Long-term Medical Trial with Time-Dependent Dropout and Event Rates

Ref. Wu M, Fisher M, and DeMets D: Sample Sizes for Long-Term Medical Trials with Time-Dependent Dropout and Event Rates. Controlled Clinical Trials 1:111-123, 1980.

The program computes sample size or power based on event rates adjusted for non-compliance and lag time to full treatment efficacy. The program assumes that the proportions test will be used to compare the number of events in the control and experimental groups. The particular features of this program which are extensions to the situation of comparing binomial proportions between two groups are the interval- dependent rates and a lag time to treatment efficacy. The study period is divided into equal-sized intervals. The following may then be made interval dependent: event rate, drop-out (non-adherence to therapy in the experimental group) rate, and drop-in (adoption of therapy in the control group) rate.

The following assumptions are made:

  1. All subjects are observed for the entire study period. The study period cannot be divided into an accrual and a follow-up period. However, in situations where patients are accrued over some time period, the calculations are valid if each subject is observed for the entire length of the study period; i.e. there is no administrative censoring.
  2. The instantaneous event rate for the control group is constant within an interval, as is the instantaneous drop-out rate for the experimental group. Each of these rates may be different in the other intervals.
  3. For drop-outs, the event rate returns to the appropriate control group level in the same linear fashion as the event rate decreased before drop-out. The time required to return to the rate equals the time spent on study before drop-out.
  4. Full effect of treatment for those in the experimental group and for drop-ins is achieved in a linear fashion over the lag time.
  5. No "returns" among drop-outs or drop-ins.
  6. The user specifies the total length of the study and the number of intervals into which it is divided. After this, all aspects of the trial should be conceptualized as functions of these intervals.

The user provides the following:

  1. The length of the study period (T years) and the number of equal-sized intervals into which it is to be divided (15 or fewer). The term "year" (and later "annual") is used here to denote an arbitrary time unit.
  2. Either the control group annual exponential incidence rate or the proportion expected to experience an event over the length of the study. If the annual exponential incidence rate is entered, the exponential rate will be used to project the proportion expected to experience an event over the length of the study. When incidence rate is selected, the relationship with the experimental incidence must be specified as a risk ratio equivalent to the 'incidence ratio'.
  3. The lag time to full treatment efficacy, expressed as an integral multiple of intervals.
  4. The percentage of relative reduction in the proportion of events, after attainment of full treatment efficacy. (e.g., .3 reduction in .8 results in .56)
  5. The proportions expected to drop-in and to drop-out in the control and treated groups, respectively, during the entire study period.
  6. The event, drop-out, and drop-in patterns; that is, for each of the three, the weighted percent of the total proportion expected to occur in each of the intervals. For example, suppose we expect 10% (.10) total drop-out in a 2 year trial divided into 4 intervals, with 50% of the drop-out expected to occur in interval 1, 40% in interval 2, 10% in interval 3 and none in interval 4. The pattern entered (on a single line) is: .5 .4 .1 0.

Then given two of the following:

  1. One-sided significance level
  2. Power
  3. Sample size
the program can compute the third.