# Continuous Marker Prognostic Power

### Program Code

The program is written in R.

### Description

This program uses simulation to approximate the power of a test of a marker association with survival data assuming an exponential regression model. The calculations include a proportional hazards test using the model where the log-hazard ratio is linear in a continuous marker distribution. The choice of distributions include uniform, normal, and lognormal. All covariates are mean centered for the analysis. The magnitude of the effect size can be specified based on a unit of the prognostic factor or the interquartile range of the marker.

It is useful to note that if the underlying distribution is assumed to be uniform, and the proportional hazards model regression log-hazard ratio is linear in the marker, then effect size associated with the interquartile range will correspond to approximately same effect as a marker dichotomized at the median (or other quantile).

In addition to power calculations, the approximate confidence intervals of parameter effects and expected number of events are also presented.

The impact of the continuous prognostic model assumptions in terms of the expected events can be helpful in gaging assumptions on expected events relative to the overall trial design.

### Input Items

The user is prompted to enter values for the following items. For items that have initial default values set, the default values are given in parentheses.

• Alpha Level (0.05): α, the 1 or 2-sided significance level
• Accrual Period: Length of the accrual period
• Follow-up Period: Length of the follow-up period, i.e. the time from end of accrual to analysis
• Hazard Rate (marker value = 0): The hazard rate at marker value equal to 0
• Effect Size (Hazard Ratio per Unit): The hazard ratio associated with a one unit change in the marker
• Unit (Interquartile Range): Unit of marker effect is either Interquartile Range or Unstandardized
• Marker distribution (Uniform): Either standard uniform, or normal (standardized to a mean of zero), or lognormal
• Log-Normal Distribution Parameter (1): The spread/shape parameter (sigma) for each of the distributions. It is fixed to 1 for uniform and normal and defaulted to 1 for log-normal. The log-normal density is specified as $$f(x) = {e^{-(log(x) - \mu)^2 / (2 \sigma^2)} \over \sqrt {2\pi} \sigma x}$$ where $\mu$ and $\sigma$ are the mean and standard deviation of the logarithm of the normal random variable.

### Output Items

• Power, depending on the selected calculation
• Continuous Marker analysis: estimated power, and estimated effect and empirical histogram simulated expected confidence interval

### Note

This is a simulation-based calculation and therefore may take several seconds to calculate depending both on the number of simulations and sample size. For better approximations, increase the number of simulations.