Running CompARE

The Input Panel provides the user with a number of input parameters to chose from. The input panel of CompARE-Time to Event is composed of three different tabs:

Endpoints
Correlation
Alpha and Power
Follow-up

Endpoints

This is the main part of the input parameters.

Endpoint 1
- Death. If the Endpoint 1 is a terminating event or not. It is relevant in the competing risk setting.
- \(\bf{p_1^{(0)}}\): Probability of observing the event for the endpoint 1 in control group
- \(\bf{HR_1}\): Expected Hazard Ratio for the Endpoint 1
- \(\bf{\lambda_1}\): Risk over time. It defines the common (for both groups) shape parameter of the Weibull distribution used to model the time to event:
  - Constant (default) \(\rightarrow \lambda_1^{(0)}=\lambda_1^{(1)}=1\)
  - Increasing \(\rightarrow \lambda_1^{(0)}=\lambda_1^{(1)}=2\)
  - Decreasing \(\rightarrow \lambda_1^{(0)}=\lambda_1^{(1)}=0.5\)
Endpoint 2
- Death. If the Endpoint 2 is a terminating event or not. It is relevant in the competing risk setting.
- \(\bf{p_2^{(0)}}\): Probability of observing the event for the endpoint 2 in control group
- \(\bf{HR_2}\): Expected Hazard Ratio for the Endpoint 2
- \(\bf{\lambda_2}\): Risk over time. It defines the common (for both groups) shape parameter of the Weibull distribution used to model the time to event.
  - Constant (default) \(\rightarrow \lambda_2^{(0)}=\lambda_2^{(1)}=1\)
  - Increasing \(\rightarrow \lambda_2^{(0)}=\lambda_2^{(1)}=2\)
  - Decreasing \(\rightarrow \lambda_2^{(0)}=\lambda_2^{(1)}=0.5\)

Formulas

Let \(T_1\) and \(T_2\) be a random variable for time to event with Weibull distribution and let \(\tau\) be the time of follow-up, then the probabilities of observing the event are:

\(p_1^{(0)}=P(T_1<\tau)\) in case that Endpoint 2 was not death.
\(p_1^{(0)}=P(T_1<\tau | T_1<T_2)\) in case that Endpoint 2 was death.
\(p_1^{(0)}=P(T_2<\tau)\) in case that Endpoint 1 was not death.
\(p_1^{(0)}=P(T_2<\tau | T_2<T_1)\) in case that Endpoint 1 was death.

Hazard Ratios for the components are assumed constant over time:

\(HR_1=\frac{\lambda_1^{(1)}}{\lambda_1^{(0)}}\)
\(HR_2=\frac{\lambda_2^{(1)}}{\lambda_2^{(0)}}\)

Correlation

Set the strength of correlation between endpoints by means of Spearman’s \(\rho\) or Kendall’s \(\tau\) correlation coefficients:

Correlation. The value of the correlation in terms of the type of coefficient specified in the next input parameter.
- Very Strong (\(\rho=0.9\))
- Strong (\(\rho=0.7\))
- Moderate (\(\rho=0.5\))
- Weak (\(\rho=0.3\))
- Very Weak (\(\rho=0.15\))
- No correlation (\(\rho=0\))
Type. Spearman’s \(\rho\) (default) or Kendall’s \(\tau\). The relation between both is bijective.
Copula. To determine the joint distribution for both endpoints, copulas are used. The user can choose among 5 implemented types of copulas. For more information about copulas, see reference [1]

Remarks

It is assumed that the correlation between endpoints is the same in both arms.
It is assumed that the correlation between endpoints should be positive since it is the usual situation.
When there is no correlation (or weak correlation) then \(\rho = 0\) (or close to \(0\)).
Since most of the times, information regarding to correlation is unavailable, CompARE produces plots to visualize how much the correlation impacts on the results.
When some copula does not allow some ccorrelation value, the program replaces too small (large) values by lower (upper) bound. For instance, for FGM Spearman’s \(\rho\) must be in [-1/3,1/3]; a value of 0.5 will be considered as 1/3.

Alpha and Power

Set the type I and type II errors as well as the formula to perform sample size calculations:

Significance level (\(\alpha\)). Probability of detecting some treatment effect when it does not exist.
Power (\(1 - \beta\)). Probability of detecting some treatment effect when it exists.
Formula. Two methods are implemented for computing sample size assuming constant HRs over time
- Schoendfeld (default) [2]
- Freedman. Freedman’s formula predicts the highest power for the logrank test when the sample size ratio of the two groups equals the reciprocal of the hazard ratio [3,4]

Formulas

The number of events are estimated as follows:

Schoendfeld \(\rightarrow\) \(E=\frac{4\cdot(Z_{1-\alpha} + Z_{\beta})^2}{\big(ln(HR)\big)^2}\)
Freedman \(\rightarrow\) \(E=\frac{(Z_{1-\alpha} + Z_{\beta})^2\cdot(1+HR)^2}{(1-HR)^2}\)

Follow-up

Define the study times:

Follow-up
- Time. Amount of time that the patients will be followed. Default=1
- Units. Years, months (default) or days
Recruitment (not implemented yet)
- Time. Amount of time that will be spent in enrolling the patients. Default=1
- Units. Years, months (default) or days

References

Plana O. Using Gumbel copula to assess the efficiency of the main endpoint in a randomized clinical trial and comparison with Frank copula. Master thesis, 2012. Available here: https://upcommons.upc.edu/bitstream/handle/2099.1/17975/memoria.pdf
Schoenfeld D. Sample-size formula for the proportional-hazards regression model. Biometrics 1983;39:499-503
Freedman LS. Tables of the number of patients required in clinical trials using the logrank test. Statistics in Medicine 1982, 1, 121-129
Hsieh FY. Comparing sample size formulae for trials with unbalanced allocation using the logrank test. Stat Med 1992;11:1091-8.

Help Document: The Input Panel

CompARE Team

Running CompARE

Endpoints

Formulas

Correlation

Remarks

Alpha and Power

Formulas

Follow-up

References