#### Description:

In this tutorial, we’ll show you how to enter parameter ranges and quickly create power and sample size curves using PASS. In our first example, we’ll show you how to generate sample size curves for the logrank test. In the second example, we’ll demonstrate how to create power curves using the tests for 2 correlated proportions procedure.

For the first example, let’s suppose we’d like to determine the sample size required for a parallel, two-group, equal-sample-allocation design to compare the hazard rate of a new treatment with that of the current treatment using a two-sided, logrank test. We’ll assume that about 60% of the patients in each group will experience the event of interest. Let’s find the sample size required for 90% power to detect a hazard ratio of 2 at an alpha-level of 0.05.

The results indicate that the required sample size is 73 patients in each group, for a total of 146 patients overall.
This calculation was pretty straightforward, but what if we’re not confident in the assumption that 60% of patients will actually experience the event. What if we believe that the correct value could be anywhere from 45% to 70%? PASS allows you to easily enter several values for the input parameters. If there are just a few values, you can simply input them directly, with numbers separated by spaces. When there are many, it is easier to enter a range statement. In this example, we’ll enter the phrase 0.45 to 0.7 by 0.05 for the control event probability to generate the series. We’ll now re-run the report.

Notice that now several sample size calculations have been made, one for each combination of the input parameters. The numeric report lists the sample size required for each scenario. The sample size plot gives a nice graphical representation of these calculations. The vertical axis on the plot always displays the parameter that was solved for, which, in this case, was sample size or N. As you’d expect, the required sample size decreases as the event proportion increases. The total sample size ranges from 195 when the event proportion is 0.45 to 125 when the event proportion is 0.70.

Let’s further complicate things with some additional requirements. Suppose now that we want to study the sample sizes required to achieve 80% and 90% power for hazard ratios ranging from 1.6 to 3. We’ll enter the power parameters directly since there are only two, and use a range statement for the hazard ratios. Now generate the report.

Once again, the numeric report lists the required sample size for each combination of the parameters. This time it is much longer because we have varied three of the input items. The plots summarize the table, making it much easier to see the trends.

By default, PASS automatically chooses where to place each varying parameter in the plot. As was mentioned earlier, the solve for parameter is always plotted on the vertical axis. Additionally, the parameter with the most distinct values is plotted on the horizontal axis and the parameter with the second-most values is assigned to the legend. Separate plots are created for each combination of the remaining parameters.
In this case, PASS automatically chose to place hazard ratio on the horizontal-axis and event proportion in the legend and created separate plots for each of the values of entered for power, 0.8 and 0.9.

You can override this default behavior by going back to the procedure window and clicking on the Plots tab. You can’t change the Vertical Axis parameter, but you can change where the other parameters are placed. Let’s force PASS to plot the event proportions on the horizontal axis and hazard ratio in the legend.

Keep in mind that you can always change the formatting on the plot by clicking on the large plot format button. Let’s go ahead and change the vertical axis minimum and maximum so that all plots have the same scale for comparison and also make a few changes to the symbols. Now, we rerun the report.

As you can see, the customized plots have been created with the axes as specified. General trends are apparent from the plots: smaller sample sizes are required to detect larger hazard ratios. Also, the sample sizes increase as the desired power increases.

So far, we’ve been looking at sample size curves, but power curves are just as easy to create in PASS. We’ll use the Tests for Two Correlated Proportions procedure to create a power curve. In order to create a power curve, we need to solve for power instead of sample size. Let’s investigate sample sizes between 20 and 400 and proportion differences of 0.1, 0.15, and 0.2. Before we run the report, let’s click on the Plots tab and select the option that allows us to edit the graph while the procedure is running. We’ll do this to add a reference line so we can see where each curve achieves 90% power. Now, we click the green button to generate the plot.

On the scatter plot format window that appears, we’ll click on the Grid Lines tab and include a Grid line from the Y axis at 0.9. Let’s make the line thick and red so we can see it easily. Notice that the plot preview is updated automatically. Click OK to save the settings and generate the plot.

The plot shows the relationship between sample size and power among the 3 differences for this test. The horizontal line makes it very easy to compare the sample sizes required for 90% power. Sample sizes of about 330, 150, and 80 are required to detect proportion differences of 0.1, 0.15, and 0.2 respectively.

Looking at the plot another way, we can conclude that if you had only 100 individuals in a study, then your power would be only about 37% to detect a proportion difference of 0.1, but you’d have well over 90% power to detect a difference of 0.2.