# 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

1. 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
2. Schoenfeld D. Sample-size formula for the proportional-hazards regression model. Biometrics 1983;39:499-503
3. Freedman LS. Tables of the number of patients required in clinical trials using the logrank test. Statistics in Medicine 1982, 1, 121-129
4. Hsieh FY. Comparing sample size formulae for trials with unbalanced allocation using the logrank test. Stat Med 1992;11:1091-8.