Transplant Statistics: Annual Report

Technical Notes and Analytic Methods
Calculation of Individual Expected Survival

The tables of coefficients (see www.ustransplant.org ) can be used to calculate the expected graft or patient survival at one month, one year, and 34 months for a specific patient.

Suppose that p is the number of patient characteristics listed in the appropriate model table and x_ij is the specific value of the j^th characteristic for the i^th patient. For patient i, we calculate

where b_j is the j^th coefficient from the model.

For a categorical characteristic, the value is 1 if the patient falls into the category and 0 otherwise. For linear ischemia time, subtract the average ischemia time (given above) from the patient's ischemia time (in hours) before multiplying by the corresponding b_i. For quadratic ischemia time, subtract the average ischemia time (given above) from the patient's ischemia time (in hours) and square it before multiplying by the corresponding b_i. For HLA mismatch values 1-6, subtract 3.5 from the patient's HLA mismatch value before multiplying by the corresponding b_i.

The Cox model yields estimates of the coefficients b_j and the baseline survival curve, S₀(t). For patient i, with characteristics

we calculate

the expected survival at time point t₀ (t₀ = 1 month, 1 year, or 34 months). Then

is the expected survival at time t₀ for this facility.

We also calculate

Then -ln(S_i(t_i)) is equal to the expected number of events for patient i during follow-up and

is the expected number of events during follow-up at the facility.

NOTATION:

• N = number of transplants accrued
  
• t₀ = reporting time point (e.g., one month, one year, 34 months)
  
• t_i = end of follow-up for patient i during period. Note t_i < t₀ for patients who die or are lost to follow-up during the period.
  
• S_i(t₀) = fraction surviving to end of period adjusted to characteristics of patient i
  
• S_Avg(t₀) = expected fraction surviving to end of period adjusted to average characteristics of patients at the facility