PROC SEVERITY: Censoring and Truncation :: SAS/ETS(R) 9.22 User's Guide

The SEVERITY Procedure

Censoring and Truncation

One of the key features of PROC SEVERITY is that it enables you to specify whether the severity event’s magnitude is observable and if it is observable, then whether the exact value of the magnitude is known. If an event is unobservable when the magnitude is in certain intervals, then it is referred to as a truncation effect. If the exact magnitude of the event is not known, but it is known to have a value in a certain interval, then it is referred to as a censoring effect.

PROC SEVERITY allows a severity event to be left-truncated and right-censored. An event is said to be left-truncated if it is observed only when $\text{[math]}$ , where $\text{[math]}$ denotes the random variable for the magnitude and $\text{[math]}$ denotes a random variable for the truncation threshold. An event is said to be right-censored if it is known that the magnitude is $\text{[math]}$ , but the exact value of $\text{[math]}$ is not known. $\text{[math]}$ is a random variable for the censoring limit.

PROC SEVERITY assumes that the input data is given as a triplet $\text{[math]}$ , where $\text{[math]}$ is the number of observations (in a BY group), $\text{[math]}$ is the observed value (magnitude) of the response (event) variable, $\text{[math]}$ is the left-truncation threshold, and $\text{[math]}$ is a right-censoring indicator. If $\text{[math]}$ is equal to one of the values specified in the RIGHTCENSORED= option (or 0 if no indicator value is specified), then it indicates that $\text{[math]}$ is right-censored. In that case, the censoring limit $\text{[math]}$ is assumed to be equal to the recorded value $\text{[math]}$ . If $\text{[math]}$ is not equal to one of the indicator values or has a missing value, then $\text{[math]}$ is assumed to be the exact event value; that is, the observation is uncensored.

If the global left-truncation threshold $\text{[math]}$ is specified by using the LEFTTRUNCATED= option, then $\text{[math]}$ . If $\text{[math]}$ for some $\text{[math]}$ , then that observation is ignored and a warning is written to the SAS log. A missing value for $\text{[math]}$ indicates that the observation is not left-truncated.

If the global right-censoring limit $\text{[math]}$ is specified by using the RIGHTCENSORED= option, then $\text{[math]}$ is compared with $\text{[math]}$ . If $\text{[math]}$ , then $\text{[math]}$ to indicate exact (uncensored) observation; otherwise, $\text{[math]}$ to indicate right-censored observation. Note that the case of $\text{[math]}$ is considered as right-censored, because it is assumed that the actual event magnitude is greater than $\text{[math]}$ . However, it gets recorded as $\text{[math]}$ . If $\text{[math]}$ for some observation, then it is reduced to the limit ( $\text{[math]}$ ) and a warning is written to the SAS log.

Specification of right-censoring and left-truncation affects the likelihood of the data (see the section Likelihood Function) and how the empirical distribution function (EDF) is estimated (see the section Empirical Distribution Function Estimation Methods).

Probability of Observability

For left-truncated data, PROC SEVERITY also enables you to provide additional information in the form of probability of observability by using the PROBOBSERVED= option. It is defined as the probability that the underlying severity event gets observed (and recorded) for the specified left-truncation threshold value. For example, if you specify a value of 0.75, then for every 75 observations recorded above a specified threshold, 25 more events have happened with a severity value less than or equal to the specified threshold. Although the exact severity value of those 25 events is not known, PROC SEVERITY can use the information about the number of those events.

In particular, for each left-truncated observation, PROC SEVERITY assumes a presence of $\text{[math]}$ additional observations with $\text{[math]}$ . These additional observations are then used for computing the likelihood (see the section Probability of Observability and Likelihood) and an unconditional estimate of the empirical distribution function (see the section EDF Estimates and Left-Truncation).

Note: This procedure is experimental.

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