#### Description:

There are a variety of procedures in PASS for examining sample size and power for the comparison of two survival curves using the Logrank test. The procedures differ by the assumptions that are made about the survival curves, the research article the procedure is based upon, and the way the survival parameters are specified.

In PASS, there are four procedures that allow complete flexibility of survival curve, accrual, and loss-to-follow-up specification for the Logrank test. These four procedures differ only in the way the survival rates are specified. Hazard rates, median survival times, proportion surviving, and mortality rates are each simple functions of any of the others. You may wish to use the procedure that specifies the survival rates in a way that is reported in previous studies or your preliminary studies. Or you can convert among rates using the Survival Parameter Conversion Tool. In this video, we will use the procedure where Hazard Rates are specified directly.

There will be two examples that follow. In the first example, the study design will be less complex than the second. The purpose of the first example will be to give a feel of the basic parameters that are needed to estimate a sample size for a Logrank test. The second example will extend the first example to examine the flexibility of this PASS procedure to account for more complex aspects of the designs that commonly occur.

Suppose we are planning a study where a new minimally invasive surgery technique is to be compared to an existing more invasive technique. The purpose of using PASS in this instance is to determine the number of subjects that will be needed in each group to produce a Logrank test with a power of 90% to detect a difference in hazard rates of the two techniques. Specifically, we will assume that the true yearly hazard rate of subjects where the existing technique is used, is 0.45, and the true yearly hazard rate of subjects using the new technique, is 0.3. In this first example, we will assume that an equal number of subjects will be available in each group. Further, all subjects will be available at a single start time, and will be followed for 3 years.

To determine the sample size for this scenario in PASS, the Logrank Tests (Input Hazard Rates) procedure is used. We wish to solve for Sample Size, and a two-sided test will be used. The Power is set to 0.9, and the alpha level is 0.05. The number of subjects to be allocated to each group is equal.

The Control group hazard rate is then entered directly. The treatment group hazard rate can be entered as a hazard ratio, which would here be about 0.667, or directly as a hazard rate.

The accrual time is zero, and the total time is 3 years.

We then press the Calculate button to obtain the result. The total number of subjects needed is 384, or 192 per group. The anticipated number of events in this scenario is shown in the events section below.

Since it is not likely known in advance, we may want to consider a range of hazard rates for the new surgery technique. We will examine the sample sizes needed for hazard rates between 0.1 and 0.4.

The comparison of the sample sizes needed is most easily viewed using the sample size curve at the end of the report.
The number of needed subjects increases dramatically as the hazard rate of the treatment group approaches that of the control group, particularly if the treatment hazard rate is greater than 0.3.

For the second example, we will now extend the first example to some additional commonly encountered nuances of survival curve comparison designs.

Suppose, now, that obtaining subjects to receive the new minimally invasive technique is much more difficult than obtaining subjects to receive the existing surgery technique. We expect to obtain half as many treatment group subjects as control group subjects. In PASS, we can specify this by changing the Group Allocation to ‘Enter R’, and then setting R to 0.5, since this is the ratio of the number of treatment individuals to the number of control individuals. In this case, the total number of required subjects goes from 384 to 425. We can see both results by entering 0.5 and 1 for R.
Next we will change the time units we are using from years to months, so that we can enter more specific details of the anticipated study. To change from 3 years to 36 months, we can first change the Total Time to 36. Because the hazard rate is relative to the time units, we will adjust the hazard rates accordingly. In this case, the hazard rates are divided by 12 to become monthly rates. The survival parameter conversion tool can be used, or we can simply divide 0.45 and 0.3 by 12. We can then calculate the resulting sample size requirement to verify that it is indeed the same to follow subjects for 3 years, as it is to follow them for 36 months. Using 36 months as the total time, however, allows us more detailed specification of values within a year.

Now suppose that we anticipate enrolling subjects over the course of an accrual year, with two additional years of follow-up after the end of the accrual year. In this case, subjects that enter the study at the very beginning of the accrual period will be followed for 3 years or 36 months, while subjects that enter at the end of the accrual period will be followed for two years, or 24 months. To do this we enter 12 for the accrual time, to indicate 12 months of accrual.

If we leave the accrual pattern at ‘Uniform or Equal’, the subjects will be enrolled evenly across the 12 months. But suppose we know that accrual for the first three months will be high, and then gradually slow down. We can then use Non-Uniform accrual entry with the spreadsheet. The values in each of 12 rows of column 1 of the spreadsheet represent the relative amount of accrual for that month. The software will divide by the total of this column so that the relative weights sum to one.

We can also specify individual hazard rates for the control and treatment groups across the 36 months. Perhaps we anticipate a very high monthly hazard rate for the first four months after surgery, followed by a stable hazard rate for the remainder of the 36 months.

We now consider the control and treatment subjects that are lost to follow-up during the course of the study. If we expect to lose to follow-up about 10% per year, we can use the survival parameter conversion tool to find the corresponding monthly values. This value could be specified individually for each of the 36 months, but we assume in this case that the drop-out rates will be the same throughout the study. Thus, the single value is entered for the Controls lost and for the Treatments Lost.

We can see by running the procedure that the total number of subjects needed to obtain 90% power to detect a difference in the two survival curves is 422, with 281 receiving the control invasive surgery technique, and 141 receiving the new minimally invasive surgery.

We could also enter multiple values for any of the parameters if we wish to see the effect on the sample size. For example, to compare the required sample size for two proportions of treatments lost, we simply enter the two values and press ‘Calculate’.