Censoring in statistical models of health status: what happens when one can do better than '1'.

A number of measures of health-related quality of life (HRQL) are subject to ceiling and/or floor effects. In a paper in this issue of Quality of Life Research, Austin, Escobar and Kopec (AEK) [1] draw attention to ceiling effects in the context of utility indices. Analysts often use multi-attribute utility functions to translate categorical information on multiple dimensions of health status into a single summary score. The health utilities index mark 3 (HU13) system, for example, describes 972,000 health states, formed by the different combinations of levels of capacity for its eight attributes: vision, hearing, speech, ambulation, dexterity, emotion, cognition and pain and discomfort. The multi-attribute utility scoring function then translates the categorical information into a score on the conventional scale in which dead = 0.00 and the absence of any disability (normal health) = 1.00. For instance, an individual who requires corrective lenses (Level 2 vision) but has normal capacity for each of the other seven attributes would be assigned a score of 0.97. In some populations, there are likely to be individuals whose health status exceeds the normal level of capacity defined by the HUI3. For example, the HUI3 would assign normal vision capacity to an individual 'able to see well enough to read ordinary newsprint and recognize a friend on the other side of the street, without glasses or contact lenses', even if the individual could also read ordinary newsprint from across the street. In statistical parlance, the reported visual capacity of such an individual would be censored at the normal level of visual capacity: all we can infer from their reported capacity is that it is at least as good as normal. AEK consider the implications of censoring for the estimation of the effects of age on HRQL, as measured by the HUI3 score, although their arguments are potentially equally valid for the estimation of the effects of clinical interventions, socio-economic and other individual characteristics on censored utility scores. Figure 1 should elucidate the problems inherent in estimating the relationship between HRQL and age. The solid dots represent actual observations on subjects' HUI3 scores and age, such as those available in a population health survey, whereas the clear dots represent hypothetical observations one would observe if the HRQL measurement instrument did not censor observations on subjects with supra-normal HRQL at 1. Because of censoring, observations above the ceiling value of 1 (such as point 'a') are recorded in the data as being 1. Suppose that one suspected that the relationship between HRQL and age is linear; in this case, there are two unknown parameters to be estimated the position (a) and slope (a) in the linear model: HRQL = a + f x Age + c. The term ? reflects the random variation in HRQL not explained by Age. The most widely used estimation method, ordinary least-squares regression, would estimate these parameters by finding the line that minimized the sum of squared vertical distances between each of the observed data points and the line. This line is labeled 'estimated' in Figure 1. Compare this to the line which would be estimated if we could observe the actual HRQL values of the censored observations; this is labeled 'actual'. While we wish we could estimate the 'actual' line, we end up with a different 'estimated' line. Specifically, we find that the estimated slope (a), representing the effect of a 1 year change in age on HRQL, is smaller