# Right censoring and Left censoring

Wikipedia gives the following definitions:

Right censoring: a data point is above a certain value but it is unknown by how much.
Left censoring: a data point is below a certain value but it is unknown by how much.

In these definitions, what is meant by:

• "data point"
• "certain value", and
• "how much"

In general, What is right and left censoring?

Is the statement below true :

"In Right censoring, we only have the lower bound for censored value."

What would be the analogous statement for left censoring?

As the survival tag is used I'll add an answer offering some examples with a survival analysis flavour.

Data point

By a data point, we just mean some observation, i.e. the outcome of one or more variables. For instance, we might have the following in a data set: person 1 in our study is a male and dies at age 58. We could consider that a data point. But in your example it's clear that the data point only consists of the outcome of one variable, e.g. 58.

Right-censoring

If we're modelling the time to failure there's an obvious reason for censoring, namely, that we don't necessarily have time to wait for all subjects to fail. Say we're testing the effect of children's vaccines. If we were to conduct a randomized trial, the last of our subjects would die a hundred years or more from now. This naturally introduces censoring, in this case right-censoring, as we would at some point have to say "we don't know how much longer this person will live, we only know that she's still alive". Right-censoring might also occur if people in the randomized trial are lost to follow-up, e.g. they might want to discontinue their participation in the study or move away. These are examples of right-censoring, basically we're interested in the longevity of our subjects but due to practical circumstances we only have censored observations, meaning that for some subjects we will never know when they die, only that at some point in time (the censoring time) they were still alive. Thus, we know that for a censored individual, the data point (time of death) is larger than a certain value (the censoring time).

Left-censoring

As an example of left-censoring, consider the following. Let's say that some baboon troop always sleeps in the trees. We want to estimate at what time in the morning they descend from the trees, and let's assume that they do descend every day. We follow them for a number days, however, we like sleeping in, meaning that some days they descend before we even arrive on the scene. If we arrive at 9 a.m. on day $x$ and the baboons already descended, we have left-censored data. We want to know when they descended, but all we have is an upper limit (9 a.m.), because we know that at our time of arrival they had already descended. Analogously, we now know that the data point (time of descent on day $x$) is smaller than a certain value (9. a.m.).

This example is taken from

Andersen, P. K., Borgan, Ø., Gill, R. D., and Keiding, N. (1993), Statistical Models based on Counting Processes, Springer Series in Statistics, Springer-Verlag, New York.

This book provides a mathematical definition of censoring and is probably not the first book on survival analysis one should get. However, it also has some intuitive examples, as the above.

• But , say, one baboon have not descended yet when we arrive at the place (9 a.m , certain value) . So we started to observe when it will descend . Then isn't we have the data point (time of descent) above certain value (9 a.m) ? But still it is left-censored data . Did wikipedia give the definition more generally ?
– ABC
Commented May 28, 2015 at 0:36
• I assumed the entire troop to descend at one time, all together. But it doesn't make much of a difference. If we arrive after the descent we only know a upper bound on the time of descent (namely our arrival time), thus this data point (time of descent on the specific day) is left-censored. If we arrive before the descent, that data point will not be censored (unless we get tired of waiting and leave before the descent, in that case we have right-censored data, assuming that they do descent every day).
– swmo
Commented May 28, 2015 at 8:09
• Another example of right censored data is time interval data where we don't know it's beginning. This is often misconstrued as right-censored data (there are several examples on forums and mailing lists with this misconception). Commented Jul 29, 2015 at 5:47

Suppose I own a bar where I have bands play. The bar is pretty small, so only 150 people can see a show at any one time (this is key). I sell tickets to the shows, so my accounting data would look like this:

date     band               price   tickets_sold
10/01/14 Texas Instruments  $20 2 10/02/14 Unkind Donuts$30     150
...
03/02/15 The Capybaras      $15 120  A data point is just a row in this table. Suppose the variable I want to consider is demand for tickets. The demand for the first show is not censored. Only two people wanted to see Texas Instruments at \$20 and 148 tickets went unsold. I know the demand at \$20 exactly: the 2 tickets that sold. However, the demand variable in censored in the second row because the show sold out. I know that at least 150 people wanted to see Unkind Donuts at \$30 per ticket, but just how many people that got turned away without a ticket is unknown to me, so I don't know demand exactly. All I know is the lower bound of 150.

Now suppose I wanted to measure attendance at the third show instead. We could count people at the door, but for the sake of this example let's assume that my bouncer is bad at arithmetic. We know that some people will buy tickets and then not come. This means that the attendance is at most 120 since that is how many tickets sold. That is the upper bound on attendance for The Capybaras, which is left-censored.

A common misconception with left censoring is classification of a time interval data point where you don't know it's beginning. Many think this is left censored, but it is actually right censored since we have a lower bound on the length of the interval.

A concrete example could be clinical data on the duration of "foo-pox", usually a non-terminal disease, and we are interested in the length of time it takes people to recover. The symptoms of foo-pox are easy to observe (e.g.: your teeth go green). Most people in our study know exactly when that started and when it ended.

The classical example of right censored data in this type of study are subjects who either still had foo-pox at the end of the study or still had foo-pox when they disappeared ("lost to followup") during the study (lets assume we know the start date of the disease for these people). For these people we have a lower bound on the duration, hence their data is right censored. This is intuitively "right censored" as we don't know the right hand end of the time period.

The problem is when we don't know the start date of the time period (people who live alone and don't have a mirror, so don't know when their teeth went green). Are these left or right censored? Many erroneosly think that the left end of the time period is unknown, therefore left censored. This is an unfortunate result of the terminology, which I guess developed in the absence of this kind of censoring. For these people, we have a lower bound on the time period (we know they had foo-pox at least from when their neighbour mentioned their green teeth until they got better or the study ended and they were still ill), thus their data is right censored.

• I think your answer is a bit confusing. In the start, you talk about censoring a time interval, later about censoring a time period (a single number). In the last paragraph, you might equally well state that data are intervals from time of onset of foo-pox until death. In that case you would have left-censored intervals in your example, as you don't necessarily know the exact time of onset, only an upper bound. Equally well, you could (as you do) look at duration of foo-pox, in which case you could get a lower bound on the duration, thus have right-censoring.
– swmo
Commented Jul 29, 2015 at 7:37
• How would you code this in the data then, for example in an R Surv object? Would the event of a on-the-left right censored record be "remission" or "event occurance", while the event of an on-the-right right censored record be "right censored/no occurance/no remission"? Also, it seems that the hazard rates must be different between the two types of right-truncated events, since these should be modeled as a function from the onset of the disease? What type of model would handle this? Commented Nov 15, 2016 at 23:05
• @AllenWang Afraid I'm not too familliar with R Surv objects, but I expect their terminology is consistent, so if you're careful to understand and follow it, you should be fine. As for hazard rates, there is no difference, in both cases, you only have a lower bound on the event duration. Commented Nov 17, 2016 at 16:55