Missing Data

There are three types of missing data:

Missing Completely at Random (MCAR)

Data are missing completely at random (MCAR) when the probability of the data being missing is unrelated to any data, observed or unobserved.

Examples are

  • data were accidentally recorded for the wrong team
  • a file got corrupted

When data are MCAR, you can ignore the missingness and use standard tencniques. The only impact is that your data has a smaller sample size and thus less power than you had original planned. Thus when missing data are MCAR, this is the least problematic.

Missing at Random (MAR)

Data are missing at random (MAR) when the probability of missingness is related to observed data, but not due to the missing data itself.

Examples are

  • injury status is not reported for older players
  • kicker is not asked to kick a field goal due to the distance

When the data are MAR, you can model the missingness based on the observed data (explanatory variables or other .

Missing Not at Random (MNAR)

Data are missing not at random (MNAR) when the probability of missingness depends on the value of the missing data itself.

Examples are

  • salaries that are too high or too low are not reported
  • a player underreports training load to avoid being rested

When the data are MNAR, you will need to model the missingness directly and your inferences about the quantities you care about will depend on the assumptions you make in the missingness models. Thus, MNAR data are the most challenging to deal of the missingness types.

Two types of MNAR data that are extremely common are censoring and truncation.

Censoring

Data are censored when you know the data exists, but you only know a range for the data. There are types of censored data:

  • right censored
  • left censored
  • interval censored

Data are right censored when you know the data are above a certain value. Examples are salary is $100,000+, marathon time is greater than some time when the runner drops out, and career duration is at least X years.

Data are left censored when you know the data are below a certain value. Examples are reaction time when your test has limit of 0.2s and someone reacts in less than that amount of time and you start recording data after an event/career has started.

Data are interval censored when you know the data are within an interval. Examples a periodic testing that report where you know something occurred between the tests but not precisely when it occurred and an injury occurs between two games but it is not precisely known when between those two games.

Truncation

Data are truncated when there is no reporting data outside of some limits. In these situations, you will not know if data outside of these limits even exists or how many data points there are outside of those limits. Examples are marathon data only reporting the top 20 runners, players will only be included in an analysis if they have played 10 games, and reactions times above a limit are excluded from the data set.

Summary

Censored data needs to be dealt with carefully. It is not simply enough to say “I think the data are MCAR and proceed as normal.”