Transplant Statistics: Annual Report

Technical Notes and Analytic Methods
Computing Expected Patient and Graft Survival

Model Fitting Methods

Survival models are adjusted for patient characteristics. The model used for each organ is adjusted for patient characteristics specific to that organ, so we refer to the list of characteristics generically with the notation x. Individual patients are numbered sequentially and we refer generically to the i^th patient. The specific values of the characteristics for patient i are denoted by x_i. Based on a model, we calculate S_i(t), the probability of survival to time t for patients with characteristics x_i. The probability of survival at time point t₀ for patient i is S_i(t₀). The average survival for the N accrued transplant patients at the center is calculated as (1/n) SS_i(t₀) (Zucker). The expected number of events during follow-up for each patient was calculated as –ln(S_i(t_i)), where S_i(t_i) is the survival curve adjusted to the characteristics of patient i and t_I is the follow-up time for that patient (SAS/STAT User's Guide, Andersen, Collett). The expected number of events is S–ln(S_i(t_i)) for the N transplants during the follow-up times for the patients at this center.

The models included patient characteristics determined to be important in the previous Center-Specific Reports (UNOS, October 2000). In addition, if one level of a characteristic was included, the others also were included. For example, if black recipient was included in a model in the past, then the current model included each of the race groups. Also, if an interaction was included in previous reports, then the main effects for that interaction were also included.

The Model Description Tables, located at www.ustransplant.org, indicate the value of the coefficient for each characteristic in each of the models (beta) as well as the corresponding standard error and a p-value indicating if the coefficient is significantly different than 0. The relative risk (RR) for mortality or graft loss associated with a particular patient characteristic, compared with the reference group for that characteristic, can be calculated as RR=exp(beta). For continuous variables, this is interpreted as the RR associated with one unit higher value (e.g., for ischemia time, it would be the RR associated with one hour longer time). However, keep in mind that these models are estimated for the purposes of adjustment, not for interpretation of coefficients. Some standard errors are large, which reflects uncertainty in the interpretation of the corresponding covariate but does not adversely affect the accuracy of the adjusted estimate.

Missing Data

For age, ethnicity, gender, medical condition, on life support, creatinine, recipient on ventilator, partial liver transplant, double kidney transplant, VAD, IABP, and ECMO, patients with missing information are included in the reference group. For race, missing is combined with other and unknown race. For diagnosis, missing is combined with other. For years since onset of diabetes, ABO compatibility, HLA mismatch, pretransplant transfusions, peak PRA, duct management, and ischemia time, there is a separate category for patients with missing values.

Diagnosis

In models adjusting for diagnosis, the diagnosis groups and reference groups differ by organ and, in some cases, by age group.

Ischemia Time

Ischemia time refers to cold ischemia time for kidney and liver transplants, total time for heart transplants,and maximum time for lung transplants. In all models, ischemia time was measured in hours. Ischemia time was included in the models as both a linear and a quadratic term, as well as an indicator for missing ischemia time. Ischemia time was centered on the average ischemia time by organ, which are shown below.

Heart — 3.0 hours
Lung — 4.6 hours
Liver — 8.8 hours
Kidney — 15.1 hours

HLA Mismatch

For models adjusting for HLA mismatch, there is a term for missing mismatch information, one for 0 mismatch, and a linear term for mismatch 1-6. The linear HLA mismatch term was centered on 3.5.