Response shift in the presence of missing data

The authors of these three papers [1–3] have undertaken a very challenging task by simultaneously attempting to address two very difficult problems: response shift (RS) and missing data. Part of the challenge is that both approaches often rely on making untestable assumptions. These papers also illustrate other challenges. The first challenge is that there is no gold standard for detecting RS. This is illustrated in two of the articles. Sajobi et al. [2] and Guilleux et al. [3] both include different methods for estimating RS. The challenge is to figure out how much of the variability of the results is due to the different methods for detecting RS versus differences in the methods for handling missing data. For example, Guilleux et al. [3] contrast IRT and SEM methods. It would be interesting to know how consistent these methods are when the data are complete, then when the data are missing at random (MAR), and finally, when the data are not MAR. The second challenge is to conceptually and explicitly identify the linkage between the RS and the missing data mechanisms, and the interaction between them. This is critical to understanding the results of any analyses as well as for utilizing the appropriate method for imputing missing data. For example, Verdam et al. [1] examine a model where the changes in associations measured in a structural equation model are associated with survival (or number of assessments). The premise and the implied underlying hypotheses are valid. However, it is difficult to interpret the results from the tables. A creative graphical presentation would facilitate both understanding of the proposed model and interpretation. In Sajobi et al. [2], the RS model hypothesized that differences in stroke severity would be associated with reprioritization by caregivers of the domains of the SF-12 over a 6-month period. Missing data was observed to be associated with greater severity of the stroke. What does this suggest, however, in terms of the association of the missing caregiver SF-12 scores with the initial severity? Do the missing values reflect a weaker (or stronger) association between stroke severity with some domains relative to other domains? The third challenge is to adapt the concepts of missing data that were originally adapted for assessing means to those that assess associations. It may be necessary to consider a conceptual variation on the classical definitions of the missing data proposed by Little and Rubin [4]. Most methods used to identify RS examine relationships between two or more variables over different conditions. For example, in Sajobi et al. [2], recalibration is assessed based on the relationship between caregiver SF-36 domain scores (y) and an indicator of stroke severity (x). For the purposes of this discussion, I will designate the relationship as the correlation between x and y, qxy. If either x or y are missing, then qxy is missing. When the probability that qxy is missing is unrelated to qxy or other data z, then the value is missing completely at random (MCAR). In this case, the estimate of the relationship is the same for the individuals with missing or observed data (q xy 1⁄4 q xy ) and the observed data can be used to obtain an unbiased estimate of the relationship for the entire sample. If missingness is related to observed values of qxy or other data z, then the value is MAR. In this case, estimating the relationship is unbiased for certain methods (e.g., maximum likelihood) when the observed qxy or other data z are included in the estimation process. Finally, if the probability is related to the unobserved values of qxy, then the value is missing not D. L. Fairclough (&) Department of Biostatistics and Informatics, Colorado School of Public Health, Aurora, CO, USA e-mail: Diane.Fairclough@ucdenver.edu