lower limit of this. If we were to calculate an average like described then the resulting figure would be an underestimate. For example, we would estimate that the average time of infection in our hospital example is smaller than it actually is because we wouldn't be taking in to account the time after the patients leave hospital and before they get infected (if at all).
If we do take this censoring into account when we're trying to get our answer this means that things get more involved and complicated. One approach would be to sensibly fill-in the missing values using what information we do have at our disposal. Another is to change how we do the calculation. For example, the mean estimate for the censored time can be calculated using the probabilities of not yet having an infection at certain times, something called a survival time.
When the models are bigger and complex coming up with an alternative method, like the survival time formula, becomes even more difficult and the filling-in or imputing method begins to look more appealling. This can mean imputing lots and lots of missing values, over and over again, so that we get an idea about how the values we're coming up with are affecting the output. If you think this sounds like a job for brute force you'd be dead right.