Almost always in clinical research the log-rank test (Mantel-Haenszel test) is employed to compare the equality of two survival curves. Is there a good way to decide whether other tests may be more sensitive in detecting differences between the groups? For example, if the Kaplan-Meier curves cross as opposed to being roughly parallel over time? Does the total number of events influence the choice of the test statistic? What if we are also adjusting for a covariate, does this situation affect which test should be used to compare the two survival curves?
There are two ways to address the problem. I would assume that one could develop some very complex mathematical proof. That is not what I am good at. My approach would be to use random number generators to create data sets that have specific properties, or specific ranges of properties. I would then analyze them in different ways and compare the results. Compare model power, and error rates.
The benefit of using fake data is that you know the correct answer. If treatment A is rnorm(200, mean=1, sd=10) and treatment 2 is rnorm(200, mean=1.01, sd=10) then you know that the treatments are significantly different with a difference of 0.01, and that they satisfy all the assumptions of normality.
There are a number of ways to do this. If survival at time t is St, then you could generate a bunch of simulated survival curves by a simple loop where incrementing the loop decreases survival. Alternatively, you could simulate raw data. I start with an array of 1,000,000 cells. At each time interval there is a certain probability that a cell will die. This cellular automata type of simulation can be as complex as desired. A simple version is Conway's game of life (https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life). In your case I might give every cell a life score normal(mean=100, sd=30). At each time step some of this life is lost. A cell dies when it gets to zero. Since you want to compare two survival curves, you would have two arrays each of 1M cells. Array 2 could get a life boost to simulate the effect of a new drug. The level of boost would be another random number.
The effect of sample size is relatively easy to deal with. Think of it this way. There is the underlying distribution. This distribution is sampled many times to get the current population. The current population is sampled to gather experimental data. Now repeat sampling (with replacement) of the current population to get the effect of experimental design on the results. Each experiment gives a mean and standard deviation. But these are both estimates. So there is a mean and standard deviation about (the experimental mean) and there is a mean and standard deviation about (the experimental standard deviation).
Hi David,
There are several tests to compare KM curves. The log-rank test is perhaps the most popular because it provides the most power if the proportional hazards assumption is true. It weights all events equally (e.g., 1) and evaluates the difference between the observed and expected events at each event time, so it evaluates KM uniformly across time. The Wilcoxon test, on the other hand, weights each comparison by the number at-risk at each event time. So the Wilcoxon test is more powerful at detecting departures early in the study than later. Other tests use different types of weighting (e.g., Tarone-Ware uses the square-root of n).
Crossing KM curves presents a problem. (I assume that there is clear crossing, rather than random crossing that may occur when two curves are nearly identical or arising from small samples). Crossing is a clear violation of the proportional hazards assumption. This may happen if a treatment has early benefit, but not later benefit -- e.g.., the hazards change with time in one group, but not the other, or due to a time-varying covariate. Depending on the reason why the hazards are crossing, (or more generally, why they are not proportion), one approach is to compare curves within specific time segments, when the hazards may be proportional. Another approach to consider is to use a stratified test or Cox regression that accommodates time-varying covariates (again, depending on the problem).
Some great texts that cover this are David Collett, "Modelling Survival Data in Medical Research" and Hosmer and Lemeshow, "Applied Survival Analysis"
Paul
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